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In computer science, a"let" expressionassociates afunctiondefinition with a restrictedscope.
The"let" expressionmay also be defined in mathematics, where it associates a Boolean condition with a restricted scope.
The "let" expression may be considered as alambda abstractionapplied to a value. Within mathematics, a let expression may also be considered as aconjunctionof expressions, within anexistential quantifierwhich restricts the scope of the variable.
The let expression is present in many functional languages to allow the local definition of expression, for use in defining another expression. The let-expression is present in some functional languages in two forms; let or "let rec". Let rec is an extension of the simple let expression which uses thefixed-point combinatorto implementrecursion.
Dana Scott'sLCF language[1]was a stage in the evolution of lambda calculus into modern functional languages. This language introduced the let expression, which has appeared in most functional languages since that time.
The languagesScheme,[2]ML, and more recentlyHaskell[3]have inherited let expressions from LCF.
Stateful imperative languages such asALGOLandPascalessentially implement a let expression, to implement restricted scope of functions, in block structures.[citation needed]
A closely related "where" clause, together with its recursive variant "where rec", appeared already inPeter Landin'sThe mechanical evaluation of expressions.[4]
A "let" expression defines a function or value for use in another expression. As well as being a construct used in many functional programming languages, it is a natural language construct often used in mathematical texts. It is an alternate syntactical construct for a where clause.
Let
and
in
where
and
In both cases the whole construct is an expression whose value is 5. Like theif-then-elsethe type returned by the expression is not necessarily Boolean.
A let expression comes in 4 main forms,
In functional languages theletexpression defines functions which may be called in the expression. The scope of the function name is limited to the let expression structure.
In mathematics, the let expression defines a condition, which is a constraint on the expression. The syntax may also support the declaration of existentially quantified variables local to the let expression.
The terminology, syntax and semantics vary from language to language. InScheme,letis used for the simple form andlet recfor the recursive form. In MLletmarks only the start of a block of declarations withfunmarking the start of the function definition. In Haskell,letmay be mutually recursive, with the compiler figuring out what is needed.
Alambda abstractionrepresents a function without a name. This is asource of the inconsistencyin the definition of a lambda abstraction. However lambda abstractions may be composed to represent a function with a name. In this form the inconsistency is removed. The lambda term,
is equivalent to defining the functionf{\displaystyle f}byfx=y{\displaystyle f\ x=y}in the expressionz{\displaystyle z}, which may be written as theletexpression;
The let expression is understandable as a natural language expression. The let expression represents the substitution of a variable for a value. The substitution rule describes the implications of equality as substitution.
Inmathematicstheletexpression is described as the conjunction of expressions. In functional languages the let expression is also used to limit scope. In mathematics scope is described by quantifiers. The let expression is a conjunction within an existential quantifier.
whereEandFare of type Boolean.
Theletexpression allows the substitution to be applied to another expression. This substitution may be applied within a restricted scope, to a sub expression. The natural use of the let expression is in application to a restricted scope (calledlambda dropping). These rules define how the scope may be restricted;
whereFisnot of type Boolean. From this definition the following standard definition of a let expression, as used in a functional language may be derived.
For simplicity the marker specifying the existential variable,x:{\displaystyle x:}, will be omitted from the expression where it is clear from the context.
To derive this result, first assume,
then
Using the rule of substitution,
so for allL,
LetLX=(X=K){\displaystyle L\ X=(X=K)}whereKis a new variable. then,
So,
But from the mathematical interpretation of a beta reduction,
Here if y is a function of a variable x, it is not the same x as in z. Alpha renaming may be applied. So we must have,
so,
This result is represented in a functional language in an abbreviated form, where the meaning is unambiguous;
Here the variablexis implicitly recognised as both part of the equation defining x, and the variable in the existential quantifier.
A contradiction arises if E is defined byE=¬{\displaystyle E=\neg }. In this case,
becomes,
and using,
This is false if G is false. To avoid this contradictionFis not allowed to be of type Boolean. For BooleanFthe correct statement of the dropping rule uses implication instead of equality,
It may appear strange that a different rule applies for Boolean than other types. The reason for this is that the rule,
only applies whereFis Boolean. The combination of the two rules creates a contradiction, so where one rule holds, the other does not.
Let expressions may be defined with multiple variables,
then it can be derived,
so,
TheEta reductiongives a rule for describing lambda abstractions. This rule along with the two laws derived above define the relationship between lambda calculus and let expressions.
To avoid thepotential problemsassociated with themathematical definition,Dana Scottoriginally defined theletexpression from lambda calculus. This may be considered as the bottom up, or constructive, definition of theletexpression, in contrast to the top down, or axiomatic mathematical definition.
The simple, non recursiveletexpression was defined as beingsyntactic sugarfor the lambda abstraction applied to a term. In that definition,
The simpleletexpression definition was then extended to allow recursion using thefixed-point combinator.
Thefixed-point combinatormay be represented by the expression,
This representation may be converted into a lambda term. A lambda abstraction does not support reference to the variable name, in the applied expression, soxmust be passed in as a parameter tox.
Using the eta reduction rule,
gives,
A let expression may be expressed as a lambda abstraction using,
gives,
This is possibly the simplest implementation of a fixed point combinator in lambda calculus. However one beta reduction gives the more symmetrical form of Curry's Y combinator.
The recursiveletexpression called "let rec" is defined using the Y combinator for recursive let expressions.
This approach is then generalized to support mutual recursion. A mutually recursive let expression may be composed by rearranging the expression to remove any and conditions. This is achieved by replacing multiple function definitions with a single function definition, which sets a list of variables equal to a list of expressions. A version of the Y combinator, called theY*poly-variadic fix-point combinator[5]is then used to calculate fixed point of all the functions at the same time. The result is a mutually recursive implementation of theletexpression.
A let expression may be used to represent a value that is a member of a set,
Under function application, of one let expression to another,
But a different rule applies for applying the let expression to itself.
There appear no simple rule for combining values. What is required is a general form of expression that represents a variable whose value is a member of a set of values. The expression should be based on the variable and the set.
Function application applied to this form should give another expression in the same form. In this way any expression on functions of multiple values may be treated as if it had one value.
It is not sufficient for the form to represent only the set of values. Each value must have a condition that determines when the expression takes the value. The resulting construct is a set of pairs of conditions and values, called a "value set". Seenarrowing of algebraic value sets.
Meta-functionswill be given that describe the conversion betweenlambdaandletexpressions. A meta-function is a function that takes a program as a parameter. The program is data for the meta-program. The program and the meta program are at different meta-levels.
The following conventions will be used to distinguish program from the meta program,
For simplicity the first rule given that matches will be applied. The rules also assume that the lambda expressions have been pre-processed so that each lambda abstraction has a unique name.
The substitution operator is also used. The expressionL[G:=S]{\displaystyle L[G:=S]}means substitute every occurrence ofGinLbySand return the expression. The definition used is extended to cover the substitution of expressions, from the definition given on theLambda calculuspage. The matching of expressions should compare expressions for alpha equivalence (renaming of variables).
The following rules describe how to convert from a lambda expression to aletexpression, without altering the structure.
Rule 6 creates a unique variable V, as a name for the function.
For example, theY combinator,
is converted to,
These rules reverse the conversion described above. They convert from aletexpression to a lambda expression, without altering the structure. Not all let expressions may be converted using these rules. The rules assume that the expressions are already arranged as if they had been generated byde-lambda.
There is no exact structural equivalent in lambda calculus forletexpressions that have free variables that are used recursively. In this case some addition of parameters is required. Rules 8 and 10 add these parameters.
Rules 8 and 10 are sufficient for two mutually recursive equations in theletexpression. However they will not work for three or more mutually recursive equations. The general case needs an extra level of looping which makes the meta function a little more difficult. The rules that follow replace rules 8 and 10 in implementing the general case. Rules 8 and 10 have been left so that the simpler case may be studied first.
For example, theletexpression obtained from theY combinator,
is converted to,
For a second example take the lifted version of theY combinator,
is converted to,
For a third example the translation of,
is,
For a forth example the translation of,
is,
which is the famousy combinator.
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Inphilosophy, anactionis something anagentdoes. Actions contrast with events which merely happen to someone and are typically performed for apurposeand guided by anintention.[1][2]The first question in thephilosophy of actionis to determine how actions differ from other forms of behavior, likeinvoluntary reflexes.[3][4]According toLudwig Wittgenstein, it involves discovering "What is left over if I subtract the fact that my arm goes up from the fact that I raise my arm".[5]A common response to this question focuses on the agent's intentions. So driving a car is an action since the agent intends to do so, butsneezingis a mere behavior since it happens independent of the agent's intention. The dominant theory of the relation between the intention and the behavior iscausalism:[1]driving the car is an action because it iscausedby the agent's intention to do so. On this view, actions are distinguished from other events by their causal history.[2]Causalist theories includeDonald Davidson's account, which defines actions as bodily movements caused by intentions in the right way, and volitionalist theories, according to whichvolitionsform a core aspect of actions. Non-causalist theories, on the other hand, often see intentions not as the action's cause but as a constituent of it.
An important distinction among actions is between non-basic actions, which are done by doing something else, and basic actions, for which this is not the case. Most philosophical discussions of actions focus on physical actions in the form of bodily movements. But many philosophers consider mental actions to be a distinct type of action that has characteristics quite different from physical actions. Deliberations and decisions are processes that often precede and lead to actions. Actions can be rational or irrational depending on the reason for which they are performed. The problem of responsibility is closely related to the philosophy of actions since we usually hold people responsible for what they do.
Conceptions of action try to determine what all actions have in common or what their essential features are. Causalist theories, likeDonald Davidson's account or standard forms of volitionalism, hold that causal relations between the agent's mental states and the resulting behavior are essential to actions. According to Davidson, actions are bodily movements that are caused by intentions in the right way. Volitionalist theories include the notion of volitions in their account of actions. Volitions are understood as forms of summoning of means within one's power and are different from merely intending to do something later. Non-causalists, on the other hand, deny that intentions or similar states cause actions.
The most well-known account of action, sometimes simply referred to as thestandard account, is due to Davidson, who holds that actions are bodily movements that are caused by intentions.[6]Davidson explains the intentions themselves in terms ofbeliefsanddesires.[1]For example, the action of flipping a light switch rests, on the one hand, on the agent's belief that this bodily movement would turn on the light and, on the other hand, on the desire to have light.[7]Because of its reliance on psychological states and causal relations, this position is considered to be aHumean theory of action.[8]According to Davidson, it is not just the bodily behavior that counts as the action but also the consequences that follow from it. So the movement of the finger flipping the switch is part of the action as well as the electrons moving through the wire and the light bulb turning on. Some consequences are included in the action even though the agent did not intend them to happen.[2][4]It is sufficient that what the agent does "can be described under an aspect that makes it intentional".[9][4]So, for example, if flipping the light switch alerts the burglar then alerting the burglar is part of the agent's actions.[1]In an example fromAnscombe's manuscriptIntention, pumping water can also be an instance of poisoning the inhabitants.[10]
One difficulty with theories of action that try to characterize actions in terms of causal relations between mental states and bodily movements, so-calledcausalist theories, is what has been referred to aswaywardcausal chains.[3]A causal chain iswaywardif the intention caused its goal to realize but in a very unusual way that was not intended, e.g. because the skills of the agent are not exercised in the way planned.[1]For example, a rock climber forms the intention to kill the climber below him by letting go of the rope. A wayward causal chain would be that, instead of opening the holding hand intentionally, the intention makes the first climber so nervous that the rope slips through his hand and thus leads to the other climber's death.[11]Davidson addresses this issue by excluding cases of wayward causation from his account since they are not examples of intentional behavior in the strict sense. So bodily behavior only constitutes an action if it was caused by intentionsin the right way.
One important objection to Davidson's theory of actions is that it does not account for the agent's role in the production of action. This role could include reflecting on what to do, choosing an alternative and then carrying it out.[6]Another objection is that mere intentions seem to be insufficient to cause actions, that other additional elements, namely volitions or tryings, are necessary. For example, asJohn Searlehas pointed out, there seems to be a causal gap between intending to do something and actually doing it, which needs an act of the will to be overcome.[6]
Volitionalistsaim to overcome these shortcomings of Davidson's account by including the notion ofvolitionortryingin their theory of actions.[6]Volitionsandtryingsare forms of affirming something, likeintentions. They can be distinguished from intentions because they are directed at executing a course of action in the here and now, in contrast to intentions, which involve future-directed plans to do something later.[6]Some authors also distinguishvolitions, as acts of the will, fromtryings, as the summoning of means within one's power.[6][12]But it has been argued that they can be treated as a unified notion since there is no important difference between the two for the theory of action because they play the same explanatory role.[13]This role includes both the experiential level,[4]involving the trying of something instead of merely intending to do so later, and the metaphysical level, in the form of mental causation bridging the gap between mental intention and bodily movement.[14][6]
Volitionalismas a theory is characterized by three core theses: (1) that every bodily action is accompanied by a trying, (2) that tryings can occur without producing bodily movements and (3) that in the case of successful tryings, the trying is the cause of the bodily movement.[6][4]The central idea of the notion oftryingis found in the second thesis. It involves the claim that some of our tryings lead to successful actions while others arise without resulting in an action.[15]But even in an unsuccessful case there is still something: it is different from not trying at all.[6]For example, a paralyzed person, after having received a new treatment, may test if the treatment was successful by trying to move her legs. But trying and failing to move the legs is different from intending to do it later or merely wishing to do it: only in the former case does the patient learn that the treatment was unsuccessful.[6]There is a sense in which tryings either take place or not, but cannot fail, unlike actions, whose success is uncertain.[15][3]This line of thought has led some philosophers to suggest that the trying itself is an action: a special type of action calledbasic action.[1]But this claim is problematic since it threatens to lead to avicious regress: if something is an action because it was caused by a volition then we would have to posit one more volition in virtue of which the first trying can be regarded as an action.[3][16]
An influential criticism of the volitional explanations of actions is due toGilbert Ryle, who argued that volitions are eitheractive, in which case the aforementioned regress is inevitable, or they are not, in which case there would be no need to posit them as an explanatorily inert "ghost in the machine".[4]But it has been suggested that this constitutes afalse dilemma: that volitions can play an explanatory role without leading to avicious regress.John Stuart Mill, for example, avoids this problem by holding that actions are composed of two parts: a volition and the bodily movement corresponding to it.[4]
Volitions can also be used to explain how the agent knows about her own action. This knowledge about what one is doing or trying to do is available directly through introspection: the agent does not need to observe her behavior through sensory perception to arrive at this knowledge, unlike an external observer.[1][4]The experience of agency involved in volitions can be distinguished from the experience of freedom, which involves the additional aspect of having various alternative routes of action to choose from.[4]But volition is possible even if there are no additional alternatives.[4]
Volitionalists usually hold that there is a causal relation between volitions and bodily movements.[6]Critics have pointed out that this position threatens to alienate us from our bodies since it introduces a strict distinction between our agency and our body, which is not how things appear to us.[6][17]One way to avoid this objection is to hold that volitions constitute bodily movements, i.e. are an aspect of them, instead of causing them.[17]Another response able to soften this objection is to hold that volitions are not just the initial triggers of the bodily movements but that they are continuous activities guiding the bodily movements while they are occurring.[6][18]
Non-causalistoranti-causalisttheories deny that intentions or similar statescauseactions.[19][20][21]They thereby opposecausalisttheories like Davidson's account or standard forms of volitionalism. They usually agree that intentions are essential to actions.[22]This brings with it the difficulty of accounting for the relation between intentions and actions in a non-causal way.[19]Some suggestions have been made on this issue but this is still an open problem since none of them have gathered significant support. The teleological approach, for example, holds that this relation is to be understood not in terms ofefficient causationbut in terms offinal "causation".[23]One problem with this approach is that the two forms of causation do not have to be incompatible. Few theorists deny that actions are teleological in the sense of being goal-oriented. But the representation of a goal in the agent's mind may act as an efficient cause at the same time.[19]Because of these problems, most of the arguments for non-causalism are negative: they constitute objections pointing out why causalist theories are unfeasible.[19][24]Important among them are arguments from wayward causation: that behavior only constitutes an action if it was caused by an intention in the right way, not in any way. This critique focuses on difficulties causalists have faced in explicitly formulating how to distinguish between proper and wayward causation.[25]
An important challenge to non-causalism is due to Davidson.[22][25]As he points out, we usually have many differentreasonsfor performing the same action. But when we perform it, we often perform it for one reason but not for another.[25][24]For example, one reason for Abdul to go for cancer treatment is that he has prostate cancer, another is that they have his favorite newspaper in the waiting area. Abdul is aware of both of these reasons, but he performs this action only because of the former reason. Causalist theories can account for this fact through causal relation: the former but not the latter reason causes the action. The challenge to non-causalist theories is to provide a convincing non-causalexplanationof this fact.[25][24]
The problem ofindividuationconcerns the question of whether two actions are identical or of how actions should be counted. For example, on April 14, 1865,John Wilkes Boothboth pulled the trigger of his gun, fired a shot andkilled Abraham Lincoln. On afine-grainedtheory ofindividuation, the pulling, the firing and the killing are three distinct actions.[3]In its most extreme form, there is one distinct action for every action type.[4]So, for example, since "singing" and "singing loudly" are two different action types, someone who sings loudly performs at least these two distinct actions.[3]This kind of view has the unintuitive consequence that even the most simple exercises of agency result in a vast number of actions. Theories ofcoarse-grainedindividuation of actions, on the other hand, hold that events that constitute each other or cause each other are to be counted as one action.[3][2]On this view, the action of pulling the trigger is identical to the action of firing the gun and to the action of killing Lincoln. So in doing all of these things, Booth performed only one action. One intuition in favor of this view is that we often do one thing by doing another thing:[2]we shoot the gun by pulling the trigger or we turn on the light by flipping the switch. One argument against this view is that the different events may happen at different times.[4]For example, Lincoln died of his injuries the following day, so a significant time after the shooting. This raises the question of how to explain that two events happening at different times are identical.[4]
An important distinction among actions is betweenbasicandnon-basic actions. This distinction is closely related to the problem of individuation since it also depends on the notion of doing one thingbyorin virtue ofdoing another thing, like turning on a light by flipping a switch.[26][27][28]In this example, the flipping of the switch is more basic than the turning-on of the light. But the turning-on of the light can itself constitute another action, like the action of alerting the burglar. It is usually held that the chain or hierarchy of actions composed this way has a fundamental level at which it stops.[26][4]The action at this fundamental level is called abasic action: it is not done by doing something else.[3]For this reason,basic actionsare simple while non-basic actions are complex.[26]
It is often assumed that bodily movements arebasic actions, like the pressing of one's finger against the trigger, while the consequences of these movements, like the firing of the gun, arenon-basic actions.[3]But it seems that bodily movements are themselves constituted by other events (muscle contractions)[4]which are themselves constituted by other events (chemical processes). However, it appears that these more basic events are not actions since they are not under our direct volitional control.[1][4]One way to solve these complications is to hold thatbasic actionscorrespond to the most simple commands we can follow.[26]This position excludes most forms of muscle contractions and chemical processes from the list of basic actions since we usually cannot follow the corresponding commands directly. What counts as a basic action, according to this view, depends on the agent's skills.[26]So contracting a given muscle is a basic action for an agent who has learned to do so. For something to be a basic action it is not just important what the agent can do but what the agent actually does. So raising one's right hand may only count as a basic action if it is done directly through the right hand. If the agent uses her left hand to lift the right hand then the raising of the right hand is not a basic action anymore.[1][4]
A contrasting view identifies basic actions not with bodily movements but with mental volitions.[1]One motivation for this position is that volitions are the most direct element in the chain of agency: they cannot fail, unlike bodily actions, whose success is initially uncertain.[3]One argument against this position is that it may lead to avicious regressif it is paired with the assumption that an earlier volition is needed in order for the first volition to constitute an action.[16]This is whyvolitionistsoften hold that volitions cause actions or are parts of actions but are not full actions themselves.
Philosophers have investigated the concept of actions mostly in regard to physical actions, which are usually understood in terms of bodily movements.[9][16]It is not uncommon among philosophers to understand bodily movements as the only form of action.[6]Some volitionists, on the other hand, claim that all actions are mental because they consist in volitions. But this position involves various problems, as explained in the corresponding section above. However, there is a middle path possible between these two extreme positions that allows for the existence of both physical and mental actions.[16]Various mental events have been suggested as candidates for non-physical actions, like imagining, judging or remembering.[16]
One influential account of mental action comes fromGalen Strawson, who holds that mental actions consist in "triggering the delivery of content to one's field of consciousness".[16][29]According to this view, the events of imagining, judging or remembering are not mental actions strictly speaking but they can be the products of mental actions.[16]Mental actions, in the strict sense, areprefatoryorcatalytic: they consist in preparing the mind for these contents to arise.[29]They foster hospitable conditions but cannot ensure that the intended contents will appear.[16]Strawson uses the analogy of jumping off a wall, in which the jumping itself (corresponding to the triggering) is considered an action, but the falling (corresponding to the entertaining of a content) is not an action anymore since it is outside the agent's control.[16][29]Candace L. Upton and Michael Brent object that this account of mental actions is not complete.[16]Taking their lead from mental activities taking place duringmeditation, they argue that Strawson's account leaves out various forms of mental actions, like maintaining one's attention on an object or removing a content from consciousness.[16]
One reason for doubting the existence of mental actions is that mental events often appear to be involuntary responses to internal or external stimuli and therefore not under our control.[16]Another objection to the existence of mental actions is that the standard account of actions in terms of intentions seems to fail for mental actions. The problem here is that the intention to think about something already needs to include the content of the thought. So the thought is no longer needed since the intention already "thinks" the content. This leads to a vicious regress since another intention would be necessary to characterize the first intention as an action.[16]An objection not just to mental actions but to the distinction between physical and mental actions arises from the difficulty of finding strict criteria to distinguish the two.[30]
Deliberationsanddecisionsare relevant for actions since they frequently precede the action. It is often the case that several courses of action are open to the agent.[3]In such cases, deliberation performs the function of evaluating the different options by weighing the reasons for and against them. Deciding then is the process of picking one of these alternatives and forming an intention to perform it, thereby leading toward an action.[3][31]
Explanationscan be characterized as answers to why-questions.[32][33]Explanations of actions are concerned with why the agent performed the action. The most straightforward answer to this question cites the agent's desire. For example, John went to the fridgebecausehe had a desire for ice cream. The agent's beliefs are another relevant feature for action explanation.[3]So the desire to have ice cream does not explain that John went to the fridge unless it is paired with John's belief that there is ice cream in the fridge. The desire together with the belief is often referred to as thereasonfor the action.[3][4]Causalist theoriesof action usually hold that this reason explains the action because itcausesthe action.[3][6]
Behavior that does not have a reason is not an action since it is not intentional. Every action has a reason but not every action has a good reason. Only actions with good reasons are consideredrational.[34]For example, John's action of going to the fridge would be considered irrational if his reason for this is bad, e.g. because his belief that there is ice cream in the fridge is merely based onwishful thinking.[35]
The problem ofresponsibilityis closely related to the philosophy of actions since we usually hold people responsible for what they do. But in one sense the problem of responsibility is wider since we can be responsible not just for doing something but for failing to do something, so-calledomissions.[3][2][4]For example, a pedestrian witnessing a terrible car accident may be morally responsible for calling an ambulance and for providing help directly if possible. Additionally to what the agent did, it is also relevant what the agent could have done otherwise, i.e. what powers and capacities the agent had.[36]The agent's intentions are also relevant for responsibility, but we can be responsible for things we did not intend. For example, a chain smoker may have a negative impact on the health of the people around him. This is a side-effect of his smoking that is not part of his intention. The smoker may still be responsible for this damage, either because he was aware of this side-effect and decided to ignore it or because he should have been aware of it, so-callednegligence.[37]
In the theory ofenactivism, perception is understood to besensorimotorin nature. That is, we carry out actions as an essential part of perceiving the world.Alva Noëstates:
'We move our eyes, head and body in taking in what is around us... [we]: crane our necks, peer, squint, reach for our glasses or draw near to get a better look...'...'Perception is a mode of activity on the part of the whole animal...It cannot be represented in terms of merely passive, and internal, processes...'[38]
Some philosophers (e.g.Donald Davidson[39]) have argued that the mental states the agent invokes as justifying his action are physical states that cause the action.[citation needed]Problems have been raised for this view because the mental states seem to be reduced to mere physical causes.[citation needed]Their mental properties don't seem to be doing any work.[citation needed]If the reasons an agent cites as justifying his action, however, are not the cause of the action, they must explain the action in some other way or be causally impotent.[citation needed]Those who hold the belief that mental properties are reducible to physical properties are known as token-identity reductionists.[40]Some have disagreed with the conclusion that this reduction means the mental explanations are causally impotent while still maintaining that the reduction is possible.[41]For example, Dretske has put forward the viewpoint of reasons as structuring causes.[41]This viewpoint maintains that the relation, intentional properties that are created in the process of justifying one's actions are causally potent in that the process is an instance of action.[41]When considering that actions are causally potent, Dretske claims that the process of justifying one's actions is necessarily part of the causal system.[41]Others have objected to the belief that mental states can cause physical action without asserting that mental properties can be reduced to physical properties.[42]Such individuals suggest that mental states are epiphenomenal, in that they have no impact on physical states, but are nonetheless distinct entities (seeepiphenomenalism).[43]
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Value theory, also calledaxiology, studies the nature, sources, and types ofvalues. It is a branch ofphilosophyand an interdisciplinary field closely associated withsocial sciencessuch aseconomics,sociology,anthropology, andpsychology.
Value is the worth of something, usually understood as a degree that covers both positive and negative magnitudes corresponding to the termsgoodandbad. Values influence many human endeavors related toemotion,decision-making, andaction. Value theorists distinguish various types of values, like the contrast betweenintrinsic and instrumental value. An entity hasintrinsic valueif it is good in itself, independent of external factors. An entity has instrumental value if it is useful as a means leading to other good things. Other classifications focus on the type of benefit, including economic, moral, political, aesthetic, and religious values. Further categorizations cover attributive, predicative, personal, impersonal, and agent-relative values.
Valuerealistsstate that values exist asobjectivefeatures of reality. Anti-realists reject this, with some seeing values as subjective human creations and others viewing value statements as meaningless. Regarding the sources of value,hedonistsargue that onlypleasurehas intrinsic value, whereas desire theorists discussdesiresas the ultimate source of value.Perfectionism, another approach, emphasizes the cultivation of characteristic human abilities. Valuepluralismidentifies diverse sources of intrinsic value, raising the issue of whether values belonging to different types are comparable. Value theorists employ variousmethods of inquiry, ranging fromreliance on intuitionsandthought experimentsto thedescription of first-person experienceand the analysis of language.
Ethics, a related field, focuses primarily onnormativeconcepts of right behavior, whereas value theory exploresevaluativeconcepts about what is good. In economics,theories of valueare frameworks to assess and explain the economic value ofcommodities. Sociology and anthropology examine values as aspects of societies and cultures, reflecting dominant preferences and beliefs. Psychologists tend to understand values as abstractmotivationalgoals that shape an individual'spersonality. The roots of value theory lie inantiquityas reflections on thehighest goodthat humans should pursue. Diverse traditions contributed to this area of thought during themedievalandearly modern periods, but it was only established as a distinct discipline in the late 19th and early 20th centuries.
Value theory, also known asaxiologyandtheory of values, is the systematic study ofvalues. As a branch ofphilosophy, it examines which things are good and what it means for something to be good. It distinguishes different types of values and explores how they can be measured and compared. This field also studies whether values are a fundamental aspect of reality and how they influence phenomena such asemotion,desire, decision, andaction.[2]Value theory is relevant to many human endeavors because values are guiding principles that underlie the political, economic, scientific, and personal spheres.[3]It analyzes and evaluates phenomena such aswell-being,utility,beauty, human life,knowledge,wisdom,freedom,love, andjustice.[4]
The precise definition of value theory is debated and some theorists rely on alternative characterizations. In a broad sense,value theoryis a catch-all label that encompasses all philosophical disciplines studying evaluative and normative topics. According to this view, value theory is one of the main branches of philosophy and includesethics,aesthetics,social philosophy,political philosophy, andphilosophy of religion.[5]A similar broad characterization sees value theory as a multidisciplinary area of inquiry that integrates research from fields likesociology,anthropology,psychology, andeconomicsalongside philosophy.[6]In a narrow sense, value theory is a subdiscipline of ethics that is particularly relevant to the school ofconsequentialismsince it determines how to assess the value of consequences.[7]
The wordaxiologyhas its origin in theancient Greektermsἄξιος(axios, meaning'worthy'or'of value') andλόγος(logos, meaning'study'or'theory of').[8]Even though the roots of value theory reach back to theancient period, this area of thought was only conceived as a distinct discipline in the late 19th and early 20th centuries, when the termaxiologywas coined.[9]The termsvalue theoryandaxiologyare usually used as synonyms but some philosophers distinguish between them. According to one characterization, axiology is a subfield of value theory that limits itself to theories about which things are valuable and how valuable they are.[10][a]The termtimologyis an older and less common synonym.[12]
Value is the worth, usefulness, or merit of something.[b]Value theorists examine the expressions used to describe and compare values, calledevaluative terms.[15]They are further interested in the types or categories of values. The proposed classifications overlap and are based on factors like the source, beneficiary, and function of the value.[16]
Values are expressed through evaluative terms. For example, the wordsgood,best,great, andexcellentconvey positive values, whereas words likebadandterribleindicate negative values.[15]Value theorists distinguish between thin andthick evaluative terms. Thin evaluative terms, likegoodandbad, express pure evaluations without any additional descriptive content.[c]They contrast with thick evaluative terms, likecourageousandcruel, which provide more information by expressing other qualities, such ascharacter traits, in addition to the evaluation.[18]Values are often understood as degrees that cover positive and negative magnitudes corresponding to good and bad. The termvalueis sometimes restricted to positive degrees to contrast with the termdisvaluefor negative degrees. The wordsbetterandworseare used to compare degrees, but it is controversial whether a quantitative comparison is always possible.[19]Evaluationis the assessment or measurement of value, often employed to compare the benefits of different options to find the most advantageous choice.[20]
Evaluative terms are sometimes distinguished fromnormativeor deontic terms. Normative terms, likeright,wrong, andobligation, prescribe actions or other states by expressing what ought to be done or what is required.[21]Evaluative terms have a wider scope because they are not limited to what people can control or are responsible for. For instance, involuntary events like digestion and earthquakes can have a positive or negative value even if they are not right or wrong in a strict sense.[22]Despite the distinction, evaluative and normative concepts are closely related. For example, the value of the consequences of an action may influence its normative status—whether the action is right or wrong.[23]
A thing has intrinsic or final value if it is good in itself or good for its own sake, independent of external factors or outcomes. A thing has extrinsic or instrumental value if it is useful or leads to other good things, serving as a means to bring about a desirable end. For example, tools like microwaves or money have instrumental value due to the useful functions they perform.[24]In some cases, the thing produced this way has itself instrumental value, like when using money to buy a microwave. This can result in a chain of instrumentally valuable things in which each link gets its value by causing the following link. Intrinsically valuable things stand at the endpoint of these chains and ground the value of all the preceding links.[25]
One suggestion to distinguish between intrinsic and instrumental value, proposed byG. E. Moore, relies on athought experimentthat imagines the valuable thing in isolation from everything else. In such a situation, purely instrumentally valuable things lose their value since they serve no purpose while purely intrinsically valuable things remain valuable.[26][d]According to a common view,pleasureis one of the sources of intrinsic value. Other suggested sources includedesiresatisfaction,virtue,life,health,beauty,freedom, andknowledge.[28]
Intrinsic and instrumental value are not exclusive categories. As a result, a thing can have both intrinsic and instrumental value if it is both good in itself while also leading to other good things.[29]In a similar sense, a thing can have different instrumental values at the same time, both positive and negative ones. This is the case if some of its consequences are good while others are bad. The total instrumental value of a thing is the value balance of all its consequences.[30]
Because instrumental value depends on other values, it is an open question whether it should be understood as a value in a strict sense. For example, the overall value of a chain of causes leading to an intrinsically valuable thing remains the same if instrumentally valuable links are added or removed without affecting the intrinsically valuable thing. The observation that the overall value does not change is sometimes used as an argument that the things added or removed do not have value.[31]
Traditionally, value theorists have used the termsintrinsic valueandfinal valueinterchangeably, just like the termsextrinsic valueandinstrumental value. This practice has been questioned in the 20th century based on the idea that they are similar but not identical concepts. According to this view, a thing has intrinsic value if the source of its value is anintrinsic property, meaning that the value does not depend on how the thing is related to other objects. Extrinsic value, by contrast, depends onexternal relations. This view sees instrumental value as one type of extrinsic value based on externalcausal relations. At the same time, it allows that there are other types of non-instrumental extrinsic value that result from external non-causal relations. Final value is understood as what is valued for its own sake, independent of whether intrinsic or extrinsic properties are responsible.[32][e]
Another distinction relies on the contrast between absolute and relative value. Absolute value, also calledvaluesimpliciter, is a form of unconditional value. A thing has relative value if its value is limited to certain considerations or viewpoints.[34]
One form of relative value is restricted to the type of an entity, expressed in sentences like "That is a good knife" or "Jack is a good thief". This form is known asattributivegoodnesssince the word "good" modifies the meaning of another term. To be attributively good as a certain type means to possess qualities characteristic of that type. For instance, a good knife is sharp and a good thief has the skill of stealing without getting caught. Attributive goodness contrasts withpredicativegoodness. The sentence "Pleasure is good" is an example since the wordgoodis used as a predicate to talk about the unqualified value of pleasure.[35]Attributive and predicative goodness can accompany each other, but this is not always the case. For instance, being a good thief is not necessarily a good thing.[36]
Another type of relative value restricts goodness to a specific person. Known aspersonal value,[f]it expresses what benefits a particular person, promotes theirwelfare, or is in their interest. For example, a poem written by a child may have personal value for the parents even if the poem lacks value for others. Impersonal value, by contrast, is good in general without restriction to any specific person or viewpoint.[38]Some philosophers, like Moore, reject the existence of personal values, holding that all values are impersonal. Others have proposed theories about the relation between personal and impersonal value. The agglomerative theory says that impersonal value is nothing but the sum of all personal values. Another view understands impersonal value as a specific type of personal value taken from the perspective of the universe as a whole.[39]
Agent-relative value is sometimes contrasted with personal value as another person-specific limitation of the evaluative outlook. Agent-relative values affect moral considerations about what a person is responsible for or guilty of. For example, if Mei promises to pick Pedro up from the airport then an agent-relative value obligates Mei to drive to the airport. This obligation is in place even if it does not benefit Mei, in which case there is an agent-relative value without a personal value. Inconsequentialism,[g]agent-relative values are often discussed in relation toethical dilemmas. One dilemma revolves around the question of whether an individual should murder an innocent person if this prevents the murder of two innocent people by a different perpetrator. The agent-neutral perspective tends to affirm this idea since one murder is preferable to two. The agent-relative perspective tends to reject this conclusion, arguing that the initial murder should be avoided since it negatively impacts the agent-relative value of the individual committing it.[41]
Traditionally, most value theorists see absolute value as the main topic of value theory and focus their attention on this type. Nonetheless, some philosophers, likePeter GeachandPhilippa Foot, have argued that the concept of absolute value by itself is meaningless and should be understood as one form of relative value.[42]
Other classifications of values have been proposed without a widely accepted main classification.[43]Some focus on the types of entities that have value. They include distinct categories for entities like things, the environment, individuals, groups, and society. Another subdivision pays attention to the type of benefit involved and encompasses material, economic, moral, social, political, aesthetic, and religious values. Classifications by the beneficiary of the value distinguish between self- and other-oriented values.[44]
A historically influential approach identifies three spheres of value:truth, goodness, and beauty.[h]For example, theneo-KantianphilosopherWilhelm Windelbandcharacterizes them as the highest goals ofconsciousness, withthoughtaiming at truth,willaiming at goodness, andemotionaiming at beauty. A similar view, proposed by theChinese philosopherZhang Dainian, says that the value of truth belongs to knowledge, the value of goodness belongs to behavior, and the value of beauty belongs to art.[46]This three-fold distinction also plays a central role in the philosophies ofFranz BrentanoandJürgen Habermas.[47]Other suggested types of values include objective, subjective, potential, actual, contingent, necessary, inherent, and constitutive values.[48]
Valuerealismis the view that values have mind-independent existence.[49][i]This means thatobjectivefacts determine what has value, irrespective of subjective beliefs and preferences.[50]According to this view, the evaluative statement "That act is bad" is as objectively true or false as the empirical statement "That act causes distress".[51]
Realists often analyze values aspropertiesof valuable things.[52]For example, stating that kindness is good asserts that kindness possesses the property of goodness. Value realists disagree about what type of property is involved.Naturalistssay that value is a natural property. Natural properties, like size and shape, can be known throughempirical observationand are studied by the natural sciences.Non-naturalistsreject this view but agree that values are real. They say that values differ significantly from empirical properties and belong to another realm of reality. According to one view, they are known through rational or emotional intuition rather than empirical observation.[53]
Another disagreement among realists is about whether the entity carrying the value is a concreteindividualor astate of affairs.[54]For instance, the name "Bill" refers to an individual while the sentence "Bill is pleased" refers to a state of affairs. States of affairs are complex entities that combine other entities, like the individual "Bill" and the property "pleased". Some value theorists hold that the value is a property directly of Bill while others contend that it is a property of the state of affairs that Bill is pleased.[55]This distinction affects various disputes in value theory. In some cases, a value is intrinsic according to one view and extrinsic according to the other.[56]
Value realism contrasts withanti-realism, which comes in various forms. In its strongest version, anti-realism rejects the existence of values in any form, claiming that value statements are meaningless.[57][j]Between these two positions, there are various intermediate views. Some anti-realists accept that value claims have meaning but deny that they have atruth value,[k]a position known asnon-cognitivism. For example,emotivistssay that value claims express emotional attitudes, similar to how exclamations like "Yay!" or "Boo!" express emotions rather than stating facts.[60][l]
Cognitivistscontend that value statements have a truth value.Error theoristsdefend anti-realism based on this view by stating that all value statements are false because there are no values.[62]Another view accepts the existence of values but denies that they are mind-independent. According to this view, themental statesof individuals determine whether an object has value, for instance, because individuals desire it.[63]A similar view is defended byexistentialistslikeJean-Paul Sartre, who argued that values are human creations that endow the world with meaning.[64]Subjectivist theories say that values are relative to each subject, whereas more objectivist outlooks hold that values depend onmindin general rather than on the individual mind.[65]A different position accepts that values are mind-independent but holds that they are reducible to other facts, meaning that they are not a fundamental part of reality. One form ofreductionismmaintains that a thing is good if it is fitting to favor this thing, regardless of whether people actually favor it, a position known as thefitting-attitude theory of value. The buck-passing account, a closely related reductive view, argues that a thing is valuable if people have reasons to treat the thing in certain ways. These reasons come from other features of the valuable thing. The strongest form of realism says that value is a fundamental part of reality and cannot be reduced to other aspects.[66]
Various theories about the sources of value have been proposed. They aim to clarify what kinds of things are intrinsically good.[67]The historically influential theory ofhedonism[m]states that how people feel is the only source of value. More specifically, it says thatpleasureis the only intrinsic good andpainis the only intrinsic evil.[69]According to this view, everything else only has instrumental value to the extent that it leads to pleasure or pain, including knowledge, health, and justice. Hedonists usually understand the termpleasurein a broad sense that covers all kinds of enjoyable experiences, including bodily pleasures of food and sex as well as more intellectual or abstract pleasures, like the joy of reading a book or happiness about a friend's promotion. Pleasurable experiences come in degrees, and hedonists usually associate their intensity and duration with the magnitude of value they have.[70][n]
Many hedonists identify pleasure and pain as symmetric opposites, meaning that the value of pleasure balances out the disvalue of pain if they have the same intensity. However, some hedonists reject this symmetry and give more weight to avoiding pain than to experiencing pleasure.[72]Although it is widely accepted that pleasure is valuable, the hedonist claim that it is the only source of value is controversial.[73]Welfarism, a closely related theory, understandswell-beingas the only source of value. Well-being is what is ultimately good for a person, which can include other aspects besides pleasure, such as health,personal growth, meaningfulrelationships, and a sense of purpose in life.[74]
Desire theories offer a slightly different account, stating that desire satisfaction[o]is the only source of value.[p]This theory overlaps with hedonism because many people desire pleasure and because desire satisfaction is often accompanied by pleasure. Nonetheless, there are important differences: people desire a variety of other things as well, like knowledge, achievement, and respect; additionally, desire satisfaction may not always result in pleasure.[77]Some desire theorists hold that value is a property of desire satisfaction itself, while others say that it is a property of the objects that satisfy a desire.[78]One debate in desire theory concerns whether every desire is a source of value. For example, if a person has a false belief that money makes them happy, it is questionable whether the satisfaction of their desire for money is a source of value. To address this consideration, some desire theorists say that a desire can only provide value if a fully informed and rational person would have it, thereby excluding misguided desires from being a source of value.[79]
Perfectionismidentifies the realization ofhuman natureand the cultivation of characteristic human abilities as the source of intrinsic goodness. It covers capacities and character traits belonging to the bodily, emotional, volitional, cognitive, social, artistic, and religious fields. Perfectionists disagree about which human excellences are the most important. Many are pluralistic in recognizing a diverse array of human excellences, such as knowledge, creativity, health, beauty, free agency, and moral virtues like benevolence and courage.[80]According to one suggestion, there are two main fields of human goods: theoretical abilities responsible for understanding the world and practical abilities responsible for interacting with it.[81]Some perfectionists provide an ideal characterization of human nature as the goal of human flourishing, holding that human excellences are those aspects that promote the realization of this goal. This view is exemplified inAristotle's focus onrationalityas the nature and ideal state of human beings.[82]Non-humanistic versions extend perfectionism to the natural world in general, arguing that excellence as a source of intrinsic value is not limited to the human realm.[83]
Monisttheories of value assert that there is only a single source of intrinsic value. They agree that various things have value but maintain that all fundamentally good things belong to the same type. For example, hedonists hold that nothing but pleasure has intrinsic value, while desire theorists argue that desire satisfaction is the only source of fundamental goodness.Pluralistsreject this view, contending that a simple single-value system is too crude to capture the complexity of the sphere of values. They say that diverse sources of value exist independently of one another, each contributing to the overall value of the world.[84]
One motivation for value pluralism is the observation that people value diverse types of things, including happiness, friendship, success, and knowledge.[85]This diversity becomes particularly prominent when people face difficult decisions between competing values, such as choosing between friendship and career success.[86]In such cases, value pluralists can argue that the different items have different types of value. Since monists accept only one source of intrinsic value, they may provide a different explanation by proposing that some of the valuable items only have instrumental value but lack intrinsic value.[87]
Pluralists have proposed various accounts of how their view affects practical decisions. Rational decisions often rely on value comparisons to determine which course of action should be pursued.[89]Some pluralists discuss a hierarchy of values reflecting the relative importance and weight of different value types to help people promote higher values when faced with difficult choices.[90]For example, philosopherMax Schelerranks values based on how enduring and fulfilling they are into the levels of pleasure, utility, vitality, culture, and holiness. He asserts that people should not promote lower values, like pleasure, if this comes at the expense of higher values.[88][q]
Radical pluralists reject this approach, putting more emphasis on diversity by holding that different types of values are not comparable with each other. This means that each value type is unique, making it impossible to determine which one is superior.[92][r]Some value theorists use radical pluralism to argue that value conflicts are inevitable, that the gain of one value cannot always compensate for the loss of another, and that someethical dilemmasare irresolvable.[94]For example, philosopherIsaiah Berlinapplied this idea to the values oflibertyandequality, arguing that a gain in one cannot make up for a loss in the other. Similarly, philosopherJoseph Razsaid that it is often impossible to compare the values of career paths, like when choosing between becoming alawyeror aclarinetist.[95]The termsincomparabilityandincommensurabilityare often used as synonyms. However, philosophers likeRuth Changdistinguish them. According to this view, incommensurability means that there is no common measure to quantify values of different types. Incommensurable values may or may not be comparable. If they are, it is possible to say that one value is better than another, but it is not possible to quantify how much better it is.[96]
Several controversies surround the question of how the intrinsic value of awholeis determined by the intrinsic values of its parts. According to the additivity principle, the intrinsic value of a whole is simply the sum of the intrinsic values of its parts. For example, if a virtuous person becomes happy then the intrinsic value of the happiness is simply added to the intrinsic value of the virtue, thereby increasing the overall value.[97]
Various counterexamples to the additivity principle have been proposed, suggesting that the relation between parts and wholes is more complex. For instance,Immanuel Kantargued that if a vicious person becomes happy, this happiness, though good in itself, does not increase the overall value. On the contrary, it makes things worse, according to Kant, since viciousness should not be rewarded with happiness. This situation is known as anorganic unity—a whole whose intrinsic value differs from the sum of the intrinsic values of its parts.[99]Another perspective, calledholism about value, asserts that the intrinsic value of a thing depends on its context. Holists can argue that happiness has positive intrinsic value in the context of virtue and negative intrinsic value in the context of vice. Atomists reject this view, saying that intrinsic value is context-independent.[100]
Theories of value aggregation provide concrete principles for calculating the overall value of an outcome based on how positively or negatively each individual is affected by it. For example, if a government implements a new policy that affects some people positively and others negatively, theories of value aggregation can be used to determine whether the overall value of the policy is positive or negative. Axiologicalutilitarianismaccepts the additivity principle, saying that the total value is simply the sum of all individual values.[101]Axiologicalegalitariansare not only interested in the sum total of value but also in how the values are distributed. They argue that an outcome with a balanced advantage distribution is better than an outcome where some benefit a lot while others benefit little, even if the two outcomes have the same sum total.[102]Axiologicalprioritariansare particularly concerned with the benefits of individuals who are worse off. They say that providing advantages to people in need has more value than providing the same advantages to others.[102]
Another debate addresses themeaning of life, investigating whether life or existence as a whole has a higher meaning or purpose.[103]Naturalistviews argue that the meaning of life is found within the physical world, either as objective values that are true for everyone or as subjective values that vary according to individual preferences. Suggested fields where humans find meaning include exercisingfreedom, committing oneself to a cause, practicingaltruism, engaging in positivesocial relationships, or pursuing personalhappiness.[104]Supernaturalists, by contrast, propose that meaning lies beyond the natural world. For example, various religions teach thatGodcreated the world for a higher purpose, imbuing existence with meaning. A related outlook argues that immortalsoulsserve as sources of meaning by being connected to atranscendent realityand evolvingspiritually.[105]Existential nihilistsreject both naturalist and supernaturalist explanations by asserting that there is no higher purpose. They suggest that life is meaningless, with the consequence that there is no higher reason to continue living and that all efforts, achievements, happiness, and suffering are ultimately pointless.[106]
Formal axiology is a theory of value initially developed by philosopherRobert S. Hartman. This approach treats axiology as aformal science, akin tologicandmathematics. It usesaxiomsto give an abstract definition of value, understanding it not as a property of things but as a property of concepts. Value measures the extent to which an entity fulfills its concept. For example, a good car has all the desirable qualities of cars, like a reliable engine and effective brakes, whereas a bad car lacks many. Formal axiology distinguishes between three fundamental value types: intrinsic values apply to people; extrinsic values apply to things, actions, and social roles; systemic values apply to conceptual constructs. Formal axiology examines how these value types form a hierarchy and how they can be measured.[107]
Value theorists employ variousmethodsto conduct their inquiries, justify theories, and measure values.Intuitionistsrely onintuitionsto assess evaluative claims. In this context, an intuition is an immediate apprehension or understanding of aself-evidentclaim, meaning that its truth can be assessed withoutinferringit from another observation.[108]Value theorists often rely onthought experimentsto gain this type of understanding. Thought experiments are imagined scenarios that exemplify philosophical problems. Philosophers usecounterfactual reasoningto evaluate possible consequences and gain insight into underlying problems.[109]For example, philosopherRobert Nozickimagines anexperience machinethat can virtually simulate an ideal life. Based on his contention that people would not want to spend the rest of their lives in this pleasurable simulation, Nozick argues against thehedonistclaim that pleasure is the only source of intrinsic value. According to him, the thought experiment shows that the value of an authentic connection to reality is not reducible to pleasure.[110][s]
Phenomenologistsprovide a detailed first-person description of theexperienceof values. They closely examine emotional experiences, ranging from desire, interest, and preference to feelings in the form of love and hate. However, they do not limit their inquiry to these phenomena, asserting that values permeate experience at large.[111]A key aspect of the phenomenological method is tosuspend preconceived ideas and judgmentsto understand the essence of experiences as they present themselves to consciousness.[112]
The analysis of concepts andordinary languageis another method of inquiry. By examining terms and sentences used to talk about values, value theorists aim to clarify their meanings, uncover crucial distinctions, and formulate arguments for and against axiological theories.[113]For instance, a prominent dispute betweennaturalistsandnon-naturalistshinges on theconceptual analysisof the termgood, in particular, whether its meaning can be analyzed through natural terms, likepleasure.[114][t]
In thesocial sciences, value theorists face the challenge of measuring the evaluative outlook of individuals and groups. Specifically, they aim to determine personal value hierarchies, for example, whether a person gives more weight to truth than to moral goodness or beauty.[116]They distinguish between direct and indirect measurement methods. Direct methods involve asking people straightforward questions about what things they value and which value priorities they have. This approach assumes that people are aware of their evaluative outlook and able to articulate it accurately. Indirect methods do not share this assumption, asserting instead that values guide behavior and choices on an unconscious level. Consequently, they observe how people decide and act, seeking to infer the underlying value attitudes responsible for picking one course of action rather than another.[117]
Various catalogs orscales of valueshave been proposed to measure value priorities. TheRokeach Value Surveyconsiders a total of 36 values divided into two groups: instrumental values, like honesty and capability, which serve as means to promote terminal values, such as freedom and family security. It asks participants to rank the values based on their impact on the participants' lives, aiming to understand the relative importance assigned to each of them. TheSchwartz theory of basic human valuesis a modification of the Rokeach Value Survey that seeks to provide a more cross-cultural and universal assessment. It arranges the values in a circular manner to reflect that neighboring values are compatible with each other, such as openness to change and self-enhancement, while values on opposing sides may conflict with each other, such as openness to change and conservation.[118]
Ethics and value theory are overlapping fields of inquiry. Ethics studiesmoralphenomena, focusing on how people should act or which behaviors are morally right.[119]Value theory investigates the nature, sources, and types of values in general.[2]Some philosophers understand value theory as a subdiscipline of ethics. This is based on the idea that what people should do is affected by value considerations but not necessarily limited to them.[7]Another view sees ethics as a subdiscipline of value theory. This outlook follows the idea that ethics is concerned with moral values affecting what people can control, whereas value theory examines a broader range of values, including those beyond anyone's control.[120]Some perspectives contrast ethics and value theory, asserting that thenormativeconcepts examined by ethics are distinct from the evaluative concepts examined by value theory.[23]Axiological ethicsis a subfield of ethics examining the nature and role of values from a moral perspective, with particular interest in determining which ends are worth pursuing.[121]
The ethical theory ofconsequentialismcombines the perspectives of ethics and value theory, asserting that the rightness of an action depends on the value of its consequences. Consequentialists compare possible courses of action, saying that people should follow the one leading to the best overall consequences.[122]The overall consequences of an action are the totality of its effects, or how it impacts the world by starting a causal chain of events that would not have occurred otherwise.[123]Distinct versions of consequentialism rely on different theories of the sources of value.Classical utilitarianism, a prominent form of consequentialism, says that moral actions produce the greatest amount ofpleasurefor the greatest number of people. It combines a consequentialist outlook on right action with ahedonistoutlook on pleasure as the only source of intrinsic value.[124]
Economics is asocial sciencestudying how goods and services are produced, distributed, and consumed, both from the perspective of individual agents and societal systems.[125]Economists view evaluations as a driving force underlying economic activity. They use the notion ofeconomic valueand related evaluative concepts to understand decision-making processes, resource allocation, and the impact of policies. The economic value or benefit of acommodityis the advantage it provides to aneconomic agent, often measured in terms of what people arewilling to payfor it.[126]
Economic theories of value are frameworks to explain how economic value arises and which factors influence it. Prominent frameworks include the classicallabor theory of valueand the neo-classicalmarginal theory of value.[127]The labor theory, initially developed by the economistsAdam SmithandDavid Ricardo, distinguishes betweenuse value—the utility or satisfaction a commodity provides—andexchange value—the proportion at which one commodity can be exchanged with another.[128]It focuses on exchange value, which it says is determined by theamount of labor required to produce the commodity. In its simplest form, it directly correlates exchange value to labor time. For example, if the time needed to hunt a deer is twice the time needed to hunt a beaver then one deer is worth two beavers.[129]The philosopherKarl Marxextended the labor theory of value in various ways. He introduced the concept ofsurplus value, which goes beyond the time and resources invested to explain howcapitalistscan profit from the labor of their employees.[130]
The marginal theory of value focuses on consumption rather than production. It says that the utility of a commodity is the source of its value. Specifically, it is interested inmarginal utility, the additional satisfaction gained from consuming one more unit of the commodity. Marginal utility often diminishes if many units have already been consumed, leading to a decrease in the exchange value of commodities that are abundantly available.[131]Both the labor theory and the marginal theory were later challenged by theSraffian theory of value.[132]
Sociology studies social behavior, relationships, institutions, and society at large.[133]In their analyses and explanations of these phenomena, some sociologists use the concept of values to understand issues likesocial cohesionandconflict, the norms and practices people follow, andcollective action. They usually understand values as subjective attitudes possessed by individuals and shared in social groups. According to this view, values are beliefs or priorities about goals worth pursuing that guide people to act in certain ways. For example, societies that value education may invest substantial resources to ensure high-quality schooling. This subjective conception of values as aspects of individuals and social groups contrasts with the objective conceptions of values more prominent in economics, which understand values as aspects of commodities.[134]
Shared values can help unite people in the pursuit of a common cause, fostering social cohesion. Value differences, by contrast, may divide people into antagonistic groups that promote conflicting projects. Some sociologists employ value research to predict how people will behave. Given the observation that someone values the environment, they may conclude that this person is more likely torecycleor support pro-environmental legislation.[135]One approach to this type of research usesvalue scales, such as theRokeach Value Surveyand theSchwartz theory of basic human values, to measure the value outlook of individuals and groups.[136]
Anthropology also studies human behavior and societies but does not limit itself to contemporary social structures, extending its focus to humanity both past and present.[137]Similar to sociologists, many anthropologists understand values as social representations of goals worth pursuing. For them, values are embedded in mental structures associated with culture and ideology about what is desirable. A slightly different approach in anthropology focuses on the practical side of values, holding that values are constantly created through human activity.[138]
Anthropological value theoristsuse values to compare cultures.[139]They can be employed to examine similarities as universal concerns present in every society. For example, anthropologistClyde Kluckhohnand sociologistFred Strodtbeckproposed a set of value orientations found in every culture.[140]Values can also be used to analyze differences between cultures and value changes within a culture. AnthropologistLouis Dumontfollowed this idea, suggesting that the cultural meaning systems in distinct societies differ in their value priorities. He argued that values are ordered hierarchically around a set of paramount values that trump all other values. For example, Dumont analyzed thetraditional Indian caste systemas a cultural hierarchy based on the value of purity, extending from the pureBrahminsto the "untouchable"Dalits.[141]
The contrast betweenindividualism and collectivismis an influential topic in cross-cultural value research. Individualism promotes values associated with theautonomyof individuals, such asself-directedness, independence, and the fulfillment of personal goals. Collectivism gives priority to group-related values, like cooperation,conformity, and foregoing personal advantages for the sake of collective benefits. As a rough simplification, it is often suggested that individualism is more prominent inWestern cultures, whereas collectivism is more commonly observed inEastern cultures.[142]
As the study ofmental phenomenaand behavior, psychology contrasts with sociology and anthropology by focusing more on the perspective of individuals than the broader social and cultural contexts.[143]Psychologists tend to understand values as abstractmotivationalgoals or general principles about what matters.[144]From this perspective, values differ from specific plans andintentionssince they are stable evaluative tendencies not bound to concrete situations.[145]
Various psychological theories of values establish a close link between an individual's evaluative outlook and theirpersonality.[146]An early theory, formulated by psychologistsPhilip E. VernonandGordon Allport, understands personality as a collection of aspects unified by a coherentvalue system. It distinguishes between six personality types corresponding to the value spheres of theory, economy, aesthetics, society, politics, and religion. For example, people with theoretical personalities place special importance on thevalue of knowledgeand the discovery oftruth.[147]Influenced by Vernon and Allport, psychologistMilton Rokeachconceptualized values as enduring beliefs about what goals and conduct are preferable. He divided values into the categories of instrumental and terminal values. He thought that a central aspect of personality lies in how people prioritize the values within each category.[148]PsychologistShalom Schwartzrefined this approach by linking values to emotion and motivation. He explored how value rankings affect decisions in which the values of different options conflict.[149]
The origin of value theory lies in theancient period, with early reflections on the good life and the ends worth pursuing.[150]Socrates(c.469–399 BCE)[151]identified the highest good as the right combination ofknowledge,pleasure, andvirtue, holding that active inquiry is associated with pleasure while knowledge of the Good leads to virtuous action.[152]Plato(c.428–347 BCE)[153]conceivedthe Goodas a universal and changeless idea. It is the highest form in histheory of forms, acting as the source of all other forms and the foundation of reality and knowledge.[154]Aristotle(384–322 BCE)[155]saweudaimoniaas the highest good and ultimate goal of human life. He understood eudaimonia as a form of happiness or flourishing achieved through the exercise of virtues in accordance withreason, leading to the full realization of human potential.[156]Epicurus(c.341–271 BCE) proposed a nuancedegoistichedonism, stating that personal pleasure is the greatest good while recommending moderation to avoid the negative effects of excessive desires and anxiety about the future.[157]According to theStoics, a virtuous life following nature and reason is the highest good. They thought that self-mastery andrationalitylead to a pleasantequanimityindependent of external circumstances.[158]Influenced by Plato,Plotinus(c.204/5–270 CE) held that the Good is the ultimate principle of reality from which everything emanates. For him,evilis not a distinct opposing principle but merely a deficiency or absence ofbeingresulting from a missing connection to the Good.[159]
In ancientIndian philosophy, the idea that people are trapped in acycle of rebirthsarose around 600 BCE.[161]Many traditions adopted it, arguing that liberation from this cycle is the highest good.[162]Hindu philosophydistinguishes thefour fundamental valuesofduty,economic wealth,sensory pleasure, andliberation.[163]ManyHindu schools of thoughtprioritize the value of liberation.[164]A similar outlook is found in ancientBuddhist philosophy, starting between the sixth and the fifth centuries BCE, where the cessation ofsufferingthrough the attainment ofNirvanais considered the ultimate goal.[165]Inancient China,Confucius(c.551–479 BCE)[166]explored the role ofself-cultivationin leading a virtuous life, viewinggeneral benevolence towards humanityas the supreme virtue.[160]In comparing the highest virtue to water,Laozi(6th century BCE)[u]emphasized the importance of living in harmony with thenatural order of the universe.[168]
Religious teachings influenced value theory in themedieval period. EarlyChristian thinkers, such asAugustine of Hippo(354–430 CE),[170]adapted the theories of Plato and Plotinus into a religious framework. They identified God as the ultimate source of existence and goodness, seeing evil as a mere lack or privation of good.[171]Drawing onAristotelianism, Christian philosopherThomas Aquinas(1224–1274 CE)[172]said that communion with the divine, achieved through abeatific visionof God, is thehighest endof humans.[173]InArabic–Persian philosophy,al-Farabi(c.878–950 CE)[174]asserted that the supreme form of human perfection is an intellectual happiness, reachable in theafterlifeby developing the intellect to its fullest potential.[175]Avicenna(980–1037 CE)[176]also regarded the intellect as the highest human faculty. He thought that a contemplative life prepares humans for the greatest good, which is only attained in the afterlife when humans are free from bodily distractions.[177]In Indian philosophy,Adi Shankara(c.700–750 CE)[178]taught that liberation, the highest human end, is reached by realizing that theselfis the same asultimate realityencompassing all of existence.[169]In Chinese thought, the earlyneo-ConfucianphilosopherHan Yu(768–824 CE) identified the sage as an ideal role model who, through self-cultivation, achieves personal integrity expressed in harmony between theory and action in daily life.[179]
In theearly modern period,Thomas Hobbes(1588–1679)[180]understood values as subjective phenomena that depend on a person's interests. He examined how the interests of individuals can be aggregated to guide political decisions.[181]David Hume(1711–1776)[182]agreed with Hobbes's subjectivism, exploringhow values differ from objective facts.[183]Immanuel Kant(1724–1804)[184]asserted that the highest good is happiness in proportion to moral virtue. He emphasized the primacy of virtue by respecting the moral law and the inherent value of people, adding that moral virtue is ideally, but not always, accompanied by personal happiness.[185]Jeremy Bentham(1748–1832)[186]andJohn Stuart Mill(1806–1873)[187]formulatedclassical utilitarianism, combining ahedonisttheory about value with aconsequentialisttheory about right action.[188]Hermann Lotze(1817–1881)[189]developed a philosophy of values, holding that values make the world meaningful as an ordered whole centered around goodness.[190]Influenced by Lotze, theneo-KantianphilosopherWilhelm Windelband(1848–1915)[191]understood philosophy as a theory of values, claiming that universal values determine the principles that all subjects should follow, including the norms of knowledge and action.[192]Friedrich Nietzsche(1844–1900)[193]held that values are human creations. He criticized traditional values in general and Christian values in particular, calling for arevaluation of all valuescentered on life-affirmation, power, and excellence.[194]
In the early 20th century,PragmatistphilosopherJohn Dewey(1859–1952)[195]defended axiologicalnaturalism. He distinguished values from value judgments, adding that the skill of correct value assessment must be learned through experience.[196][v]G. E. Moore(1873–1958)[198]developed and refined various axiological concepts, such as organic unity and the contrast between intrinsic and extrinsic value. He defendednon-naturalismabout the nature of values andintuitionismabout the knowledge of values.[199]W. D. Ross(1877–1971)[200]accepted and further elaborated on Moore's intuitionism, using it to formulate an axiological pluralism.[201][w]R. B. Perry(1876–1957)[203]andD. W. Prall(1886–1940)[204]articulated systematic theories of value based on the idea that values originate in affective states such as interest and liking.[205]Robert S. Hartman(1910–1973)[206]developed formal axiology, saying that values measure the level to which a thing embodies its ideal concept.[207]A. J. Ayer(1910–1989)[208]proposed anti-realism about values, arguing thatvalue statements merely expressthe speaker's approval or disapproval.[209]A different type of anti-realism, introduced byJ. L. Mackie(1917–1981),[210]suggests thatall value assertions are falsesince no values exist.[211]G. H. von Wright(1916–2003)[212]provided aconceptual analysisof the termgoodby distinguishing different meanings or varieties of goodness, such as the technical goodness of a good driver and the hedonic goodness of a good meal.[213]
Incontinental philosophy,Franz Brentano(1838–1917)[215]formulated an early version of the fitting-attitude theory of value, saying that a thing is good if it is fitting to have a positive attitude towards it, such as love.[214]In the 1890s, his studentsAlexius Meinong(1853–1920)[216]andChristian von Ehrenfels(1859–1932)[217]conceived the idea of a general theory of values.[218]Edmund Husserl(1859–1938),[216]another of Brentano's students, developedphenomenologyand applied this approach to the study of values.[219]Following Husserl's approach,Max Scheler(1874–1928) andNicolai Hartmann(1882–1950) each proposed a comprehensive system ofaxiological ethics.[220]Asserting that values have objective reality, they explored how different value types form a hierarchy and examined the problems of value conflicts and right decisions from this hierarchical perspective.[221]Martin Heidegger(1889–1976)[222]criticized value theory, claiming that it rests on a mistakenmetaphysicalperspective by understanding values as aspects of things.[223]ExistentialistphilosopherJean-Paul Sartre(1905–1980)[224]suggested that values do not exist by themselves but are actively created, emphasizing the role of humanfreedom, responsibility, andauthenticityin the process.[225]
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Bradley's regressis a philosophical problem concerning the nature ofrelations. It is named afterF. H. Bradleywho discussed the problem in his 1893 bookAppearance and Reality. It bears a close kinship to the issue of theunity of the proposition.
Bradley raises the problem while discussing thebundle theoryof objects, according to which anobjectis merely a "bundle" ofproperties. This theory raises the question of how the various properties that together comprise an object are related when they in fact comprise an object. More generally, the question that arises is what has to be the case for any two things to be related. Bradley's Regress appears to show that the notion of two things being related generates aninfinite regress.
Suppose, for example, thatarespectsb. Thisstate of affairsseems to involve three things:a,b, and the relation of respecting. For the state of affairs ofarespectingbto obtain, it doesn't, however, suffice that these three things (a,b, and the relation of respecting) exist. They must also be related in some way. What is required, we might say, is thataandb"stand in" the relation of respecting. But now we seem to have another state of affairs: the state of affairs ofaandbstanding in the relation of respecting. This state of affairs in turn seems to involve four things:a,b, the relation of respecting, and the relation of standing in. Again, however, for it to be the case thataandbstand in the relation of respecting, it doesn't suffice that these four items exist. They must also be related in some way. What is required, we might now say, is thata,b, and the relation of respecting stand in the relation of standing in. And so on,ad infinitum.
InAppearance and Reality, Bradley seems to conclude that the regress should lead us to abandon the idea that relations are "independently real". One way to take this suggestion is as recommending that in the case of a respecting b, we are dealing with a state of affairs that has only two constituents: a and b. It does not, in addition, involve a third item, "the relation of respecting", to which a and b must then bear some further relation ("standing in").
A different option is to accept that the regress is real, but to deny that it is avicious regress.
A third option, taken byP.F. StrawsonandGustav Bergmann, is to deny the proposition that instantiation is a relation.Gottlob Fregewent even further by rejecting instantiation altogether.William F. Vallicellacriticized both options; according to Vallicella, both options fail because they cannot explain why objects and properties are connected.[1]
Michael Della Roccauses a version of Bradley's regress to argue in favor of strictmonism, which denies that relations or distinctions are intelligible. On his view, "if we are to retain the notion of substance or being at all, then, instead of individuated, differentiated substances or beings, we should accept only undifferentiated substance or being that stands in no relations of distinction, either internal or external. There is simply substance or being. Similarly, there is simply action, there is simply knowledge, there is simply meaning. And, of course, there is no distinction between being, action, knowledge, and meaning."[2]
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Thechicken or the eggcausalitydilemmais commonly stated as the question, "which came first: thechickenor theegg?" The dilemma stems from the observation that all chickens hatch from eggs and all chicken eggs are laid by chickens. "Chicken-and-egg" is a metaphoric adjective describing situations where it is not clear which of two events should be considered thecauseand which should be considered theeffect, to express a scenario ofinfinite regress, or to express the difficulty of sequencing actions where each seems to depend on others being done first.Plutarchposed the question as a philosophical matter in his essay "The Symposiacs", written in the 1st century CE.[1][2]
The question represents an ancient folk paradox addressing the problem of origins andfirst cause.[3]Aristotle, writing in the fourth century BCE, concluded that this was an infinite sequence, with no true origin.[3]Plutarch, writing four centuries later, specifically highlighted this question as bearing on a "great and weighty problem (whether the world had a beginning)".[4]In the fifth century CE,Macrobiuswrote that while the question seemed trivial, it "should be regarded as one of importance".[4]
By the end of the 16th century, the well-known question seemed to have been regarded as settled in the Christian world, based on the origin story of theBible. In describing the creation of animals, it allows for a first chicken that did not come from an egg. However, laterEnlightenmentphilosophers began to question this solution.[4]Carlo Datiin the mid 17th-century published an erudite satire on the subject.[5]
Although the question is typically used metaphorically,evolutionary biologyprovides literal answers, made possible by the Darwinian principle that speciesevolveover time, and thus that chickens had ancestors that were not chickens,[4]similar to a view expressed by the Greek philosopherAnaximanderwhen addressing the paradox.[3]
If the question refers to eggs in general, the egg came first. The firstamnioteegg – that is, a hard-shelled egg that could be laid on land, rather than remaining in water like the eggs of fish or amphibians – appeared around 312 million years ago.[6]In contrast, chickens are domesticated descendants ofred junglefowland probably arose little more than eight thousand years ago, at most.[7]
If the question refers tochickeneggs specifically, the answer is still the egg, but the explanation is more complicated.[8]The process by which the chicken arose through the interbreeding and domestication of multiple species of wild jungle fowl is poorly understood, and the point at which this evolving organism became a chicken is a somewhat arbitrary distinction. Whatever criteria one chooses, an animal nearly identical to the modern chicken (i.e., aproto-chicken) laid a fertilized egg that had DNA making it a modern chicken due to mutations in the mother's ovum, the father's sperm, or the fertilisedzygote.[9][4][10][11]
It has been suggested that the actions of aproteinfound in modern chicken eggs may make the answer different.[10][11]In the uterus, chickens produce ovocleidin-17 (OC-17), which causes the formation of the thickenedcalcium carbonateshell around their eggs. Because OC-17 is expressed by the hen and not the egg, the bird in which the protein first arose, though having hatched from a non-reinforced egg, would then have laid the first egg having such a reinforced shell: the chicken would have preceded this first 'modern' chicken egg.[10][11]However, the presence of OC-17 or a homolog in other species, such as turkeys[12]and finches[13]suggests that such eggshell-reinforcing proteins are common to all birds,[14]and thus long predate the first chickens.
On 24 July 2024, two men began a chicken-or-egg debate at a party with alcohol in Indonesia. One man became so emotionally enraged he left and returned with a knife, stabbing the other 15 times and killing him. It is uncertain what side of the argument — chicken or egg — the killer took.[15][16][17]
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Theunmoved mover(Ancient Greek:ὃ οὐ κινούμενον κινεῖ,romanized:ho ou kinoúmenon kineî,lit.'that which moves without being moved')[1]orprime mover(Latin:primum movens) is a concept advanced byAristotleas a primarycause(orfirst uncaused cause)[2]or "mover" of all the motion in theuniverse.[3]As is implicit in the name, theunmoved movermoves other things, but is not itself moved by any prior action. In Book 12 (Ancient Greek:Λ) of hisMetaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfectcontemplation: self-contemplation. He also equates this concept with theactive intellect. This Aristotelian concept had its roots incosmologicalspeculations of the earliest Greekpre-Socratic philosophers[4]and became highly influential and widely drawn upon inmedieval philosophyandtheology.St. Thomas Aquinas, for example, elaborated on the unmoved mover in theFive Ways.
Aristotle argues, in Book 8 of thePhysicsand Book 12 of theMetaphysics, "that there must be an immortal, unchanging being, ultimately responsible for all wholeness and orderliness in the sensible world."[5]
In thePhysics(VIII 4–6) Aristotle finds "surprising difficulties" explaining even commonplace change, and in support of his approach of explanation byfour causes, he required "a fair bit of technical machinery".[6]This "machinery" includespotentiality and actuality,hylomorphism,the theory of categories, and "an audacious and intriguing argument, that the bare existence of change requires the postulation of a first cause, an unmoved mover whose necessary existence underpins the ceaseless activity of the world of motion".[7]Aristotle's "first philosophy", orMetaphysics("afterthePhysics"), develops his peculiar theology of the prime mover, asπρῶτον κινοῦν ἀκίνητον: an independent divine eternal unchanging immaterial substance.[8]
Aristotle adopted the geometrical model ofEudoxus of Cnidusto provide a general explanation of the apparent wandering of theclassical planetsarising from uniform circular motions ofcelestial spheres.[9]While the number of spheres in the model itself was subject to change (47 or 55), Aristotle's account ofaether, and ofpotentiality and actuality, required an individual unmoved mover for each sphere.[10]
Simpliciusargues that the first unmoved mover is a cause not only in the sense of being a final cause—which everyone in his day, as in ours, would accept—but also in the sense of being an efficient cause (1360. 24ff.), and his masterAmmoniuswrote a whole book defending the thesis (ibid. 1363. 8–10). Simplicius's arguments include citations ofPlato's views in theTimaeus—evidence not relevant to the debate unless one happens to believe in the essential harmony of Plato and Aristotle—and inferences from approving remarks which Aristotle makes about the role ofNousinAnaxagoras, which require a good deal of reading between the lines. But he does point out rightly that the unmoved mover fits the definition of an efficient cause—"whence the first source of change or rest" (Phys. II. 3, 194b29–30; Simpl. 1361. 12ff.). The examples which Aristotle adduces do not obviously suggest an application to the first unmoved mover, and it is at least possible that Aristotle originated his fourfold distinction without reference to such an entity. But the real question is whether his definition of the efficient cause includes the unmoved mover willy-nilly. One curious fact remains: that Aristotle never acknowledges the alleged fact that the unmoved mover is an efficient cause (a problem of which Simplicius is well aware: 1363. 12–14)...[11]
Despite their apparent function in the celestial model, the unmoved movers were afinal cause,notanefficient causefor the movement of the spheres;[12]they were solely a constant inspiration,[13]and even if taken for an efficient causepreciselydue to being a final cause,[14]the nature of the explanation is purely teleological.[15]
The unmoved mover, if they were anywhere, were said to fill the outer void beyond the sphere of fixed stars:
It is clear then that there is neither place, nor void, nor time, outside the heaven. Hence whatever is there, is of such a nature as not to occupy any place, nor does time age it; nor is there any change in any of the things which lie beyond the outermost motion; they continue through their entire duration unalterable and unmodified, living the best and most self sufficient of lives… From [the fulfilment of the whole heaven] derive the being and life which other things, some more or less articulately but other feebly, enjoy.[16]
The unmoved mover is an immaterial substance (separate and individual beings), having neither parts nor magnitude. As such, it would be physically impossible for them to move material objects of any size by pushing, pulling, or collision. Because matter is, for Aristotle, a substratum in which a potential to change can be actualized, any potentiality must be actualized in an eternal being, but it must not be still because continuous activity is essential for all forms of life. This immaterial form of activity must be intellectual and cannot be contingent upon sensory perception if it is to remain uniform; therefore, eternal substance must think only of thinking itself and exist outside the starry sphere, where even the notion of place is undefined for Aristotle. Their influence on lesser beings is purely the result of an "aspiration or desire,"[17]and each aetheric celestial sphere emulates one of the unmoved movers, as best it can, byuniform circular motion. The first heaven, the outmost sphere of fixed stars, is moved by a desire to emulate the prime mover (first cause),[18][note 1]about whom, the subordinate movers suffer an accidental dependency.
Many of Aristotle's contemporaries complained that oblivious, powerless gods are unsatisfactory.[8]Nonetheless, it was a life which Aristotle enthusiastically endorsed as one most enviable and perfect, the unembellished basis of theology. As the whole of nature depends on the inspiration of the eternal unmoved movers, Aristotle was concerned with establishing the metaphysical necessity of the perpetual motions of the heavens. Through the Sun's seasonal action upon the terrestrial spheres, the cycles of generation and corruption give rise to allnaturalmotion as efficient cause.[15]The intellect,nous, "or whatever else it be that is thought to rule and lead us by nature, and to have cognizance of what is noble and divine" is the highest activity, according to Aristotle (contemplation or speculative thinking,theōríā). It is also the most sustainable, pleasant, self-sufficient activity;[19]something which is aimed at for its own sake. (Unlike politics and warfare, it does not involve doing things we'd rather not do, but rather something we do at our leisure.) This aim is not strictly human: to achieve it means to live following not mortal thoughts but something immortal and divine within humans. According to Aristotle, contemplation is the only type of happy activity that it would not be ridiculous to imagine the gods having. In Aristotle's psychology and biology, the intellect is thesoul(see alsoeudaimonia).
According toGiovanni Reale, the first Unmoved Mover is a living, thinking, andpersonalGod who "possesses the theoretical knowledge alone or in the highest degree...knows not only Himself, but all things in their causes and first principles."[20]
In Book VIII of hisPhysics,[21]Aristotle examines the notions of change or motion, and attempts to show by a challenging argument, that the mere supposition of a 'before' and an 'after', requires afirst principle. He argues that in the beginning, if the cosmos had come to be, its first motion would lack an antecedent state; and, asParmenidessaid, "nothing comes from nothing". Thecosmological argument, later attributed to Aristotle, thereby concludes that God exists. However, if the cosmos had a beginning, Aristotle argued, it would require anefficientfirst cause, a notion that Aristotle took to demonstrate a critical flaw.[22][23][24]
But it is a wrong assumption to suppose universally that we have an adequate first principle in virtue of the fact that something always is so ... ThusDemocritusreduces the causes that explain nature to the fact that things happened in the past in the same way as they happen now: but he does not think fit to seek for a first principle to explain this 'always' ... Let this conclude what we have to say in support of our contention that there never was a time when there was not motion, and never will be a time when there will not be motion.
The purpose of Aristotle'scosmological argumentthat at least one eternal unmoved mover must exist is to support everyday change.[26]
Of things that exist, substances are the first. But if substances can, then all things can perish... and yet, time and change cannot. Now, the only continuous change is that of place, and the only continuous change of place is circular motion. Therefore, there must be an eternal circular motion and this is confirmed by the fixed stars which are moved by the eternal actual substance that's purely actual.[27]
In Aristotle's estimation, an explanation without the temporalactuality and potentialityof an infinite locomotive chain is required for an eternal cosmos with neither beginning nor end: an unmoved eternal substance for whom thePrimum Mobile[note 2]turns diurnally, whereby all terrestrial cycles are driven by day and night, the seasons of the year, the transformation of the elements, and the nature of plants and animals.[10]
Aristotle begins by describing substance, of which he says there are three types: the sensible, subdivided into the perishable, which belongs to physics, and the eternal, which belongs to "another science." He notes that sensible substance is changeable and that there are several types of change, including quality and quantity, generation and destruction, increase and diminution, alteration, and motion. Change occurs when one given state becomes something contrary to it: that is to say, what exists potentially comes to exist actually (seepotentiality and actuality). Therefore, "a thing [can come to be], incidentally, out of that which is not, [and] also all things come to be out of that which is, but ispotentially, and is not actually." That by which something is changed is the mover, that which is changed is the matter, and that into which it is changed is the form.[citation needed]
Substance is necessarily composed of different elements. The proof for this is that there are things that are different from each other and that all things are composed of elements. Since elements combine to form composite substances, and because these substances differ from each other, there must be different elements: in other words, "b or a cannot be the same as ba."[citation needed]
Near the end ofMetaphysics, BookΛ, Aristotle introduces a surprising question, asking "whether we have to suppose one such [mover] or more than one, and if the latter, how many."[28]Aristotle concludes that the number of all the movers equals the number of separate movements, and we can determine these by considering the mathematical science most akin to philosophy, i.e., astronomy. Although the mathematicians differ on the number of movements, Aristotle considers that the number ofcelestial sphereswould be 47 or 55. Nonetheless, he concludes hisMetaphysics, BookΛ, with a quotation from theIliad: "The rule of many is not good; one ruler let there be."[29][30]
John Burnet(1892) noted[31]
The Neoplatonists were quite justified in regarding themselves as the spiritual heirs of Pythagoras; and, in their hands, philosophy ceased to exist as such, and became theology. And this tendency was at work all along; hardly a single Greek philosopher was wholly uninfluenced by it. PerhapsAristotlemight seem to be an exception; but it is probable that, if we still possessed a few such "exoteric" works as theProtreptikosin their entirety, we should find that the enthusiastic words in which he speaks of the "blessed life" in theMetaphysicsand in theEthics(Nicomachean Ethics)were less isolated outbursts of feeling than they appear now. In later days,Apollonios of Tyanashowed in practice what this sort of thing must ultimately lead to. Thetheurgyandthaumaturgyof the late Greek schools were only the fruit of the seed sown by the generation which immediately preceded the Persian War.
Aristotle's principles of being (see section above) influencedAnselm's view of God, whom he called "that than which nothing greater can be conceived." Anselm thought God did not feel emotions such as anger or love but appeared to do so through our imperfect understanding. The incongruity of judging "being" against something that might not exist may have led Anselm to his famous ontological argument for God's existence.
Manymedievalphilosophers used the idea of approaching a knowledge of God through negative attributes. For example, we should not say that God exists in the usual sense of the term; all we can safely say is that God is not nonexistent. We should not say that God is wise, but we can say that God is not ignorant (i.e., in some way, God has some properties of knowledge). We should not say that God is One, but we can state that there is no multiplicity in God's being.
Many later Jewish, Islamic, and Christian philosophers accepted Aristotelian theological concepts. KeyJewish philosophersincludedibn Tibbon,Maimonides, andGersonides, among many others. Their views of God are considered mainstream by many Jews of all denominations, even today. Preeminent among Islamic philosophers who were influenced by Aristotelian theology areAvicennaandAverroes. In Christian theology, the key philosopher influenced by Aristotle was undoubtedlyThomas Aquinas. There had been earlier Aristotelian influences within Christianity (notably Anselm), but Aquinas (who, incidentally, found his Aristotelian influence via Avicenna, Averroes, and Maimonides) incorporated extensive Aristotelian ideas throughout his theology. Through Aquinas and theScholastic Christian theologyof which he was a significant part, Aristotle became "academic theology's great authority in the thirteenth century"[32]and influenced Christian theology that became widespread and deeply embedded. However, notable Christian theologians rejected[a]Aristotelian theological influence, especially the first generation of Christian Reformers,[b]most notablyMartin Luther.[33][34][35]In subsequent Protestant theology, Aristotelian thought quickly reemerged inProtestant scholasticism.
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Inepistemology, theMünchhausen trilemmais athought experimentintended to demonstrate the theoretical impossibility ofprovinganytruth, even in the fields oflogicandmathematics, without appealing to acceptedassumptions. If it is asked how any givenpropositionis known to be true, proof in support of that proposition may be provided. Yet that same question can be asked of that supporting proof and any subsequent supporting proof. The Münchhausen trilemma is that there are only three ways of completing a proof:
Thetrilemma, then, is having to choose one of three equally unsatisfying options.
The nameMünchhausen-Trilemmawas coined by the German philosopherHans Albertin 1968 in reference to atrilemmaof "dogmatismversusinfinite regressversuspsychologism" used byKarl Popper.[1]It is a reference to the problem of "bootstrapping", based on the story ofBaron Munchausen(in German, "Münchhausen") pulling himself and the horse on which he was sitting out of amireby his own hair. Like Munchausen, who cannot make progress because he has no solid ground to stand on, any purported justification of all knowledge must fail, because it must start from a position of no knowledge, and therefore cannot make progress. It must either start with some knowledge, as withdogmatism, not start at all, as withinfinite regress, or be acircular argument, justified only by itself and have no solid foundation, much like the absurdity of Münchhausen pulling himself out of the mire without any independent support. In contemporaryepistemology, advocates ofcoherentismare supposed to accept the "circular" horn of the trilemma;foundationalistsrely on theaxiomaticargument. The view that accepts infinite regress is calledinfinitism.
It is also known asAgrippa's trilemmaor theAgrippan trilemma[2]after a similar argument reported bySextus Empiricus, which was attributed toAgrippa the SkepticbyDiogenes Laërtius. Sextus' argument, however, consists of five (not three) "modes".
Popper, inLogic of Scientific Discovery, mentions neither Sextus nor Agrippa but instead attributes his trilemma to German philosopherJakob Friedrich Fries, leading some to call itFries's trilemmaas a result.[3]
Jakob Friedrich Fries formulated a similar trilemma in which statements can be accepted either:[4]
The first two possibilities are rejected by Fries as unsatisfactory, requiring his adopting the third option.Karl Popperargued that a way to avoid the trilemma was to use an intermediate approach incorporating some dogmatism, some infinite regress, and some perceptual experience.[5]
The argument proposed byHans Albertruns as follows: All of the only three possible attempts to get a certain justification must fail:
An English translation of a quote from the original German text by Albert is as follows:[6]
Here, one has a mere choice between:
Albert stressed repeatedly that there is no limitation of the Münchhausen trilemma to deductive conclusions. The verdict concerns also inductive, causal, transcendental, and all otherwise structured justifications. They all will be in vain.
Therefore, certain justification is impossible to attain. Once having given up the classical idea of certain knowledge, one can stop the process of justification where one wants to stop, presupposed one is ready to start critical thinking at this point, always anew if necessary.
This trilemma rounds off the classical problem ofjustification in the theory of knowledge.
The failure to prove exactly any truth, as expressed by the Münchhausen trilemma, does not have to lead to the dismissal of objectivity, as withrelativism. One example of an alternative is thefallibilismof Karl Popper and Hans Albert, accepting thatcertaintyis impossible but that it is best to get as close as possible to truth while remembering our uncertainty.
In Albert's view, the impossibility of proving any certain truth is not in itself a certain truth. After all, one needs to assume some basic rules of logical inference to derive his result, and in doing so, must either abandon the pursuit of "certain" justification, as above, or attempt to justify these rules, etc. He suggests that it has to be taken as true as long as nobody has come forward with a truth that is scrupulously justified as a certain truth. Several philosophers defied Albert's challenge; his responses to such criticisms can be found in his long addendum to hisTreatise on Critical Reasonand later articles.
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https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma
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"The Unreality of Time" is the best-known philosophical work ofUniversity of CambridgeidealistJ. M. E. McTaggart(1866–1925). In the argument, first published as a journal article inMindin 1908, McTaggart argues that time is unreal because our descriptions of time are either contradictory, circular, or insufficient. A slightly different version of the argument appeared in 1927 as one of the chapters in the second volume of McTaggart's most well known book,The Nature of Existence.[1]
The argument for the unreality of time is popularly treated as a stand-alone argument that does not depend on any significant metaphysical principles (e.g. as argued byC. D. Broad1933 and L. O. Mink 1960). R. D. Ingthorsson disputes this, and argues that the argument can only be understood as an attempt to draw out certain consequences of the metaphysical system that McTaggart presents in the first volume ofThe Nature of Existence.[2]
It is helpful to consider the argument as consisting of three parts. In the first part, McTaggart offers a phenomenological analysis of the appearance of time, in terms of the now famous A-series and B-series (see below for detail). In the second part, he argues that a conception of time as only forming a B-series but not an A-series is an inadequate conception of time because the B-series does not contain any notion of change. The A-series, on the other hand, appears to contain change and is thus more likely to be an adequate conception of time. In the third and final part, he argues that the conception of time forming an A-series is contradictory and thus nothing can be like an A-series. Since the A-series and the B-series exhaust possible conceptions of how reality can be temporal, and neither is adequate, the conclusion McTaggart reaches is that reality is not temporal at all.
To frame his argument, McTaggart initially offers a phenomenological analysis of how time appears to us in experience. McTaggart claims that time appears in the form of events standing in temporal positions, of which there are two kinds. On the one hand events are earlier than and later than each other, and on the other hand they are future, present, and past, and continually changing their position in terms of futurity, presentness, and pastness. The two kinds of temporal positions each represent events in time as standing in a certain order which McTaggart chooses to call the A-series and the B-series. The A-series represents the series of positions determined as future, present, and past, and which continuously pass from the distant future towards the present, and through the present into the remote past. The B-series represents the series of positions determined as earlier than or later than each other.
The determinations of the B-series hold between the events in time, and never change. If an event ever is earlier or later than some other event, then their respective position in time never changes. The determinations of the A-series must hold to something outside of time, something that does not itself change its position in time, but in relation to which the events in time pass from being future, present, and past. McTaggart does not suggest the present, or NOW, as this something whose position in time is fixed and unchanging instead claiming that it will be difficult to identify any such entity (seeing as it is outside time). Broad explains that McTaggart believed that the difficulty of identifying this entity was serious enough in its own right to be persuaded that time is unreal, but thinks that the contradiction of the A-series is still more convincing; for that reason he leaves this particular difficulty aside.[3]
McTaggart argues that the conception of time as only forming a B-series is inadequate because the B-series does not change, andchangeis of the essence of time. If any conception ofrealityrepresents it as changeless, then this is a conception of an atemporal reality. The B-series does not change because earlier-later relationships never change (e.g. the year 2010 is always later than 2000). The events that form a B-series must therefore also form an A-series in order to count as being in time, i.e. they must pass from future to present, and from present to past, in order to change.
The A-series and B-series are not mutually exclusive. If events form an A-series they automatically also form a B-series (anything in the present is earlier than anything in the future, and later than everything past). The question is not therefore whether time forms an A- or a B-series; the question is whether time forms both an A-series and a B-series, or only a B-series.
The proponents of the B-view of time typically respond by arguing that even if events do not change their positions in the B-series, it does not follow that there can be no change in the B-series. This conclusion only follows if it is assumed thateventsare the only entities that can change. There can be change in the B-series in the form of objects bearing different properties at different times.[4]
The suggestion that the B-view of time can escape the problem by appealing to particulars that endure through time and have different properties at different times is controversial in its own right, but it is generally assumed that this is a controversy that has nothing to do with McTaggart. Instead it is treated as a separate issue, the question of whether things can endure in B-time. However, as Ingthorsson has argued, McTaggart does discuss variation in the properties of persistent entities in the 1st Volume of The Nature of Existence, and there comes to the conclusion that variation in the properties of things between times is not change but mere variation between the temporal parts of things.[5]
Attacking the A-series, McTaggart argues that any event in the A-series is past, present,andfuture, which is contradictory in that each of those properties excludes the other two. McTaggart admits that the contradictory nature of the A-series may not be obvious, because it would appear that events never are simultaneously future, present, and past, but only successively so. However, there is a contradiction, he insists, because any attempt to explain why they are future, present, and past,at different timesis (i)circularbecause we would need to describe the successive order of those "different times" again by invoking the determinations of being future, presentorpast, and (ii) this in turn will inevitably lead to avicious infinite regress. Thevicious infinite regressarises, because to explain why the second appeal to future, present, and past, doesn't lead again to the same difficulty all over, we need to explain that they in turn apply successively and thus we must again explain that succession by appeal to future, present, and past, and there is no end to such an explanation. It is the validity of the argument in favour of a vicious infinite regress that has received the most attention in 20th Centuryphilosophy of time.
In the later version of the argument, inThe Nature of Existence,[6]McTaggart no longer advances the circularity objection. This is, arguably, because by then he has come to treattenseas a simple and indefinable notion, and thus cannot contend that the terms need to be explained at all in order to be applied. He now instead argues that even if it is admitted that they are simple and indefinable, and thus can be applied without further analysis, they still lead to contradiction.
Philosophers who favour the B-view of time tend to find McTaggart's argument against the A-series to demonstrate conclusively that tense involves a contradiction.[7]On the other hand, philosophers who favour the A-view of time struggle to see why the argument should be considered to have any force. Two of the most commonly invoked objections are, first, that McTaggart is mistaken about thephenomenologyof time; that he is claiming to see a contradiction in the appearance of time, where none is apparent.[8]Second, that McTaggart is mistaken about thesemanticsof tensed discourse. The idea here is that claims like "Mis present, has been future, and will be past" can only imply a contradiction if it is interpreted as saying thatMis all at once future in the past, present in the present, and also past in the future. This interpretation has been criticized as absurd, on the grounds that expressions such as “has been” and “will be” refer not to howMis at present, but to howMonce was and is no longer, or how it will be but is not yet. Hence it is wrong to think of the expression as an attribution toMof futurity, presentness, and pastness, all at once.[9]
Ingthorsson has argued that the reason for this incommensurability between the proponents of the A-views and B-views is found in the prevailing view that McTaggart's argument is a stand-alone argument. If it is read in that way, the proponents of each view will understand the argument against the background of their respective views of time, and come to incompatible conclusions.[2]Indeed, on closer scrutiny it will be found that McTaggart explicitly claims that in "The Unreality of Time" he is inquiring whetherrealitycan have the characteristics it appears to have in experience (notably being temporal and material) given his earlier conclusions about what reality must really be like in Absolute Reality[clarification needed]. In the introduction to the 2nd Volume ofThe Nature of Existence, he says:
Starting from our conclusionsas to the general nature of the existent, as reached in the earlier Books,we shall have to ask, firstly which of these characteristics can really be possessed by what is existent, and which of them, in spite of theprimâ facieappearance to the contrary, cannot be possessed by anything existent (1927: sect. 295).
And he continues:
It will be possible to show that,having regard to the general nature of the existent as previously determined, certain characteristics, that we consider here for the first time, cannot be true of the existent (1927: sect. 298).
As Ingthorsson notes, the most central result of McTaggart's earlier inquiry into the general nature of the existent in Absolute Reality, an inquiry McTaggart claims is based entirely on a priori arguments (i.e. such as do not rely on any empirical observations), is thatexistenceandrealitycoincide and have no degrees: either something exists and thus is real, or it does not. It immediately follows that for the future and past to be real, they must exist. This is why he interprets the statement "Mis present, has been future, and will be past" as a statement aboutMexisting in the present bearing the property of being present, and existing in the past bearing the property of being future, and existing in the future bearing the property of being past. This interpretation of the expression, if correct, does say thatMis future, present, and past, which is contradictory. However, since it starts from the premise that the future and past can only be real by existing, then it remains to show that this is what the A-view of time assumes.
Having come to the conclusion that reality can neither form an A- nor a B-series, despite appearances to the contrary, then McTaggart finds it necessary to explain what the world is really like such that it appears to be different from what it appears to be. Here is where the C-series comes into play. McTaggart does not say much about the C-series in the original journal article, but inThe Nature of Existencehe devotes six whole chapters to discuss it.[10]
The C-series is rarely given much attention. When it is mentioned, it is described as "an expression synonymous with 'B-series' when the latter is shorn of its temporal connotations".[11]There is a grain of truth in this, but there is more to the C-series than this. Stripping the temporal features from the B-series only gives what the C-series and B-series haveminimally in common, notably the constituents of the series and the formal characteristics of being linear, asymmetric, and transitive. However, the C-series has features that the B-series does not have. The constituents of the C-series arementalstates (a consequence of McTaggart's argument in Ch. 34 ofThe Nature of Existencethat reality cannot really be material), which are related to each other on the basis of their conceptual content in terms of beingincluded inandinclusive of.[12]These atemporal relations are meant to provide what the earlier/later than relation cannot, notably explain why an illusion of change and temporal succession can arise in an atemporal reality.
McTaggart's argument has had an enormous influence on thephilosophy of time. His phenomenological analysis of the appearance of time has been accepted as good and true even by those who firmly deny the end conclusion that time is unreal. For instance, J. S. Findlay (1940) andA. Prior(1967) took McTaggart's phenomenological analysis as their point of departure in the development of moderntense logic.
McTaggart's characterisation of the appearance of time in terms of the A-series and B-series served to sharpen the contrast between the two emerging and rival views of time that we now know as the A-series and B-views of time. The assumption is that the A-view, in accepting the reality oftense, represent time as being like an A-series, and that the B-view, in rejecting the reality oftense, represent time as being like a B-series.
McTaggart’s two objections to the conception of time as forming an A-series and a B-series remain central challenges for contemporary theories of time. Specifically, the A-theory faces the problem of potential contradiction, while the B-theory is often criticized for its difficulty in accounting forchange.
The controversy about McTaggart's argument for the unreality of time continues unabated.[13]
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Parmenides(Greek:Παρμενίδης) is one of thedialoguesofPlato. It is widely considered to be one of the most challenging and enigmatic ofPlato's dialogues.[1][2][3]TheParmenidespurports to be an account of a meeting between the two great philosophers of theEleatic school,ParmenidesandZeno of Elea, and a youngSocrates. The occasion of the meeting was the reading by Zeno of his treatise defending Parmenideanmonismagainst those partisans of plurality who asserted that Parmenides' supposition that there is aonegives rise to intolerable absurdities and contradictions. The dialogue is set during a supposed meeting between Parmenides and Zeno of Elea in Socrates' hometown of Athens. This dialogue is chronologically the earliest of all as Socrates is only nineteen years old here. It is also notable that he takes the position of the student here while Parmenides serves as the lecturer.
Most scholars agree that the dialogue does not record historic conversations, and is most likely an invention by Plato.[4]
The heart of the dialogue opens with a challenge by Socrates to the elder and revered Parmenides and Zeno. Employing his customary method of attack, thereductio ad absurdum, Zeno has argued that if as the pluralists say things are many, then they will be both like and unlike; but this is an impossible situation, for unlike things cannot be like, nor like things unlike. But this difficulty vanishes, says Socrates, if we are prepared to make the distinction betweensensibleson one hand andForms, in which sensibles participate, on the other. Thus one and the same thing can be both like and unlike, or one and many, by participating in the Forms of Likeness and Unlikeness, of Unity and Plurality; I am one man, and as such partake of the Form ofUnity, but I also have many parts and in this respect I partake of the Form ofPlurality. There is no problem in demonstrating that sensible things may have opposite attributes; what would cause consternation, and earn the admiration of Socrates, would be if someone were to show that the Forms themselves were capable of admitting contrary predicates.
At this point, Parmenides takes over as Socrates' interlocutor and dominates the remainder of the dialogue. After establishing that Socrates himself has made the distinction between Forms and sensibles, Parmenides asks him what sorts of Form he is prepared to recognize. Socrates replies that he has no doubt about the existence of mathematical, ethical and aesthetic Forms (e.g., Unity, Plurality,Goodness,Beauty), but is unsure of Forms ofMan,FireandWater; he is almost certain, though admits to some reservations, that undignified objects likehair,mudanddirtdo not have Forms. Parmenides suggests that when he is older and more committed to philosophy, he will consider all the consequences of his theory, even regarding seemingly insignificant objects like hair and mud.
For the remainder of the first part of the dialogue, Parmenides draws Socrates out on certain aspects of the Theory of Forms and in the process brings to bear fiveargumentsagainst the theory.
Argument 1. (130e–131e)If particular things come to partake of the Form of Beauty or Likeness or Largeness they thereby become beautiful or like or large. Parmenides presses Socrates on how precisely many particulars can participate in a single Form. On one hand, if the Form as a whole is present in each of its many instances, then it would as a whole be in numerically different places, and thus separate from itself. Socrates suggests that the Form might be like a day, and thus present in many things at once. Parmenides counters that this would be little different from a single sail covering a number of people, wherein different parts touch different individuals; consequently, the Form is many.
Argument 2. (132a–b)Socrates' reason for believing in the existence of a single Form in each case is that when he views a number of (say) large things, there appears to be a single character which they all share,viz.the character of Largeness. But considering the series of large things; x, y, z, Largeness itself, the latter is also in some sense considered to be large, and if all members of this series partake of a single Form, then there must be another Largeness in which large things and the first Form of Largeness partake. But if this second Form of Largeness is also large, then there should be a third Form of Largeness over the large things and the first two Forms, and so onad infinitum. Hence, instead of there being one Form in every case, we are confronted with an indefinite number. This Largeness regress is commonly known under the name given to it by Aristotle, the famous Third Man Argument (TMA).
Argument 3. (132b–c)To the suggestion that each Form is athoughtexisting in asoul, thus maintaining the unity of the Form, Parmenides replies that a thought must be a thought of something that is a Form. Thus we still have to explain the participation relation. Further, if things share in Forms which are no more than thoughts, then either things consist of thoughts and think, or else they are thoughts, yet do not think.
Argument 4. (132c–133a)Socrates now suggests that the Forms are patterns in nature (παραδείγματαparadeigmata"paradigms") of which the many instances are copies or likenesses. Parmenides argues that if the many instances are like the Forms, then the Forms are like their instances. Yet if things are like, then they come to be like by participating in Likeness; therefore Likeness is like the likeness in concrete things, and another regress is generated.
Argument 5. (133a–134e)Called the "great difficulty [ἀπορία]"(133a) by Parmenides, the theory of Forms arises as a consequence of the assertion of the separate existence of the Forms. Forms do not exist in our world but have their being with reference to one another in their own world. Similarly, things of our world are related among themselves, but not to Forms. Just as Mastership has its being relative to Slavery, so mastership in our world has its being relative to slavery in our world. No terrestrial master is master of Slave itself, and no terrestrial master-slave relation has any relationship to the ideal Master-Slave relation. And so it is withknowledge. All our knowledge is such with respect to our world, not to the world of the Forms, while ideal Knowledge is knowledge of the things not of our world but of the world of the Forms. Hence, we cannot know the Forms. What is more, the gods who dwell in the divine world, can have no knowledge of us, and nor can their ideal mastership rule us.
In spite of Socrates' inability to defend the theory against Parmenides' arguments, in the following transitional section of the dialogue Parmenides himself appears to advocate the theory. He insists that without Forms there can be no possibility of dialectic, and that Socrates was unable to uphold the theory because he has been insufficiently exercised. There follows a description of the kind of exercise, or training, that Parmenides recommends.
The remainder of the dialogue is taken up with an actual performance of such an exercise, where a youngAristoteles(later a member of theThirty Tyrants, not to be confused with Plato's eventual studentAristotle), takes the place of Socrates as Parmenides' interlocutor.
This difficult second part of the dialogue is generally agreed to be one of the most challenging, and sometimes bizarre, pieces in the whole of the Platonic corpus. One of the second part is "whether Plato was committed to any of the arguments developed in the second part of the dialogue."[5]It consists of an unrelenting series of difficult and subtle arguments, where the exchange is stripped of all but the bare essentials of the arguments involved. Gone are the drama and colour we are accustomed to from earlier dialogues.
The second part of the dialogue can be divided thus:
Hypothesis/Deduction n. 1 (137c-142a): If it is one. The one cannot be made up of parts, because then the one would be made of many. Nor can it be a whole, because wholes are made of parts. Thus the one has no parts and is not a whole. It has not a beginning, a middle nor an end because these are parts, it is therefore unlimited. It has no shape because it is neither linear nor circular: a circle has parts all equidistant from the centre, but the one has no parts nor a centre; It is not a line because a line has a middle and two extremes, which the one cannot have. Thus the one has no shape. The one cannot be in anything nor in itself. If it was in another it would be all surrounded and by what it is inside and would be touched at many parts by what contains it, but the one has no parts and thus cannot be inside something else. If it were in itself it would contain itself, but if it is contained then it is different from what contains it and thus the one would be two. The one cannot move because movement is change or change in position. It cannot change because it has no parts to change. If it moves position it moves either circularly or linearly. If it spins in place its outer part revolves around its middle but the one has neither. If it moves its position it moves through something else, which it cannot be inside. Thus the one does not move. The one must be itself and cannot be different from it. The one does not take part in the flowing of time so it is imperishable.
Hypothesis/Deduction n. 2 (142b–155e): If the one is. The one is, it must be and it is part of being. The one is part of being and vice versa. Being is a part of the one, the one is a whole that is a group of sections. The one does not participate in the being, so it must be a single part. Being is unlimited and is contained in everything, however big or small it is. So, since the one is part of being, it is divided into as many parts as being, thus it is unfinished. The parts are themselves sections of a whole, the whole is delimited, confirming the presence of a beginning, a centre, and an end. Therefore, since the centre is itself at the same distance from the beginning and the end, the one must have a form: linear, spherical, or mixed. If the whole is in some of its parts, it will be the plus into the minus, and different from itself. The one is also elsewhere, it is stationary and in movement at the same time.
The Appendix to the First Two Deductions 155e–157b
Hypothesis/Deduction n. 3 (157b–159b): If the one is not. If the one is not it participates in everything different from it, so everything is partially one. Similarity, dissimilarity, bigness, equality and smallness belong to it since the one is similar to itself but dissimilar to anything that is, but it can be big or small as regards dissimilarity and equal as concerns similarity. So the one participates of non-being and also of being because you can think of it. Therefore, the one becomes and perishes and, since it participates of non-being, stays. The one removes from itself the contraries so that it is unnameable, not disputable, not knowable or sensible or showable. The other things appear one and many, limited and unlimited, similar and dissimilar, the same and completely different, in movement and stationary, and neither the first nor the latter thing since they are different from the one and other things. Eventually they are not. So if the one is not, being is not.
A satisfactory characterisation of this part of the dialogue has eluded scholars since antiquity. Many thinkers have tried, among themCornford,Russell,Ryle, andOwen; but few would accept without hesitation any of their characterisations as having got to the heart of the matter. Recent interpretations of the second part have been provided by Miller (1986), Meinwald (1991), Sayre (1996), Allen (1997), Turnbull (1998), Scolnicov (2003), and Rickless (2007). It is difficult to offer even a preliminary characterisation, since commentators disagree even on some of the more rudimentary features of any interpretation. Benjamin Jowett did maintain in the introduction to his translation of the book that the dialogue was certainly not a Platonic refutation of the Eleatic doctrine. In fact, it could well be an Eleatic assessment of the theory of Forms.[6]It might even mean that the Eleatic monist doctrine wins over the pluralistic contention of Plato.[citation needed]The discussion, at the very least, concerns itself with topics close to Plato's heart in many of the later dialogues, such as Being, Sameness, Difference, and Unity; but any attempt to extract a moral from these passages invites contention.
The structure of the remainder of the dialogue:
The Fourth Deduction 159b–160b
The Fifth Deduction 160b–163b
The Sixth Deduction 163b–164b
The Seventh Deduction 164b–165e
The Eighth Deduction 165e–166c
Plato'stheory of Forms, as it is presented in such dialogues as thePhaedo,Republicand the first part of theParmenides, seems committed to the following principles: "F" stands for anyForm("appearance, property")—formais a Boethian translation for εἶδος (eidos), which is the word that Plato used. Plato, in theParmenides, uses the example "greatness" (μέγεθος) for "F-ness"; Aristotle uses the example "man".[7]
However, the TMA shows that these principles are mutually contradictory, as long as there is a plurality of things that are F:
(In what follows, μέγας [megas; "great"] is used as an example; however, the argumentation holds for any F.)
Begin, then, with the assumption that there is a plurality of great things, say (A, B, C). By one-over-many, there is a form of greatness (say, G1) by virtue of partaking of which A, B, and C are great. By self-predication, G1 is great.
But then we can add G1 to (A, B, C) to form a new plurality of great things: (A, B, C, G1). By one-over-many, there is a form of greatness (say, G2) by virtue of partaking of which A, B, C, and G1 are great. But in that case G1 partakes of G2, and by Non-Self-Partaking, G1 is not identical to G2. So there are at least two forms of greatness, G1 and G2. This already contradicts Uniqueness, according to which there is exactly one (and hence no more than one) form of greatness.
But it gets worse for the theory of Forms. For by Self-Predication, G2 is great, and hence G2 can be added to (A, B, C, G1) to form a new plurality of great things: (A, B, C, G1, G2). By One-Over-Many, there is a form of greatness (say, G3) by virtue of partaking of which A, B, C, G1, and G2 are great. But in that case G1 and G2 both partake of G3, and by Non-Self-Partaking, neither of G1 and G2 is identical to G3. So there must be at least three forms of greatness, G1, G2, and G3.
Repetition of this reasoning shows that there is an infinite hierarchy of forms of greatness, with each form partaking of the infinite number of forms above it in the hierarchy. According to Plato, anything that partakes of many things must itself be many. So each form in the infinite hierarchy of forms of greatness is many. But then, given Purity and One/Many, it follows that each form in the infinite hierarchy of forms of greatness is not one. This contradicts Oneness.
The third man argument was furthered byAristotle(Metaphysics990b17–1079a13, 1039a2;Sophistic Refutations178b36ff.) who, rather than using the example of "greatness" (μέγεθος), used the example of a man (hence the name of the argument) to explain this objection to the theory, which he attributes to Plato; Aristotle posits that if a man is a man because he partakes in the form of man, then a third form would be required to explain how man and the form of man are both man, and so on,ad infinitum.
TheParmenideswas the frequent subject of commentaries byNeoplatonists. Important examples include those ofProclusand ofDamascius, and an anonymous 3rd or 4th commentary possibly due toPorphyry. The 13th century translation of Proclus' commentary by Dominican friarWilliam of Moerbekestirred subsequent medieval interest (Klibansky, 1941). In the 15th century, Proclus' commentary influenced the philosophy ofNicolas of Cusa, and NeoplatonistsGiovanni Pico della MirandolaandMarsilio Ficinopenned major commentaries. According to Ficino:
While Plato sprinkled the seeds of all wisdom throughout all his dialogues, yet he collected the precepts of moral philosophy in the books on theRepublic, the whole of science in theTimaeus, and he comprehended the whole of theology in theParmenides. And whereas in the other works he rises far above all other philosophers, in this one he seems to surpass even himself and to bring forth this work miraculously from the adytum of the divine mind and from the innermost sanctum of philosophy. Whosoever undertakes the reading of this sacred book shall first prepare himself in a sober mind and detached spirit, before he makes bold to tackle the mysteries of this heavenly work. For here Plato discusses his own thoughts most subtly: how the One itself is the principle of all things, which is above all things and from which all things are, and in what manner it is outside everything and in everything, and how everything is from it, through it, and toward it. (in Klibansky, 1941)
Some scholars (includingGregory Vlastos) believe that the third man argument is a "record of honest perplexity". Other scholars think that Plato means us to reject one of the premises that produces the infinite regress (namely, One-Over-Many, Self-Predication, or Non-Self-Partaking). But it is also possible to avoid the contradictions by rejecting Uniqueness and Purity (while accepting One-Over-Many, Self-Predication, and Non-Self-Partaking).
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"What the Tortoise Said to Achilles",[1]written byLewis Carrollin 1895 for the philosophical journalMind,[1]is a brief allegorical dialogue on the foundations oflogic.[1]The titlealludesto one ofZeno's paradoxes of motion,[2]in whichAchillescould never overtake thetortoisein a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into aninfinite regression.[1]
The discussion begins by considering the following logical argument:[1][3]
The tortoise accepts premisesAandBas true but not the hypothetical:
The Tortoise claims that it is not "under any logical necessity to acceptZas true". The tortoise then challenges Achilles to force it logically to acceptZas true. Instead of searching the tortoise’s reasons for not acceptingC, Achilles asks it to acceptC, which it does. After which, Achilles says:
The tortoise responds, "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"[1][3]
Again, instead of requesting reasons for not acceptingD, he asks the tortoise to acceptD. And again, it is "quite willing to grant it",[1][3]but it still refuses to accept Z. It then tells Achilles to write into his book,
Following this, the Tortoise says: "until I’ve granted that [i.e.,E], of course I needn’t grant Z. So it's quite a necessary step".[1]With a touch of sadness, Achilles sees the point.[1][3]
The story ends by suggesting that the list of premises continues to grow without end, but without explaining the point of the regress.[1][3]
Lewis Carroll was showing that there is a regressive problem that arises frommodus ponensdeductions.
Or, in words: propositionP(is true) impliesQ(is true), and givenP, thereforeQ.
The regress problem arises because a prior principle is required to explain logical principles, heremodus ponens, and oncethatprinciple is explained,anotherprinciple is required to explainthatprinciple. Thus, if the argumentative chain is to continue, the argument falls into infinite regress. However, if a formal system is introduced wherebymodus ponensis simply arule of inferencedefined within the system, then it can be abided by simply by reasoning within the system. That is not to say that the user reasoning according to this formal system agrees with these rules (consider, for example, theconstructivist's rejection of thelaw of the excluded middleand thedialetheist's rejection of thelaw of noncontradiction). In this way, formalising logic as a system can be considered as a response to the problem of infinite regress:modus ponensis placed as a rule within the system, the validity ofmodus ponensis eschewed without the system.
In propositional logic, the logical implication is defined as follows:
P implies Q if and only if the propositionnot P or Qis atautology.
Hencemodus ponens, [P ∧ (P → Q)] ⇒ Q, is a valid logical conclusion according to the definition of logical implication just stated. Demonstrating the logical implication simply translates into verifying that the compound truth table produces a tautology. But the tortoise does not accept on faith the rules of propositional logic that this explanation is founded upon. He asks that these rules, too, be subject to logical proof. The tortoise and Achilles do not agree on any definition of logical implication.
In addition, the story hints at problems with the propositional solution. Within the system of propositional logic, no proposition or variable carries any semantic content. The moment any proposition or variable takes on semantic content, the problem arises again because semantic content runs outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise.
Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between theconditional connectiveand theimplication relation. These logicians use the phrasenot p or qfor the conditional connective and the termimpliesfor an asserted implication relation.
Several philosophers have tried to resolve Carroll's paradox.Bertrand Russelldiscussed the paradox briefly in§ 38 ofThe Principles of Mathematics(1903), distinguishing betweenimplication(associated with the form "ifp, thenq"), which he held to be a relation betweenunassertedpropositions, andinference(associated with the form "p, thereforeq"), which he held to be a relation betweenassertedpropositions; having made this distinction, Russell could deny that the Tortoise's attempt to treatinferringZfromAandBas equivalent to, or dependent on, agreeing to thehypothetical"IfAandBare true, thenZis true."
Peter Winch, aWittgensteinianphilosopher, discussed the paradox inThe Idea of a Social Science and its Relation to Philosophy(1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learningto dosomething" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the properapplicationof rules governing a form of human activity cannot itself be summed up with a set offurtherrules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).
Carroll's dialogue is apparently the first description of an obstacle toconventionalismabout logical truth,[4]later reworked in more sober philosophical terms byW.V.O. Quine.[5]
Lewis Carroll (April 1895). "What the Tortoise Said to Achilles".Mind.IV(14):278–280.doi:10.1093/mind/IV.14.278.
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Zeno's paradoxesare a series ofphilosophicalargumentspresented by theancient GreekphilosopherZeno of Elea(c. 490–430 BC),[1][2]primarily known through the works ofPlato,Aristotle, and later commentators likeSimplicius of Cilicia.[2]Zeno devised these paradoxes to support his teacherParmenides's philosophy ofmonism, which posits that despite our sensory experiences, reality is singular and unchanging. The paradoxes famously challenge the notions of plurality (theexistenceof many things), motion, space, and time by suggesting they lead tological contradictions.
Zeno's work, primarily known fromsecond-hand accountssince hisoriginal textsare lost, comprises forty "paradoxes of plurality," which argue against thecoherenceof believing in multiple existences, and several arguments against motion and change.[2]Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility inspaceandtime.[1][2]In this paradox, Zeno argues that a swift runner likeAchillescannot overtake a slower movingtortoisewith a head start, because thedistancebetween them can be infinitely subdivided, implying Achilles would require aninfinitenumber of steps to catch the tortoise.[1][2]
These paradoxes have stirred extensive philosophical and mathematical discussion throughouthistory,[1][2]particularly regarding the nature of infinity and the continuity of space and time. Initially,Aristotle's interpretation, suggesting a potential rather than actual infinity, was widely accepted.[1]However, modern solutions leveraging the mathematical framework ofcalculushave provided a different perspective, highlighting Zeno's significant early insight into the complexities of infinity and continuous motion.[1]Zeno's paradoxes remain a pivotal reference point in the philosophical and mathematical exploration of reality, motion, and the infinite, influencing both ancient thought and modern scientific understanding.[1][2]
The origins of the paradoxes are somewhat unclear, but they are generally thought to have been developed to supportParmenides' doctrine ofmonism, that all of reality is one, and thatall change is impossible, that is, that nothing everchanges in locationor in any other respect.[1][2]Diogenes Laërtius, citingFavorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.[3]Modern academicsattribute the paradox to Zeno.[1][2]
Many of these paradoxes argue that contrary to the evidence of one's senses,motionis nothing but anillusion.[1][2]InPlato'sParmenides(128a–d), Zeno is characterized as taking on the project of creating theseparadoxesbecause other philosophers claimed paradoxes arise when considering Parmenides' view. Zeno's arguments may then be early examples of a method of proof calledreductio ad absurdum, also known asproof by contradiction. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one."[4]Plato hasSocratesclaim that Zeno and Parmenides were essentially arguing exactly the same point.[5]They are also credited as a source of thedialecticmethod used by Socrates.[6]
Some of Zeno's nine surviving paradoxes (preserved inAristotle'sPhysics[7][8]andSimplicius'scommentary thereon) are essentially equivalent to one another. Aristotle offered a response to some of them.[7]Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite.[9]However, none of the original ancient sources has Zeno discussing the sum of any infinite series.Simpliciushas Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding thesum, but rather withfinishinga task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?[10][11][12][13]
Three of the strongest and most famous—that of Achilles and the tortoise, theDichotomyargument, and that of an arrow in flight—are presented in detail below.
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
SupposeAtalantawishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be represented as:
This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.[14]
This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be anillusion.[15]
This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in anasymptote. It is also known as theRace Courseparadox.
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
In the paradox ofAchilles and the tortoise,Achillesis in a footrace with a tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy.[16]It lacks, however, the apparent conclusion of motionlessness.
If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.[17]
In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that at any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.[18]It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.
Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.[19]
Aristotle gives three other paradoxes.
From Aristotle:
If everything that exists has a place, place too will have a place, and so onad infinitum.[20]
Description of the paradox from theRoutledge Dictionary of Philosophy:
The argument is that a single grain ofmilletmakes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion.[21]
Aristotle's response:
Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.[22]
Description from Nick Huggett:
This is aParmenideanargument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.[23]
From Aristotle:
... concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.[24]
An expanded account of Zeno's arguments, as presented by Aristotle, is given inSimplicius'scommentaryOn Aristotle's Physics.[25][2][1]
According to Angie Hobbs of The University of Sheffield, this paradox is intended to be considered together with the paradox of Achilles and the Tortoise, problematizing the concept of discrete space & time where the other problematizes the concept of infinitely divisible space & time.[26]
According toSimplicius,Diogenes the Cynicsaid nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions.[25][2]To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Throughout history several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.
Aristotle(384 BC–322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.[27][failed verification][28]Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").[29]Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."[30]Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."[31][32][33]
Some mathematicians and historians, such asCarl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which moderncalculusprovides a mathematical solution.[34]Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With theepsilon-deltadefinition oflimit,WeierstrassandCauchydeveloped a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.[35][36]
Somephilosophers, however, say that Zeno's paradoxes and their variations (seeThomson's lamp) remain relevantmetaphysicalproblems.[10][11][12]While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown[10]and Francis Moorcroft[11]hold that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."[10]
An alternative conclusion, proposed byHenri Bergsonin his 1896 bookMatter and Memory, is that, while the path is divisible, the motion is not.[37][38]
In 2003, Peter Lynds argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.[39][40][41]Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. Nick Huggett argues that Zeno isassuming the conclusionwhen he says that objects that occupy the same space as they do at rest must be at rest.[19]
Based on the work ofGeorg Cantor,[42]Bertrand Russelloffered a solution to the paradoxes, what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is just change in position over time.[43][44]
Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According toHermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem".[45][46]According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry.Jean Paul Van Bendegemhas argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.[34][47]
In 1977,[48]physicistsE. C. George Sudarshanand B. Misra discovered that the dynamical evolution (motion) of aquantum systemcan be hindered (or even inhibited) throughobservationof thesystem.[49]This effect is usually called the "Quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.[50]
In the field of verification and design oftimedandhybrid systems, the system behaviour is calledZenoif it includes an infinite number of discrete steps in a finite amount of time.[51]Someformal verificationtechniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.[52][53]Insystems designthese behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.[54]
Roughly contemporaneously during theWarring States period(475–221 BCE),ancient Chinesephilosophers from theSchool of Names, a school of thought similarly concerned with logic and dialectics, developed paradoxes similar to those of Zeno. The works of the School of Names have largely been lost, with the exception of portions of theGongsun Longzi. The second of the Ten Theses ofHui Shisuggests knowledge of infinitesimals:That which has no thickness cannot be piled up; yet it is a thousand li in dimension.Among the many puzzles of his recorded in theZhuangziis one very similar to Zeno's Dichotomy:
"If from a stick a foot long you every day take the half of it, in a myriad ages it will not be exhausted."
The Mohist canonappears to propose a solution to this paradox by arguing that in moving across a measured length, the distance is not covered in successive fractions of the length, but in one stage. Due to the lack of surviving works from the School of Names, most of the other paradoxes listed are difficult to interpret.[56]
"What the Tortoise Said to Achilles",[57]written in 1895 byLewis Carroll, describes a paradoxical infinite regress argument in the realm of pure logic. It uses Achilles and the Tortoise as characters in a clear reference to Zeno's paradox of Achilles.[58]
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Originally,fallibilism(fromMedieval Latin:fallibilis, "liable to error") is the philosophical principle thatpropositionscan be accepted even though they cannot be conclusively proven orjustified,[1][2]or that neitherknowledgenorbeliefiscertain.[3]The term was coined in the late nineteenth century by the American philosopherCharles Sanders Peirce, as a response tofoundationalism. Theorists, following Austrian-British philosopherKarl Popper, may also refer to fallibilism as the notion that knowledge might turn out to be false.[4]Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision.[5]Fallibilism is often juxtaposed withinfallibilism.
According to philosopherScott F. Aikin, fallibilism cannot properly function in the absence ofinfinite regress.[6]The term, usually attributed toPyrrhonistphilosopherAgrippa, is argued to be the inevitable outcome of all human inquiry, since every proposition requires justification.[7]Infinite regress, also represented within theregress argument, is closely related to theproblem of the criterionand is a constituent of theMünchhausen trilemma. Illustrious examples regarding infinite regress are thecosmological argument,turtles all the way down, and thesimulation hypothesis. Many philosophers struggle with the metaphysical implications that come along with infinite regress. For this reason, philosophers have gotten creative in their quest to circumvent it.
Somewhere along the seventeenth century, English philosopherThomas Hobbesset forth the concept of "infinite progress". With this term, Hobbes had captured the human proclivity to strive forperfection.[8]Philosophers likeGottfried Wilhelm Leibniz,Christian Wolff, andImmanuel Kant, would elaborate further on the concept. Kant even went on to speculate thatimmortalspecies should hypothetically be able to develop their capacities to perfection.[9]
Already in 350 B.C.E, Greek philosopherAristotlemade a distinction between potential andactual infinities. Based on his discourse, it can be said that actual infinities do not exist, because they are paradoxical. Aristotle deemed it impossible for humans to keep on adding members tofinite setsindefinitely. It eventually led him to refute some ofZeno's paradoxes.[10]Other relevant examples of potential infinities includeGalileo's paradoxand the paradox ofHilbert's hotel. The notion that infinite regress and infinite progress only manifest themselves potentially pertains to fallibilism. According to philosophy professor Elizabeth F. Cooke, fallibilism embraces uncertainty, and infinite regress and infinite progress are not unfortunate limitations on humancognition, but rather necessary antecedents forknowledge acquisition. They allow us to live functional and meaningful lives.[11]
In the mid-twentieth century, several important philosophers began to critique the foundations oflogical positivism. In his workThe Logic of Scientific Discovery(1934), Karl Popper, the founder of critical rationalism, argued that scientific knowledge grows from falsifying conjectures rather than anyinductiveprinciple and that falsifiability is the criterion of a scientific proposition. The claim that all assertions are provisional and thus open to revision in light of newevidenceis widely taken for granted in thenatural sciences.[12]
Furthermore, Popper defended his critical rationalism as anormativeand methodological theory, that explains howobjective, and thus mind-independent, knowledge ought to work.[13]Hungarian philosopherImre Lakatosbuilt upon the theory by rephrasing the problem of demarcation as theproblem of normative appraisal. Lakatos' and Popper's aims were alike, that is finding rules that could justify falsifications. However, Lakatos pointed out that critical rationalism only shows how theories can be falsified, but it omits how our belief in critical rationalism can itself be justified. The belief would require an inductively verified principle.[14]When Lakatos urged Popper to admit that the falsification principle cannot be justified without embracing induction, Popper did not succumb.[15]Lakatos' critical attitude towardsrationalismhas become emblematic for his so calledcritical fallibilism.[16][17]While critical fallibilism strictly opposesdogmatism, critical rationalism is said to require a limited amount of dogmatism.[18][19]Though, even Lakatos himself had been a critical rationalist in the past, when he took it upon himself to argue against the inductivist illusion thataxiomscan be justified by the truth of their consequences.[16]In summary, despite Lakatos and Popper picking one stance over the other, both have oscillated between holding a critical attitude towards rationalism as well as fallibilism.[15][17][18][20]
Fallibilism has also been employed by philosopherWillard V. O. Quineto attack, among other things, the distinction betweenanalytic and synthetic statements.[21]British philosopherSusan Haack, following Quine, has argued that the nature of fallibilism is often misunderstood, because people tend to confuse falliblepropositionswith fallibleagents. She claims that logic is revisable, which means that analyticity does not exist and necessity (ora priority) does not extend to logical truths. She hereby opposes the conviction that propositions in logic are infallible, while agents can be fallible.[22]Critical rationalistHans Albertargues that it is impossible to prove any truth with certainty, not only in logic, but also in mathematics.[23]
InProofs and Refutations: The Logic of Mathematical Discovery(1976), philosopherImre Lakatosimplementedmathematical proofsinto what he called Popperian "critical fallibilism".[24]Lakatos's mathematical fallibilism is the general view that all mathematicaltheoremsare falsifiable.[25]Mathematical fallibilism deviates from traditional views held by philosophers likeHegel, Peirce, and Popper.[16][25]Although Peirce introduced fallibilism, he seems to preclude the possibility of our being mistaken in our mathematical beliefs.[2]Mathematical fallibilism appears to uphold that even though a mathematical conjecture cannot be proven true, we may consider some to be good approximations or estimations of the truth. This so calledverisimilitudemay provide us withconsistencyamidst an inherentincompletenessin mathematics.[26]Mathematical fallibilism differs fromquasi-empiricism, to the extent that the latter does not incorporateinductivism, a feature considered to be of vital importance to the foundations ofset theory.[27]
In thephilosophy of mathematics, a central tenet of fallibilism isundecidability(which bears resemblance to the notion ofisostheneia, or "equal veracity").[25]Two distinct types of the word "undecidable" are currently being applied. The first one relates, most notably, to thecontinuum hypothesis, which was proposed by mathematicianGeorg Cantorin 1873.[28][29]The continuum hypothesis represents a tendency for infinite sets to allow for undecidable solutions — solutions which are true in oneconstructible universeand false in another. Both solutions are independent from the axioms inZermelo–Fraenkel set theorycombined with theaxiom of choice(also called ZFC). This phenomenon has been labeled theindependence of the continuum hypothesis.[30]Both the hypothesis and its negation are thought to be consistent with the axioms of ZFC.[31]Many noteworthy discoveries have preceded the establishment of the continuum hypothesis.
In 1877, Cantor introduced thediagonal argumentto prove that thecardinalityof two finite sets is equal, by putting them into aone-to-one correspondence.[32]Diagonalization reappeared inCantors theorem, in 1891, to show that thepower setof anycountable setmust have strictly higher cardinality.[33]The existence of the power set was postulated in theaxiom of power set; a vital part of Zermelo–Fraenkel set theory. Moreover, in 1899,Cantor's paradoxwas discovered. It postulates thatthere is no set of all cardinalities.[33]Two years later,polymathBertrand Russellwould invalidate the existence of theuniversal setby pointing towardsRussell's paradox, which implies thatno set can contain itself as an element (or member). The universal set can be confuted by utilizing either theaxiom schema of separationor theaxiom of regularity.[34]In contrast to the universal set, a power set does not contain itself. It was only after 1940 that mathematicianKurt Gödelshowed, by applying inter alia thediagonal lemma, that the continuum hypothesis cannot be refuted,[28]and after 1963, that fellow mathematicianPaul Cohenrevealed, through the method offorcing, that the continuum hypothesis cannot be proved either.[30]In spite of the undecidability, both Gödel and Cohen suspected dependence of the continuum hypothesis to be false. This sense of suspicion, in conjunction with a firm belief in the consistency of ZFC, is in line with mathematical fallibilism.[35]Mathematical fallibilists suppose that new axioms, for example theaxiom of projective determinacy, might improve ZFC, but that these axioms will not allow for dependence of the continuum hypothesis.[36]
The second type of undecidability is used in relation tocomputability theory(or recursion theory) and applies not solely to statements but specifically todecision problems; mathematical questions of decidability. Anundecidable problemis a type ofcomputational problemin which there arecountably infinitesets of questions, each requiring aneffective methodto determine whether an output is either "yes or no" (or whether a statement is either "true or false"), but where there cannot be anycomputer programorTuring machinethat will always provide the correct answer. Any program would occasionally give a wrong answer or run forever without giving any answer.[37]Famous examples ofundecidable problemsare thehalting problem, theEntscheidungsproblem, and the unsolvability of theDiophantine equation. Conventionally, an undecidable problem is derived from arecursive set, formulated inundecidable language, and measured by theTuring degree.[38][39]Undecidability, with respect tocomputer scienceandmathematical logic, is also calledunsolvabilityornon-computability.
Undecidability and uncertainty are not one and the same phenomenon. Mathematical theorems which can be formally proved, will, according to mathematical fallibilists, nevertheless remain inconclusive.[40]Take for example proof of the independence of the continuum hypothesis or, even more fundamentally, proof of the diagonal argument. In the end, both types of undecidability add further nuance to fallibilism, by providing these fundamentalthought-experiments.[41]
Fallibilism should not be confused with local or globalskepticism, which is the view that some or all types of knowledge are unattainable.
But the fallibility of our knowledge — or the thesis that all knowledge is guesswork, though some consists of guesses which have been most severely tested — must not be cited in support of scepticism or relativism. From the fact that we can err, and that a criterion of truth which might save us from error does not exist, it does not follow that the choice between theories is arbitrary, or non-rational: that we cannot learn, or get nearer to the truth: that our knowledge cannot grow.
Fallibilism claims that legitimate epistemic justifications can lead to false beliefs, whereasacademic skepticismclaims that no legitimate epistemic justifications exist (acatalepsy). Fallibilism is also different to epoché, a suspension of judgement, often accredited toPyrrhonian skepticism.
Nearly all philosophers today are fallibilists in some sense of the term.[3]Few would claim that knowledge requires absolute certainty, or deny that scientific claims are revisable, though in the 21st century some philosophers have argued for some version of infallibilist knowledge.[42][43][44]Historically, many Western philosophers fromPlatotoSaint AugustinetoRené Descarteshave argued that some human beliefs are infallibly known.John Calvinespoused a theological fallibilism towards others beliefs.[45][46]Plausible candidates for infallible beliefs include logical truths ("Either Jones is a Democrat or Jones is not a Democrat"), immediate appearances ("It seems that I see a patch of blue"), and incorrigible beliefs (i.e., beliefs that are true in virtue of being believed, such as Descartes' "I think, therefore I am"). Many others, however, have taken even these types of beliefs to be fallible.[22]
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Finitismis aphilosophy of mathematicsthat accepts the existence only offinitemathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g.,infinite sets) are accepted as existing.
The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While allnatural numbersare accepted as existing, thesetof all natural numbers is not considered to exist as a mathematical object. Thereforequantificationover infinite domains is not considered meaningful. The mathematical theory often associated with finitism isThoralf Skolem'sprimitive recursive arithmetic.
The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase whenGeorg Cantorin 1874 introduced what is now callednaive set theoryand used it as a base for his work ontransfinite numbers. When paradoxes such asRussell's paradox,Berry's paradoxand theBurali-Forti paradoxwere discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects.
One position was theintuitionistic mathematicsthat was advocated byL. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Another position was endorsed byDavid Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led toHilbert's programof proving bothconsistencyandcompletenessof set theory using finitistic means as this would imply that adding ideal mathematical objects isconservativeover the finitistic part. Hilbert's views are also associated with theformalist philosophy of mathematics. Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due toKurt Gödel'sincompleteness theorems. However,Harvey Friedman'sgrand conjecturewould imply that most mathematical results are provable using finitistic means.
Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work withPaul Bernayssome experts such asTait (1981)have argued thatprimitive recursive arithmeticcan be considered an upper bound on what Hilbert considered finitistic mathematics.[1]
As a result of Gödel's theorems, as it became clear that there is no hope of proving both the consistency and completeness of mathematics, and with the development of seemingly consistentaxiomatic set theoriessuch asZermelo–Fraenkel set theory, most modern mathematicians do not focus on this topic.
In her bookThe Philosophy of Set Theory,Mary Tilescharacterized those who allowpotentially infiniteobjects asclassical finitists, and those who do not allow potentially infinite objects asstrict finitists: for example, a classical finitist would allow statements such as "every natural number has asuccessor" and would accept the meaningfulness ofinfinite seriesin the sense oflimitsof finite partial sums, while a strict finitist would not. Historically, the written history of mathematics was thus classically finitist until Cantor created the hierarchy oftransfinitecardinalsat the end of the 19th century.
Leopold Kroneckerremained a strident opponent to Cantor's set theory:[2]
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.God created the integers; all else is the work of man.
Reuben Goodsteinwas another proponent of finitism. Some of his work involved building up toanalysisfrom finitist foundations.
Although he denied it, much ofLudwig Wittgenstein's writing on mathematics has a strong affinity with finitism.[4]
If finitists are contrasted withtransfinitists(proponents of e.g.Georg Cantor's hierarchy of infinities), then alsoAristotlemay be characterized as a finitist. Aristotle especially promoted thepotential infinityas a middle option between strict finitism andactual infinity(the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinitecardinalandordinalnumbers, which have nothing to do with the things in nature):
But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.
Ultrafinitism(also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.
Towards the end of the 20th centuryJohn Penn Mayberrydeveloped a system of finitary mathematics which he called "Euclidean Arithmetic". The most striking tenet of his system is a complete and rigorous rejection of the special foundational status normally accorded to iterative processes, including in particular the construction of the natural numbers by the iteration "+1". Consequently Mayberry is in sharp dissent from those who would seek to equate finitary mathematics withPeano arithmeticor any of its fragments such asprimitive recursive arithmetic.
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Perspectivism(also calledperspectivalism) is theepistemological principlethatperceptionof andknowledgeof something are always bound to the interpretiveperspectivesof those observing it. While perspectivismdoes notregard all perspectives and interpretations as being of equaltruthorvalue, it holds that no one has access to an absolute view of the world cut off from perspective.[1]Instead, all suchviewingoccurs from some point of view which in turn affects how things are perceived. Rather than attempt todetermine truth by correspondenceto things outside any perspective, perspectivism thus generally seeks to determine truth by comparing and evaluating perspectives among themselves.[1]Perspectivism may be regarded as an early form ofepistemological pluralism,[2]though in some accounts includes treatment ofvalue theory,[3]moral psychology,[4]andrealist metaphysics.[5]
Early forms of perspectivism have been identified in the philosophies ofProtagoras,Michel de Montaigne, andGottfried Leibniz. However, its first major statement is considered to beFriedrich Nietzsche's development of the concept in the 19th century,[2][4]influenced byGustav Teichmüller's use of the term some years prior.[6]For Nietzsche, perspectivism takes the form of a realistantimetaphysics[7]while rejecting both thecorrespondence theory of truthand the notion that thetruth-valueof a belief always constitutes its ultimate worth-value.[3]The perspectival conception ofobjectivityused by Nietzsche sees the deficiencies of each perspective as remediable by anasymptoticstudy of the differences between them. This stands in contrast toPlatonicnotions in which objective truth is seen to reside in a wholly non-perspectival domain.[4]
According to Alexander Nehamas, perspectivism is often misinterpreted as a form ofrelativism, whereby we acknowledge the true virtue of fully rejecting the 'Law of excluded middle' regarding a particular proposition.[3]Lacewing Michael adds that although perspectivism doesn't accede to an objective view of the world that is detached from our subjectivity, our assessment of reality can still approach "objectivity" subjectively and asymptotically.[8]Nehamas also describes how perspectivism does not prohibit someone from holding some interpretations to be definitively true. It only alerts us that we cannot objectively determine the truth from outside our perspective.[3][9]The idea that perspectivism is an absolutely true thesis, is calledweak perspectivismby Brian Lightbody.[9]
The basic principle that things are perceived differently from different perspectives (or that perspective determines one's limited andunprivileged accessto knowledge) has sometimes been accounted as a rudimentary, uncontentious form of perspectivism.[10]The basic practice of comparing contradictory perspectives to one another may also be considered one such form of perspectivism(See also:Intersubjectivity),[11]as may the entirephilosophical problemof how true knowledge is to penetrate one's perspectival limitations.[12]
In Western languages, scholars have found perspectivism in the philosophies ofHeraclitus(c.540–c.480 BCE),Protagoras(c.490–c.420 BCE),Michel de Montaigne[3][13](1533 – 1592 CE), andGottfried Leibniz[2](1646 – 1716 CE). The origins of perspectivism have also been found to lie also withinRenaissancedevelopments inphilosophy of artand its artistic notion ofperspective.[14]In Asian languages, scholars have found perspectivism inBuddhist,[15]Jain,[16]andDaoisttexts.[17]Anthropologists have found a kind of perspectivism in the thinking of someindigenous peoples.[18]Some theologians believeJohn Calvininterpreted various scriptures in a perspectivist manner.[19]
The Western origins of perspectivism can be found in thepre-Socratic philosophiesofHeraclitus[20]andProtagoras.[2]In fact, a major cornerstone ofPlato's philosophy is his rejection and opposition to perspectivism—this forming a principal element of hisaesthetics,ethics,epistemology, andtheology.[21]The antiperspectivism of Plato made him a central target of critique for later perspectival philosophers such as Nietzsche.[22]
Montaigne's philosophy presents in itself a perspectivism less as a doctrinaire position than as a core philosophical approach put into practice. Inasmuch as no one can occupy aGod's-eye view, Montaigne holds that no one has access to a view which is totally unbiased, which does notinterpretaccording to its own perspective. It is instead only the underlyingpsychological biaseswhich view one's own perspective as unbiased.[13]In a passage from his "Of Cannibals", he writes:
Men of intelligence notice more things and view them more carefully, but they [interpret] them; and to establish and substantiate their interpretation, they cannot refrain from altering the facts a little. They never present things just as they are but twist and disguise them to conform to the point of view from which they have seen them; and to gain credence for their opinion and make it attractive, they do not mind adding something of their own, or extending and amplifying.[23]
In his works, Nietzsche makes a number of statements on perspective which at times contrast each other throughout the development of his philosophy. Nietzsche's perspectivism begins by challenging the underlying notions of 'viewing from nowhere', 'viewing from everywhere', and 'viewing without interpreting' as being absurdities.[22]Instead, allviewingis attached to some perspective, and all viewers are limited in some sense to the perspectives at their command.[24]InThe Genealogy of Moralshe writes:
Let us be on guard against the dangerous old conceptual fiction that posited a 'pure, will-less, painless, timeless knowing subject'; let us guard against the snares of such contradictory concepts as 'pure reason', 'absolute spirituality', 'knowledge in itself': these always demand that we should think of an eye that is completely unthinkable, an eye turned in no particular direction, in which the active and interpreting forces, through which alone seeing becomes seeingsomething, are supposed to be lacking; these always demand of the eye an absurdity and a nonsense. There isonlya perspective seeing,onlya perspective knowing; and themoreaffects we allow to speak about one thing, themoreeyes, different eyes, we can use to observe one thing, the more complete will our 'concept' of this thing, our 'objectivity' be.[25]
In this, Nietzsche takes acontextualistapproach which rejects anyGod's-eye viewof the world.[26]This has been further linked to his notion of thedeath of Godand the dangers of a resultingrelativism. However, Nietzsche's perspectivism itself stands in sharp contrast to any such relativism.[3]In outlining his perspectivism, Nietzsche rejects those who claim everything to be subjective, by disassembling the notion of the subject as itself a mere invention and interpretation.[27]He further states that, since the two are mutually dependent on each other, the collapse of the God's-eye view causes also the notion of thething-in-itselfto fall apart with it. Nietzsche views this collapse to reveal, through hisgenealogicalproject, that all that has been considered non-perspectival knowledge, the entire tradition of Western metaphysics, has itself been only a perspective.[24][26]His perspectivism and genealogical project are further integrated into each other in addressing the psychological drives that underlie various philosophical programs and perspectives, as a form of critique.[4]Here, contemporary scholarKen Gemesviews Nietzsche's perspectivism to above all be a principle ofmoral psychology, rejecting interpretations of it as an epistemological thesis outrightly.[4]It is through this method of critique that the deficiencies of various perspectives can be alleviated—through a critical mediation of the differences between them rather than any appeals to the non-perspectival.[4][13]In a posthumously published aphorism fromThe Will to Power, Nietzsche writes:
"Everything is subjective," you say; but even this is interpretation. The "subject" is not something given, it is something added and invented and projected behind what there is.—Finally, is it necessary to posit an interpreter behind the interpretation? Even this is invention, hypothesis.
In so far as the word "knowledge" has any meaning, the world is knowable; but it isinterpretableotherwise, it has no meaning behind it, but countless meanings.—"Perspectivism."
It is our needs that interpret the world; our drives and their For and Against. Every drive is a kind of lust to rule; each one has its perspective that it would like to compel all the other drives to accept as a norm.[27]
While Nietzsche does not plainly reject truth and objectivity, he does reject the notions ofabsolutetruth,externalfacts, andnon-perspectivalobjectivity.[4][22]
Despite receiving much attention withincontemporary philosophy, there is no academic consensus on Nietzsche's conception of truth.[28]While his perspectivism presents a number of challenges regarding the nature of truth, its more controversial element lies in its questioning of thevalueof truth.[3]Contemporary scholars Steven D. Hales and Robert C. Welshon write that:
Nietzsche's writings on truth are among the most elusive and difficult ones in his corpus. One indication of their obscurity is that on an initial reading he appears either blatantly inconsistent in his use of the words 'true' and 'truth', or subject to inexplicable vacillations on the value of truth.[29]
In the 20th century, perspectivism was discussed separately byJosé Ortega y Gasset[30]andKarl Jaspers.[31]Ortega's perspectivism, replaced his previous position that "man is completely social". His reversal is prominent in his workVerdad y perspectiva("Truth and perspective"), where he explained that "each man has a mission of truth" and that what he sees of reality no other eye sees.[32]He explained:
From different positions two people see the same surroundings. However, they do not see the same thing. Their different positions mean that the surroundings are organized in a different way: what is in the foreground for one may be in the background for another. Furthermore, as things are hidden one behind another, each person will see something that the other may not.[33]
Ortega also maintained that perspective is perfected by the multiplication of its viewpoints.[34]He noted that war transpires due to the lack of perspective and failure to see the larger contexts of the actions among nations.[34]Ortega also cited the importance of phenomenology in perspectivism as he argued against speculation and the importance of concrete evidence in understanding truth and reality.[35]In this discourse, he highlighted the role of "circumstance" in finding out the truth since it allows us to understand realities beyond ourselves.[35]
During the 21st century, perspectivism has led a number of developments withinanalytic philosophy[36]andphilosophy of science,[37]particularly under the early influence ofRonald Giere,Jay Rosenberg,Ernest Sosa, and others.[38]This contemporary form of perspectivism, also known as scientific perspectivism, is more narrowly focused than prior forms—centering on the perspectival limitations ofscientific models,theories,observations, and focused interest, while remaining more compatible for example withKantian philosophyand correspondence theories of truth.[38][39]Furthermore, scientific perspecitivism has come to address a number of scientific fields such asphysics,biology,cognitive neuroscience, andmedicine, as well asinterdisciplinarityandphilosophy of time.[38]Studies of perspectivism have also been introduced intocontemporary anthropology, initially through the influence ofEduardo Viveiros de Castroand his research intoindigenous cultures of South America.[18]
Contemporary types of perspectivism include:
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Infinite regressis aphilosophicalconcept to describe a series of entities. Each entity in the series depends on its predecessor, following arecursiveprinciple. For example, theepistemic regressis a series of beliefs in which thejustificationof each belief depends on the justification of the belief that comes before it.
Aninfinite regress argumentis an argument against a theory based on the fact that this theory leads to an infinite regress. For such an argument to be successful, it must demonstrate not just that the theory in question entails an infinite regress but also that this regress isvicious. There are different ways in which a regress can be vicious. The most serious form of viciousness involves acontradictionin the form ofmetaphysical impossibility. Other forms occur when the infinite regress is responsible for the theory in question being implausible or for its failure to solve the problem it was formulated to solve.
Traditionally, it was often assumed without much argument that each infinite regress is vicious but this assumption has been put into question in contemporary philosophy. While some philosophers have explicitly defended theories with infinite regresses, the more common strategy has been to reformulate the theory in question in a way that avoids the regress. One such strategy isfoundationalism, which posits that there is a first element in the series from which all the other elements arise but which is not itself explained this way. Another way iscoherentism, which is based on a holistic explanation that usually sees the entities in question not as a linear series but as an interconnected network.
Infinite regress arguments have been made in various areas of philosophy. Famous examples include thecosmological argumentandBradley's regress.
Aninfinite regressis an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor.[1]This principle can often be expressed in the following form:XisFbecauseXstands inRtoYandYisF.XandYstand for objects,Rstands for a relation andFstands for a property in the widest sense.[1][2]In the epistemic regress, for example, a belief is justified because it is based on another belief that is justified. But this other belief is itself in need of one more justified belief for itself to be justified and so on.[3]Or in the cosmological argument, an event occurred because it was caused by another event that occurred before it, which was itself caused by a previous event, and so on.[1][4]This principle by itself is not sufficient: it does not lead to a regress if there is noXthat isF. This is why an additional triggering condition has to be fulfilled: there has to be anXthat isFfor the regress to get started.[5]So the regress starts with the fact thatXisF. According to the recursive principle, this is only possible if there is a distinctYthat is alsoF. But in order to account for the fact thatYisF, we need to posit aZthat isFand so on. Once the regress has started, there is no way of stopping it since a new entity has to be introduced at each step in order to make the previous step possible.[1]
Aninfinite regress argumentis an argument against a theory based on the fact that this theory leads to an infinite regress.[1][5]For such an argument to be successful, it has to demonstrate not just that the theory in question entails an infinite regress but also that this regress isvicious.[1][4]The mere existence of an infinite regress by itself is not a proof for anything.[5]So in addition to connecting the theory to a recursive principle paired with a triggering condition, the argument has to show in which way the resulting regress is vicious.[4][5]For example, one form ofevidentialismin epistemology holds that a belief is only justified if it is based on another belief that is justified. An opponent of this theory could use an infinite regress argument by demonstrating (1) that this theory leads to an infinite regress (e.g. by pointing out the recursive principle and the triggering condition) and (2) that this infinite regress is vicious (e.g. by showing that it is implausible given the limitations of the human mind).[1][5][3][6]In this example, the argument has a negative form since it only denies that another theory is true. But it can also be used in a positive form to support a theory by showing that its alternative involves a vicious regress.[3]This is how thecosmological argumentfor the existence of God works: it claims that positing God's existence is necessary in order to avoid an infinite regress of causes.[1][4][3]
For aninfinite regress argumentto be successful, it has to show that the involved regress isvicious.[3]Anon-viciousregress is calledvirtuousorbenign.[5]Traditionally, it was often assumed without much argument that each infinite regress is vicious but this assumption has been put into question in contemporary philosophy. In most cases, it is not self-evident whether an infinite regress is vicious or not.[5]Thetruth regressconstitutes an example of an infinite regress that is not vicious: if the proposition "P" is true, then the proposition that "It is true that P" is also true and so on.[4]Infinite regresses pose a problem mostly if the regress concerns concrete objects.Abstract objects, on the other hand, are often considered to be unproblematic in this respect. For example, the truth-regress leads to an infinite number of true propositions or thePeano axiomsentail the existence of infinitely manynatural numbers. But these regresses are usually not held against the theories that entail them.[4]
There are different ways in which a regress can be vicious. The most serious type of viciousness involves acontradictionin the form ofmetaphysical impossibility.[4][1][7]Other types occur when the infinite regress is responsible for the theory in question being implausible or for its failure to solve the problem it was formulated to solve.[4][7]The vice of an infinite regress can be local if it causes problems only for certain theories when combined with other assumptions, or global otherwise. For example, an otherwise virtuous regress is locally vicious for a theory that posits a finite domain.[1]In some cases, an infinite regress is not itself the source of the problem but merely indicates a different underlying problem.[1]
Infinite regresses that involvemetaphysical impossibilityare the most serious cases of viciousness. The easiest way to arrive at this result is by accepting the assumption thatactual infinitiesare impossible, thereby directly leading to a contradiction.[5]This anti-infinitists position is opposed to infinity in general, not just specifically to infinite regresses.[1]But it is open to defenders of the theory in question to deny this outright prohibition on actual infinities.[5]For example, it has been argued that only certain types of infinities are problematic in this way, like infinite intensive magnitudes (e.g. infinite energy densities).[4]But other types of infinities, like infinite cardinality (e.g. infinitely many causes) or infinite extensive magnitude (e.g. the duration of the universe's history) are unproblematic from the point of view of metaphysical impossibility.[4]While there may be some instances of viciousness due to metaphysical impossibility, most vicious regresses are problematic because of other reasons.[4]
A more common form of viciousness arises from the implausibility of the infinite regress in question. This category often applies to theories about human actions, states or capacities.[4]This argument is weaker than the argument from impossibility since it allows that the regress in question is possible. It only denies that it is actual.[1]For example, it seems implausible due to the limitations of the human mind that there are justified beliefs if this entails that the agent needs to have an infinite amount of them. But this is not metaphysically impossible, e.g. if it is assumed that the infinite number of beliefs are onlynon-occurrent or dispositionalwhile the limitation only applies to the number of beliefs one is actually thinking about at one moment.[4]Another reason for the implausibility of theories involving an infinite regress is due to the principle known asOckham's razor, which posits that we should avoid ontological extravagance by not multiplying entities without necessity.[8]Considerations of parsimony are complicated by the distinction between quantitative and qualitative parsimony: concerning how many entities are posited in contrast to how many kinds of entities are posited.[1]For example, thecosmological argumentfor the existence of God promises to increasequantitativeparsimony by positing that there is one first cause instead of allowing an infinite chain of events. But it does so by decreasingqualitativeparsimony: it posits God as a new type of entity.[4]
Another form of viciousness applies not to the infinite regress by itself but to it in relation to the explanatory goals of a theory.[4][7]Theories are often formulated with the goal of solving a specific problem, e.g. of answering the question why a certain type of entity exists. One way how such an attempt can fail is if the answer to the question already assumes in disguised form what it was supposed to explain.[4][7]This is akin to theinformal fallacyofbegging the question.[2]From the perspective of a mythological world view, for example, one way to explain why the earth seems to be at rest instead of falling down is to hold that it rests on the back of a giant turtle. In order to explain why the turtle itself is not in free fall, another even bigger turtle is posited and so on, resulting in a world that isturtles all the way down.[4][1]Despite its shortcomings in clashing with modern physics and due to its ontological extravagance, this theory seems to be metaphysically possible assuming that space is infinite. One way to assess the viciousness of this regress is to distinguish betweenlocalandglobalexplanations.[1]Alocalexplanation is only interested in explaining why one thing has a certain property through reference to another thing without trying to explain this other thing as well. Aglobalexplanation, on the other hand, tries to explain why there are any things with this property at all.[1]So as a local explanation, the regress in the turtle theory is benign: it succeeds in explaining why the earth is not falling. But as a global explanation, it fails because it has to assume rather than explain at each step that there is another thing that is not falling. It does not explain why nothing at all is falling.[1][4]
It has been argued that infinite regresses can be benign under certain circumstances despite aiming at global explanation. This line of thought rests on the idea of thetransmissioninvolved in the vicious cases:[9]it is explained thatXisFbecauseYisFwhere thisFwas somehow transmitted fromYtoX.[1]The problem is that to transfer something, it first must be possessed, so the possession is presumed rather than explained. For example, in trying to explain why one's neighbor has the property of being the owner of a bag of sugar, it is revealed that this bag was first in someone else's possession before it was transferred to the neighbor and that the same is true for this and every other previous owner.[1]This explanation is unsatisfying since ownership is presupposed at every step. In non-transmissive explanations, however,Yis still the reason forXbeingFandYis alsoFbut this is just seen as a contingent fact.[1][9]This line of thought has been used to argue that the epistemic regress is not vicious. From aBayesianpoint of view, for example, justification or evidence can be defined in terms of one belief raising the probability that another belief is true.[10][11]The former belief may also be justified but this is not relevant for explaining why the latter belief is justified.[1]
Philosophers have responded to infinite regress arguments in various ways. The criticized theory can be defended, for example, by denying that an infinite regress is involved.Infinitists, on the other hand, embrace the regress but deny that it is vicious.[6]Another response is to modify the theory in order to avoid the regress. This can be achieved in the form offoundationalismor ofcoherentism.
Traditionally, the most common response isfoundationalism.[1]It posits that there is a first element in the series from which all the other elements arise but which is not itself explained this way.[12]So from any given position, the series can be traced back to elements on the most fundamental level, which the recursive principle fails to explain. This way an infinite regress is avoided.[1][6]This position is well-known from its applications in the field of epistemology.[1]Foundationalist theories of epistemic justification state that besides inferentially justified beliefs, which depend for their justification on other beliefs, there are also non-inferentially justified beliefs.[12]The non-inferentially justified beliefs constitute the foundation on which the superstructure consisting of all the inferentially justified beliefs rests.[13]Acquaintance theories, for example, explain the justification of non-inferential beliefs through acquaintance with the objects of the belief. On such a view, an agent is inferentially justified to believe that it will rain tomorrow based on the belief that the weather forecast told so. They are non-inferentially justified in believing that they are in pain because they are directly acquainted with the pain.[12]So a different type of explanation (acquaintance) is used for the foundational elements.
Another example comes from the field ofmetaphysicsconcerning the problem ofontological hierarchy. One position in this debate claims that some entities exist on a more fundamental level than other entities and that the latter entities depend on or are grounded in the former entities.[14]Metaphysical foundationalismis the thesis that these dependence relations do not form an infinite regress: that there is a most fundamental level that grounds the existence of the entities from all other levels.[1][15]This is sometimes expressed by stating that the grounding-relation responsible for this hierarchy iswell-founded.[15]
Coherentism, mostly found in the field of epistemology, is another way to avoid infinite regresses.[1]It is based on a holistic explanation that usually sees the entities in question not as a linear series but as an interconnected network. For example, coherentist theories of epistemic justification hold that beliefs are justified because of the way they hang together: they cohere well with each other.[16]This view can be expressed by stating that justification is primarily a property of the system of beliefs as a whole. The justification of a single belief is derivative in the sense that it depends on the fact that this belief belongs to a coherent whole.[1]Laurence BonJouris a well-known contemporary defender of this position.[17][18]
Aristotleargued that knowing does not necessitate an infinite regress because some knowledge does not depend on demonstration:
Some hold that owing to the necessity of knowing the primary premises, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premises. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand – they say – the series terminates and there are primary premises, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premises, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premises are true. The other party agrees with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premises is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premises from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition, we maintain that besides scientific knowledge there is its original source which enables us to recognize the definitions.[19][20]
Gilbert Ryleargues in the philosophy of mind thatmind-body dualismis implausible because it produces an infinite regress of "inner observers" when trying to explain how mental states are able to influence physical states.[citation needed]
Media related toInfinite regressat Wikimedia Commons
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Relativismis a family ofphilosophicalviews which deny claims toobjectivitywithin a particular domain and assert that valuations in that domain are relative to the perspective of an observer or the context in which they are assessed.[1]
There are many different forms of relativism, with a great deal of variation in scope and differing degrees of controversy among them.[2]Moral relativismencompasses the differences in moral judgments among people and cultures.[3]Epistemic relativismholds that there are no absolute principles regarding normativebelief,justification, orrationality, and that there are only relative ones.[4]Alethic relativism(also factual relativism) is the doctrine that there are noabsolute truths, i.e., that truth is always relative to some particular frame of reference, such as a language or a culture (cultural relativism), whilelinguistic relativismasserts that a language's structures influence a speaker's perceptions.[5][6]Some forms of relativism also bear a resemblance tophilosophical skepticism.[7]Descriptive relativismseeks to describe the differences among cultures and people without evaluation, while normative relativism evaluates the word truthfulness of views within a given framework.
Anthropological relativismrefers to amethodologicalstance, in which the researcher suspends (or brackets) their own cultural prejudice while trying to understand beliefs or behaviors in their contexts. This has become known asmethodological relativism, and concerns itself specifically with avoidingethnocentrismor the application of one's own cultural standards to the assessment of other cultures.[8]This is also the basis of the so-called "emic" and "etic" distinction, in which:
Philosophical relativism, in contrast, asserts that the truth of a proposition depends on the metaphysical, or theoretical frame, or the instrumental method, or the context in which the proposition is expressed, or on the person, groups, or culture who interpret the proposition.[9]
Methodological relativism and philosophical relativism can exist independently from one another, but most anthropologists base their methodological relativism on that of the philosophical variety.[10]
The concept of relativism also has importance both forphilosophersand foranthropologistsin another way. In general, anthropologists engage in descriptive relativism ("how things are" or "how things seem"), whereas philosophers engage innormativerelativism ("how things ought to be"), although there is some overlap (for example, descriptive relativism can pertain to concepts, normative relativism to truth).
Descriptive relativism assumes that certain cultural groups have different modes of thought, standards of reasoning, and so forth, and it is the anthropologist's task to describe, but not to evaluate the validity of these principles and practices of a cultural group. It is possible for an anthropologist in his or her fieldwork to be a descriptive relativist about some things that typically concern the philosopher (e.g., ethical principles) but not about others (e.g., logical principles). However, the descriptive relativist's empirical claims about epistemic principles, moral ideals and the like are often countered by anthropological arguments that such things are universal, and much of the recent literature on these matters is explicitly concerned with the extent of, and evidence for, cultural or moral or linguistic or human universals.[11]
The fact that the various species of descriptive relativism are empirical claims may tempt the philosopher to conclude that they are of little philosophical interest, but there are several reasons why this is not so. First, some philosophers, notably Kant, argue that certain sorts of cognitive differences between human beings (or even all rational beings) are impossible, so such differences could never be found to obtain in fact, an argument that places a priori limits on what empirical inquiry could discover and on what versions of descriptive relativism could be true. Second, claims about actual differences between groups play a central role in some arguments for normative relativism (for example, arguments for normative ethical relativism often begin with claims that different groups in fact have different moral codes or ideals). Finally, the anthropologist's descriptive account of relativism helps to separate the fixed aspects of human nature from those that can vary, and so a descriptive claim that some important aspect of experience or thought does (or does not) vary across groups of human beings tells us something important about human nature and the human condition.
Normative relativism concerns normative orevaluativeclaims that modes of thought, standards of reasoning, or the like are only right or wrong relative to a framework. 'Normative' is meant in a general sense, applying to a wide range of views; in the case of beliefs, for example, normative correctness equals truth. This does not mean, of course, that framework-relative correctness or truth is always clear, the first challenge being to explain what it amounts to in any given case (e.g., with respect to concepts, truth, epistemic norms). Normative relativism (say, in regard to normative ethical relativism) therefore implies that things (say, ethical claims) are not simply true in themselves, but only havetruth valuesrelative to broader frameworks (say, moral codes). (Many normative ethical relativist arguments run from premises about ethics to conclusions that assert the relativity of truth values, bypassing general claims about the nature of truth, but it is often more illuminating to consider the type of relativism under question directly.)[12]
In Englishcommon law, two (perhaps three) separate standards of proof are recognized:
Relationismis the theory that there are only relations between individual entities, and no intrinsic
properties. Despite the similarity in name, it is held by some to be a position distinct from relativism—for instance, because "statements about relational properties [...] assert an absolute truth about things in the world".[14]On the other hand, others wish to equate relativism, relationism and evenrelativity, which is a precise theory of relationships between physical objects:[15]Nevertheless, "This confluence of relativity theory with relativism became a strong contributing factor in the increasing prominence of relativism".[16]
Whereas previous investigations of science only sought sociological or psychological explanations of failed scientific theories or pathological science, the 'strong programme' is more relativistic, assessing scientific truth and falsehood equally in a historic and cultural context.
A common argument[17][18][19]against relativism suggests that it inherentlyrefutes itself: the statement "all is relative" classes either as a relative statement or as an absolute one. If it is relative, then this statement does not rule out absolutes. If the statement isabsolute, on the other hand, then it provides an example of an absolute statement, proving that not all truths are relative. However, this argument against relativism only applies to relativism that positions truth as relative–i.e. epistemological/truth-value relativism. More specifically, it is only extreme forms of epistemological relativism that can come in for this criticism as there are many epistemological relativists[who?]who posit that some aspects of what is regarded as factually "true" are not universal, yet still accept that other universal truths exist (e.g.gas lawsor moral laws).
Another argument against relativism posits the existence ofnatural law. Simply put, the physical universe works under basic principles: the "Laws of Nature". Some contend that a natural moral law may also exist, for example as argued by,Immanuel KantinCritique of Practical Reason,Richard DawkinsinThe God Delusion(2006)[20]and addressed byC. S. LewisinMere Christianity(1952).[21]Dawkins said "I think we face an equal but much more sinister challenge from the left, in the shape of cultural relativism - the view that scientific truth is only one kind of truth and it is not to be especially privileged".[22]PhilosopherHilary Putnam,[23]among others,[24]states that some forms of relativism make it impossible to believe one is in error. If there is no truth beyond an individual'sbeliefthat something is true, then an individual cannot hold their own beliefs to be false or mistaken. A related criticism is that relativizing truth to individuals destroys the distinction between truth and belief.
PhilosopherDonald Davidsonpresented an influential critique of conceptual relativism in his 1974 essayOn the Very Idea of a Conceptual Scheme. Conceptual relativism is the idea that different people or even entire communities could make sense of the world in radically different,incommensurable(meaning untranslatable) ways. Davidson attacks what he believes to be the entire framework which makes conceptual relativism intelligible, namely scheme–content dualism, which is the idea that all knowledge is the result of a subjective scheme imposing one's concepts onto objective content from the world. In refuting scheme–content dualism, Davidson shows that knowledge of one's scheme of concepts is necessarily inseparable from one's knowledge of the world, and so translation between different people or communities is always possible in principle.[25]
According to Belgian philosopher of scienceMaarten Boudry, relativism is rarely applied consistently. In an opinion piece, he argues that no one truly acts according to the belief that truth is relative. Even self-proclaimed relativists, he suggests, do not genuinely believe their own slogans and catchphrases. They become indignant when falsely accused of a crime and laugh at those who claim the Earth is flat. Boudry contends that people abandon their relativism when it really matters—for instance, when visiting a doctor for cancer screening or boarding a plane, trusting in the laws of physics. He argues that relativism about truth is not so much a sincere conviction as it is an empty slogan or a convenient rhetorical device people deploy when it suits them. Boudry refers to this phenomenon as “occasional relativism,” highlighting what he sees as the casual and opportunistic nature of such relativist claims.[26]
Sophistsare considered the founding fathers of relativism inWestern philosophy. Elements of relativism emerged among theSophistsin the 5th centuryBC. Notably, it wasProtagoraswho coined the phrase, "Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not." The thinking of the Sophists is mainly known through their opponent,Plato. In a paraphrase from Plato's dialogueTheaetetus, Protagoras said: "What is true for you is true for you, and what is true for me is true for me."[27][28][29]
Bernard Crick, a British political scientist and advocate of relativism, suggested inIn Defence of Politics(1962) that moral conflict between people is inevitable. He thought that onlyethicscan resolve such conflict, and when that occurs in public it results inpolitics. Accordingly, Crick saw the process ofdispute resolution,harms reduction,mediationorpeacemakingas central to all of moral philosophy. He became an important influence onfeministsand later on theGreens.
Philosopher of sciencePaul Feyerabendis often considered to be a relativist, although he denied being one.[30]
Feyerabend argued that modern science suffers from being methodologically monistic (the belief that only a single methodology can producescientific progress).[31]Feyerabend summarises his case inAgainst Methodwith the phrase "anything goes".[32]
Thomas Kuhn's philosophy of science, as expressed inThe Structure of Scientific Revolutions, is often interpreted as relativistic. He claimed that, as well as progressing steadily and incrementally ("normal science"), science undergoes periodic revolutions or "paradigm shifts", leaving scientists working in different paradigms with difficulty in even communicating. Thus the truth of a claim, or the existence of a posited entity, is relative to the paradigm employed. However, it is not necessary for him to embrace relativism because every paradigm presupposes the prior, building upon itself through history and so on. This leads to there being a fundamental, incremental, and referential structure of development which is not relative but again, fundamental.
But Kuhn rejected the accusation of being a relativist later in his postscript:
Some have argued that one can also read Kuhn's work as essentially positivist in its ontology: the revolutions he posits are epistemological, lurching toward a presumably 'better' understanding of an objective reality through the lens presented by the new paradigm. However, a number of passages inStructuredo indeed appear to be distinctly relativist, and to directly challenge the notion of an objective reality and the ability of science to progress towards an ever-greater grasp of it, particularly through the process of paradigm change.
George LakoffandMark Johnsondefine relativism inMetaphors We Live Byas the rejection of bothsubjectivismandmetaphysical objectivismin order to focus on the relationship between them, i.e. themetaphorby which we relate our current experience to our previous experience. In particular, Lakoff and Johnson characterize "objectivism" as a "straw man", and, to a lesser degree, criticize the views ofKarl Popper,KantandAristotle.[page needed]
In his bookInvariances,Robert Nozickexpresses a complex set of theories about the absolute and the relative. He thinks the absolute/relative distinction should be recast in terms of an invariant/variant distinction, where there are many things a proposition can be invariant with regard to or vary with. He thinks it is coherent for truth to be relative, and speculates that it might vary with time. He thinks necessity is an unobtainable notion, but can be approximated by robust invariance across a variety of conditions—although we can never identify a proposition that is invariant with regard to everything. Finally, he is not particularly warm to one of the most famous forms of relativism,moral relativism, preferring an evolutionary account.
Joseph Margolisadvocates a view he calls "robust relativism" and defends it in his booksHistoried Thought, Constructed World, Chapter 4 (California, 1995) andThe Truth about Relativism(Blackwell, 1991). He opens his account by stating that our logics should depend on what we take to be the nature of the sphere to which we wish to apply our logics. Holding that there can be no distinctions which are not "privileged" between thealethic, theontic, and theepistemic, he maintains that amany-valued logicjust might be the most apt foraestheticsorhistorysince, because in these practices, we are loath to hold to simplebinary logic; and he also holds that many-valued logic is relativistic. (This is perhaps an unusual definition of "relativistic". Compare with his comments on "relationism".) To say that "True" and "False" are mutually exclusive and exhaustive judgements onHamlet, for instance, really does seem absurd. A many-valued logic—with its values "apt", "reasonable", "likely", and so on—seems intuitively more applicable to interpretingHamlet. Where apparent contradictions arise between such interpretations, we might call the interpretations "incongruent", rather than dubbing either of them "false", because using many-valued logic implies that a measured value is a mixture of two extreme possibilities. Using the subset of many-valued logic,fuzzy logic, it can be said that various interpretations can be represented by membership in more than one possible truth set simultaneously. Fuzzy logic is therefore probably the best mathematical structure for understanding "robust relativism" and has been interpreted byBart Koskoas philosophically being related to Zen Buddhism.
It wasAristotlewho held that relativism implies that we should, sticking with appearances only, end up contradicting ourselves somewhere if we could apply all attributes to allousiai(beings). Aristotle, however, made non-contradiction dependent upon hisessentialism. If his essentialism is false, then so too is his ground for disallowing relativism. (Subsequent philosophers have found other reasons for supporting the principle of non-contradiction.)[clarification needed]
Beginning withProtagorasand invokingCharles Sanders Peirce, Margolis shows that the historic struggle to discredit relativism is an attempt to impose an unexamined belief in the world's essentially rigid rule-like nature. Plato and Aristotle merely attacked "relationalism"—the doctrine of true for l or true for k, and the like, where l and k are different speakers or different worlds—or something similar (most philosophers would call this position "relativism"). For Margolis, "true" means true; that is, the alethic use of "true" remains untouched. However, in real world contexts, and context is ubiquitous in the real world, we must apply truth values. Here, in epistemic terms, we mighttout courtretire "true" as an evaluation and keep "false". The rest of our value-judgements could be graded from "extremely plausible" down to "false". Judgements which on a bivalent logic would be incompatible or contradictory are further seen as "incongruent", although one may well have more weight than the other. In short, relativistic logic is not, or need not be, the bugbear it is often presented to be. It may simply be the best type of logic to apply to certain very uncertain spheres of real experiences in the world (although some sort of logic needs to be applied in order to make that judgement). Those who swear bybivalent logicmight simply be the ultimate keepers of the great fear of the flux.[citation needed]
PhilosopherRichard Rortyhas a somewhatparadoxicalrole in the debate over relativism: he is criticized for his relativistic views by many commentators, but has always denied that relativism applies to much anybody, being nothing more than a Platonic scarecrow. Rorty claims, rather, that he is apragmatist, and that to construe pragmatism as relativism is tobeg the question.
Rorty takes adeflationaryattitude totruth, believing there is nothing of interest to be said about truth in general, including the contention that it is generally subjective. He also argues that the notion ofwarrantor justification can do most of the work traditionally assigned to the concept of truth, and that justificationisrelative; justification is justification to an audience, for Rorty.
InContingency, Irony, and Solidarityhe argues that the debate between so-called relativists and so-called objectivists is beside the point because they do not have enough premises in common for either side to prove anything to the other.
In his bookMage Lokaya(My World), 1986,Nalin de Silvacriticized the basis of the established western system of knowledge, and its propagation, which he refers as "domination throughout the world".He explained in this book that mind independent reality is impossible and knowledge is not found but constructed. Further he has introduced and developed the concept of "Constructive Relativism" as the basis on which knowledge is constructed relative to the sense organs, culture and the mind completely based onAvidya.[41]
The term "relativism" often comes up in debates overpostmodernism,poststructuralismandphenomenology. Critics of these perspectives often identify advocates with the label "relativism". For example, theSapir–Whorf hypothesisis often considered a relativist view because it posits that linguistic categories and structures shape the way people view the world.Stanley Fishhas defended postmodernism and relativism.[42]
These perspectives do not strictly count as relativist in the philosophical sense, because they express agnosticism on the nature of reality and makeepistemologicalrather thanontologicalclaims. Nevertheless, the term is useful to differentiate them fromrealistswho believe that the purpose of philosophy, science, or literary critique is to locate externally true meanings. Important philosophers and theorists such asMichel Foucault,Max Stirner, political movements such aspost-anarchismorpost-Marxismcan also be considered as relativist in this sense - though a better term might besocial constructivist.
The spread and popularity of this kind of "soft" relativism varies between academic disciplines. It has wide support inanthropologyand has a majority following in cultural studies. It also has advocates in political theory and political science, sociology, andcontinental philosophy(as distinct from Anglo-American analytical philosophy). It has inspired empirical studies of the social construction of meaning such as those associated with labelling theory, which defenders can point to as evidence of the validity of their theories (albeit risking accusations ofperformative contradictionin the process). Advocates of this kind of relativism often also claim that recent developments in the natural sciences, such as Heisenberg'suncertainty principle,quantum mechanics,chaos theoryandcomplexity theoryshow that science is now becoming relativistic. However, many scientists who use these methods continue to identify as realist orpost-positivist, and some sharply criticize the association.[43][44]
Madhyamaka Buddhism, which forms the basis for manyMahayanaBuddhist schools and which was founded byNāgārjuna.[45]Nāgārjuna taught the idea of relativity. In the Ratnāvalī, he gives the example that shortness exists only in relation to the idea of length. The determination of a thing or object is only possible in relation to other things or objects, especially by way of contrast. He held that the relationship between the ideas of "short" and "long" is not due to intrinsic nature (svabhāva). This idea is also found in the Pali Nikāyas and Chinese Āgamas, in which the idea of relativity is expressed similarly: "That which is the element of light ... is seen to exist on account of [in relation to] darkness; that which is the element of good is seen to exist on account of bad; that which is the element of space is seen to exist on account of form."[46]
Madhyamaka Buddhism discerns two levels of truth: relative and ultimate. Thetwo truths doctrinestates that there areRelativeor conventional, common-sense truth, which describes our daily experience of a concrete world, andUltimatetruth, which describes the ultimate reality assunyata, empty of concrete and inherent characteristics. Conventional truth may be understood, in contrast, as "obscurative truth" or "that which obscures the true nature". It is constituted by the appearances of mistaken awareness. Conventional truth would be the appearance that includes a duality of apprehender and apprehended, and objects perceived within that. Ultimate truth is the phenomenal world free from the duality of apprehender and apprehended.[47]
TheCatholic Church, especially underJohn Paul IIandPope Benedict XVI, has identified relativism as one of the most significant problems for faith and morals today.[48]
According to the Church and to some theologians,[who?]relativism, as a denial of absolute truth, leads to moral license and a denial of the possibility ofsinand ofGod. Whether moral or epistemological, relativism constitutes a denial of the capacity of the human mind and reason to arrive at truth. Truth, according to Catholic theologians and philosophers (following Aristotle) consists ofadequatio rei et intellectus, thecorrespondenceof the mind and reality. Another way of putting it states that themindhas the same form as reality. This means when the form of the computer in front of someone (the type, color, shape, capacity, etc.) is also the form that is in their mind, then what they know is true because their mind corresponds to objective reality.
The denial of an absolute reference, of anaxis mundi, denies God, who equates to Absolute Truth, according to these Christian theologians. They link relativism tosecularism, an obstruction of religion inhuman life.
Pope Leo XIII(1810–1903) was the first known Pope to use the word "relativism", in his encyclicalHumanum genus(1884). Leo condemnedFreemasonryand claimed that its philosophical and political system was largely based on relativism.[49]
John Paul IIwrote inVeritatis Splendor
InEvangelium Vitae(The Gospel of Life), he says:
In April 2005, in his homily during Mass prior to the conclave which would elect him asPope, thenCardinal Joseph Ratzingertalked about the world "moving towards a dictatorship of relativism":
On June 6, 2005, Pope Benedict XVI told educators:
Then during theWorld Youth Dayin August 2005, he also traced to relativism the problems produced by the communist and sexual revolutions, and provided a counter-counter argument.[52]
Pope Francisrefers inEvangelii gaudiumto two forms of relativism, "doctrinal relativism" and a "practical relativism" typical of "our age".[53]The latter is allied to "widespread indifference" to systems of belief.[54]
Mahavira(599-527 BC), the 24thTirthankaraofJainism, developed a philosophy known asAnekantavada. John Koller describesanekāntavādaas "epistemological respect for view of others" about the nature of existence, whether it is "inherently enduring or constantly changing", but "not relativism; it does not mean conceding that all arguments and all views are equal".[55]
InSikhismtheGurus(spiritual teachers) have propagated the message of "many paths" leading to theone Godand ultimatesalvationfor all souls who tread on the path ofrighteousness. They have supported the view that proponents of all faiths can, by doing good and virtuous deeds and by remembering theLord, certainly achieve salvation. The students of the Sikh faith are told to accept all leading faiths as possible vehicles for attaining spiritual enlightenment provided the faithful study, ponder and practice the teachings of their prophets and leaders. The holy book of theSikhscalled theSri Guru Granth Sahibsays: "Do not say that the Vedas, the Bible and the Koran are false. Those who do not contemplate them are false."Guru Granth Sahibpage 1350;[56]later stating: "The seconds, minutes, and hours, days, weeks and months, and the various seasons originate from the one Sun; O nanak, in just the same way, the many forms originate from the Creator."Guru Granth Sahibpage 12,13.
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Acorner reflectoris aretroreflectorconsisting of three mutuallyperpendicular,intersectingflat reflective surfaces. It reflectswavesincident from any direction directly towards the source, buttranslated. The three intersecting surfaces often are triangles (forming atetrahedron) or may have square shapes. Radar corner reflectors made of metal are used to reflect radio waves fromradarsets. Optical corner reflectors, calledcorner cubesorcube corners, made of three-sided glassprisms, are used insurveyingandlaser ranging.
The incoming ray is reflected three times, once by each surface, which results in a reversal of direction.[1][2]To see this, the three corresponding normal vectors of the corner's perpendicular sides can be considered to form abasis(arectangular coordinate system) (x,y,z) in which to represent the direction of an arbitrary incoming ray,[a,b,c]. When the ray reflects from the first side, sayx, the ray'sxcomponent,a, is reversed to −awhile theyandzcomponents are unchanged, resulting in a direction of[−a,b,c]. Similarly, when reflected from sideyand finally from sidez, thebandccomponents are reversed. Therefore, the ray direction goes from[a,b,c]to[−a,b,c]to[−a, −b,c]to[−a, −b, −c], and it leaves the corner reflector with all three components of direction exactly reversed. The distance travelled, relative to a plane normal to the direction of the rays, is also equal for any ray entering the reflector, regardless of the location where it first reflects.[citation needed]
Radar corner reflectors are designed to reflect themicrowaveradio wavesemitted byradarsets back toward the radar antenna. This causes them to show a strong "return" on radar screens. A simple corner reflector consists of three conducting sheet metal or screen surfaces at 90° angles to each other, attached to one another at the edges, forming a "corner". These reflect radio waves coming from in front of them back parallel to the incoming beam. To create a corner reflector that will reflect radar waves coming from any direction, 8 corner reflectors are placed back-to-back in anoctahedron(diamond) shape. The reflecting surfaces must be larger than severalwavelengthsof the radio waves to function.[3]
In maritime navigation they are placed onbridgeabutments,buoys,shipsand, especially,lifeboats, to ensure that these show up strongly on ship radar screens. Corner reflectors are placed on the vessel's masts at a height of at least 4.6 m (15 feet) above sea level (giving them an approximate minimumhorizondistance of 8 kilometers or 4.5nautical miles).Marine radarusesX-bandmicrowaves with wavelengths of 2.5–3.75 cm (1–1.5 inches), so small reflectors less than 30 cm (12 inches) across are used. In aircraft navigation, corner reflectors are installed on ruralrunways, to make them show up on aircraft radar.
An object that has multiple reflections from smooth surfaces produces a radar return of greater magnitude than might be expected from the physical size of the object. This effect was put to use on theADM-20 Quail, a small decoy missile which had the sameradar cross sectionas aB-52.
The corner reflector is not the only efficient radar reflector design; otherretroreflectordesigns have also seen use.Luneburg lens, for example, are used on theADM-141 TALD.[4]
Inoptics, corner reflectors typically consist of threemirrorsor reflectiveprismfaces which return an incidentlight beamin the opposite direction. Insurveying,retroreflectorprisms are commonly used as targets for long-range electronic distance measurement using atotal station.
Five arrays of optical corner reflectors have been placed on theMoonfor use byLunar Laser Ranging experimentsobserving alaser'stime-of-flightto measure the Moon's orbit more precisely than was possible before. The three largest were placed byNASAas part of theApollo program, and theSoviet Unionbuilt two smaller ones into theLunokhod rovers.
Automobileandbicycletail lights are molded with arrays of small corner reflectors, with different sections oriented for viewing from different angles. Reflectivepaintfor visibility at night usually containsretroreflective spherical beads.
Thin plastic with microscopic corner reflector structures can be used astape, on signs, or sewn or molded ontoclothing.
Corner reflectors can also occur accidentally.Tower blockswithbalconiesare often accidentalacousticcorner reflectors and return a distinctiveechoto an observer making a sharp sound noise, such as a hand clap, nearby.
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Akaleidoscope(/kəˈlaɪdəskoʊp/) is anoptical instrumentwith two or more reflecting surfaces (ormirrors) tilted to each other at anangle, so that one or more (parts of) objects on one end of these mirrors are shown as asymmetricalpattern when viewed from the other end, due to repeatedreflection. These reflectors are usually enclosed in atube, often containing on one end a cell with loose, colored pieces of glass or other transparent (and/oropaque) materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing view being presented.
The term "kaleidoscope" was coined by its Scottish inventorDavid Brewster.[1]It is derived from theAncient Greekwordκαλός(kalos), "beautiful, beauty",[2]εἶδος(eidos), "that which is seen: form, shape"[3]andσκοπέω(skopeō), "to look to, to examine",[4]hence "observation of beautiful forms".[5]It was first published in the patent that was granted on July 10, 1817.[6]
Multiple reflection by two or more reflecting surfaces has been known since antiquity and was described as such byGiambattista della Portain hisMagia Naturalis(1558–1589). In 1646,Athanasius Kircherdescribed an experiment with a construction of two mirrors, which could be opened and closed like a book and positioned in various angles, showing regular polygon figures consisting of reflected aliquot sectors of 360°.Richard Bradley'sNew Improvements in Planting and Gardening(1717) described a similar construction to be placed on geometrical drawings to show an image with multiplied reflection. However, an optimal configuration that produces the full effects of the kaleidoscope was not recorded before 1815.[7]
In 1814,Sir David Brewsterconducted experiments onlight polarizationby successive reflections between plates of glass and first noted "the circular arrangement of the images of a candle round a center, and the multiplication of the sectors formed by the extremities of the plates of glass". He forgot about it, but noticed a more impressive version of the effect during further experiments in February 1815. A while later, he was impressed by the multiplied reflection of a bit of cement that was pressed through at the end of a triangular glass trough, which appeared more regular and almost perfectly symmetrical in comparison to the reflected objects that had been situated further away from the reflecting plates in earlier experiments. This triggered more experiments to find the conditions for the most beautiful and symmetrically perfect conditions. An early version had pieces of colored glass and other irregular objects fixed permanently and was admired by some Members of theRoyal Society of Edinburgh, includingSir George Mackenziewho predicted its popularity. A version followed in which some of the objects and pieces of glass could move when the tube was rotated. The last step, regarded as most important by Brewster, was to place the reflecting panes in a draw tube with a concave lens to distinctly introduce surrounding objects into the reflected pattern.[7]
Brewster thought his instrument to be of great value in "all the ornamental arts" as a device that creates an "infinity of patterns". Artists could accurately delineate the produced figures of the kaleidoscope by means of the solar microscope (a type ofcamera obscuradevice),magic lanternorcamera lucida. Brewster believed it would at the same time become a popular instrument "for the purposes of rational amusement". He decided to apply for apatent.[7]British patent no. 4136 "for a new Optical Instrument called "The Kaleidoscope" for exhibiting and creating beautiful Forms and Patterns of great use in all the ornamental Arts" was granted in July 1817.[6][8]Unfortunately, the manufacturer originally engaged to produce the product had shown one of the patent instruments to some of the London opticians to see if he could get orders from them. Soon the instrument was copied and marketed before the manufacturer had prepared any number of kaleidoscopes for sale. An estimated two hundred thousand kaleidoscopes sold inLondonandParisin just three months. Brewster figured at most a thousand of these were authorized copies that were constructed correctly, while the majority of the others did not give a correct impression of his invention. Because so relatively few people had experienced a proper kaleidoscope or knew how to apply it to ornamental arts, he decided to publicize a treatise on the principles and the correct construction of the kaleidoscope.[7]
It was thought that the patent was reduced in a Court of Law since its principles were supposedly already known. Brewster stated that the kaleidoscope was different because the particular positions of the object and of the eye, played a very important role in producing the beautiful symmetrical forms. Brewster's opinion was shared by several scientists, includingJames Watt.[9]
Philip Carpenteroriginally tried to produce his own imitation of the kaleidoscope, but was not satisfied with the results. He decided to offer his services to Brewster as manufacturer.[10]Brewster agreed and Carpenter's models were stamped "sole maker". Realizing that the company could not meet the level of demand, Brewster gained permission from Carpenter in 1818 for the device to be made by other manufacturers. In his 1819Treatise on the KaleidoscopeBrewster listed more than a dozen manufacturers/sellers of patent kaleidoscopes.[7]Carpenter's company would keep on selling kaleidoscopes for 60 years.[11]
In 1987, kaleidoscope artist Thea Marshall, working with theWillamette Science and Technology Center, a science museum located inEugene,Oregon, designed and constructed a 1,000-square-foot (93 m2) traveling mathematics and science exhibition titledKaleidoscopes: Reflections of Science and Art. With funding from theNational Science Foundation,[12]and circulated under the auspices of the Smithsonian Institution Traveling Exhibition Service (SITES[13]), the exhibition appeared in 15 science museums over a three-year period, reaching more than one million visitors in the United States and Canada. Interactive exhibit modules enabled visitors to better understand and appreciate how kaleidoscopes function.
David Brewster defined several variables in his patent and publications:
In his patent, Brewster perceived two forms for the kaleidoscope:
In hisTreatise on the Kaleidoscope(1819) he described the basic form with an object cell:
Brewster also developed several variations:
Brewster also imagined another application for the kaleidoscope:
Manufacturers and artists have created kaleidoscopes with a wide variety of materials and in many shapes. A few of these added elements that were not previously described by inventor David Brewster:
Cozy Baker (d. October 19, 2010)—founder of the Brewster Kaleidoscope Society—collected kaleidoscopes and wrote books about many of the artists making them in the 1970s through 2001. Her bookKaleidoscope Artistry[15]is a limited compendium of kaleidoscope makers, containing pictures of the interior and exterior views of contemporary artworks. Baker is credited with energizing a renaissance in kaleidoscope-making in the US; she spent her life putting kaleidoscope artists and galleries together so they would know each other and encourage each other.[16]
In 1999, a short-lived magazine dedicated to kaleidoscopes—Kaleidoscope Review—was published, covering artists, collectors, dealers, events, and including how-to articles. This magazine was created and edited by Brett Bensley, at that time a well-known kaleidoscope artist and resource on kaleidoscope information. Changed name toThe New Kaleidoscope Review, and then switched to a video presentation on YouTube, "The Kaleidoscope Maker".
In her albumThe Rise and Fall of a Midwest Princess,Chappell Roanreleased the song Kaleidoscope, which compares love to the unpredictable nature of the kaleidoscope.[17]
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Amirror image(in a plane mirror) is areflectedduplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As anoptical effect, it results fromspecular reflectionoff from surfaces of lustrous materials, especially amirrororwater. It is also a concept ingeometryand can be used as aconceptualizationprocess for 3D structures.
Ingeometry, the mirror image of an object ortwo-dimensional figureis thevirtual imageformed byreflectionin aplane mirror; it is of the same size as the original object, yet different, unless the object or figure hasreflection symmetry(also known as aP-symmetry).
Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside-out. If we first look at an object that is effectively two-dimensional (such as the writing on a card) and then turn the card to face a mirror, the object turns through an angle of 180° and we see a left-right reversal in the mirror. In this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror and face an object that is in front of the mirror. Then we compare the object with its reflection by turning ourselves 180°, towards the mirror. Again we perceive a left-right reversal due to a change in our orientation. So, in these examples the mirror does not actually cause the observed reversals.
The concept of reflection can be extended tothree-dimensionalobjects, including the inside parts, even if they are nottransparent. The term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the direction perpendicular to the mirror surface. In physics, mirror images are investigated in the subject calledgeometrical optics. More fundamentally in geometry and mathematics they form the principal objects ofCoxeter grouptheory andreflection groups.
In chemistry, two versions (isomers) of a molecule, one a "mirror image" of the other, are calledenantiomersif they are not "superposable" (the correct technical term, though the term "superimposable" is also used) on each other. That is an example ofchirality. In general, an object and its mirror image are calledenantiomorphs.
If a point of an object has coordinates (x,y,z) then the image of this point (as reflected by a mirror in they,zplane) has coordinates (−x,y,z). Thus reflection is a reversal of the coordinate axis perpendicular (normal) to the mirror's surface. Although a plane mirror reverses an object only in the direction normal to the mirror surface, this turns the entire three-dimensional image seen in the mirror inside-out, so there is aperceptionof a left-right reversal. Hence, the reversal is somewhat misleadingly called a "lateral inversion". The perception of a left-right reversal is geometrically explained by the fact that a three-dimensional object seen in a mirror is an inside-out version of the actual object, like a glove stripped off the left hand and turned into a right-hand glove, but there is still some confusion about the explanation amongst psychologists. The psychology of the perceived left-right reversal is discussed in "Much ado about mirrors" by ProfessorMichael Corballis(see "external links", below).
Reflection in a mirrordoesresult in a change inchirality, more specifically from a right-handed to a left-handed coordinate system (or vice versa). If one looks in a mirror two axes (up-down and left-right) coincide with those in the mirror, but the third axis (front-back) is reversed.
If a person stands side-on to a mirror, left and right hands will be reverseddirectlyby the mirror, because the person's left-right axis is then normal to the mirror plane. However, it is important to understand that there arealwaysonly two enantiomorphs, the object and its inside-out image. Therefore, no matter how the object is oriented towards the mirror, all the resulting images are fundamentally identical (as Corballis explains in his paper "Much ado about mirrors", mentioned above).
In the picture of the mountain reflected in the lake (photograph top right), the reversal normal to the reflecting surface is obvious. Notice that there is no obvious front-back or left-right of the mountain. In the example of the urn and mirror (photograph to right), the urn is fairly symmetrical front-back (and left-right). Thus, no obvious reversal of any sort can be seen in the mirror image of the urn.
A mirror image appears more obviously three-dimensional if the observer moves, or if the image is viewed usingbinocular vision. This is because the relative position of objects changes as the observer's perspective changes, or is differently viewed with each eye.[1]
Looking through a mirror from different positions (but necessarily with the point of observation restricted to the halfspace on one side of the mirror) is like looking at the 3D mirror image of space; without further mirrors only the mirror image of the halfspace before the mirror is relevant; if there is another mirror, the mirror image of the other halfspace is too.
A mirror does not just produce an image of what would be there without it; it also changes the light distribution in the halfspace in front of and behind the mirror. A mirror hanging on the wall makes the room brighter because additional light sources appear in the mirror image. However, the appearance of additional light does not violate theconservation of energyprinciple, because some light no longer reaches behind the mirror, as the mirror simply re-directs the light energy. In terms of the light distribution, the virtual mirror image has the same appearance and the same effect as a real, symmetrically arranged half-space behind a window (instead of the mirror). Shadows may extend from the mirror into the halfspace before it, and vice versa.
Inmirror writinga text is deliberately displayed as its mirror image, in order to be read through a mirror. For example, emergency vehicles such asambulancesor fire engines often display a label (e.g. "AMBULANCE") on their front end with the text reversed, so that drivers of vehicles in front of them can read the words right way round in therear-view mirror. Somemovie theatersalso use mirror writing in aRear Window Captioning Systemused to assist individuals withhearing impairmentsin watching films.
In the case of two mirrors, in planes at an angle α, looking through both from the sector which is the intersection of the two halfspaces, is like looking at a version of the world rotated by an angle of 2α; the points of observations and directions of looking for which this applies correspond to those for looking through a frame like that of the first mirror, and a frame at the mirror image with respect to the first plane, of the second mirror. If the mirrors have vertical edges then the left edge of the field of view is the plane through the right edge of the first mirror and the edge of the second mirror which is on the right when looked at directly, but on the left in the mirror image.
In the case of two parallel mirrors, looking through both at once is like looking at a version of the world which is translated by twice the distance between the mirrors, in the direction perpendicular to them, away from the observer. Since the plane of the mirror in which one looks directly is beyond that of the other mirror, one always looks at an oblique angle, and the translation just mentioned has not only a component away from the observer, but also one in a perpendicular direction. The translated view can also be described by a translation of the observer in opposite direction. For example, with a verticalperiscope, the shift of the world is away from the observer and down, both by the length of the periscope, but it is more practical to consider the equivalent shift of the observer: up, and backward.
It is also possible to create anon-reversing mirrorby placing twofirst surface mirrorsat 90º to give an image which is not reversed.
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https://en.wikipedia.org/wiki/Mirror_image
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Anoptical cavity,resonating cavityoroptical resonatoris an arrangement ofmirrorsor other optical elements that confineslight wavessimilarly to how acavity resonatorconfines microwaves. Optical cavities are a major component oflasers, surrounding thegain mediumand providingfeedbackof the laser light. They are also used inoptical parametric oscillatorsand someinterferometers. Light confined in the cavity reflects multiple times, producingmodeswith certainresonance frequencies. Modes can be decomposed intolongitudinal modesthat differ only in frequency andtransverse modesthat have different intensity patterns across the cross section of the beam. Many types of optical cavities producestanding wavemodes.
Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them. Flat mirrors are not often used because of the difficulty of aligning them to the needed precision. The geometry (resonator type) must be chosen so that the beam remains stable, i.e. the size of the beam does not continually grow with multiple reflections. Resonator types are also designed to meet other criteria such as a minimum beam waist or having no focal point (and therefore no intense light at a single point) inside the cavity.
Optical cavities are designed to have a largeQ factor, meaning a beam undergoes many oscillation cycles with littleattenuation.[1]In the regime of high Q values, this is equivalent to the frequencyline widthbeing small compared to the resonant frequency of the cavity.
Light confined in a resonator will reflect multiple times from the mirrors, and due to the effects ofinterference, only certain patterns andfrequenciesof radiation will be sustained by the resonator, with the others being suppressed by destructive interference. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable. These are known as themodesof the resonator.[2]
Resonator modes can be divided into two types:longitudinal modes, which differ in frequency from each other; andtransverse modes, which may differ in both frequency and theintensitypattern of the light. The basic, or fundamental transverse mode of a resonator is aGaussian beam.
The most common types of optical cavities consist of two facing plane (flat) or spherical mirrors. The simplest of these is the plane-parallel orFabry–Pérotcavity, consisting of two opposing flat mirrors.[3][4][5][6][7][8][9]While simple, this arrangement is rarely used in large-scale lasers due to the difficulty of alignment; the mirrors must be aligned parallel within a fewseconds of arc, or "walkoff" of the intracavity beam will result in it spilling out of the sides of the cavity. However, this problem is much reduced for very short cavities with a small mirror separation distance (L< 1 cm). Plane-parallel resonators are therefore commonly used in microchip andmicrocavitylasers andsemiconductor lasers. In these cases, rather than using separate mirrors, a reflectiveoptical coatingmay be directly applied to the laser medium itself. The plane-parallel resonator is also the basis of theFabry–Pérot interferometer.
For a resonator with two mirrors with radii of curvatureR1andR2, there are a number of common cavity configurations. If the two radii are equal to half the cavity length (R1=R2=L/2), a concentric or spherical resonator results. This type of cavity produces adiffraction-limitedbeam waist in the centre of the cavity, with large beam diameters at the mirrors, filling the whole mirror aperture. Similar to this is the hemispherical cavity, with one plane mirror and one mirror of radius equal to the cavity length.
A common and important design is the confocal resonator, with mirrors of equal radii to the cavity length (R1=R2=L).[10][11][12][13][14][15]This design produces the smallest possible beam diameter at the cavity mirrors for a given cavity length, and is often used in lasers where the purity of the transverse mode pattern is important.
A concave-convex cavity has one convex mirror with a negative radius of curvature. This design produces no intracavity focus of the beam, and is thus useful in very high-power lasers where the intensity of the light might be damaging to the intracavity medium if brought to a focus.
Less common resonator types includeoptical ring resonatorsandwhispering-gallery moderesonators, in which a resonance is formed by waves moving in a closed loop rather than reflecting between two mirrors.
Only certain ranges of values forR1,R2, andLproduce stable resonators in which periodic refocussing of the intracavity beam is produced. If the cavity is unstable, the beam size will grow without limit, eventually growing larger than the size of the cavity mirrors and being lost. By using methods such asray transfer matrix analysis, it is possible to calculate a stability criterion:[16]
Values which satisfy the inequality correspond to stable resonators.
The stability can be shown graphically by defining a stability parameter,gfor each mirror:
and plottingg1againstg2as shown. Areas bounded by the lineg1g2= 1 and the axes are stable. Cavities at points exactly on the line are marginally stable; small variations in cavity length can cause the resonator to become unstable, and so lasers using these cavities are in practice often operated just inside the stability line.
A simple geometric statement describes the regions of stability: A cavity is stable if the line segments between the mirrors and their centers of curvature overlap, but one does not lie entirely within the other.
In the confocal cavity, if a ray is deviated from its original direction in the middle of the cavity, its displacement after reflecting from one of the mirrors is larger than in any other cavity design. This preventsamplified spontaneous emissionand is important for designing high power amplifiers with good beam quality.
If the optical cavity is not empty (e.g., a laser cavity which contains the gain medium), the value ofLneeds to be adjusted to account for the index of refraction of the medium. Optical elements such as lenses placed in the cavity alter the stability and mode size. In addition, for most gain media, thermal and other inhomogeneities create a variable lensing effect in the medium, which must be considered in the design of the laser resonator.
Practical laser resonators may contain more than two mirrors; three- and four-mirror arrangements are common, producing a "folded cavity". Commonly, a pair of curved mirrors form one or more confocal sections, with the rest of the cavity being quasi-collimatedand using plane mirrors. The shape of the laser beam depends on the type of resonator: The beam produced by stable, paraxial resonators can be well modeled by aGaussian beam. In special cases the beam can be described as a single transverse mode and the spatial properties can be well described by the Gaussian beam, itself. More generally, this beam may be described as a superposition of transverse modes. Accurate description of such a beam involves expansion over some complete, orthogonal set of functions (over two-dimensions) such asHermite polynomialsor theInce polynomials. Unstable laser resonators on the other hand, have been shown to produce fractal shaped beams.[17]
Some intracavity elements are usually placed at a beam waist between folded sections. Examples includeacousto-optic modulatorsforcavity dumpingandvacuumspatial filtersfortransverse modecontrol. For some low power lasers, the laser gain medium itself may be positioned at a beam waist. Other elements, such asfilters,prismsanddiffraction gratingsoften need large quasi-collimated beams.
These designs allow compensation of the cavity beam'sastigmatism, which is produced byBrewster-cutelements in the cavity. A Z-shaped arrangement of the cavity also compensates forcomawhile the 'delta' or X-shaped cavity does not.
Out of plane resonators lead to rotation of the beam profile and more stability. The heat generated in the gain medium leads to frequency drift of the cavity, therefore the frequency can be actively stabilized by locking it to unpowered cavity. Similarly the pointing stability of a laser may still be improved by spatial filtering by anoptical fibre.
Precise alignment is important when assembling an optical cavity. For best output power and beam quality, optical elements must be aligned such that the path followed by the beam is centered through each element.
Simple cavities are often aligned with an alignment laser—a well-collimated visible laser that can be directed along the axis of the cavity. Observation of the path of the beam and its reflections from various optical elements allows the elements' positions and tilts to be adjusted.
More complex cavities may be aligned using devices such as electronicautocollimatorsandlaser beam profilers.
Optical cavities can also be used as multipass optical delay lines, folding a light beam so that a long path-length may be achieved in a small space. A plane-parallel cavity with flat mirrors produces a flat zigzag light path, but as discussed above, these designs are very sensitive to mechanical disturbances and walk-off. When curved mirrors are used in a nearly confocal configuration, the beam travels on a circular zigzag path. The latter is called a Herriott-type delay line. A fixed insertion mirror is placed off-axis near one of the curved mirrors, and a mobile pickup mirror is similarly placed near the other curved mirror. A flat linear stage with one pickup mirror is used in case of flat mirrors and a rotational stage with two mirrors is used for the Herriott-type delay line.
The rotation of the beam inside the cavity alters thepolarizationstate of the beam. To compensate for this, a single pass delay line is also needed, made of either a three or two mirrors in a 3d respective 2d retro-reflection configuration on top of a linear stage. To adjust for beam divergence a second car on the linear stage with two lenses can be used. The two lenses act as a telescope producing a flat phase front of aGaussian beamon a virtual end mirror.
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Recursionoccurs when the definition of a concept or process depends on a simpler or previous version of itself.[1]Recursion is used in a variety of disciplines ranging fromlinguisticstologic. The most common application of recursion is inmathematicsandcomputer science, where afunctionbeing defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur.
A process that exhibits recursion isrecursive.Video feedbackdisplays recursive images, as does aninfinity mirror.
In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:
For example, the following is a recursive definition of a person'sancestor. One's ancestor is either:
TheFibonacci sequenceis another classic example of recursion:
Many mathematical axioms are based upon recursive rules. For example, the formal definition of thenatural numbersby thePeano axiomscan be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."[2]By this base case and recursive rule, one can generate the set of all natural numbers.
Other recursively defined mathematical objects includefactorials,functions(e.g.,recurrence relations),sets(e.g.,Cantor ternary set), andfractals.
There are various more tongue-in-cheek definitions of recursion; seerecursive humor.
Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be 'recursive'.[3]
To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, while the running of a procedure involves actually following the rules and performing the steps.
Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
When a procedure is thus defined, this immediately creates the possibility of an endless loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the procedure can complete.
Even if it is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of the procedure; this requires some administration as to how far various simultaneous instances of the procedures have progressed. For this reason, recursive definitions are very rare in everyday situations.
LinguistNoam Chomsky, among many others, has argued that the lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as the time available to utter one), can be explained as the consequence of recursion in natural language.[4][5]
This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence:Dorothy thinks witches are dangerous, in which the sentencewitches are dangerousoccurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion.
This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length:Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another.[6]Over the years, languages in general have proved amenable to this kind of analysis.
The generally accepted idea that recursion is an essential property of human language has been challenged byDaniel Everetton the basis of his claims about thePirahã language. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this.[7]Literaryself-referencecan in any case be argued to be different in kind from mathematical or logical recursion.[8]
Recursion plays a crucial role not only in syntax, but also innatural language semantics. The wordand, for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that is suitably flexible,andis typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one.[9]
Arecursive grammaris aformal grammarthat contains recursiveproduction rules.[10]
Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving acircular definitionorself-reference, in which the putative recursive step does not get closer to a base case, but instead leads to aninfinite regress. It is not unusual for such books to include a joke entry in their glossary along the lines of:
A variation is found on page 269 in theindexof some editions ofBrian KernighanandDennis Ritchie's bookThe C Programming Language; the index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found inLet's talk Lispby Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and inSoftware Toolsby Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears inThe UNIX Programming Environmentby Kernighan and Pike. It did not appear in the first edition ofThe C Programming Language. The joke is part of thefunctional programmingfolklore and was already widespread in the functional programming community before the publication of the aforementioned books.[12][13]
Another joke is that "To understand recursion, you must understand recursion."[11]In the English-language version of the Google web search engine, when a search for "recursion" is made, the site suggests "Did you mean:recursion."[14]An alternative form is the following, fromAndrew Plotkin:"If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer toDouglas Hofstadterthan you are; then ask him or her what recursion is."
Recursive acronymsare other examples of recursive humor.PHP, for example, stands for "PHP Hypertext Preprocessor",WINEstands for "WINE Is Not an Emulator",GNUstands for "GNU's not Unix", andSPARQLdenotes the "SPARQL Protocol and RDF Query Language".
The canonical example of a recursively defined set is given by thenatural numbers:
In mathematical logic, thePeano axioms(or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematicianRichard Dedekindand by the Italian mathematicianGiuseppe Peano. The Peano Axioms define the natural numbers referring to a recursive successor function and addition and multiplication as recursive functions.
Another interesting example is the set of all "provable" propositions in anaxiomatic systemthat are defined in terms of aproof procedurewhich is inductively (or recursively) defined as follows:
Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with a collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating theCantor setis a subdivision rule, as isbarycentric subdivision.
Afunctionmay be recursively defined in terms of itself. A familiar example is theFibonacci numbersequence:F(n) =F(n− 1) +F(n− 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this caseF(0) = 0 andF(1) = 1.
Applying the standard technique ofproof by casesto recursively defined sets or functions, as in the preceding sections, yieldsstructural induction— a powerful generalization ofmathematical inductionwidely used to derive proofs inmathematical logicand computer science.
Dynamic programmingis an approach tooptimizationthat restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is theBellman equation, which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step).
Inset theory, this is a theorem guaranteeing that recursively defined functions exist. Given a setX, an elementaofXand a functionf:X→X, the theorem states that there is a unique functionF:N→X{\displaystyle F:\mathbb {N} \to X}(whereN{\displaystyle \mathbb {N} }denotes the set of natural numbers including zero) such that
for any natural numbern.
Dedekind was the first to pose the problem of unique definition of set-theoretical functions onN{\displaystyle \mathbb {N} }by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?"[15]
Take two functionsF:N→X{\displaystyle F:\mathbb {N} \to X}andG:N→X{\displaystyle G:\mathbb {N} \to X}such that:
whereais an element ofX.
It can be proved bymathematical inductionthatF(n) =G(n)for all natural numbersn:
By induction,F(n) =G(n)for alln∈N{\displaystyle n\in \mathbb {N} }.
A common method of simplification is to divide a problem into subproblems of the same type. As acomputer programmingtechnique, this is calleddivide and conquerand is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach isdynamic programming. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.
A classic example of recursion is the definition of thefactorialfunction, given here inPythoncode:
The function calls itself recursively on a smaller version of the input(n - 1)and multiplies the result of the recursive call byn, until reaching thebase case, analogously to the mathematical definition of factorial.
Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to the problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is inparsersfor programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program.
Recurrence relationsare equations which define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain a non-recursive definition (e.g., aclosed-form expression).
Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity of instructions. The main disadvantage is that the memory usage of recursive algorithms may grow very quickly, rendering them impractical for larger instances.
Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example isRomanesco broccoli.[16]
Authors use the concept ofrecursivityto foreground the situation in which specificallysocialscientists find themselves when producing knowledge about the world they are always already part of.[17][18]According to Audrey Alejandro, “as social scientists, the recursivity of our condition deals with the fact that we are both subjects (as discourses are the medium through which we analyse) and objects of the academic discourses we produce (as we are social agents belonging to the world we analyse).”[19]From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise ofreflexiveefforts:
we are socialised into discourses and dispositions produced by the socio-political order we aim to challenge, a socio-political order that we may, therefore, reproduce unconsciously while aiming to do the contrary. The recursivity of our situation as scholars – and, more precisely, the fact that the dispositional tools we use to produce knowledge about the world are themselves produced by this world – both evinces the vital necessity of implementing reflexivity in practice and poses the main challenge in doing so.
Recursion is sometimes referred to inmanagement scienceas the process of iterating through levels of abstraction in large business entities.[20]A common example is the recursive nature of managementhierarchies, ranging fromline managementtosenior managementviamiddle management. It also encompasses the larger issue ofcapital structureincorporate governance.[21]
TheMatryoshka dollis a physical artistic example of the recursive concept.[22]
Recursion has been used in paintings sinceGiotto'sStefaneschi Triptych, made in 1320. Its central panel contains the kneeling figure of Cardinal Stefaneschi, holding up the triptych itself as an offering.[23][24]This practice is more generally known as theDroste effect, an example of theMise en abymetechnique.
M. C. Escher'sPrint Gallery(1956) is a print which depicts a distorted city containing a gallery whichrecursivelycontains the picture, and soad infinitum.[25]
The filmInceptionhas colloquialized the appending of the suffix-ceptionto a noun to jokingly indicate the recursion of something.[26]
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In the mathematical theory ofdynamical systems, anirrational rotationis amap
whereθis anirrational number. Under the identification of acirclewithR/Z, or with the interval[0, 1]with the boundary points glued together, this map becomes arotationof acircleby a proportionθof a full revolution (i.e., an angle of2πθradians). Sinceθis irrational, the rotation has infiniteorderin thecircle groupand the mapTθhas noperiodic orbits.
Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map
The relationship between the additive and multiplicative notations is the group isomorphism
It can be shown thatφis anisometry.
There is a strong distinction in circle rotations that depends on whetherθis rational or irrational. Rational rotations are less interesting examples of dynamical systems because ifθ=ab{\displaystyle \theta ={\frac {a}{b}}}andgcd(a,b)=1{\displaystyle \gcd(a,b)=1}, thenTθb(x)=x{\displaystyle T_{\theta }^{b}(x)=x}whenx∈[0,1]{\displaystyle x\in [0,1]}. It can also be shown thatTθi(x)≠x{\displaystyle T_{\theta }^{i}(x)\neq x}when1≤i<b{\displaystyle 1\leq i<b}.
Irrational rotations form a fundamental example in the theory ofdynamical systems. According to theDenjoy theorem, every orientation-preservingC2-diffeomorphism of the circle with an irrationalrotation numberθistopologically conjugatetoTθ. An irrational rotation is ameasure-preservingergodic transformation, but it is notmixing. ThePoincaré mapfor the dynamical system associated with theKronecker foliationon atoruswith angleθ>is the irrational rotation byθ.C*-algebrasassociated with irrational rotations, known asirrational rotation algebras, have been extensively studied.
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Incomputational mathematics, aniterative methodis amathematical procedurethat uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which thei-th approximation (called an "iterate") is derived from the previous ones.
A specific implementation withterminationcriteria for a given iterative method likegradient descent,hill climbing,Newton's method, orquasi-Newton methodslikeBFGS, is analgorithmof an iterative method or amethod of successive approximation. An iterative method is calledconvergentif the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however,heuristic-based iterative methods are also common.
In contrast,direct methodsattempt to solve the problem by a finite sequence of operations. In the absence ofrounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equationsAx=b{\displaystyle A\mathbf {x} =\mathbf {b} }byGaussian elimination). Iterative methods are often the only choice fornonlinear equations. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.[1]
If an equation can be put into the formf(x) =x, and a solutionxis an attractivefixed pointof the functionf, then one may begin with a pointx1in thebasin of attractionofx, and letxn+1=f(xn) forn≥ 1, and the sequence {xn}n≥ 1will converge to the solutionx. Herexnis thenth approximation or iteration ofxandxn+1is the next orn+ 1 iteration ofx. Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings. (For example,x(n+1)=f(x(n)).) If the functionfiscontinuously differentiable, a sufficient condition for convergence is that thespectral radiusof the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.
In the case of asystem of linear equations, the two main classes of iterative methods are thestationary iterative methods, and the more generalKrylov subspacemethods.
Stationary iterative methods solve a linear system with anoperatorapproximating the original one; and based on a measurement of the error in the result (the residual), form a "correction equation" for which this process is repeated. While these methods are simple to derive, implement, and analyze, convergence is only guaranteed for a limited class of matrices.
Aniterative methodis defined by
and for a given linear systemAx=b{\displaystyle A\mathbf {x} =\mathbf {b} }with exact solutionx∗{\displaystyle \mathbf {x} ^{*}}theerrorby
An iterative method is calledlinearif there exists a matrixC∈Rn×n{\displaystyle C\in \mathbb {R} ^{n\times n}}such that
and this matrix is called theiteration matrix.
An iterative method with a given iteration matrixC{\displaystyle C}is calledconvergentif the following holds
An important theorem states that for a given iterative method and its iteration matrixC{\displaystyle C}it is convergent if and only if itsspectral radiusρ(C){\displaystyle \rho (C)}is smaller than unity, that is,
The basic iterative methods work bysplittingthe matrixA{\displaystyle A}into
and here the matrixM{\displaystyle M}should be easilyinvertible.
The iterative methods are now defined as
or, equivalently,
From this follows that the iteration matrix is given by
Basic examples of stationary iterative methods use a splitting of the matrixA{\displaystyle A}such as
whereD{\displaystyle D}is only the diagonal part ofA{\displaystyle A}, andL{\displaystyle L}is the strict lowertriangular partofA{\displaystyle A}.
Respectively,U{\displaystyle U}is the strict upper triangular part ofA{\displaystyle A}.
Linear stationary iterative methods are also calledrelaxation methods.
Krylov subspace methods[2]work by forming abasisof the sequence of successive matrix powers times the initial residual (theKrylov sequence).
The approximations to the solution are then formed by minimizing the residual over the subspace formed.
The prototypical method in this class is theconjugate gradient method(CG) which assumes that the system matrixA{\displaystyle A}issymmetricpositive-definite.
For symmetric (and possibly indefinite)A{\displaystyle A}one works with theminimal residual method(MINRES).
In the case of non-symmetric matrices, methods such as thegeneralized minimal residual method(GMRES) and thebiconjugate gradient method(BiCG) have been derived.
Since these methods form a basis, it is evident that the method converges inNiterations, whereNis the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practiceNcan be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of thespectrumof the operator.
The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such asGMRES(alternatively,preconditionedKrylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.
Mathematical methods relating to successive approximation include:
Jamshīd al-Kāshīused iterative methods to calculate the sine of 1° andπinThe Treatise of Chord and Sineto high precision.
An early iterative method forsolving a linear systemappeared in a letter ofGaussto a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest[citation needed].
The theory of stationary iterative methods was solidly established with the work ofD.M. Youngstarting in the 1950s. The conjugate gradient method was also invented in the 1950s, with independent developments byCornelius Lanczos,Magnus HestenesandEduard Stiefel, but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well forpartial differential equations, especially the elliptic type.
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Inmathematics, therotation numberis aninvariantofhomeomorphismsof thecircle.
It was first defined byHenri Poincaréin 1885, in relation to theprecessionof theperihelionof aplanetary orbit. Poincaré later proved a theorem characterizing the existence ofperiodic orbitsin terms ofrationalityof the rotation number.
Suppose thatf:S1→S1{\displaystyle f:S^{1}\to S^{1}}is an orientation-preservinghomeomorphismof thecircleS1=R/Z.{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} .}Thenfmay beliftedto ahomeomorphismF:R→R{\displaystyle F:\mathbb {R} \to \mathbb {R} }of the real line, satisfying
for every real numberxand every integerm.
Therotation numberoffis defined in terms of theiteratesofF:
Henri Poincaréproved that the limit exists and is independent of the choice of the starting pointx. The liftFis unique modulo integers, therefore the rotation number is a well-defined element ofR/Z.{\displaystyle \mathbb {R} /\mathbb {Z} .}Intuitively, it measures the average rotation angle along theorbitsoff.
Iff{\displaystyle f}is a rotation by2πN{\displaystyle 2\pi N}(where0<N<1{\displaystyle 0<N<1}), then
and its rotation number isN{\displaystyle N}(cf.irrational rotation).
The rotation number is invariant undertopological conjugacy, and even monotone topologicalsemiconjugacy: iffandgare two homeomorphisms of the circle and
for a monotone continuous maphof the circle into itself (not necessarily homeomorphic) thenfandghave the same rotation numbers. It was used by Poincaré andArnaud Denjoyfor topological classification of homeomorphisms of the circle. There are two distinct possibilities.
The rotation number iscontinuouswhen viewed as a map from the group of homeomorphisms (withC0topology) of the circle into the circle.
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Inmathematics,Sharkovskii's theorem(also spelledSharkovsky,Sharkovskiy,ŠarkovskiiorSarkovskii), named afterOleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result aboutdiscrete dynamical systems.[1]One of the implications of the theorem is that if a discrete dynamical system on thereal linehas aperiodic pointof period 3, then it must have periodic points of every other period.
For someintervalI⊂R{\displaystyle I\subset \mathbb {R} }, suppose thatf:I→I{\displaystyle f:I\to I}is acontinuous function. The numberx{\displaystyle x}is called aperiodic point of periodm{\displaystyle m}iff(m)(x)=x{\displaystyle f^{(m)}(x)=x}, wheref(m){\displaystyle f^{(m)}}denotes theiterated functionobtained by composition ofm{\displaystyle m}copies off{\displaystyle f}. The numberx{\displaystyle x}is said to haveleast periodm{\displaystyle m}if, in addition,f(k)(x)≠x{\displaystyle f^{(k)}(x)\neq x}for all0<k<m{\displaystyle 0<k<m}. Sharkovskii's theorem concerns the possible least periods of periodic points off{\displaystyle f}. Consider the following ordering of the positiveintegers, sometimes called the Sharkovskii ordering:[2]357911…(2n+1)⋅20…3⋅25⋅27⋅29⋅211⋅2…(2n+1)⋅21…3⋅225⋅227⋅229⋅2211⋅22…(2n+1)⋅22…3⋅235⋅237⋅239⋅2311⋅23…(2n+1)⋅23…⋮…2n…24232221{\displaystyle {\begin{array}{cccccccc}3&5&7&9&11&\ldots &(2n+1)\cdot 2^{0}&\ldots \\3\cdot 2&5\cdot 2&7\cdot 2&9\cdot 2&11\cdot 2&\ldots &(2n+1)\cdot 2^{1}&\ldots \\3\cdot 2^{2}&5\cdot 2^{2}&7\cdot 2^{2}&9\cdot 2^{2}&11\cdot 2^{2}&\ldots &(2n+1)\cdot 2^{2}&\ldots \\3\cdot 2^{3}&5\cdot 2^{3}&7\cdot 2^{3}&9\cdot 2^{3}&11\cdot 2^{3}&\ldots &(2n+1)\cdot 2^{3}&\ldots \\&\vdots \\\ldots &2^{n}&\ldots &2^{4}&2^{3}&2^{2}&2&1\end{array}}}
It consists of:
This ordering is atotal order: every positive integer appears exactly once somewhere on this list. However, it is not awell-order. In a well-order, every subset would have an earliest element, but in this order there is no earliest power of two.
Sharkovskii's theorem states that iff{\displaystyle f}has a periodic point of least periodm{\displaystyle m}, andm{\displaystyle m}precedesn{\displaystyle n}in the above ordering, thenf{\displaystyle f}has also a periodic point of least periodn{\displaystyle n}.
One consequence is that iff{\displaystyle f}has only finitely many periodic points, then they must all have periods that are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.
Sharkovskii's theorem does not state that there arestablecycles of those periods, just that there are cycles of those periods. For systems such as thelogistic map, thebifurcation diagramshows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture.
The assumption of continuity is important. Without this assumption, the discontinuouspiecewise linear functionf:[0,3)→[0,3){\displaystyle f:[0,3)\to [0,3)}defined as:f:x↦{x+1for0≤x<2x−2for2≤x<3{\displaystyle f:x\mapsto {\begin{cases}x+1&\mathrm {for\ } 0\leq x<2\\x-2&\mathrm {for\ } 2\leq x<3\end{cases}}}for which every value has period 3, would be a counterexample. Similarly essential is the assumption off{\displaystyle f}being defined on an interval. Otherwisef:x↦(1−x)−1{\displaystyle f:x\mapsto (1-x)^{-1}}, which is defined on real numbers except the one:R∖{1},{\displaystyle \mathbb {R} \setminus \{1\},}and for which every non-zero value has period 3, would be a counterexample.
Sharkovskii also proved the converse theorem: everyupper setof the above order is the set of periods for some continuous function from an interval to itself. In fact all such sets of periods are achieved by the family of functionsTh:[0,1]→[0,1]{\displaystyle T_{h}:[0,1]\to [0,1]},x↦min(h,1−2|x−1/2|){\displaystyle x\mapsto \min(h,1-2|x-1/2|)}forh∈[0,1]{\displaystyle h\in [0,1]}, except for the empty set of periods which is achieved byT:R→R{\displaystyle T:\mathbb {R} \to \mathbb {R} },x↦x+1{\displaystyle x\mapsto x+1}.[3][4]
On the other hand, with additional information on the combinatorial structure of the interval map acting on the points in a periodic orbit, a period-n point may force period-3 (and hence all periods). Namely, if the orbit type (the cyclic permutation generated by the map acting on the points in the periodic orbit) has a so-called stretching pair, then this implies the existence of a periodic point of period-3. It can be shown (in an asymptotic sense) that almost all cyclic permutations admit at least one stretching pair, and hence almost all orbit types imply period-3.[5]
Tien-Yien LiandJames A. Yorkeshowed in 1975 that not only does the existence of a period-3 cycle imply the existence of cycles of all periods, but in addition it implies the existence of an uncountable infinitude of points that never map to any cycle (chaotic points)—a property known asperiod three implies chaos.[6]
Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find acircle mapwith periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit. For example,Peter Kloedenshowed that Sharkovskii's theorem holds for triangular mappings, i.e., mappings for which the componentfidepends only on the firsticomponentsx1,..., xi.[7]
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Inmathematics, arecurrence relationis anequationaccording to which then{\displaystyle n}th term of asequenceof numbers is equal to some combination of the previous terms. Often, onlyk{\displaystyle k}previous terms of the sequence appear in the equation, for a parameterk{\displaystyle k}that is independent ofn{\displaystyle n}; this numberk{\displaystyle k}is called theorderof the relation. If the values of the firstk{\displaystyle k}numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.
Inlinear recurrences, thenth term is equated to alinear functionof thek{\displaystyle k}previous terms. A famous example is the recurrence for theFibonacci numbers,Fn=Fn−1+Fn−2{\displaystyle F_{n}=F_{n-1}+F_{n-2}}where the orderk{\displaystyle k}is two and the linear function merely adds the two previous terms. This example is alinear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend onn.{\displaystyle n.}For these recurrences, one can express the general term of the sequence as aclosed-form expressionofn{\displaystyle n}. As well,linear recurrences with polynomial coefficientsdepending onn{\displaystyle n}are also important, because many commonelementary functionsandspecial functionshave aTaylor serieswhose coefficients satisfy such a recurrence relation (seeholonomic function).
Solving a recurrence relation means obtaining aclosed-form solution: a non-recursive function ofn{\displaystyle n}.
The concept of a recurrence relation can be extended tomultidimensional arrays, that is,indexed familiesthat are indexed bytuplesofnatural numbers.
Arecurrence relationis an equation that expresses each element of asequenceas a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form
where
is a function, whereXis a set to which the elements of a sequence must belong. For anyu0∈X{\displaystyle u_{0}\in X}, this defines a unique sequence withu0{\displaystyle u_{0}}as its first element, called theinitial value.[1]
It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.
This defines recurrence relation offirst order. A recurrence relation oforderkhas the form
whereφ:N×Xk→X{\displaystyle \varphi :\mathbb {N} \times X^{k}\to X}is a function that involveskconsecutive elements of the sequence.
In this case,kinitial values are needed for defining a sequence.
Thefactorialis defined by the recurrence relation
and the initial condition
This is an example of alinear recurrence with polynomial coefficientsof order 1, with the simple polynomial (inn)
as its only coefficient.
An example of a recurrence relation is thelogistic mapdefined by
for a given constantr.{\displaystyle r.}The behavior of the sequence depends dramatically onr,{\displaystyle r,}but is stable when the initial conditionx0{\displaystyle x_{0}}varies.
The recurrence of order two satisfied by theFibonacci numbersis the canonical example of a homogeneouslinear recurrencerelation with constant coefficients (see below). The Fibonacci sequence is defined using the recurrence
withinitial conditions
Explicitly, the recurrence yields the equations
etc.
We obtain the sequence of Fibonacci numbers, which begins
The recurrence can be solved by methods described below yieldingBinet's formula, which involves powers of the two roots of the characteristic polynomialt2=t+1{\displaystyle t^{2}=t+1}; thegenerating functionof the sequence is therational function
A simple example of a multidimensional recurrence relation is given by thebinomial coefficients(nk){\displaystyle {\tbinom {n}{k}}}, which count the ways of selectingk{\displaystyle k}elements out of a set ofn{\displaystyle n}elements.
They can be computed by the recurrence relation
with the base cases(n0)=(nn)=1{\displaystyle {\tbinom {n}{0}}={\tbinom {n}{n}}=1}. Using this formula to compute the values of all binomial coefficients generates an infinite array calledPascal's triangle. The same values can also be computed directly by a different formula that is not a recurrence, but usesfactorials, multiplication and division, not just additions:
The binomial coefficients can also be computed with a uni-dimensional recurrence:
with the initial value(n0)=1{\textstyle {\binom {n}{0}}=1}(The division is not displayed as a fraction for emphasizing that it must be computed after the multiplication, for not introducing fractional numbers).
This recurrence is widely used in computers because it does not require to build a table as does the bi-dimensional recurrence, and does involve very large integers as does the formula with factorials (if one uses(nk)=(nn−k),{\textstyle {\binom {n}{k}}={\binom {n}{n-k}},}all involved integers are smaller than the final result).
Thedifference operatoris anoperatorthat mapssequencesto sequences, and, more generally,functionsto functions. It is commonly denotedΔ,{\displaystyle \Delta ,}and is defined, infunctional notation, as
It is thus a special case offinite difference.
When using the index notation for sequences, the definition becomes
The parentheses aroundΔf{\displaystyle \Delta f}andΔa{\displaystyle \Delta a}are generally omitted, andΔan{\displaystyle \Delta a_{n}}must be understood as the term of indexnin the sequenceΔa,{\displaystyle \Delta a,}and notΔ{\displaystyle \Delta }applied to the elementan.{\displaystyle a_{n}.}
Givensequencea=(an)n∈N,{\displaystyle a=(a_{n})_{n\in \mathbb {N} },}thefirst differenceofaisΔa.{\displaystyle \Delta a.}
Thesecond differenceisΔ2a=(Δ∘Δ)a=Δ(Δa).{\displaystyle \Delta ^{2}a=(\Delta \circ \Delta )a=\Delta (\Delta a).}A simple computation shows that
More generally: thekth differenceis defined recursively asΔk=Δ∘Δk−1,{\displaystyle \Delta ^{k}=\Delta \circ \Delta ^{k-1},}and one has
This relation can be inverted, giving
Adifference equationof orderkis an equation that involves thekfirst differences of a sequence or a function, in the same way as adifferential equationof orderkrelates thekfirstderivativesof a function.
The two above relations allow transforming a recurrence relation of orderkinto a difference equation of orderk, and, conversely, a difference equation of orderkinto recurrence relation of orderk. Each transformation is theinverseof the other, and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation.
For example, the difference equation
is equivalent to the recurrence relation
in the sense that the two equations are satisfied by the same sequences.
As it is equivalent for a sequence to satisfy a recurrence relation or to be the solution of a difference equation, the two terms "recurrence relation" and "difference equation" are sometimes used interchangeably. SeeRational difference equationandMatrix difference equationfor example of uses of "difference equation" instead of "recurrence relation"
Difference equations resemble differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations, and therefore recurrence relations.
Summation equationsrelate to difference equations asintegral equationsrelate to differential equations. Seetime scale calculusfor a unification of the theory of difference equations with that of differential equations.
Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are aboutn{\displaystyle n}-dimensional grids. Functions defined onn{\displaystyle n}-grids can also be studied with partial difference equations.[2]
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:
there is also a nice method to solve it:[3]
Let
Then
If we apply the formula toan+1=(1+hfnh)an+hgnh{\displaystyle a_{n+1}=(1+hf_{nh})a_{n}+hg_{nh}}and take the limith→0{\displaystyle h\to 0}, we get the formula for first orderlinear differential equationswith variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.
Many homogeneous linear recurrence relations may be solved by means of thegeneralized hypergeometric series. Special cases of these lead to recurrence relations for theorthogonal polynomials, and manyspecial functions. For example, the solution to
is given by
theBessel function, while
is solved by
theconfluent hypergeometric series. Sequences which are the solutions oflinear difference equations with polynomial coefficientsare calledP-recursive. For these specific recurrence equations algorithms are known which findpolynomial,rationalorhypergeometricsolutions.
Furthermore, for the general non-homogeneous linear recurrence relation with constant coefficients, one can solve it based on variation of parameter.[4]
A first order rational difference equation has the formwt+1=awt+bcwt+d{\displaystyle w_{t+1}={\tfrac {aw_{t}+b}{cw_{t}+d}}}. Such an equation can be solved by writingwt{\displaystyle w_{t}}as a nonlinear transformation of another variablext{\displaystyle x_{t}}which itself evolves linearly. Then standard methods can be used to solve the linear difference equation inxt{\displaystyle x_{t}}.
The linear recurrence of orderd{\displaystyle d},
has thecharacteristic equation
The recurrence isstable, meaning that the iterates converge asymptotically to a fixed value, if and only if theeigenvalues(i.e., the roots of the characteristic equation), whether real or complex, are all less thanunityin absolute value.
In the first-order matrix difference equation
with state vectorx{\displaystyle x}and transition matrixA{\displaystyle A},x{\displaystyle x}converges asymptotically to the steady state vectorx∗{\displaystyle x^{*}}if and only if all eigenvalues of the transition matrixA{\displaystyle A}(whether real or complex) have anabsolute valuewhich is less than 1.
Consider the nonlinear first-order recurrence
This recurrence islocally stable, meaning that itconvergesto a fixed pointx∗{\displaystyle x^{*}}from points sufficiently close tox∗{\displaystyle x^{*}}, if the slope off{\displaystyle f}in the neighborhood ofx∗{\displaystyle x^{*}}is smaller thanunityin absolute value: that is,
A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuousftwo adjacent fixed points cannot both be locally stable.
A nonlinear recurrence relation could also have a cycle of periodk{\displaystyle k}fork>1{\displaystyle k>1}. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
withf{\displaystyle f}appearingk{\displaystyle k}times is locally stable according to the same criterion:
wherex∗{\displaystyle x^{*}}is any point on the cycle.
In achaoticrecurrence relation, the variablex{\displaystyle x}stays in a bounded region but never converges to a fixed point or an attracting cycle; any fixed points or cycles of the equation are unstable. See alsologistic map,dyadic transformation, andtent map.
When solving anordinary differential equationnumerically, one typically encounters a recurrence relation. For example, when solving theinitial value problem
withEuler's methodand a step sizeh{\displaystyle h}, one calculates the values
by the recurrence
Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in thediscretizationarticle.
Some of the best-known difference equations have their origins in the attempt to modelpopulation dynamics. For example, theFibonacci numberswere once used as a model for the growth of a rabbit population.
Thelogistic mapis used either directly to model population growth, or as a starting point for more detailed models of population dynamics. In this context, coupled difference equations are often used to model the interaction of two or morepopulations. For example, theNicholson–Bailey modelfor a host-parasiteinteraction is given by
withNt{\displaystyle N_{t}}representing the hosts, andPt{\displaystyle P_{t}}the parasites, at timet{\displaystyle t}.
Integrodifference equationsare a form of recurrence relation important to spatialecology. These and other difference equations are particularly suited to modelingunivoltinepopulations.
Recurrence relations are also of fundamental importance inanalysis of algorithms.[5][6]If analgorithmis designed so that it will break a problem into smaller subproblems (divide and conquer), its running time is described by a recurrence relation.
A simple example is the time an algorithm takes to find an element in an ordered vector withn{\displaystyle n}elements, in the worst case.
A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons isn{\displaystyle n}.
A better algorithm is calledbinary search. However, it requires a sorted vector. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by
thetime complexityof which will beO(log2(n)){\displaystyle O(\log _{2}(n))}.
Indigital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise ininfinite impulse response(IIR)digital filters.
For example, the equation for a "feedforward" IIRcomb filterof delayT{\displaystyle T}is:
wherext{\displaystyle x_{t}}is the input at timet{\displaystyle t},yt{\displaystyle y_{t}}is the output at timet{\displaystyle t}, andα{\displaystyle \alpha }controls how much of the delayed signal is fed back into the output. From this we can see that
etc.
Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.[7][8]In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (interest rate, realGDP, etc.) in terms of past and current values of other variables.
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TheAbel equation, named afterNiels Henrik Abel, is a type offunctional equationof the form
or
The forms are equivalent whenαisinvertible.horαcontrol theiterationoff.
The second equation can be written
Takingx=α−1(y), the equation can be written
For a known functionf(x), a problem is to solve the functional equation for the functionα−1≡h, possibly satisfying additional requirements, such asα−1(0) = 1.
The change of variablessα(x)= Ψ(x), for arealparameters, brings Abel's equation into the celebratedSchröder's equation,Ψ(f(x)) =sΨ(x).
The further changeF(x) = exp(sα(x))intoBöttcher's equation,F(f(x)) =F(x)s.
The Abel equation is a special case of (and easily generalizes to) thetranslation equation,[1]
e.g., forω(x,1)=f(x){\displaystyle \omega (x,1)=f(x)},
The Abel functionα(x)further provides the canonical coordinate forLie advective flows(one parameterLie groups).
Initially, the equation in the more general form[2][3]was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.[4][5][6]
In the case of a linear transfer function, the solution is expressible compactly.[7]
The equation oftetrationis a special case of Abel's equation, withf= exp.
In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
and so on,
The Abel equation has at least one solution onE{\displaystyle E}if and only iffor allx∈E{\displaystyle x\in E}and alln∈N{\displaystyle n\in \mathbb {N} },fn(x)≠x{\displaystyle f^{n}(x)\neq x}, wherefn=f∘f∘...∘f{\displaystyle f^{n}=f\circ f\circ ...\circ f}, is the functionfiteratedntimes.[8]
We have the following existence and uniqueness theorem[9]: Theorem B
Leth:R→R{\displaystyle h:\mathbb {R} \to \mathbb {R} }beanalytic, meaning it has a Taylor expansion. To find: real analytic solutionsα:R→C{\displaystyle \alpha :\mathbb {R} \to \mathbb {C} }of the Abel equationα∘h=α+1{\textstyle \alpha \circ h=\alpha +1}.
A real analytic solutionα{\displaystyle \alpha }exists if and only if both of the following conditions hold:
The solution is essentially unique in the sense that there exists a canonical solutionα0{\displaystyle \alpha _{0}}with the following properties:
{α0+β∘α0|β:R→Ris analytic, with period 1}.{\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.}
Analytic solutions (Fatou coordinates) can be approximated byasymptotic expansionof a function defined bypower seriesin the sectors around aparabolic fixed point.[10]The analytic solution is unique up to a constant.[11]
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Böttcher's equation, named afterLucjan Böttcher, is thefunctional equation
where
Thelogarithmof this functional equation amounts toSchröder's equation.
Solution offunctional equationis afunctioninimplicit form.
Lucian Emil Böttchersketched a proof in 1904 on the existence of solution: an analytic functionFin a neighborhood of the fixed pointa, such that:[1]
This solution is sometimes called:
The complete proof was published byJoseph Rittin 1920,[3]who was unaware of the original formulation.[4]
Böttcher's coordinate (the logarithm of theSchröder function) conjugatesh(z)in a neighbourhood of the fixed point to the functionzn. An especially important case is whenh(z)is a polynomial of degreen, anda= ∞ .[5]
One can explicitly compute Böttcher coordinates for:[6]
For the function h and n=2[7]
the Böttcher function F is:
Böttcher's equation plays a fundamental role in the part ofholomorphic dynamicswhich studiesiterationofpolynomialsof onecomplex variable.
Global properties of the Böttcher coordinate were studied byFatou[8][9]andDouadyandHubbard.[10]
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Inmathematics,tetration(orhyper-4) is anoperationbased oniterated, or repeated,exponentiation. There is no standardnotationfor tetration, thoughKnuth's up arrow notation↑↑{\displaystyle \uparrow \uparrow }and the left-exponentxb{\displaystyle {}^{x}b}are common.
Under the definition as repeated exponentiation,na{\displaystyle {^{n}a}}meansaa⋅⋅a{\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}}, wherencopies ofaare iterated via exponentiation, right-to-left, i.e. the application of exponentiationn−1{\displaystyle n-1}times.nis called the "height" of the function, whileais called the "base," analogous to exponentiation. It would be read as "thenth tetration ofa". For example, 2 tetrated to 4 (or the fourth tetration of 2) is42=2222=224=216=65536{\displaystyle {^{4}2}=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536}.
It is the nexthyperoperationafterexponentiation, but beforepentation. The word was coined byReuben Louis Goodsteinfromtetra-(four) anditeration.
Tetration is also defined recursively as
allowing for theholomorphicextension of tetration tonon-natural numberssuch asreal,complex, andordinal numbers, which was proved in 2017.
The two inverses of tetration are calledsuper-rootandsuper-logarithm, analogous to thenth rootand the logarithmic functions. None of the three functions areelementary.
Tetration is used for thenotation of very large numbers.
The first fourhyperoperationsare shown here, with tetration being considered the fourth in the series. Theunary operationsuccession, defined asa′=a+1{\displaystyle a'=a+1}, is considered to be the zeroth operation.
Importantly, nested exponents are interpreted from the top down:abc{\displaystyle a^{b^{c}}}meansa(bc){\displaystyle a^{\left(b^{c}\right)}}and not(ab)c.{\displaystyle \left(a^{b}\right)^{c}.}
Succession,an+1=an+1{\displaystyle a_{n+1}=a_{n}+1}, is the most basic operation; while addition (a+n{\displaystyle a+n}) is a primary operation, for addition of natural numbers it can be thought of as a chained succession ofn{\displaystyle n}successors ofa{\displaystyle a}; multiplication (a×n{\displaystyle a\times n}) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involvingn{\displaystyle n}numbers ofa{\displaystyle a}. Exponentiation can be thought of as a chained multiplication involvingn{\displaystyle n}numbers ofa{\displaystyle a}and tetration (na{\displaystyle ^{n}a}) as a chained power involvingn{\displaystyle n}numbersa{\displaystyle a}. Each of the operations above are defined by iterating the previous one;[1]however, unlike the operations before it, tetration is not anelementary function.
The parametera{\displaystyle a}is referred to as thebase, while the parametern{\displaystyle n}may be referred to as theheight. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positivereala>0{\displaystyle a>0}and non-negativeintegern≥0{\displaystyle n\geq 0}, we can definena{\displaystyle \,\!{^{n}a}}recursively as:[1]
The recursive definition is equivalent to repeated exponentiation fornaturalheights; however, this definition allows for extensions to the other heights such as0a{\displaystyle ^{0}a},−1a{\displaystyle ^{-1}a}, andia{\displaystyle ^{i}a}as well – many of these extensions are areas of active research.
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.
Owing in part to some shared terminology and similarnotational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:
In the first two expressionsais thebase, and the number of timesaappears is theheight(add one forx). In the third expression,nis theheight, but each of the bases is different.
Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either meaniteratedpowersor iteratedexponentials.
There are many different notation styles that can be used to express tetration. Some notations can also be used to describe otherhyperoperations, while some are limited to tetration and have no immediate extension.
One notation above uses iterated exponential notation; this is defined in general as follows:
There are not as many notations for iterated exponentials, but here are a few:
Because of the extremely fast growth of tetration, most values in the following table are too large to write inscientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate. Usually, the limit that can be calculated in a numerical calculation program such asWolfram Alphais 3↑↑4, and the number of digits up to 3↑↑5 can be expressed.
(106.00225×103,638,334,640,023)
Remark:Ifxdoes not differ from 10 by orders of magnitude, then for allk≥3,mx=exp10kz,z>1⇒m+1x=exp10k+1z′withz′≈z{\displaystyle k\geq 3,~^{m}x=\exp _{10}^{k}z,~z>1~\Rightarrow ~^{m+1}x=\exp _{10}^{k+1}z'{\text{ with }}z'\approx z}. For example,z−z′<1.5⋅10−15forx=3=k,m=4{\displaystyle z-z'<1.5\cdot 10^{-15}{\text{ for }}x=3=k,~m=4}in the above table, and the difference is even smaller for the following rows.
Tetration can be extended in two different ways; in the equationna{\displaystyle ^{n}a\!}, both the baseaand the heightncan be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to differentdomains, includingn0{\displaystyle {^{n}0}}, complex functions such asni{\displaystyle {}^{n}i}, and heights of infiniten, the more limited properties of tetration reduce the ability to extend tetration.
The exponential00{\displaystyle 0^{0}}is not consistently defined. Thus, the tetrationsn0{\displaystyle \,{^{n}0}}are not clearly defined by the formula given earlier. However,limx→0nx{\displaystyle \lim _{x\rightarrow 0}{}^{n}x}is well defined, and exists:[10]
Thus we could consistently definen0=limx→0nx{\displaystyle {}^{n}0=\lim _{x\rightarrow 0}{}^{n}x}. This is analogous to defining00=1{\displaystyle 0^{0}=1}.
Under this extension,00=1{\displaystyle {}^{0}0=1}, so the rule0a=1{\displaystyle {^{0}a}=1}from the original definition still holds.
Sincecomplex numberscan be raised to powers, tetration can be applied tobasesof the formz=a+bi(whereaandbare real). For example, innzwithz=i, tetration is achieved by using theprincipal branchof thenatural logarithm; usingEuler's formulawe get the relation:
This suggests a recursive definition forn+1i=a′+b′igiven anyni=a+bi:
The following approximate values can be derived:
Solving the inverse relation, as in the previous section, yields the expected0i= 1and−1i= 0, with negative values ofngiving infinite results on the imaginary axis.[citation needed]Plotted in thecomplex plane, the entire sequence spirals to the limit0.4383 + 0.3606i, which could be interpreted as the value wherenis infinite.
Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers withfractaland symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.[citation needed]
Tetration can be extended toinfiniteheights; i.e., for certainaandnvalues inna{\displaystyle {}^{n}a}, there exists a well defined result for an infiniten. This is because for bases within a certain interval, tetration converges to a finite value as the height tends toinfinity. For example,222⋅⋅⋅{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdot ^{\cdot ^{\cdot }}}}}}converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:
In general, the infinitely iterated exponentialxx⋅⋅⋅{\displaystyle x^{x^{\cdot ^{\cdot ^{\cdot }}}}\!\!}, defined as the limit ofnx{\displaystyle {}^{n}x}asngoes to infinity, converges fore−e≤x≤e1/e, roughly the interval from 0.066 to 1.44, a result shown byLeonhard Euler.[11]The limit, should it exist, is a positive real solution of the equationy=xy. Thus,x=y1/y. The limit defining the infinite exponential ofxdoes not exist whenx>e1/ebecause the maximum ofy1/yise1/e. The limit also fails to exist when0 <x<e−e.
This may be extended to complex numberszwith the definition:
whereWrepresentsLambert's W function.
As the limity=∞x(if existent on the positive real line, i.e. fore−e≤x≤e1/e) must satisfyxy=ywe see thatx↦y=∞xis (the lower branch of) the inverse function ofy↦x=y1/y.
We can use the recursive rule for tetration,
to prove−1a{\displaystyle {}^{-1}a}:
Substituting −1 forkgives
Smaller negative values cannot be well defined in this way. Substituting −2 forkin the same equation gives
which is not well defined. They can, however, sometimes be considered sets.[12]
Forn=1{\displaystyle n=1}, any definition of−11{\displaystyle \,\!{^{-1}1}}is consistent with the rule because
Alinear approximation(solution to the continuity requirement, approximation to the differentiability requirement) is given by:
hence:
and so on. However, it is only piecewise differentiable; at integer values ofx, the derivative is multiplied bylna{\displaystyle \ln {a}}. It is continuously differentiable forx>−2{\displaystyle x>-2}if and only ifa=e{\displaystyle a=e}. For example, using these methodsπ2e≈5.868...{\displaystyle {}^{\frac {\pi }{2}}e\approx 5.868...}and−4.30.5≈4.03335...{\displaystyle {}^{-4.3}0.5\approx 4.03335...}
A main theorem in Hooshmand's paper[6]states: Let0<a≠1{\displaystyle 0<a\neq 1}. Iff:(−2,+∞)→R{\displaystyle f:(-2,+\infty )\rightarrow \mathbb {R} }is continuous and satisfies the conditions:
thenf{\displaystyle f}is uniquely determined through the equation
where(x)=x−[x]{\displaystyle (x)=x-[x]}denotes the fractional part ofxandexpa[x]{\displaystyle \exp _{a}^{[x]}}is the[x]{\displaystyle [x]}-iterated functionof the functionexpa{\displaystyle \exp _{a}}.
The proof is that the second through fourth conditions trivially imply thatfis a linear function on[−1, 0].
The linear approximation to natural tetration functionxe{\displaystyle {}^{x}e}is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:
Iff:(−2,+∞)→R{\displaystyle f:(-2,+\infty )\rightarrow \mathbb {R} }is acontinuous functionthat satisfies:
thenf=uxp{\displaystyle f={\text{uxp}}}. [Heref=uxp{\displaystyle f={\text{uxp}}}is Hooshmand's name for the linear approximation to the natural tetration function.]
The proof is much the same as before; the recursion equation ensures thatf′(−1+)=f′(0+),{\displaystyle f^{\prime }(-1^{+})=f^{\prime }(0^{+}),}and then the convexity condition implies thatf{\displaystyle f}is linear on(−1, 0).
Therefore, the linear approximation to natural tetration is the only solution of the equationf(x)=ef(x−1)(x>−1){\displaystyle f(x)=e^{f(x-1)}\;\;(x>-1)}andf(0)=1{\displaystyle f(0)=1}which isconvexon(−1, +∞). All other sufficiently-differentiable solutions must have aninflection pointon the interval(−1, 0).
Beyond linear approximations, aquadratic approximation(to the differentiability requirement) is given by:
which is differentiable for allx>0{\displaystyle x>0}, but not twice differentiable. For example,122≈1.45933...{\displaystyle {}^{\frac {1}{2}}2\approx 1.45933...}Ifa=e{\displaystyle a=e}this is the same as the linear approximation.[1]
Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where(a1n)n=a{\displaystyle \left(a^{\frac {1}{n}}\right)^{n}=a}. Namely,
Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degreenalso exist, although they are much more unwieldy.[1][13]
In 2017, it was proved[14]that there exists a unique functionFwhich is a solution of the equationF(z+ 1) = exp(F(z))and satisfies the additional conditions thatF(0) = 1andF(z)approaches thefixed pointsof the logarithm (roughly0.318 ± 1.337i) aszapproaches±i∞and thatFisholomorphicin the whole complexz-plane, except the part of the real axis atz≤ −2. This proof confirms a previousconjecture.[15]The construction of such a function was originally demonstrated byHellmuth Kneserin 1950 and is related to the concept ofhalf-exponential functions.[16]The complex map of this function is shown in the figure at right. The proof also works for other bases besidese, as long as the base is greater thane1e≈1.445{\displaystyle e^{\frac {1}{e}}\approx 1.445}. Subsequent work extended the construction to all complex bases.[17]
Using Kneser's tetration, example values includeπ2e≈5.82366...{\displaystyle {}^{\frac {\pi }{2}}e\approx 5.82366...},122≈1.45878...{\displaystyle {}^{\frac {1}{2}}2\approx 1.45878...}, and12e≈1.64635...{\displaystyle {}^{\frac {1}{2}}e\approx 1.64635...}
The requirement of the tetration being holomorphic is important for its uniqueness. Many functionsScan be constructed as
whereαandβare real sequences which decay fast enough to provide theconvergence of the series, at least at moderate values ofImz.
The functionSsatisfies the tetration equationsS(z+ 1) = exp(S(z)),S(0) = 1, and ifαnandβnapproach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of{α}or{β}are not zero, then functionShas multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients{α}and{β}are, the further away these singularities are from the real axis.
The extension of tetration into the complex plane is thus essential for the uniqueness; thereal-analytictetration is not unique.
Tetration can be defined forordinal numbersviatransfinite induction. For allαand allβ> 0:0α=1{\displaystyle {}^{0}\alpha =1}βα=sup({αγα:γ<β}).{\displaystyle {}^{\beta }\alpha =\sup(\{\alpha ^{{}^{\gamma }\alpha }:\gamma <\beta \})\,.}
Tetration (restricted toN2{\displaystyle \mathbb {N} ^{2}}) is not anelementary recursive function. One can prove by induction that for every elementary recursive functionf, there is a constantcsuch that
We denote the right hand side byg(c,x){\displaystyle g(c,x)}. Suppose on the contrary that tetration is elementary recursive.g(x,x)+1{\displaystyle g(x,x)+1}is also elementary recursive. By the above inequality, there is a constantcsuch thatg(x,x)+1≤g(c,x){\displaystyle g(x,x)+1\leq g(c,x)}. By lettingx=c{\displaystyle x=c}, we have thatg(c,c)+1≤g(c,c){\displaystyle g(c,c)+1\leq g(c,c)}, a contradiction.
Exponentiationhas two inverse operations;rootsandlogarithms. Analogously, theinversesof tetration are often called thesuper-root, and thesuper-logarithm(In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function3y=x{\displaystyle {^{3}}y=x}, the two inverses are the cube super-root ofyand the super-logarithm baseyofx.
The super-root is the inverse operation of tetration with respect to the base: ifny=x{\displaystyle ^{n}y=x}, thenyis annth super-root ofx(xns{\displaystyle {\sqrt[{n}]{x}}_{s}}orx4s{\displaystyle {\sqrt[{4}]{x}}_{s}}).
For example,
so 2 is the 4th super-root of 65,536(65,5364s=2){\displaystyle \left({\sqrt[{4}]{65{,}536}}_{s}=2\right)}.
The2nd-order super-root,square super-root, orsuper square roothas two equivalent notations,ssrt(x){\displaystyle \mathrm {ssrt} (x)}andxs{\displaystyle {\sqrt {x}}_{s}}. It is the inverse of2x=xx{\displaystyle ^{2}x=x^{x}}and can be represented with theLambert W function:[18]
The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true wheny=ssrt(x){\displaystyle y=\mathrm {ssrt} (x)}:
Likesquare roots, the square super-root ofxmay not have a single solution. Unlike square roots, determining the number of square super-roots ofxmay be difficult. In general, ife−1/e<x<1{\displaystyle e^{-1/e}<x<1}, thenxhas two positive square super-roots between 0 and 1 calculated using formulas:xs={eW−1(lnx);eW0(lnx)}{\displaystyle {\sqrt {x}}_{s}=\left\{e^{W_{-1}(\ln x)};e^{W_{0}(\ln x)}\right\}}; and ifx>1{\displaystyle x>1}, thenxhas one positive square super-root greater than 1 calculated using formulas:xs=eW0(lnx){\displaystyle {\sqrt {x}}_{s}=e^{W_{0}(\ln x)}}. Ifxis positive and less thane−1/e{\displaystyle e^{-1/e}}it does not have anyrealsquare super-roots, but the formula given above yields countably infinitely manycomplexones for any finitexnot equal to 1.[18]The function has been used to determine the size ofdata clusters.[19]
Atx=1{\displaystyle x=1}:
One of the simpler and faster formulas for a third-degree super-root is the recursive formula. Ify=xxx{\displaystyle y=x^{x^{x}}}then one can use:
For each integern> 2, the functionnxis defined and increasing forx≥ 1, andn1 = 1, so that thenth super-root ofx,xns{\displaystyle {\sqrt[{n}]{x}}_{s}}, exists forx≥ 1.
However, if thelinear approximation aboveis used, thenyx=y+1{\displaystyle ^{y}x=y+1}if−1 <y≤ 0, soyy+1s{\displaystyle ^{y}{\sqrt {y+1}}_{s}}cannot exist.
In the same way as the square super-root, terminology for other super-roots can be based on thenormal roots: "cube super-roots" can be expressed asx3s{\displaystyle {\sqrt[{3}]{x}}_{s}}; the "4th super-root" can be expressed asx4s{\displaystyle {\sqrt[{4}]{x}}_{s}}; and the "nth super-root" isxns{\displaystyle {\sqrt[{n}]{x}}_{s}}. Note thatxns{\displaystyle {\sqrt[{n}]{x}}_{s}}may not be uniquely defined, because there may be more than onenthroot. For example,xhas a single (real) super-root ifnisodd, and up to two ifniseven.[citation needed]
Just as with the extension of tetration to infinite heights, the super-root can be extended ton= ∞, being well-defined if1/e≤x≤e. Note thatx=∞y=y[∞y]=yx,{\displaystyle x={^{\infty }y}=y^{\left[^{\infty }y\right]}=y^{x},}and thus thaty=x1/x{\displaystyle y=x^{1/x}}. Therefore, when it is well defined,x∞s=x1/x{\displaystyle {\sqrt[{\infty }]{x}}_{s}=x^{1/x}}and, unlike normal tetration, is anelementary function. For example,2∞s=21/2=2{\displaystyle {\sqrt[{\infty }]{2}}_{s}=2^{1/2}={\sqrt {2}}}.
It follows from theGelfond–Schneider theoremthat super-rootns{\displaystyle {\sqrt {n}}_{s}}for any positive integernis either integer ortranscendental, andn3s{\displaystyle {\sqrt[{3}]{n}}_{s}}is either integer or irrational.[20]It is still an open question whether irrational super-roots are transcendental in the latter case.
Once a continuous increasing (inx) definition of tetration,xa, is selected, the corresponding super-logarithmslogax{\displaystyle \operatorname {slog} _{a}x}orloga4x{\displaystyle \log _{a}^{4}x}is defined for all real numbersx, anda> 1.
The functionslogaxsatisfies:
Other than the problems with the extensions of tetration, there are several open questions concerning tetration, particularly when concerning the relations between number systems such asintegersandirrational numbers:
For each graphHonhvertices and eachε > 0, define
Then each graphGonnvertices with at mostnh/Dcopies ofHcan be madeH-free by removing at mostεn2edges.[23]
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Induction puzzlesarelogic puzzles, which are examples ofmulti-agent reasoning, where the solution evolves along with the principle ofinduction.[1][2]
A puzzle's scenario always involves multiple players with the same reasoning capability, who go through the same reasoning steps. According to the principle of induction, a solution to the simplest case makes the solution of the next complicated case obvious. Once the simplest case of the induction puzzle is solved, the whole puzzle is solved subsequently.
Typical tell-tale features of these puzzles include any puzzle in which each participant has a given piece of information (usually ascommon knowledge) about all other participants but not themselves. Also, usually, some kind of hint is given to suggest that the participants can trust each other's intelligence — they are capable oftheory of mind(that "every participant knowsmodus ponens" is common knowledge).[3]Also, the inaction of a participant is anon-verbal communicationof that participant's lack of knowledge, which then becomes common knowledge to all participants who observed the inaction.
Themuddy children puzzleis the most frequently appearing induction puzzle in scientific literature onepistemic logic.[4][5][6]Muddy children puzzle is a variant of the well known wise men or cheating wives/husbands puzzles.[7]
Hat puzzlesare induction puzzle variations that date back to as early as 1961.[8]In many variations, hat puzzles are described in the context of prisoners.[9][10]In other cases, hat puzzles are described in the context of wise men.[11][12]
A group of attentive children is told that some of them have muddy faces. Each child can see the faces of the others, but cannot tell if his or her own face is muddy. The children are told that those with muddy faces must step forward, but any child with a clean face who steps forward will be punished. At the count of three, every child who believes that his or her face is muddy must step forward simultaneously; any child who signals to another in any way will be punished. If any child with a muddy face has not stepped forward, the process will be repeated.[4][5][13]
Assuming that each child has—and knows each of the others to have—perfect logic all children with muddy faces (X{\displaystyle X}) will step forward together on turnX{\displaystyle X}. The children have different information, depending on whether their own face is muddy or not. Each member ofX{\displaystyle X}seesX−1{\displaystyle X-1}muddy faces, and knows that those children will step forward on turnX−1{\displaystyle X-1}if they are the only muddy faces. When that does not occur, each member ofX{\displaystyle X}knows that he or she is also a member ofX{\displaystyle X}, and steps forward on turnX{\displaystyle X}. Each non-member ofX{\displaystyle X}seesX{\displaystyle X}muddy faces, and will not expect anyone to step forward until at least turnX{\displaystyle X}.
Assume there are two children, Alice and Bob, and that only Alice is muddy (X=1{\displaystyle X=1}). Alice knows that "some" children have muddy faces but that nobody else's face is muddy, meaning that her own face must be muddy and she steps forward on turn one. Bob, seeing Alice's muddy face, has no way of knowing on turn one if his own face is muddy or not until Alice steps forward (indicating that his own face must be clean). If both Alice and Bob are dirty (X=2{\displaystyle X=2}), each is in the position of Bob whenX=1{\displaystyle X=1}: neither can step forward on turn one. However, by turn two Bob knows that Alice must have seen that his face is muddy (because she did not step forward on turn one), and so he steps forward on turn two. Using the same logic, Alice also steps forward on turn two.
Assume that there is a third child, Charlie. If only Alice is muddy (X=1{\displaystyle X=1}), she will see no muddy faces and will step forward on turn one. If both Alice and Bob are muddy (X=2{\displaystyle X=2}), neither can step forward on turn one but each will know by turn two that the other saw a muddy face—which they can see is not Charlie's—so their own face must be muddy and both will step forward on turn two. Charlie, seeing two muddy faces, does not know on turn two whether his own face is muddy or not until Alice and Bob both step forward (indicating that his own face is clean). If all three are muddy (X=3{\displaystyle X=3}), each is in the position of Charlie whenX=2{\displaystyle X=2}: when two people fail to step forward on turn two, each knows that the other sees two muddy faces meaning that their own face must be muddy, and each steps forward on turn three.
It can be proven thatX{\displaystyle X}muddy children will step forward at turnX{\displaystyle X}.[4][14]
Muddy children puzzle can also be solved usingbackward inductionfromgame theory.[13]Muddy children puzzle can be represented as anextensive form gameofimperfect information. Every player has two actions — stay back and step forwards. There is amove by natureat the start of the game, which determines the children with and without muddy faces. Children do not communicate as innon-cooperative games. Every stroke is a simultaneous move by children. It is asequential gameof unlimited length. The game-theoretic solution needs some additional assumptions:
If only Alice is muddy, the last assumption makes it irrational for her to hesitate. If Alice and Bob are muddy, Alice knows that Bob's only reason of staying back after the first stroke is the apprehension to receive the big penalty of stepping forward without a muddy face. In the case withX{\displaystyle X}muddy children, receivingX{\displaystyle X}times the minor penalty is still better than the big penalty.
The King called the three wisest men in the country to his court to decide who would become his new advisor. He placed a hat on each of their heads, such that each wise man could see all of the other hats, but none of them could see their own. Each hat was either white or blue. The king gave his word to the wise men that at least one of them was wearing a blue hat; in other words, there could be one, two, or three blue hats, but not zero. The king also announced that the contest would be fair to all three men. The wise men were also forbidden to speak to each other. The king declared that whichever man stood up first and correctly announced the colour of his own hat would become his new advisor. The wise men sat for a very long time before one stood up and correctly announced the answer. What did he say, and how did he work it out?[15]
The King's Wise Menis one of the simplest induction puzzles and one of the clearest indicators to the method used.
Since there must be three blue hats, the first man to figure that out will stand up and say blue.
Alternative solution: This does not require the rule that the contest be fair to each. Rather it relies on the fact that they are all wise men, and that it takes some time before they arrive at a solution.
There can only be three scenarios: one blue hat, two blue hats or three blue hats. If there was only one blue hat, then the wearer of that hat would see two white hats, and quickly know that he has to have a blue hat, so he would stand up and announce this straight away. Since this hasn't happened, then there must be at least two blue hats. If there were two blue hats, then either one of those wearing a blue hat would look across and see one blue hat and one white hat, but not know the colour of their own hat. If the first wearer of the blue hat assumed he had a white hat, he would know that the other wearer of the blue hat would be seeing two white hats, and thus the 2nd wearer of the blue hat would have already stood up and announced he was wearing a blue hat. Thus, since this hasn't happened, the first wearer of the blue hat would know he was wearing a blue hat, and could stand up and announce this. Since either one or two blue hats is so easy to solve, and no one has stood up quickly, then they must all be wearing blue hats.
In Josephine's Kingdom every woman has to pass a logic exam before being allowed to marry.[16]Every married woman knows about the fidelity of every man in the Kingdomexceptfor her own husband, and etiquette demands that no woman should be told about the fidelity of her husband. Also, a gunshot fired in any house in the Kingdom will be heard in any other house. Queen Josephine announced that at least one unfaithful man had been discovered in the Kingdom, and that any woman knowing her husband to be unfaithful was required to shoot him at midnight following the day after she discovered his infidelity. How did the wives manage this?
Josephine's Problemis another good example of a general case.
This problem is also known as the Cheating Husbands Problem, the Unfaithful Wives Problem, the Muddy Children Problem. It is logically identical to theBlue Eyes Problem.
This problem also appears as a problem involving black hats and white hats in C. L. Liu's classic textbook 'Elements of Discrete Mathematics'.[17]
At the Secret Convention of Logicians, the Master Logician placed a band on each attendee's head, such that everyone else could see it but the person themselves could not. There were many different colours of band. The Logicians all sat in a circle, and the Master instructed them that a bell was to be rung in the forest at regular intervals: at the moment when a Logician knew the colour on his own forehead, he was to leave at the next bell. They were instructed not to speak, nor to use a mirror or camera or otherwise avoid using logic to determine their band colour. In case any impostors had infiltrated the convention, anyone failing to leave on time would be gruffly removed at the correct time. Similarly, anyone trying to leave early would be gruffly held in place and removed at the correct time. The Master reassured the group by stating that the puzzle would not be impossible for any True Logician present. How did they do it?[18]
Alice at the convention of Logiciansis general induction plus a leap of logic.
A number of players are each wearing a hat, which may be of various specified colours. Players can see the colours of at least some other players' hats, but not that of their own. With highly restricted communication or none, some of the players must guess the colour of their hat. The problem is to find a strategy for the players to determine the colours of their hats based on the hats they see and what the other players do. In some versions, they compete to be the first to guess correctly; in others, they can work out a strategy beforehand to cooperate and maximize the probability of correct guesses.[19]
One variation received some new publicity as a result of Todd Ebert's 1998Ph.D.thesisat theUniversity of California, Santa Barbara.[20]It is a strategy question about acooperative game, which has connections toalgebraic coding theory.[21]
Three players are told that each of them will receive either a red hat or a blue hat. They are to raise their hands if they see a red hat on another player as they stand in a circle facing each other. The first to guess the colour of his or her hat correctly wins.
All three players raise their hands. After the players have seen each other for a few minutes without guessing, one player announces "Red", and wins. How did the winner do it, and what is the color of everyone's hats?
First, if two people had blue hats, not everyone's hand would have been raised. Next, if player 1 had seen a blue hat on player 2 & a red hat on player 3, then player 1 would have known immediately that his own hat must be red. Thus any player who sees a blue hat can guess at once. Finally, the winner realizes that since no one guesses at once, there must be no blue hats, so every hat must be red.[22]
In the case where every player has to make a guess, but they are free to choose when to guess, there is a cooperative strategy that allows every player to guess correctly unless all the hats are the same colour. Each player should act as follows:
Suppose that in total there areBblue hats andRred hats. There are three cases.
IfB=Rthen those players wearing blue hats seeB− 1 blue hats andBred hats, so waitB− 1 seconds then correctly guess that they are wearing a blue hat. Similarly, those players wearing a red hat will waitR− 1 seconds before guessing correctly that they are wearing a red hat. So all players make a correct guess at the same time.
IfB<Rthen those wearing a blue hat will seeB− 1 blue hats andRred hats, whilst those wearing a red hat will seeBblue hats andR− 1 red hats. SinceB− 1 <B≤R− 1, those players wearing a blue hat will be the first to speak, guessing correctly that their hat is blue. The other players then guess correctly that their hat is red.
The case whereR<Bis similar.
Fourprisonersare arrested for acrime, but the judge offers tospare them from punishmentif they can solve a logic puzzle.[23]
Three of the men stand a line. A faces the wall, B faces A, and C faces B and A. A fourth man is put behind a wall. All four men wear hats; there are two black hats and two white hats, each prisoner is wearing one of the hats, and each of the prisoners see only the hats in front of him but neither on himself nor behind him. The fourth man behind the screen can't see or be seen by any other prisoner. No communication among the prisoners is allowed. If any prisoner can figure out what color hat he has on his own head with 100% certainty (without guessing) he must then announce it, and all four prisoners go free.
The prisoners know that there are only two hats of each color. So if C observes that A and B have hats of the same color, C would deduce that his own hat is the opposite color. However, if A and B have hats of different colors, then C can say nothing. The key is that prisoner B, after allowing an appropriate interval, and knowing what C would do, can deduce that if C says nothing the hats on A and B must be different; able to see A's hat, he can deduce his own hat color.
In common with many puzzles of this type, the solution relies upon the assumption that all participants are rational and intelligent enough to make the appropriate deductions.
After solving this puzzle, some insight into the nature ofcommunicationcan be gained by pondering whether the meaningful silence of prisoner C violates the "No communication" rule (given that communication is usually defined as the "transfer of information").[citation needed]
In this variant there are 3 prisoners and 3 hats. Each prisoner is assigned a random hat, either red or blue. Each person can see the hats of two others, but not their own. On a cue, they each have to guess their own hat color or pass. They win release if at least one person guessed correctly and none guessed incorrectly (passing is neither correct nor incorrect).
This puzzle doesn't have a 100% winning strategy, but can be won with a 75% chance. When considering the colors of hats as bits, this problem can be solved usingcoding theory, for example withhamming codes.[24]
In a variant of this puzzle, the prisoners know that there are 2 black hats and 2 white hats, and there is a wall in between A and B, yet the prisoners B, C & D can see who's in front of them i.e. D sees B, C and the wall, B sees the wall, and C sees B & the wall. (A again cannot be seen and is only there to wear one of the black hats.) How can they deduce the colours of all of them without communicating?
There are two cases: in the trivial case, two of the four prisoners wear the black hats. Each of the other two prisoners can see that one prisoner is wearing the off-colour hat.
In the non-trivial case, two of the four prisoners wear hats of the same colour, while A and C wear the black hats.
After a while, all four prisoners should be able to deduce that, because D and B were not able to state the colour of their own hat, A and C must be wearing the black hats.
In another variant, only three prisoners and five hats of known colours (in this example two black and three white) are involved.
The three prisoners are ordered to stand in a straight line facing the front, with A in front and C at the back. They are told that there will be two black hats and three white hats. One hat is then put on each prisoner's head; each prisoner can only see the hats of the people in front of him and not on his own. The first prisoner that is able to announce the color of his hat correctly will be released. No communication between the prisoners is allowed.
Assume that A wears a black hat:
So if A wears a black hat there will be a fairly quick response from B or C.
Assume that A wears a white hat:
In this case A, B and C would remain silent for some time, until A finally deduces that he must have a white hat because C and B have remained silent for some time.
As mentioned, there are three white hats and two black hats in total, and the three prisoners know this. In this riddle, you can assume that all three prisoners are very clever and very smart. If C could not guess the color of his own hat that is because he saw either two white hats or one of each color. If he saw two black hats, he could have deduced that he was wearing a white hat.
In this variant there are 10 prisoners and 10 hats. Each prisoner is assigned a random hat, either red or blue, but the number of each color hat is not known to the prisoners. The prisoners will be lined up single file where each can see the hats in front of him but not behind. Starting with the prisoner in the back of the line and moving forward, they must each, in turn, say only one word which must be "red" or "blue". If the word matches their hat color they are released, if not, they are killed on the spot. A sympathetic guard warns them of this test one hour beforehand and tells them that they can formulate a plan where by following the stated rules, 9 of the 10 prisoners will definitely survive, and 1 has a 50/50 chance of survival. What is the plan to achieve the goal?
The prisoners agree that if the first prisoner sees an odd number of red hats, he will say "red". This way, the nine other prisoners will know their own hat color after the prisoner behind them responds.
As before, there are 10 prisoners and 10 hats. Each prisoner is assigned a random hat, either red or blue, but the number of each color hat is not known to the prisoners. The prisoners are distributed in the room such that they can see the hats of the others but not their own. Now, they must each, simultaneously, say only one word which must be "red" or "blue". If the word matches their hat color they are released, and if enough prisoners resume their liberty they can rescue the others. A sympathetic guard warns them of this test one hour beforehand. If they can formulate a plan following the stated rules, 5 of the 10 prisoners will definitely be released and be able to rescue the others. What is the plan to achieve the goal?
The prisoners pair off. In a pair (A, B) of the prisoners A says the color he can see on the head of B, who says the opposite color he sees on the head of A. Then, if both wear hats with the same color, A is released (and B is not), if the colors are different, B is released (and A is not). In total, 5 prisoners answer correctly and 5 do not. This assumes the pair can communicate who is A and who is B, which may not be allowed.
Alternatively, the prisoners build two groups of 5. One group assumes that the number of red hats is even, the other assumes that there is an odd number of red hats. Similar to the variant with hearing, they can deduce their hat color out of this assumption. Exactly one group will be right, so 5 prisoners answer correctly and 5 do not.
Note that the prisoners cannot find a strategy guaranteeing the release of more than 5 prisoners. Indeed, for a single prisoner, there are as many distributions of hat colors where he says the correct answer than there are where he does not. Hence, there are as many distributions of hat colors where 6 or more prisoners say the correct answer than there are where 4 or fewer do so.
In this variant, acountably infinitenumber of prisoners, each with an unknown and randomly assigned red or blue hat line up single file line. Each prisoner faces away from the beginning of the line, and each prisoner can see all the hats in front of him, and none of the hats behind. Starting from the beginning of the line, each prisoner must correctly identify the color of his hat or he is killed on the spot. As before, the prisoners have a chance to meet beforehand, but unlike before, once in line, no prisoner can hear what the other prisoners say. The question is, is there a way to ensure that only finitely many prisoners are killed?
If one accepts theaxiom of choice, and assumes the prisoners each have the (unrealistic) ability to memorize anuncountably infiniteamount of information and perform computations with uncountably infinitecomputational complexity, the answer is yes. In fact, even if we allow anuncountablenumber of different colors for the hats and an uncountable number of prisoners, the axiom of choice provides a solution that guarantees that only finitely many prisoners must die provided that each prisoner can see the hats of every other prisoner (not just those ahead of them in a line), or at least that each prisoner can see all but finitely many of the other hats. The solution for the two color case is as follows, and the solution for the uncountably infinite color case is essentially the same:
The prisoners standing in line form a sequence of 0s and 1s, where 0 is taken to represent blue, and 1 is taken to represent red. Before they are put into the line, the prisoners define the followingequivalence relationover all possible sequences that they might be put into: Two sequences are equivalent if they are identical after a finite number of entries. From this equivalence relation, the prisoners get a collection of equivalence classes. Assuming the axiom of choice, there exists a set of representative sequences—one from each equivalence class. (Almost every specific valueis impossible to compute, but the axiom of choice implies thatsomeset of values exists, so we assume that the prisoners have access to anoracle.)
When they are put into their line, each prisoner can see all but a finite number of hats, and can therefore see which equivalence class theactualsequence of hats belongs to. (This assumes that each prisoner can perform anuncountably infinitenumber of comparisons to find a match, with each class comparison requiring acountably infinitenumber of individual hat-comparisons). They then proceed guessing their hat color as if they were in therepresentativesequence from the appropriate equivalence class. Because the actual sequence and the representative sequence are in the same equivalence class, their entries are the same after some finite numberNof prisoners. All prisoners after these firstNprisoners are saved.
Because the prisoners have no information about the color of their own hat and would make the same guess whichever color it has, each prisoner has a 50% chance of being killed. It may seem paradoxical that an infinite number of prisoners each have an even chance of being killed and yet it is certain that only a finite number are killed. The solution to this paradox lies in the fact that the function employed to determine each prisoner's guess is notMeasurable function.
To see this, consider the case of zero prisoners being killed. This happensif and only ifthe actual sequence is one of the selected representative sequences. If the sequences of 0s and 1s are viewed as binary representations of a real number between 0 and 1, the representative sequences form anon-measurable set. (This set is similar to aVitali set, the only difference being that equivalence classes are formed with respect to numbers with finite binary representations rather than all rational numbers.) Hence no probability can be assigned to the event of zero prisoners being killed. The argument is similar for other finite numbers of prisoners being killed, corresponding to a finite number of variations of each representative.
This variant is the same as the last one except that prisoners can hear the colors called out by other prisoners. The question is, what is the optimal strategy for the prisoners such that the fewest of them die in the worst case?
It turns out that, if one allows the prisoners to hear the colors called out by the other prisoners, it is possible to guarantee the life of every prisoner except the first, who dies with a 50% probability.
To do this, we define the same equivalence relation as above and again select a representative sequence from each equivalence class. Now, we label every sequence in each class with either a 0 or a 1. First, we label the representative sequence with a 0. Then, we label any sequence which differs from the representative sequence in an even number of places with a 0, and any sequence which differs from the representative sequence in an odd number of places with a 1. In this manner, we have labeled every possible infinite sequence with a 0 or a 1 with the important property that any two sequences which differ by only one digit have opposite labels.
Now, when the warden asks the first person to say a color, or in our new interpretation, a 0 or a 1, he simply calls out the label of the sequence he sees. Given this information, everyone after him can determine exactly what his own hat color is. The second person sees all but the first digit of the sequence that the first person sees. Thus, as far as he knows, there are two possible sequences the first person could have been labeling: one starting with a 0, and one starting with a 1. Because of our labeling scheme, these two sequences would receive opposite labels, so based on what the first person says, the second person can determine which of the two possible strings the first person saw, and thus he can determine his own hat color. Similarly, every later person in the line knows every digit of the sequence except the one corresponding to his own hat color. He knows those before him because they were called out, and those after him because he can see them. With this information, he can use the label called out by the first person to determine his own hat color. Thus, everyone except the first person always guesses correctly.
Ebert's version of the problem states that all players who guess must guess at the same predetermined time, but that not all players are required to guess. Now not all players can guess correctly, so the players win if at least one player guesses and all of those who guess do so correctly. How can the players maximise their chance of winning?
One strategy for solving this version of the hat problem employsHamming codes, which are commonly used to detect and correct errors indata transmission. The probability for winning will be much higher than 50%, depending on the number of players in the puzzle configuration: for example, a winning probability of 87.5% for 7 players.
Similar strategies can be applied to team sizes ofN= 2k−1 and achieve a win rate (2k-1)/2k. Thus the Hamming code strategy yields greater win rates for larger values ofN.
In this version of the problem, any individual guess has a 50% chance of being right. However, the Hamming code approach works by concentrating wrong guesses together onto certain distributions of hats. For some cases, all the players will guess incorrectly; whereas for the other cases, only one player will guess, but correctly. While half of all guesses are still incorrect, this results in the players winning more than 50% of the time.
A simple example of this type of solution with three players is instructive. With three players, there are eight possibilities; in two of them all players have the same colour hat, and in the other six, two players have one colour and the other player has the other colour.
The players can guarantee that they win in the latter cases (75% of the time) with the following strategy:
In the two cases when all three players have the same hat colour, they will all guess incorrectly. But in the other six cases, only one player will guess, and correctly, that his hat is the opposite of his fellow players'.[25]
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https://en.wikipedia.org/wiki/Induction_puzzles
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Proof by exhaustion, also known asproof by cases,proof by case analysis,complete inductionor thebrute force method, is a method ofmathematical proofin which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds.[1]This is a method ofdirect proof. A proof by exhaustion typically contains two stages:
The prevalence of digitalcomputershas greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof offour color theoremin 1976), though such approaches can also be challenged on the basis ofmathematical elegance.Expert systemscan be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.[2]
In theCurry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-stylepattern matching.[citation needed]
Proof by exhaustion can be used to prove that if anintegeris aperfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9.[3]
Proof:Each perfect cube is the cube of some integern, wherenis either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three cases are exhaustive:
Mathematicians prefer to avoid proofs by exhaustion with large numbers of cases, which are viewed asinelegant. An illustration as to how such proofs might be inelegant is to look at the following proofs that all modernSummer Olympic Gamesare held in years which are divisible by 4:
Proof: Thefirst modern Summer Olympicswere held in 1896, and then every 4 years thereafter (neglecting exceptional situations such as when the games' schedule were disrupted by World War I, World War II and theCOVID-19 pandemic.). Since 1896 = 474 × 4 is divisible by 4, the next Olympics would be in year 474 × 4 + 4 = (474 + 1) × 4, which is also divisible by four, and so on (this is a proof bymathematical induction). Therefore, the statement is proved.
The statement can also be proved by exhaustion by listing out every year in which the Summer Olympics were held, and checking that every one of them can be divided by four. With 28 total Summer Olympics as of 2016, this is a proof by exhaustion with 28 cases.
In addition to being less elegant, the proof by exhaustion will also require an extra case each time a new Summer Olympics is held. This is to be contrasted with the proof by mathematical induction, which proves the statement indefinitely into the future.
There is no upper limit to the number of cases allowed in a proof by exhaustion. Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving achess endgamepuzzlemight involve considering a very large number of possible positions in thegame treeof that problem.
The first proof of thefour colour theoremwas a proof by exhaustion with 1834 cases.[4]This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.
In general the probability of an error in the whole proof increases with the number of cases. A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. Other types of proofs—such as proof by induction (mathematical induction)—are considered moreelegant. However, there are some important theorems for which no other method of proof has been found, such as
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Mise en abyme(also mise-en-abîme, French "put in the abyss", [miːz ɒn əˈbɪːm]) is a transgeneric and transmedial technique that can occur in any literary genre, incomics,film,paintingor other media. It is a form of similarity and/or repetition, and hence a variant of self-reference.Mise en abymepresupposes at least two hierarchically different levels. A subordinate level 'mirrors' content or formal elements of a primary level.[1]
'Mirroring' can mean repetition, similarity or even, to a certain extent, contrast. The elements thus ‘mirrored’ can refer to form (e.g. apaintingwithin a painting) or content (e.g. a theme occurring on different levels).[1]
Mise en abymecan be differentiated according to its quantitative, qualitative and functional features. For instance, ‘mirroring’ can occur once, several times (on a lower and yet on a lower and so on level) or (theoretically) an infinite number of times (as in the reflection of an object between twomirrors, which creates the impression of a visual abyss). Further,mise en abymecan either be partial or complete (i.e. mirror part or all of the upper level) and either probable, improbable or paradoxical. It can contribute to the understanding of a work, or lay bare its artificiality.[2]
The termmise en abymederives fromheraldry. It describes the appearance of a smaller shield in the center of a larger one; see for example theRoyal coat of arms of the United Kingdom(in the form used between 1801 and 1837).[3]André Gide, in an 1893 entry into his journal,[4]was the first to write aboutmise en abymein connection with describing self-reflexive embeddings in various forms of art.
The term enters thelexiconthroughClaude-Edmonde Magny[5]who described the aesthetic effects of the device.Jean Ricardou[6]developed the concept further by outlining some of its functions. On the one hand it may confuse and disrupt the work in question, but on the other hand it may enhance understanding e.g. by pointing out the work's true meaning or intention. Lucien Dällenbach[7]continues the research in amagisterialstudy by classifying and describing various forms and functions ofmise en abyme.[8]
Mise en abymeis not restricted to a specific kind of literature or art. Therecursiveappearance of a novel within a novel, a play within a play, a picture within a picture, or a film within a film formmises en abymethat can have many different effects on the perception and understanding of theliterary textor work of art.
Mariage à-la-Mode(1743–45) is a narrative series of six socially and morally critical paintings byWilliam Hogarth. In the fourth painting,Mariage à-la-Mode 4: The Toilette,an example ofmise en abymecan be found. The man on the right is not the woman's husband, however they are clearlyflirtingand are possibly arranging a meeting at night. The paintings above their heads depict sexual scenes,foreshadowingwhat is going to happen.
Another example ofmise en abymewould be a novel within a novel, or a play within a play. In William Shakespeare'sHamletthe title character stages a play within the play (“The Murder of Gonzago”) to find out whether his uncle really murdered his father as theghostof his father has told him. It is not only a formal mirroring of a theatrical situation (a play within a play) but also a mirroring of a content element, namely of what supposedly had happened in the pre-history. Hamlet wants to find out the truth by instructing the actors to perform a play which contains striking similarities to the alleged murder of Hamlet's father. The embedded performance thus includes details from the broader plot, which illuminates athematicaspect of the play itself.
The Fall of the House of UsherbyEdgar Allan Poesports a particularly noteworthy example ofmise en abyme, a story within a story. Towards the end of the story, the narrator begins to read aloud parts of an antique volume entitledMad Tristby Sir Launcelot Canning. At first, the narrator only vaguely realizes that the sounds occurring in the embedded fictionMad Tristcan really be heard by him. Theembedded storyis subsequently more and more intertwined with the events that are happening in the embedding story, until, in a climactic scene, a supposedly dead and buried member of the House of Usher (Madeline), is about to enter the room where the recital takes place when both she and her incestuously beloved brother die in a final embrace. This fall (and the partial mirroring of the scene inMad Trist) anticipates the final fall of the House of Usher, which sinks into the tarn surrounding the building.
InThe Simpsonsthe characters frequently watch television: characters of a TV series are thus watching TV themselves. This act is amise en abyme, as we see a film within a film. However, if they started discussing what they are watching it would also be an instance ofmeta-reference(or rather themise en abymewould, as it sometimes does, have triggered metareferential reflections). Yet, as a rule,mise en abymemerely ‘mirrors’ elements from a superior level on a subordinate one, but does not necessarily trigger an analysis of them.
Here is a list of modern media that features a mise en abyss at the core of their scenario:
Books
TV series
Movies
Video Games
Mise en abymecan be easily confused withmetalepsisand metareference. These terms describe related features, asmise en abymecan be a springboard to metalepsis if there is aparadoxicalconfusion of the levels involved. If the artificiality of the mirroring device or related issues are foregrounded or discussed,mise en abymecan also be conducive tometareference.[9]
To summarise,mise en abymeis a form of similarity, repetition and hence a variant of self-reference that is not necessarily discussed within its appearing medium, it only occurs. If the occurrence is discussed, or if mise en abyme triggers reflections on the respective medium or the construction of the text for example, mise en abyme is combined with metareference.
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https://en.wikipedia.org/wiki/Mise_en_abyme_(in_literature_and_other_media)
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Gödel, Escher, Bach: an Eternal Golden Braid(abbreviated asGEB) is a 1979 nonfiction book by American cognitive scientistDouglas Hofstadter.
By exploring common themes in the lives and works of logicianKurt Gödel, artistM. C. Escher, and composerJohann Sebastian Bach, the book expounds concepts fundamental tomathematics,symmetry, andintelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. It also discussesself-referenceand formal rules,isomorphism, what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.
In response to confusion over the book's theme, Hofstadter emphasized thatGödel, Escher, Bachis not about the relationships ofmathematics, art, andmusic, but rather about howcognitionemergesfrom hidden neurological mechanisms. One point in the book presents an analogy about how individualneuronsin thebraincoordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in acolony of ants.[1][2]
Gödel, Escher, Bachwon thePulitzer Prize for General Nonfiction[3]and theNational Book Awardfor Science Hardcover.[4][a]
Gödel, Escher, Bachtakes the form of interweaving narratives. The main chapters alternate with dialogues between imaginary characters, usuallyAchilles and the tortoise, first used byZeno of Eleaand later byLewis Carrollin "What the Tortoise Said to Achilles". These origins are related in the first two dialogues, and later ones introduce new characters such as the Crab. These narratives frequently dip intoself-referenceandmetafiction.
Word playalso features prominently in the work. Puns are occasionally used to connect ideas, such as the "Magnificrab, Indeed" with Bach'sMagnificat in D; "SHRDLU, Toy of Man's Designing" with Bach's "Jesu, Joy of Man's Desiring"; and "Typographical Number Theory", or "TNT", which inevitably reacts explosively when it attempts to make statements about itself. One dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both theliquidandmusicalvarieties), which is titled "Djinn and Tonic". Sometimes word play has no significant connection, such as the dialogue "AMuOffering", which has no close affinity to Bach'sThe Musical Offering.
One dialogue in the book is written in the form of acrab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines that double as an answer to a question in the next line. Another is a sloth canon, where one character repeats the lines of another, but slower and negated.
The book contains many instances ofrecursionandself-reference, where objects and ideas speak about or refer back to themselves. One isQuining, a term Hofstadter invented in homage toWillard Van Orman Quine, referring to programs that produce their ownsource code. Another is the presence of a fictional author in the index,Egbert B. Gebstadter, a man with initials E, G, and B and a surname that partially matches Hofstadter. A phonograph dubbed "Record Player X" destroys itself by playing a record titledI Cannot Be Played on Record Player X(an analogy toGödel's incompleteness theorems), an examination ofcanonform inmusic, and a discussion of Escher'slithograph of two hands drawing each other.
To describe such self-referencing objects, Hofstadter coins the term "strange loop", a concept he examines in more depth in his follow-up bookI Am a Strange Loop. To escape many of the logical contradictions brought about by these self-referencing objects, Hofstadter discussesZenkoans. He attempts to show readers how to perceive reality outside their own experience and embrace such paradoxical questions by rejecting the premise, a strategy also called "unasking".
Elements ofcomputer sciencesuch ascall stacksare also discussed inGödel, Escher, Bach, as one dialogue describes the adventures of Achilles and the Tortoise as they make use of "pushing potion" and "popping tonic" involving entering and leaving different layers of reality. The same dialogue has a genie with a lamp containing another genie with another lamp and so on. Subsequent sections discuss the basic tenets of logic, self-referring statements, ("typeless") systems, and even programming. Hofstadter further createsBlooP and FlooP, two simpleprogramming languages, to illustrate his point.
The book is filled with puzzles, including Hofstadter'sMU puzzle, which contrasts reasoning within a defined logical system with reasoning about that system. Another example can be found in the chapter titledContracrostipunctus, which combines the wordsacrosticandcontrapunctus(counterpoint). In this dialogue between Achilles and the Tortoise, the author hints that there is a contrapunctal acrostic in the chapter that refers both to the author (Hofstadter) and Bach. This can be spelled out by taking the first word of each paragraph, to reveal "Hofstadter's Contracrostipunctus Acrostically Backwards Spells J. S. Bach". The second acrostic is found by taking the first letters of the words of the first, and reading them backwards to get "J S Bach", as the acrostic sentence self-referentially states.
Gödel, Escher, Bachwon thePulitzer Prize for General Nonfictionand theNational Book Awardfor Science Hardcover.
Martin Gardner's July 1979 column inScientific Americanstated, "Every few decades, an unknown author brings out a book of such depth, clarity, range, wit, beauty and originality that it is recognized at once as a major literary event."[5]
For Summer 2007, theMassachusetts Institute of Technologycreated an online course for high school students built around the book.[6]
In its February 19, 2010, investigative summary on the2001 anthrax attacks, theFederal Bureau of Investigationsuggested thatBruce Edwards Ivinswas inspired by the book to hide secret codes based uponnucleotide sequencesin theanthrax-laced letters he allegedly sent in September and October 2001,[7]using bold letters, as suggested on page 404 of the book.[8][9]It was also suggested that he attempted to hide the book from investigators by throwing it in the trash.[10]
In 2019, British mathematicianMarcus du Sautoycurated a series of events at London'sBarbican Centreto celebrate the book's fortieth anniversary.[11]
Hofstadter has expressed some frustration with howGödel, Escher, Bachwas received. He felt that readers did not fully grasp thatstrange loopswere supposed to be the central theme of the book, and attributed this confusion to the length of the book and the breadth of the topics covered.[12][13]
To remedy this issue, Hofstadter publishedI Am a Strange Loopin 2007, which had a more focused discussion of the idea.[13]
Hofstadter claims the idea of translating his book "never crossed [his] mind" when he was writing it—but when his publisher brought it up, he was "very excited about seeing [the] book in other languages, especially… French." He knew, however, that "there were a million issues to consider" when translating,[14]since the book relies not only on word-play, but on "structural puns" as well—writing where the form and content of the work mirror each other (such as the "Crab canon" dialogue, which reads almost exactly the same forwards as backwards).
Hofstadter gives an example of translation trouble in the paragraph "Mr. Tortoise, Meet Madame Tortue", saying translators "instantly ran headlong into the conflict between the feminine gender of the French nountortueand the masculinity of my character, the Tortoise."[14]Hofstadter agreed to the translators' suggestions of naming the French characterMadame Tortue, and the Italian versionSignorina Tartaruga.[15]Because of other troubles translators might have retaining meaning, Hofstadter "painstakingly went through every sentence ofGödel, Escher, Bach, annotating a copy for translators into any language that might be targeted."[14]
Translation also gave Hofstadter a way to add new meaning and puns. For instance, inChinese, the subtitle is not a translation ofan Eternal Golden Braid, but a seemingly unrelated phraseJí Yì Bì(集异璧, literally "collection of exotic jades"), which ishomophonictoGEBin Chinese. Some material regarding this interplay is in Hofstadter's later book,Le Ton beau de Marot, which is mainly about translation.
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https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
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Macbeth(also known asThe Tragedy of MacbethorRoman Polanski's Film of Macbeth) is a 1971historical drama filmdirected byRoman Polanski, and co-written by Polanski andKenneth Tynan. Afilm adaptationofWilliam Shakespeare's tragedy of thesame name, it tells the story of theHighlandlord who becomes King of Scotland through treachery and murder.Jon FinchandFrancesca Annisstar as thetitle characterandhis wife, noted for their relative youth as actors. Themes ofhistoric recurrence, greater pessimism and internal ugliness in physically beautiful characters are added to Shakespeare's story of moral decline, which is presented in a more realistic style.
Polanski opted to adaptMacbethas a means of coping with the highly publicizedManson Familymurder of his pregnant wife,Sharon Tate. Finding difficulty obtaining sponsorship from major studios,Playboy Enterprisesstepped in to provide funding. Following troubled shooting around theBritish Islesmired by poor weather,Macbethscreened out of competition at the1972 Cannes Film Festivaland was acommercial failurein the United States. Initially controversial for its graphic violence and nudity, the film has since garnered generally positive reviews, and was namedBest Filmby theNational Board of Reviewin 1972.
In theMiddle Ages, a Norwegian and Irish invasion of Scotland aided by the traitorousThane of Cawdor, is suppressed byMacbeth, Thane of Glamis, andBanquo. Cawdor is sentenced to death byKing Duncanwho decrees that Macbeth shall be awarded his title. Macbeth and Banquo do not hear of this news; when out riding, they happen uponThree Witches, who hail Macbeth as "Thane of Cawdor and future King", and Banquo as "lesser and greater". At their camp, nobles arrive and inform Macbeth he has been named the Thane of Cawdor, with Macbeth simultaneously awed and frightened at the prospect of usurping Duncan, in further fulfilment of the prophecy. He writes a letter toLady Macbeth, who is delighted at the news. However, she fears her husband has too much good nature, and vows to be cruel for him. Duncan names his eldest son,Malcolm, Prince of Cumberland, and thusheir apparent, to the displeasure of Macbeth and Malcolm's brotherDonalbain. The royal family and nobles then spend the night at Macbeth's castle, with Lady Macbeth greeting the King and dancing with him with duplicity.
Urged on by his wife, Macbeth steps into King Duncan's chambers after she has drugged the guards. Duncan wakes and utters Macbeth's name, but Macbeth stabs him to death. He frames the guards for the assassination and then murders them when Duncan's corpse is discovered. Fearing a conspiracy, Malcolm and Donalbain flee to England and Ireland, and the Thane ofRossrealises Macbeth will be king. Anopportunisticcourtier, he hails Macbeth atScone, while the nobleMacduffheads back to his home inFife. When Macbeth begins to fear possible usurpation by Banquo and his sonFleance, he sends two murderers to kill them, and then sends Ross as the mysteriousThird Murderer. Banquo is killed, while Fleance escapes. Macbeth, disposes of the two murderers by drowning them. After Banquo appears at a banquet as a ghost, Macbeth seeks out the witches, who are performing a nude ritual. The witches and the spirits they summon deceive Macbeth into thinking he is invincible, as he cannot be killed except by a man not born of woman and will not be defeated until "Birnam Wood comes to Dunsinane."
Ross is sent to Fife to direct the slaughter of Macduff, however, Macduff has gone to England. Ross enters Fife castle pretending to be a friend, but leaves the heavy castle doors open, allowing Macbeth's gang of murderers in to killLady Macduff, the children and servants. With nobles fleeing Scotland, Macbeth chooses a new Thane of Fife, bestowing the title on Seyton over Ross. Disappointed, Ross leaves Scotland to join Malcolm and Macduff in England. They welcome him, unaware of his treachery. The English King has allied with Malcolm, committing forces led bySiwardto overthrow Macbeth and install Malcolm on the Scottish throne.
The English forces invade, covering themselves by cutting down branches from Birnam Wood and holding them in front of their army to hide their numbers as they march on Macbeth in Dunsinane. When the forces storm the castle, Macduff confronts Macbeth, and during the sword fight, Macduff reveals he was delivered byCaesarean section. Macduff kills Macbeth by beheading him, and Ross presents the crown to Malcolm, who becomes the new King of Scotland. Meanwhile, Donalbain, out riding, encounters the witches.
James Morrison wroteMacbeth's themes of "murderous ambition" fit in with Polanski's filmography,[7]and saw similarities toOrson Welles's 1948 film version ofMacbethin downplaying psychology and reviving the "primitive edge". However, unlike Welles, Polanski chooses naturalism overexpressionism.[8]Author Ewa Mazierska wrote that, despite supposedrealismin presenting soliloquies as voiceovers, Polanski'sMacbethwas "absurdist", not depicting history as an explanation for current events, but as a "vicious circle of crimes and miseries". Eachcoronationoccurs after the predecessor is violently dispatched, and guests and hosts always betray each other, with Polanski adding Ross leaving Fife's castle doors open.[9]
Deanne Williams read the film as not only Polanski's reflections on the murder ofSharon Tate, but on wider issues such as theassassination of Martin Luther King Jr.and theVietnam War.[10]Francesca Royster similarly argues the use of English and Celtic cultures in the clothes and culture of the 1960s and 1970s, pointing to the publications ofThe Lord of the Ringsin the U.S. and the music ofLed Zeppelin, ties the film's past to the present.[11]While Playboy Enterprises role was mainly to provide funding, Williams also saw Polanski's Lady Macbeth as embodying "Playboymythos" paradoxes, at times warm and sexy, at times a domestic servant, and at timesfemme fatale.[12]
In one scene, Macbeth's court hostsbear-baiting, a form of entertainment in the Middle Ages in which a bear and dogs are pitted against each other. Williams suggested the scene communicates Macbeth and Lady Macbeth's growing callousness after taking power,[13]while Kenneth S. Rothwell and Morrison matched the scene to Shakespeare's Macbeth describing himself as "bear-like".[14][15]
Literary criticSylvan Barnetwrote that the younger protagonists suggested "contrast between a fair exterior and an ugly interior".[16]Williams compared Lady Macbeth toLady Godivain her "hair and naturalistic pallor", suggesting she could fit in at thePlayboy Mansion.[12]More "ugliness" is added by Polanski in the re-imagining of Ross, who becomes a more important character in this film.[17]As with the leads, Ross demonstrates "evil-in-beauty" as he is played by "handsomeJohn Stride".[18]
Barnet also wrote the changed ending with Donalbain meeting the witches replaced the message of "measure, time and place" with "unending treachery".[16]Film historian Douglas Brode also commented on the added ending, saying it reflected Polanski's pessimism in contradiction to Shakespeare's optimism. Likewise, Brode believed Macbeth's "Tomorrow and tomorrow and tomorrow" soliloquy becomes an articulation ofnihilismin the film, while Shakespeare did not intend it to reflect his own sentiment.[19]
DirectorRoman Polanskihad been interested in adapting a Shakespeare play since he was a student inKraków, Poland,[10]but he did not begin until after the murder of his pregnant wife,Sharon Tate, and three of the couple's mutual friends by members of theManson Familyat his house inBeverly Hillson the night of 9 August 1969. Following the murders, Polanski sank into deep depression, and was unhappy with the way the incident was depicted in the media, in which his films seemed to be blamed.[20]At the time, he was working on the filmThe Day of the Dolphin, a project that collapsed and was turned over to another director,Mike Nichols.[21]While inGstaad, Switzerland during the start of 1970, Polanski envisioned an adaptation ofMacbethand sought out his friend, British theatre criticKenneth Tynan, for his "encyclopedic knowledge of Shakespeare".[10]In turn, Tynan was interested in working with Polanski because the director demonstrated what Tynan considered "exactly the right combination of fantasy and violence".[22]
Around this time Richard Burton was discussing producing a version ofMacbethstarring Elizabeth Taylor as Lady Macbeth from a script byPaul Dehn. "I'd rather put it together than appear in it," said Burton, adding "I'm not very confident getting it on in the present conditions in the film industry."[23]
Tynan and Polanski found it challenging to adapt the text to suit the feel of the film. Tynan wrote to Polanski, saying, "the number oneMacbethproblem is to see the events of the film from his point of view".[24]During the writing process, Polanski and Tynan acted out their scenes in aBelgravia, London apartment, with Tynan as Duncan and Polanski as Macbeth.[25]As with the 1948 film version ofHamlet, the soliloquies are presented naturalistically asvoiceovernarration.[24]
In one scene Polanski and Tynan wrote, Lady Macbeth delivers hersleepwalking soliloquyin the nude. Their decision was motivated by the fact that people in this era always slept in the nude.[26]Likewise, consultations of academic research of the Middle Ages led to the depiction of the nobles, staying at Macbeth's castle, going to bed on hay and the ground, with animals present.[27]
The added importance the film gives to Ross did not appear in the first draft of the screenplay, which instead invented a new character called the Bodyguard, who also serves as theThird Murderer.[18]The Bodyguard was merged into Shakespeare's Ross.[28]The screenplay was completed by August 1970, with plans to begin filming in England in October.[29]
Paramount Pictures,Universal PicturesandMetro-Goldwyn-Mayerdeclined to finance the project, seeing a Shakespeare play as a poor fit for a director who achieved success withRosemary's Baby(1968).[30]Hugh Hefner, who publishedPlayboy, had produced a few films withPlayboy Enterprisesand was eager to make more when he met Polanski at a party.[31]The budget was set at $2.4 million.[32]
Hefner's involvement was announced in August 1970. TheEvening Standardreported "I understand that this new version will have a high content of sex and violence."[33]
In April 1971 Columbia Pictures announced they had signed a three-year deal with Playboy Enterprises to make at least four films together the first of which wasMacbeth.[34]
Due to a feeling that the characters ofMacbethwere more relatable to young people in the 1960s than experienced, elder actors, Polanski deliberately sought out "young and good-looking" actors for the parts of Macbeth andLady Macbeth.[35]Francesca AnnisandJon Finchwere 26 and 29, respectively,[36]with Tynan remarking characters over 60 were too old to be ambitious.[35]
Polanski wanted to cast eitherVictoria TennantorTuesday Weldin the role of Lady Macbeth.[37]Weld rejected the role, unwilling to perform the nude scene.[36]The role was also turned down byGlenda Jacksonwho said "I don't fancy six weeks on location, walking through that damned gorse for next to no money."[38]
Casting of the role did not happen until just before the film started.[39]Annis accepted the role after some reluctance, as she agreed the character should be older, but was easy to persuade to join the cast.[35]Annis called the film "75 percent Shakespeare and 25 percent Polanski" saying "there is nothing to get excited about" her nude scene. "I simply walk across a room."[40]
Polanski's first choice for Macbeth wasAlbert Finney, who rejected the role, after which Tynan recommendedNicol Williamson, but Polanski felt he was not attractive enough.[36]Finch was better known for appearing inHammer Film Productionspictures such asThe Vampire Loversand the television seriesCounterstrike.[35]Finch met Polanski on a Paris-London plane and auditioned several times.[41]
For the scene where theThree Witchesand numerous others perform "Double, double, toil and trouble" in the nude, Polanski had difficulty hiringextrasto perform. As a result, some of the witches are cut from cardboard.[12]Polanski had a few of the elderly extras sing "Happy Birthday to You" while naked for a video, sent to Hefner for his 45th birthday.[42]
Macbethwas filmed in various locations around the British Isles, starting inSnowdoniainWalesin October 1970.[36][21]A considerable amount of shooting took place inNorthumberlandon the northeast coast of England, includingLindisfarne Castle,Bamburgh Castleand beach, St. Aidan's Church and North Charlton Moors nearAlnwick.[43]Interior scenes were shot atShepperton Studios;[37]filming started 2 November 1970.[44]
The production was troubled by poor weather,[45]and the cast complaining of Polanski's "petulance".[46]Fight directorWilliam Hobbslikened the longrehearsalsin the rain to "training for thedecathlon".[46]
Polanski personally handled and demonstrated thepropsand rode horses before shooting, and walked into animal feces to film goats and sheep.[46]It was common for the director to snatch the camera away from his cameramen.[47]He also decided to use special effects to present the "dagger of the mind", believing viewers may be puzzled or would not enjoy it if the dagger did not appear on screen, but was merely described in the dialogue.[48]The great challenges in portraying the catapult of fireballs into the castle led to Polanski calling it "special defects".[49]
By mid January the film was behind schedule. The completion guarantors arranged forPeter Collinsonto be hired and filmed scenes in Shepperton.[50][51]
Polanski justified the film's inefficiency, blaming "shitty weather", and agreed to give up one-third of the rest of his salary, on top of which Hefner contributed another $500,000 to complete the film.[52]
For the film score, Polanski employed theThird Ear Band, a musical group which enjoyed initial success after publishing their albumAlchemyin 1969. The band composed original music for the film, by adding electronic music to hand drums,woodwindsandstrings.[26]Recordersandoboeswere also used, inspired byMedieval musicin Scotland.[53]Additionally, elements ofmusic in Indiaand the Middle East andjazzwere incorporated into the score.[54]
In the scene where King Duncan is entertained as Macbeth's castle,lutesare played, andFleancesings "Merciless Beauty" byGeoffrey Chaucer, though his lyrics did not fit the film's time frame.[55]While the score has some Middle Ages influence, this is not found in the scenes where Duncan is assassinated and Macbeth is killed. Polanski and the band usedaleatoric musicfor these scenes, to communicate chaos.[26]
In the United States, the film opened in the Playboy Theater in New York City on 20 December 1971.[56]Polanski bemoaned the release near January as "cinematic suicide" given usually low ticket sales for newly released films in that month.[24]The film opened in London in February 1972.[57]The film was screened at the1972 Cannes Film Festival, but was not entered into the main competition.[58]Polanski, Finch and Annis attended the Cannes festival in May 1972.[59]
The film was abox office bomb.[45]According toThe Hollywood Reporter, Playboy Enterprises estimated in September 1973 that it would lose $1.8 million on the film, and that it would damage the company as a whole.[60]
Total losses were $3.5 million.[61]The losses caused Shakespeare films to appear commercially risky untilKenneth BranaghdirectedHenry Vin 1989.[62]Film criticTerrence Raffertyassociated the financial failure with the varioussuperstitions surrounding the play.[24]
Upon release,Macbethreceived mixed reviews, with much negative attention on its violence, in light of theManson murders, and the nudity, blamed on itsPlayboyassociations.[24]Pauline Kaelwrote the film "reduces Shakespeare's meanings to the banal theme of 'life is a jungle'".[63]Varietystaff dismissed the film, writing, "Does Polanski'sMacbethwork? Not especially, but it was an admirable try".[64]Derek Malcolm, writing forThe Guardian, called the film shocking but not over-the-top, and Finch and Annis "more or less adequate".[57]
In contrast,Roger Ebertgave it four stars, writing it was "full of sound and fury" and "All those noble, tragic Macbeths –Orson WellesandMaurice Evansand the others – look like imposters now, and the king is revealed as a scared kid".[65]Roger Greenspun, forThe New York Times, said that despite gossip about the film, it is "neither especially nude nor unnecessarily violent", and that Finch and Annis give great performances.[56]InNew York,Judith Cristdefended the film as traditional, appropriately focusing on Macbeth's "moral deterioration", and suited for youthful audiences, and drew parallel with its blood to the title ofAkira Kurosawa's 1957Macbethfilm,Throne of Blood.[66]
Literary criticSylvan Barnetwrote that, given Shakespeare's writing, it was arguable "blood might just as well flow abundantly in a film". However, he wrote the perceived inspiration from theTheatre of Crueltyis "hard to take".[16]Troy Patterson, writing forEntertainment Weekly, gave the film a B, calling it "Shakespeare as fright show" and Annis a better fit forMelrose Place.[67]TheTime Outreview states the realistic acting did not do justice to the poetry, and the film "never quite spirals into dark, uncontrollable nightmare as the Welles version (for all its faults) does".[68]
Opinions improved with time, with filmmaker and novelistJohn Saylessaying, "I think it's a great piece of filmmaking" in 2007, and novelistMartin Amissaying, "I really think the film is without weaknesses" in 2013.[24]In his2014 Movie Guide,Leonard Maltingave the film three and a half stars, describing it as "Gripping, atmospheric and extremely violent".[69]The film holds an 80% rating on thereview aggregatorRotten Tomatoes, based on 64 reviews, with the consensus "Roman Polanski'sMacbethis unsettling and uneven, but also undeniably compelling."[70]
After a restoration bySony Pictures Entertainment, the film was placed in the Venice Classics section in the2014 Venice Film Festival.[73]InRegion 1,The Criterion Collectionreleased the film on DVD andBlu-rayin September 2014.[45]
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Meta-reference(ormetareference) is a category ofself-referencesoccurring in many media ormedia artifactslike published texts/documents, films, paintings, TV series, comic strips, or video games. It includes all references to, or comments on, a specific medium, medial artifact, or the media in general. These references and comments originate from a logically higher level (a "meta-level") within any given artifact, and draw attention to—or invite reflection about—media-related issues (e.g. the production, performance, or reception) of said artifact, specific other artifacts (as inparody), or to parts, or the entirety, of the medial system. It is, therefore, the recipient's awareness of an artifact's medial quality that distinguishes meta-reference from more general forms of self-reference. Thus, meta-reference triggers media-awareness within the recipient, who, in turn "becomes conscious of both the medial (or "fictional" in the sense of artificial and, sometimes in addition, "invented") status of the work" as well as "the fact that media-related phenomena are at issue, rather than (hetero-)references to the world outside the media."[1]Although certain devices, such asmise-en-abîme, may be conducive to meta-reference, they are not necessarily meta-referential themselves.[2]However, innately meta-referential devices (e.g.metalepsis) constitute a category of meta-references.
While meta-reference as a concept is not a new phenomenon and can be observed in very early works of art and media not tied to specific purposes (e.g. Homer's invocation of the muses at the beginning of theOdysseyin order to deliver the epic better), the term itself is relatively new.[3]Earlier discussions of meta-referential issues often opt for more specific terminology tied to the respective discipline. Notable discussions of meta-reference include, but are not limited to, William H. Gass's[4]and Robert Scholes's[5]exploration ofmetafiction, Victor Stoichita's examination of early modern meta-painting,[6]and Lionel Abel's[7]investigation ofmetatheatre. In the context of drama, meta-reference has also become colloquially known as the breaking of thefourth wall. The first study to underscore the problem resulting from the lack of cohesive terminology, as well as the necessity to acknowledge meta-reference as transmedial and trans-generic phenomenon, was published in 2007 by Hauthal et al.[8]Publications by Nöth and Bishara[9]as well as Wolf[10]followed suit, raised similar concerns, included case studies from various media, coined and helped establish the more uniform umbrella term meta-reference as define above.
While every medium has the potential for meta-reference, some media can transport meta-reference more easily than others. Media that can easily realise its meta-referential potential includes, for instance, literature, painting, and film. Although music can be meta-referential even outside the confines of lyrics, meta-reference in music is much harder to create or detect.[11][12]Music, therefore, would be a less typical medium for the occurrence of meta-reference. Nöth argues in this context that although non-verbal media can be the home of meta-reference, the contained meta-reference can only be implicit because non-verbal media can only show similarities, but never point directly (or explicitly) to meta-referential elements.[13]Others, however, argue that meta-reference is explicit as long as it is clear.
John Fowles begins chapter 13 of his novelThe French Lieutenant's Womanwith the words
ThisstoryI amtellingis allimagination. ThesecharactersI create never existed outside my own mind. If I have pretended until now to know mycharacters' mind and innermost thoughts, it is because I amwritingin [...] aconventionuniversally accepted at the time of mystory: that thenoveliststands next to God.[14][emphases added]
This is an example of explicit meta-reference because the text draws attention to the fact that the novel the recipient is reading is merely a fiction created by the author. It also foregrounds the convention that readers ofrealist fictionaccept the presence of an all-knowing narrator, and breaks it by allowing the narrator to take centre stage which invites meta-reflections by the recipient.
In American comic books published byMarvel Comics, the characterDeadpoolis aware that he is a fictional comic book character. He commonly breaks thefourth wall, to humorous effect. To other non-aware characters in the story, Deadpool's self-awareness as a comic book character appears to be a form ofpsychosis. When other characters question whether Deadpool's real name is even Wade Wilson, he jokes that his true identity depends on which writer the reader prefers.[15]
The Truman Showis a movie that contains a high degree of meta-reference. Truman, the protagonist, is unaware that he is part of a reality TV show, but the audience knows about the artificiality of both Truman's life and, by extension, the movie that is being watched. This is underscored by putting emphasis on the production process of the fictional reality TV show, which makes the audience aware of the same features being used in the movie at the time of watching. Further examples of meta-reference in the movie include spotlights falling from the sky seemingly out of the blue, or a raincloud which is curiously only raining on Truman following him around on Seahaven Beach. Both instances point to the artificiality of Truman's life as well as the film itself.
Other examples include films byMel Brooks, such asBlazing Saddles, which becomes a story about the production of the film, andSilent Movieis a silent movie about producing a silent movie. Additionally,The Muppet Movieand its sequels frequently showed characters referring to the movie script to see what should happen next.
An example of meta-reference in painting isManet's BalconybyRené Magritte. It comments on another painting,The BalconybyÉdouard Manet, by mimicking both the setting of the balcony as well as the poses of the depicted people, but places them in coffins. Thus, the recipient's attention is drawn to the fact that not only are the people in the painting long dead and only still "alive" in the representation, but arguably also that the artist (Manet) and theimpressionistpainting style are just as dead as the portrayed individuals. Furthermore, it is foregrounded that theimpressionistpainting style is just a style that may be copied, which further emphasises the fact that both works are only paintings created in a specific way.
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Print Gallery(Dutch:Prentententoonstelling) is alithographprinted in 1956 by theDutchartistM. C. Escher. It depicts a man in a gallery viewing a print of a seaport, and among the buildings in the seaport is the very gallery in which he is standing, making use of theDroste effectwith visualrecursion.[1]The lithograph has attracted discussion in both mathematical and artistic contexts. Escher consideredPrint Galleryto be among the best of his works.[2]
Bruno Ernst citesM. C. Escheras stating that he beganPrint Gallery"from the idea that it must be possible to make an annular bulge, a cyclic expansion ... without beginning or end."[3]Escher attempted to do this with straight lines, but intuitively switched to using curved lines which make the grid expand greatly as it rotates.[3][4]
In his bookGödel, Escher, Bach,Douglas Hofstadterexplains the seeming paradox embodied inPrint Galleryas astrange loopshowing three kinds of "in-ness": the gallery is physically in the town ("inclusion"); the town is artistically in the picture ("depiction"); the picture is mentally in the person ("representation").[5]
Escher's signature is on a circular void in the center of the work. In 2003, two Dutch mathematicians, Bart de Smit andHendrik Lenstra, reported a way of filling in the void by treating the work as drawn on anelliptic curveover the field ofcomplex numbers. They deem an idealized version ofPrint Gallerytocontain a copy of itself (the Droste effect), rotated clockwise by about 157.63 degrees and shrunk by a factor of about 22.58.[4]Their website further explores the mathematical structure of the picture.[6]
Print Galleryhas been discussed in relation topost-modernismby a number of writers, including Silvio Gaggi,[7]Barbara Freedman,[8]Stephen Bretzius,[9]andMarie-Laure Ryan.[10]
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Astory within a story, also referred to as anembedded narrative, is aliterary devicein which a character within astorybecomes the narrator of a second story (within the first one).[1]Multiple layers of stories within stories are sometimes callednested stories. A play may have a brief play within it, such as in Shakespeare's playHamlet; a film may show the characters watching a short film; or a novel may contain a short story within the novel. A story within a story can be used in all types of narration includingpoems, andsongs.
Stories within stories can be used simply to enhance entertainment for the reader or viewer, or can act as examples to teach lessons to other characters.[2]The inner story often has a symbolic and psychological significance for the characters in the outer story. There is often some parallel between the two stories, and the fiction of the inner story is used to reveal the truth in the outer story.[3]Often the stories within a story are used to satirize views, not only in the outer story, but also in the real world. When a story is told within another instead of being told as part of the plot, it allows the author to play on the reader's perceptions of the characters—the motives and thereliability of the storytellerare automatically in question.[2]
Stories within a story may disclose the background of characters or events, tell of myths and legends that influence the plot, or even seem to be extraneous diversions from the plot. In some cases, the story within a story is involved in the action of the plot of the outer story. In others, the inner story is independent, and could either be skipped or stand separately, although many subtle connections may be lost. Often there is more than one level of internal stories, leading to deeply-nested fiction.Mise en abymeis theFrenchterm for a similar literary device (also referring to the practice inheraldryof placing the image of a small shield on a larger shield).
The literary device of stories within a story dates back to a device known as a "frame story", where a supplemental story is used to help tell the main story. Typically, the outer story or "frame" does not have much matter, and most of the work consists of one or more complete stories told by one or more storytellers.
The earliest examples of "frame stories" and "stories within stories" were in ancient Egyptian andIndian literature, such as the Egyptian "Tale of the Shipwrecked Sailor"[4]andIndian epicslike theRamayana,Seven Wise Masters,HitopadeshaandVikrama and Vethala. InVishnu Sarma'sPanchatantra, an inter-woven series of colorful animal tales are told with one narrative opening within another, sometimes three or four layers deep, and then unexpectedly snapping shut in irregular rhythms to sustain attention. In the epicMahabharata, theKurukshetra Waris narrated by a character inVyasa'sJaya, which itself is narrated by a character inVaisampayana'sBharata, which itself is narrated by a character in Ugrasrava'sMahabharata.
BothThe Golden AssbyApuleiusandMetamorphosesbyOvidextend the depths of framing to several degrees. Another early example is theOne Thousand and One Nights(Arabian Nights), where the general story is narrated by an unknown narrator, and in this narration the stories are told byScheherazade. In many of Scheherazade's narrations, there are alsostories narrated, and even in some of these, there are some other stories.[5]An example of this is "The Three Apples", amurder mysterynarrated by Scheherazade. Within the story, after the murderer reveals himself, he narrates aflashbackof events leading up to the murder. Within this flashback, anunreliable narratortells a story to mislead the would-be murderer, who later discovers that he was misled after another character narrates the truth to him.[6]As the story concludes, the "Tale of Núr al-Dín Alí and his Son" is narrated within it. This perennially popular work can be traced back toArabic,Persian, and Indian storytelling traditions.
Mary Shelley'sFrankensteinhas a deeply nested frame story structure, that features the narration of Walton, who records the narration of Victor Frankenstein, who recounts the narration of his creation, who narrates the story of a cabin dwelling family he secretly observes. Another classic novel with a frame story isWuthering Heights, the majority of which is recounted by the central family's housekeeper to a boarder. Similarly,Roald Dahl's storyThe Wonderful Story of Henry Sugaris about a rich bachelor who finds an essay written by someone who learned to "see" playing cards from the reverse side. The full text of this essay is included in the story, and itself includes a lengthy sub-story told as a true experience by one of the essay's protagonists, Imhrat Khan.
Lewis Carroll'sAlicebooks,Alice's Adventures in Wonderland(1865) andThrough the Looking-Glass(1871), have several multiple poems that are mostly recited by several characters to the titular character. The most notable examples are "You Are Old, Father William","'Tis the Voice of the Lobster", "Jabberwocky", and "The Walrus and the Carpenter".
Chaucer'sThe Canterbury TalesandBoccaccio'sDecameronare also classic frame stories. In Chaucer'sCanterbury Tales, the characters tell tales suited to their personalities and tell them in ways that highlight their personalities. The noble knight tells a noble story, the boring character tells a very dull tale, and the rude miller tells a smutty tale.Homer'sOdysseytoo makes use of this device;Odysseus' adventures at sea are all narrated by Odysseus to the court of kingAlcinousinScheria. Other shorter tales, many of them false, account for much of theOdyssey. Many modern children's story collections are essentiallyanthologyworks connected by this device, such asArnold Lobel'sMouse Tales,Paula Fox'sThe Little Swineherd, and Phillip and Hillary Sherlock'sEars and Tails and Common Sense.
A well-known modern example of framing is the fantasy genre workThe Princess Bride(boththe bookandthe film). In the film, a grandfather is reading the story ofThe Princess Brideto his grandson. In the book, a more detailed frame story has a father editing a much longer (but fictive) work for his son, creating his own "Good Parts Version" (as the book called it) by leaving out all the parts that would bore or displease a young boy. Both the book and the film assert that the central story is from a book calledThe Princess Brideby a nonexistent author namedS. Morgenstern.
In the Welsh novelAelwyd F'Ewythr Robert(1852) see byGwilym Hiraethog, a visitor to a farm in north Wales tells the story ofUncle Tom's Cabinto those gathered around the hearth.
Sometimes a frame story exists in the same setting as the main story. On the television seriesThe Young Indiana Jones Chronicles, each episode was framed as though it were being told byIndywhen he was older (usually acted byGeorge Hall, but once byHarrison Ford). The same device of an adult narrator representing the older version of a young protagonist is used in the filmsStand by MeandA Christmas Story, and the television showThe Wonder YearsandHow I Met Your Mother.
InThe Amory Wars, a tale told through the music ofCoheed and Cambria, tells a story for the first two albums but reveals that the story is being actively written by a character called the Writer in the third. During the album, the Writer delves into his own story and kills one of the characters, much to the dismay of the main character.
The critically acclaimedBeatlesalbumSgt. Pepper's Lonely Hearts Club Bandis presented as a stage show by the fictional eponymous band, and one of its songs, "A Day in the Life", is in the form of a story within a dream. Similarly, theFugeesalbumThe Scoreis presented as the soundtrack to a fictional film, as are several other notableconcept albums, whileWyclef Jean'sThe Carnivalis presented as testimony at a trial. The majority ofAyreon's albums outline a sprawling, loosely interconnected science fiction narrative, as do the albums ofJanelle Monae.
OnTom Waits's concept albumAlice(consisting of music he wrote for the musical of the same name), most of the songs are (very) loosely inspired by bothAlice in Wonderland, and the book's real-life author,Lewis Carroll, and inspirationAlice Liddell. The song "Poor Edward", however, is presented as a story told by a narrator aboutEdward Mordrake, and the song "Fish and Bird" is presented as a retold story that the narrator heard from a sailor.
In his 1895historical novelPharaoh,Bolesław Prusintroduces a number of stories within the story, ranging in length fromvignettesto full-blown stories, many of them drawn fromancient Egyptiantexts, that further the plot, illuminatecharacters, and even inspire the fashioning of individual characters.Jan Potocki'sThe Manuscript Found in Saragossa(1797–1805) has an interlocking structure with stories-within-stories reaching several levels of depth.
Theprovenanceof the story is sometimes explained internally, as inThe Lord of the RingsbyJ. R. R. Tolkien, which depicts theRed Book of Westmarch(a story-internal version of the book itself) as a history compiled by several of the characters. ThesubtitleofThe Hobbit("There and Back Again") is depicted as part of a rejected title of this book within a book, andThe Lord of the Ringsis a part of the final title.[7]
An example of an interconnected inner story is "The Mad Trist" inEdgar Allan Poe'sFall of the House of Usher, where through somewhat mystical means the narrator's reading of the story within a story influences the reality of the story he has been telling, so that what happens in "The Mad Trist" begins happening in "The Fall of the House of Usher". Also, inDon QuixotebyMiguel de Cervantes, there are many stories within the story that influence the hero's actions (there are others that even the author himself admits are purely digressive).
Most of the first part is presented as a translation of afound manuscriptby (fictional)Cide Hamete Benengeli.
A commonly independentlyanthologisedstory is "The Grand Inquisitor" byDostoevskyfrom his longpsychological novelThe Brothers Karamazov, which is told by one brother to another to explain, in part, his view on religion and morality. It also, in a succinct way, dramatizes many of Dostoevsky's interior conflicts.
An example of a "bonus material" style inner story is the chapter "The Town Ho's Story" inHerman Melville's novelMoby-Dick; that chapter tells a fully formed story of an excitingmutinyand contains many plot ideas that Melville had conceived during the early stages of writingMoby-Dick—ideas originally intended to be used later in the novel—but as the writing progressed, these plot ideas eventually proved impossible to fit around the characters that Melville went on tocreate and develop. Instead of discarding the ideas altogether, Melville wove them into a coherent short story and had the character Ishmael demonstrate his eloquence and intelligence bytelling the storyto his impressed friends.
One of the most complicated structures of a story within a story was used byVladimir Nabokovin his novelThe Gift. There, as inner stories, function both poems and short stories by the main character Fyodor Cherdyntsev as well as the whole Chapter IV, a critical biography of NikolayChernyshevsky(also written by Fyodor). This novel is considered one of the first metanovels in literature.
With the rise ofliterary modernism, writers experimented with ways in which multiple narratives might nest imperfectly within each other. A particularly ingenious example of nested narratives isJames Merrill's 1974modernist poem"Lost in Translation".
InRabih Alameddine's novelThe Hakawati, orThe Storyteller, the protagonist describes coming home to the funeral of his father, one of a long line of traditional Arabic storytellers. Throughout the narrative, the author becomes hakawati (an Arabic word for a teller of traditional tales) himself, weaving the tale of the story of his own life and that of his family with folkloric versions of tales from Qur'an, the Old Testament, Ovid, and One Thousand and One Nights. Both the tales he tells of his family (going back to his grandfather) and the embedded folk tales, themselves embed other tales, often 2 or more layers deep.
InSue Townsend'sAdrian Mole: The Wilderness Years,Adrianwrites the bookLo! The Flat Hills of My Homeland, in which the character Jake Westmorland writes a book calledSparg of Kronk, where the character Sparg writes a book with no language.
InAnthony Horowitz'sMagpie Murders, a significant proportion of the book features a fictional but authentically formatted mystery novel by Alan Conway, titled 'Magpie Murders'. The secondary novel ends before its conclusion returning the narrative to the original, and primary, story where the protagonist and reviewer of the book attempts to find the final chapter. As this progresses characters and messages within the fictionalMagpie Murdersmanifest themselves within the primary narrative and the final chapter's content reveals the reason for its original absence.
Dreams are a common way of including stories inside stories, and can sometimes go several levels deep. Both the bookThe Arabian Nightmareand the curse of "eternal waking" from theNeil GaimanseriesThe Sandmanfeature an endless series of waking from one dream into another dream. InCharles Maturin's novelMelmoth the Wanderer, the use of vast stories-within-stories creates a sense of dream-like quality in the reader.
The 2023 Christian fictional novelJust OncebyKaren Kingsburyfeatures a series of three nested stories, all centering around the main characters of Hank and Irvel Myers:[citation needed]
This structure is also found in classic religious and philosophical texts. The structure ofThe SymposiumandPhaedo, attributed toPlato, is of a story within a story within a story. In the ChristianBible, thegospelsare accounts of the life and ministry ofJesus. However, they also include within them theparablesthat Jesus told. In more modern philosophical works,Jostein Gaarder's books often feature this device. Examples areThe Solitaire Mystery, where the protagonist receives a small book from a baker, in which the baker tells the story of a sailor who tells the story of another sailor, andSophie's World, about a girl who is actually a character in a book that is being read by Hilde, a girl in another dimension. Later on in the book Sophie questions this idea, and realizes that Hilde too could be a character in a story that in turn is being read by another.
Mahabharata, an Indian epic that is also the world's longest epic, has a nested structure.[8]
The experimental modernist works that incorporate multiple narratives into one story are quite often science fiction or science fiction influenced. These include most of the various novels written by the American authorKurt Vonnegut. Vonnegut includes the recurring characterKilgore Troutin many of his novels. Trout acts as the mysteriousscience fictionwriter who enhances the morals of the novels through plot descriptions of his stories. Books such asBreakfast of ChampionsandGod Bless You, Mr. Rosewaterare sprinkled with these plot descriptions.Stanisław Lem'sTale of the Three Storytelling Machines of King GeniusfromThe Cyberiadhas several levels of storytelling. All levels tell stories of the same person, Trurl.
House of Leavesis the tale of a man who finds a manuscript telling the story of a documentary that may or may not have ever existed, contains multiple layers of plot. The book includes footnotes and letters that tell their own stories only vaguely related to the events in the main narrative of the book, and footnotes for fake books.
Robert A. Heinlein's later books (The Number of the Beast,The Cat Who Walks Through WallsandTo Sail Beyond the Sunset) propose the idea that every real universe is a fiction in another universe. Thishypothesisenables many writers who are characters in the books to interact with their own creations.Margaret Atwood's novelThe Blind Assassinis interspersed with excerpts from a novel written by one of the main characters; the novel-within-a-novel itself contains ascience fictionstory written by one ofthatnovel's characters.
InPhilip K. Dick's novelThe Man in the High Castle, each character comes into interaction with a book calledThe Grasshopper Lies Heavy, which was written by the Man in the High Castle. As Dick's novel details a world in which theAxis Powers of World War IIhadsucceeded in dominating the known world, the novel within the novel details an alternative to this history in which the Allies overcome the Axis and bring stability to the world – a victory which itself is quite different from real history.
InRed Orc's RagebyPhilip J. Farmer, a doublyrecursive methodis used to intertwine its fictional layers. This novel is part of a science fiction series, theWorld of Tiers. Farmer collaborated in the writing of this novel with an American psychiatrist, A. James Giannini, who had previously used theWorld of Tiersseries in treating patients in group therapy. During these therapeutic sessions, the content and process of the text and novelist was discussed rather than the lives of the patients. In this way subconscious defenses could be circumvented. Farmer took the real life case-studies and melded these with adventures of his characters in the series.[9]
TheQuantum LeapnovelKnights of the Morningstaralso features a character who writes a book by that name. InMatthew Stover'sStar WarsnovelShatterpoint, the protagonistMace Windunarrates the story within his journal, while the main story is being told from thethird-person limitedpoint of view.
SeveralStar Trektales are stories or events within stories, such asGene Roddenberry'snovelizationofStar Trek: The Motion Picture,J. A. Lawrence'sMudd's Angels,John M. Ford'sThe Final Reflection,Margaret Wander Bonanno'sStrangers from the Sky(which adopts the conceit that it is a book from the future by an author called Gen Jaramet-Sauner), and J. R. Rasmussen's "Research" in the anthologyStar Trek: Strange New WorldsII.Steven Barnes's novelization of theStar Trek: Deep Space Nineepisode "Far Beyond the Stars" partners withGreg Cox'sThe Eugenics Wars: The Rise and Fall of Khan Noonien Singh(Volume Two) to tell us that the fictional story "Far Beyond the Stars" (whose setting and cast closely resembleDeep Space Nine)—and, by extension, all ofStar Trekitself—is the creation of 1950s writer Benny Russell.
The bookCloud Atlas(later adapted into a film byThe WachowskisandTom Tykwer) consisted of six interlinked stories nested inside each other in a Russian doll fashion. The first story (that of Adam Ewing in the 1850s befriending an escaped slave) is interrupted halfway through and revealed to be part of a journal being read by composer Robert Frobisher in 1930s Belgium. His own story of working for a more famous composer is told in a series of letters to his lover Rufus Sixsmith, which are interrupted halfway through and revealed to be in the possession of an investigative journalist named Luisa Rey and so on. Each of the first five tales are interrupted in the middle, with the sixth tale being told in full, before the preceding five tales are finished in reverse order. Each layer of the story either challenges the veracity of the previous layer, or is challenged by the succeeding layer. Presuming each layer to be a true telling within the overall story, a chain of events is created linking Adam Ewing's embrace of the abolitionist movement in the 1850s to the religious redemption of a post-apocalyptic tribal man over a century after the fall of modern civilization. The characters in each nested layer take inspiration or lessons from the stories of their predecessors in a manner that validates a belief stated in the sixth tale that "Our lives are not our own. We are bound to others, past and present and by each crime, and every kindness, we birth our future."
The Crying of Lot 49byThomas Pynchonhas several characters seeing a play calledThe Courier's Tragedyby the fictitiousJacobeanplaywrightRichard Wharfinger. The events of the play broadly mirror those of the novel and give the character Oedipa Maas a greater context to consider her predicament; the play concerns a feud between two rival mail distribution companies, which appears to be ongoing to the present day, and in which, if this is the case, Oedipa has found herself involved. As inHamlet, the director makes changes to the original script; in this instance, a couplet that was added, possibly by religious zealots intent on giving the play extra moral gravity, are said only on the night that Oedipa sees the play.
From what Pynchon relates, this is the only mention in the play of Thurn and Taxis' rivals' name—Trystero—and it is the seed for the conspiracy that unfurls. A significant portion ofWalter Moers'Labyrinth of Dreaming Booksis anekphrasison the subject of an epic puppet theater presentation. Another example is found inSamuel Delany'sTrouble on Triton, which features a theater company that produces elaborate staged spectacles for randomly selected single-person audiences. Plays produced by the "Caws of Art" theater company also feature in Russell Hoban's modern fable,The Mouse and His Child.Raina Telgemeier's best-sellingDramais a graphic novel about a middle-school musical production, and the tentative romantic fumblings of its cast members.
InManuel Puig'sKiss of the Spider Woman, ekphrases on various old movies, some real, and some fictional, make up a substantial portion of the narrative. InPaul Russell'sBoys of Life, descriptions of movies by director/antihero Carlos (loosely inspired by controversial directorPier Paolo Pasolini) provide a narrative counterpoint and add a touch of surrealism to the main narrative. They additionally raise the question of whether works of artistic genius justify or atone for the sins and crimes of their creators. Auster'sThe Book of Illusions(2002) and Theodore Roszak'sFlicker(1991) also rely heavily on fictional films within their respective narratives.
This dramatic device was probably first used byThomas KydinThe Spanish Tragedyaround 1587, where the play is presented before an audience of two of the characters, who comment upon the action.[10][11]From references in other contemporary works, Kyd is also assumed to have been the writer of an early, lost version ofHamlet(the so-calledUr-Hamlet), with a play-within-a-play interlude.[12]William Shakespeare'sHamletretains this device by having Hamlet ask some strolling players to performThe Murder of Gonzago. The action and characters inThe Murdermirror the murder of Hamlet's father in the main action, and Prince Hamlet writes additional material to emphasize this. Hamlet wishes to provoke the murderer, his uncle, and sums this up by saying "the play's the thing wherein I'll catch the conscience of the king." Hamlet calls this new playThe Mouse-trap(a title thatAgatha Christielater took for the long-running playThe Mousetrap). Christie's work was parodied in Tom Stoppard'sThe Real Inspector Hound, in which two theater critics are drawn into the murder mystery they are watching. The audience is similarly absorbed into the action in Woody Allen's playGod, which is about two failed playwrights in Ancient Greece. The phrase "The Conscience of the King" also became the title of aStar Trekepisode featuring a production of Hamlet which leads to the exposure of a murderer (although not a king).
The playI Hate Hamletand the movieA Midwinter's Taleare about a production ofHamlet, which in turn includes a production ofThe Murder of Gonzago, as does theHamlet-based filmRosencrantz & Guildenstern Are Dead, which even features a third-level puppet theatre version within their play. Similarly, inAnton Chekhov'sThe Seagullthere are specific allusions toHamlet: in the first act a son stages a play to impress his mother, a professional actress, and her new lover; the mother responds by comparing her son to Hamlet. Later he tries to come between them, as Hamlet had done with his mother and her new husband. The tragic developments in the plot follow in part from the scorn the mother shows for her son's play.[13]
Shakespeare adopted the play-within-a-play device for many of his other plays as well, includingA Midsummer Night's DreamandLove's Labours Lost. Almost the whole ofThe Taming of the Shrewis a play-within-a-play, presented to convinceChristopher Sly, a drunken tinker, that he is a nobleman watching a private performance, but the device has no relevance to the plot (unless Katharina's subservience to her "lord" in the last scene is intended to strengthen the deception against the tinker[14]) and is often dropped in modern productions. The musicalKiss Me, Kateis about the production of a fictitious musical,The Taming of the Shrew, based on the comedyThe Taming of the ShrewbyWilliam Shakespeare, and features several scenes from it.Pericles, Prince of Tyredraws in part on the 14th-centuryConfessio Amantis(itself a frame story) byJohn Gower, and Shakespeare has the ghost of Gower "assume man's infirmities" to introduce his work to the contemporary audience and comment on the action of the play.[15]
InFrancis Beaumont'sKnight of the Burning Pestle(c. 1608) a supposed common citizen from the audience, actually a "planted" actor, condemns the play that has just started and "persuades" the players to present something about a shopkeeper. The citizen's "apprentice" then acts, pretending to extemporise, in the rest of the play. This is a satirical tilt at Beaumont's playwright contemporaries and their current fashion for offering plays about London life.[16]
The operaPagliacciis about a troupe of actors who perform a play about marital infidelity that mirrors their own lives,[17]and composerRichard Rodney Bennettandplaywright-librettistBeverley Cross'sThe Mines of Sulphurfeatures a ghostly troupe of actors who perform a play about murder that similarly mirrors the lives of their hosts, from whom they depart, leaving them with the plague as nemesis.[18]John Adams'Nixon in China(1985–1987) features a surreal version ofMadam Mao'sRed Detachment of Women, illuminating the ascendance of human values over the disillusionment of high politics in the meeting.[19]
InBertolt Brecht'sThe Caucasian Chalk Circle, a play is staged as aparableto villagers in theSoviet Unionto justify the re-allocation of their farmland: the tale describes how a child is awarded to a servant-girl rather than its natural mother, an aristocrat, as the woman most likely to care for it well. This kind of play-within-a-play, which appears at the beginning of the main play and acts as a "frame" for it, is called an "induction". Brecht's one-act playThe Elephant Calf(1926) is a play-within-a-play performed in the foyer of the theatre during hisMan Equals Man.
InJean Giraudoux's playOndine, all of act two is a series of scenes within scenes, sometimes two levels deep. This increases thedramatic tensionand also makes more poignant the inevitable failure of the relationship between themortalHans andwater spriteOndine.
The Two-Character PlaybyTennessee Williamshas a concurrent double plot with the convention of a play within a play. Felice and Clare are siblings and are both actor/producers touringThe Two-Character Play. They have supposedly been abandoned by their crew and have been left to put on the play by themselves. The characters in the play are also brother and sister and are also named Clare and Felice.
The Mysteries, a modern reworking of the medievalmystery plays, remains faithful to its roots by having the modern actors play the sincere, naïve tradesmen and women as they take part in the original performances.[20]
Alternatively, a play might be about the production of a play, and include the performance of all or part of the play, as inNoises Off,A Chorus of Disapproval, orLilies. Similarly, the musicalMan of La Manchapresents the story of Don Quixote as an impromptu play staged in prison byQuixote's author,Miguel de Cervantes.
In most stagings of the musicalCats, which include the song "Growltiger's Last Stand" – a recollection of an old play by Gus the Theatre Cat – the character of LadyGriddlebonesings "The Ballad of Billy McCaw". (However, many productions of the show omit "Growltiger's Last Stand", and "The Ballad of Billy McCaw" has at times been replaced with a mock aria, so this metastory is not always seen.) Depending on the production, there is another musical scene called "The Awful Battle of the Pekes and the Pollices" where the Jellicles put on a show for their leader. InLestat: The Musical, there are three play within a plays. First, when Lestat visits his childhood friend, Nicolas, who works in a theater, where he discovers his love for theater; and two more when the Theater of the Vampires perform. One is used as a plot mechanism to explain the vampire god, Marius, which sparks an interest in Lestat to find him.
A play within a play occurs in the musicalThe King and I, where Princess Tuptim and the royal dancers give a performance ofSmall House of Uncle Thomas(orUncle Tom's Cabin) to their English guests. The play mirrors Tuptim's situation, as she wishes to run away from slavery to be with her lover, Lun Tha.
In stagings ofDina Rubina's playAlways the Same Dream, the story is about staging a school play based on a poem byPushkin.
Joseph Heller's 1967 playWe Bombed in New Havenis about actors engaged in a play about military airmen; the actors themselves become at times unsure whether they are actors or actual airmen.
The 1937 musicalBabes in Armsis about a group of kids putting on a musical to raise money. The central plot device was retained for the popular 1939 film version withJudy GarlandandMickey Rooney. A similar plot was recycled for the filmsWhite ChristmasandThe Blues Brothers.
The 1946 film noirThe Locketcontains a nestedflashbackstructure, with a screenplay bySheridan Gibneybased on the story "What Nancy Wanted" by Norma Barzman.
TheFrançois TruffautfilmDay for Nightis about the making of a fictitious movie calledMeet Pamela(Je vous présente Pamela) and shows the interactions of the actors as they are making this movie about a woman who falls for her husband's father. The story ofPamelainvolves lust, betrayal, death, sorrow, and change, events that are mirrored in the experiences of the actors portrayed inDay for Night. There are a wealth of other movies that revolve around the film industry itself, even if not centering exclusively on one nested film. These include the darkly satirical classicSunset Boulevardabout an aging star and her parasitic victim, and the Coen Brothers' farceHail, Caesar!
The script toKarel Reisz's movieThe French Lieutenant's Woman(1981), written byHarold Pinter, is a film-within-a-film adaptation ofJohn Fowles's book. In addition to the Victorian love story of the book, Pinter creates a present-day background story that shows a love affair between the main actors.
The Muppet Moviebegins withthe Muppetssitting down in a theater to watch the eponymous movie, whichKermit the Frogclaims to be a semi-biographical account of how they all met.
InBuster Keaton'sSherlock Jr., Keaton's protagonist actually enters into a film while it is playing in a cinema, as does the main character in theArnold SchwarzeneggerfilmThe Last Action Hero. A similar device is used in the music video for the song "Take On Me" byA-ha, which features a woman entering a pencil sketch. Conversely,Woody Allen'sPurple Rose of Cairois about a film character exiting the film to interact with the real world. Allen's earlier filmPlay it Again, Samfeatured liberal use of characters, dialogue and clips from the film classicCasablancaas a central device.
The 2002Pedro AlmodóvarfilmTalk to Her(Hable con ella) has the chief character Benigno tell a story calledThe Shrinking Loverto Alicia, a long-term comatose patient whom Benigno, a male nurse, is assigned to care for. The film presentsThe Shrinking Loverin the form of a black-and-white silent melodrama. To prove his love to a scientist girlfriend,The Shrinking Loverprotagonist drinks a potion that makes him progressively smaller. The resulting seven-minute scene, which is readily intelligible and enjoyable as a stand-alone short subject, is considerably more overtly comic than the rest ofTalk to Her—the protagonist climbs giant breasts as if they were rock formations and even ventures his way inside a (compared to him) gigantic vagina. Critics have noted thatThe Shrinking Loveressentially is a sex metaphor. Later inTalk to Her, the comatose Alicia is discovered to be pregnant and Benigno is sentenced to jail for rape.The Shrinking Loverwas named Best Scene of 2002 in theSkandies, an annual survey of online cinephiles and critics invited each year by critic Mike D'Angelo.[21]
Tropic Thunder(2008) is acomedy filmrevolving around a group ofprima donnaactors making aVietnam Warfilm (itself also namedTropic Thunder) when their fed-up writer and director decide to abandon them in the middle of the jungle, forcing them to fight their way out. The concept was perhaps[original research?]inspired by the 1986 comedyThree Amigos, where three washed-up silent film stars are expected to live out a real-life version of their old hit movies. The same idea of life being forced to imitate art is also reprised in theStar TrekparodyGalaxy Quest.
The first episode of theanimeseriesThe Melancholy of Haruhi Suzumiyaconsists almost entirely of a poorly made film that the protagonists created, complete withKyon's typical, sarcastic commentary.
Chuck Jones's 1953cartoonDuck AmuckshowsDaffy Ducktrapped in a cartoon that an unseen animator repeatedly manipulates. At the end, it is revealed that the whole cartoon was being controlled byBugs Bunny. TheDuck Amuckplot was essentially replicated in one of Jones' later cartoons,Rabbit Rampage(1955), in which Bugs Bunny turns out to be the victim of the sadistic animator (Elmer Fudd). A similar plot was also included in an episode ofNew Looney Tunes, in which Bugs is the victim, Daffy is the animator, and it was made on a computer instead of a pencil and paper. In 2007, theDuck Amucksequence was parodied onDrawn Together("Nipple Ring-Ring Goes to Foster Care").
All feature-length films byJörg ButtgereitexceptSchrammfeature a film within the film. InNekromantik, the protagonist goes to the cinema to see the fictional slasher filmVera. InDer Todesking, one of the character watches a video of the fictional Nazi exploitation filmVera – Todesengel der Gestapoand inNekromantik 2, the characters go to see a film calledMon déjeuner avec Vera, which is a parody ofLouis Malle'sMy Dinner with André.
Quentin Tarantino'sInglourious Basterdsdepicts aNazi propagandafilm calledNation's Pride, which glorifies a soldier in the German army.Nation's Prideis directed byEli Roth.
Joe Dante'sMatineedepictsMant, an early-1960s sci-fi/horror movie about a man who turns into anant. In one scene, the protagonists see aDisney-style family movie calledThe Shook-Up Shopping Cart.
The 2002 martial arts epicHeropresented the same narrative several different times, as recounted by different storytellers, but with both factual and aesthetic differences. Similarly, in the whimsical 1988Terry GilliamfilmThe Adventures of Baron Munchausen, and the 2003Tim BurtonfilmBig Fish, the bulk of the film is a series of stories told by an (extremely) unreliable narrator. In the 2006 Tarsem filmThe Fall, an injured silent-movie stuntman tellsheroic fantasystories to a little girl with a broken arm to pass time in the hospital, which the film visualizes and presents with the stuntman's voice becoming voiceover narration. The fantasy tale bleeds back into and comments on the film's "present-tense" story. There are often incongruities based on the fact that the stuntman is an American and the girl Persian—the stuntman's voiceover refers to "Indians", "a squaw" and "a teepee", but the visuals show a Bollywood-style devi and a Taj Mahal-like castle. The same conceit of an unreliable narrator was used to very different effect in the 1995 crime dramaThe Usual Suspects(which garnered an Oscar forKevin Spacey's performance).
Walt Disney's 1946 live-action drama filmSong of the Southhas three animated sequences, all based on theBr'er Rabbitstories, told as moral fables byUncle Remus(James Baskett) to seven-year-old Johnny (Bobby Driscoll) and his friends Ginny (Luana Patten) and Toby (Glenn Leedy).
The seminal 1950 Japanese filmRashomon, based on the Japanese short story "In a Grove" (1921), utilizes theflashback-within-a-flashback technique. The story unfolds in flashback as the four witnesses in the story—the bandit, the murderedsamurai, his wife, and the nameless woodcutter—recount the events of one afternoon in a grove. But it is also a flashback within a flashback, because the accounts of the witnesses are being retold by a woodcutter and a priest to a ribald commoner as they wait out a rainstorm in a ruined gatehouse.
The filmInceptionhas a deeply nested structure that is itself part of the setting, as the characters travel deeper and deeper into layers of dreams within dreams. Similarly, in the beginning of the music video for theMichael Jacksonsong "Thriller", the heroine is terrorized by her monster boyfriend in what turns out to be a film within a dream. The filmThe Grand Budapest Hotelhas four layers of narration: starting with a young girl at the author's memorial reading his book, it cuts to the old author in 1985 telling of an incident in 1968 when he, as a young author, stayed at the hotel and met the owner, old Zero. He was then told the story of young Zero and M. Gustave, from 1932, which makes up most of the narrative.
Then in 2025, The filmDog Manis a flim in a comic for theDog Manseries.
The 2001 filmMoulin Rouge!features a fictitious musical within a film, called "Spectacular Spectacular". The 1942Ernst LubitschcomedyTo Be or Not to Beconfuses the audience in the opening scenes with a play, "The Naughty Nazis", about Adolf Hitler which appears to be taking place within the actual plot of the film. Thereafter, the acting company players serve as the protagonists of the film and frequently use acting/costumes to deceive various characters in the film.Hamletalso serves as an important throughline in the film, as suggested by the title.Laurence Oliviersets the opening scene of his 1944 film ofHenry Vin thetiring roomof the oldGlobe Theatreas the actors prepare for their roles on stage. The early part of the film follows the actors in these "stage" performances and only later does the action almost imperceptibly expand to the full realism of theBattle of Agincourt. By way of increasingly more artificial sets (based on mediaeval paintings) the film finally returns to The Globe.
Mel Brooks' filmThe Producersrevolves around a scheme to make money by producing a disastrously bad Broadway musical,Springtime for Hitler.Ironically the film itself was later made into its own Broadway musical (although a more intentionally successful one). TheOutkastmusic video for the song "Roses" is a short film about a high school musical. InDiary of a Wimpy Kid, the middle-schoolers put on a play ofThe Wizard of Oz, whileHigh School Musicalis a romantic comedy about the eponymous musical itself. A high school production is also featured in the gay teen romantic comedyLove, Simon.
A 2012 Italian film,Caesar Must Die, stars real-life Italian prisoners who rehearse Shakespeare'sJulius CaesarinRebibbiaprison playingfictionalItalian prisoners rehearsing the same play in the same prison. In addition, the film itself becomes aJulius Caesaradaption of sorts as the scenes are frequently acted all around the prison, outside of rehearsals, and the prison life becomes indistinguishable from the play.[22]
The main plot device inRepo! The Genetic Operais an opera which is going to be held the night of the events of the film. All of the principal characters of the film play a role in the opera, though the audience watching the opera is unaware that some of the events portrayed are more than drama. The 1990 biopicKorczak, about the last days of a Jewish children's orphanage in Nazi occupied Poland, features an amateur production ofRabindranath Tagore'sThe Post Office, which was selected by the orphanage's visionary leader as a way of preparing his charges for their own impending death. That same production is also featured in the stage playKorczak's Children,also inspired by the same historical events.
The 1973 filmThe National Health, an adaptation of the 1969 playThe National HealthbyPeter Nichols, features a send-up of a typical American hospitalsoap operabeing shown on a television situated in an underfunded, unmistakably BritishNHShospital.
TheJim CarreyfilmThe Truman Showis about a person who grows to adulthood without ever realizing that he is the unwitting hero of the immersive eponymous television show.
InToy Story 2, the lead characterWoodylearns that he is based on the lead character of the same name of a 1950sWesternshow known asWoody's Roundup, which was seemingly cancelled due to the rise ofscience fiction, though this is eventually debunked after the final episode of the show can be seen playing.
The first example of a video game within a video game is almost certainlyTim Stryker's 1980s era[vague]text-only gameFazuul(also the world's first online multiplayer game), in which one of the objects that the player can create is a minigame. Another early use of this trope was inCliff Johnson's 1987 hitThe Fool's Errand, a thematically linked narrative puzzle game, in which several of the puzzles were semi-independent games played against NPCs.
Power Factorhas been cited as a rare example of a video game in which the entire concept is a video game within a video game: The player takes on the role of a character who is playing a "Virtual Reality Simulator", in which he in turn takes on the role of the hero Redd Ace.[23]The.hackfranchise also gives the concept a central role. It features a narrative in which internet advancements have created an MMORPG franchise called The World. Protagonists Kite andHaseotry to uncover the mysteries of the events surrounding The World. Characters in.hackare aware that they are video game characters.
More commonly, however, the video game within a video game device takes the form of mini-games that are non-plot-oriented, and optional to the completion of the game. For example, in theYakuzaandShenmuefranchises, there are playable arcade machines featuring other Sega games that are scattered throughout the game world.
InFinal Fantasy VIIthere are several video games that can be played in an arcade in the Gold Saucer theme park. InAnimal Crossing, the player can acquire individual NES emulations through various means and place them within their house, where they are playable in their entirety. When placed in the house, the games take the form of aNintendo Entertainment System. InFallout 4andFallout 76, the protagonist can find several cartridges throughout the wasteland that can be played on their pip-boy (an electronic device that exists only in the world of the game) or any terminal computer. InCeleste, there is a hidden room in which the protagonist can play the originalPICO-8prototype of the game.
In theRemedyvideo game titledMax Payne, players can chance upon a number of ongoing television shows when activating or happening upon various television sets within the game environs, depending on where they are within the unfolding game narrative. Among them areLords & Ladies,Captain Baseball Bat Boy,Dick Justiceand the pinnacle television serialAddress Unknown– heavily inspired byDavid Lynch-style film narrative, particularlyTwin Peaks,Address Unknownsometimes prophesies events or character motives yet to occur in the Max Payne narrative.
InGrand Theft Auto IV, the player can watch several TV channels which include many programs: reality shows, cartoons, and even game shows.[24]
Terrance & PhillipfromSouth Parkcomments on the levels of violence and acceptable behaviour in the media and allow criticism of the outer cartoon to be addressed in the cartoon itself. Similarly, on the long running animated sitcomThe Simpsons, Bart's favorite cartoon,Itchy and Scratchy(a parody ofTom & Jerry), often echoes the plotlines of the main show.The Simpsonsalso parodied this structure with numerous 'layers' of sub-stories in the Season 17 episode "The Seemingly Never-Ending Story".
The animated seriesSpongeBob SquarePantsfeatures numerous fictional shows, most notably,The Adventures of Mermaid Man and Barnacle Boy,which stars the titular elderly superheroesMermaid Man(Ernest Borgnine) andBarnacle Boy(Tim Conway).
On the showDear White People, theScandalparodyDefamationoffers an ironic commentary on the main show's theme of interracial relationships. Similarly, each season of theHBOshowInsecurehas featured a different fictional show, including the slavery-era soap operaDue North, the rebooted black 1990s sitcomKev'yn,and the investigative documentary seriesLooking for LaToya.
TheIrishtelevision seriesFather Tedfeatures a television show,Father Ben, which has characters and storylines almost identical to that ofFather Ted.
The television shows30 Rock,Studio 60 on the Sunset Strip,Sonny with a Chance, andKappa Mikeyfeature a sketch show within the TV show.
An extended plotline on the semi-autobiographical sitcomSeinfelddealt with the main characters developing a sitcom about their lives. The gag was reprised onCurb Your Enthusiasm, another semi-autobiographical show by and aboutSeinfeldco-creator Larry David, when the long-anticipatedSeinfeldreunion was staged entirely inside the new show.
The "USS Callister" episode of theBlack Mirroranthology television series is about a man who is obsessed with aStar Trek-like show and recreates it as part of a virtual reality game.
The concept of a film within a television series is employed in theMacrossuniverse.The Super Dimension Fortress Macross: Do You Remember Love?(1984) was originally intended as an alternative theatrical re-telling of the television seriesThe Super Dimension Fortress Macross(1982), but was later "retconned" into the Macrosscanonas a popular film within the television seriesMacross 7(1994).
TheStargate SG-1episode "Wormhole X-Treme!" features a fictional TV show with an almost identical premise toStargate SG-1. A later episode, "200", depicts ideas for a possible reboot ofWormhole X-Treme!, including using a "younger and edgier" cast, or evenThunderbirds-style puppets.
TheGleeepisode "Extraordinary Merry Christmas" features the members of New Directions starring in a black-and-white Christmas television special that is presented within the episode itself. The special is a homage to bothStar Wars Holiday Specialand the "Judy Garland Christmas Special".
The British TV seriesDon't Hug Me I'm Scared, based on theweb seriesDon't Hug Me I'm Scared, is notable for being apuppet showthat includes a fictionalclaymationTV series within the show:Grolton & Hovris, a parody ofWallace and Gromit.
Seinfeldhad a number of reoccurring fictional films, including a sci-fi film calledThe Flaming Globes of Sigmundand, most notably,Rochelle, Rochelle, a parody of artsy but exploitative foreign films. The trippy, metaphysically loopy thrillerDeath Castleis a central element of theMaster of Noneepisode "New York, I Love You".
Theseries finaleofBarryfeatures a biopic of the titular character which was calledThe Mask Collector,and its production served as the catalyst for the last 4 episodes of Barry's final season.
Stories inside stories can allow for genre changes.Arthur Ransomeuses the device to let his young characters in theSwallows and Amazonsseriesof children's books, set in the recognisable everyday world, take part in fantastic adventures of piracy in distant lands: two of the twelve books,Peter DuckandMissee Lee(and some would includeGreat Northern?as a third), are adventures supposedly made up by the characters.[25]Similarly, the film version ofChitty Chitty Bang Banguses a story within a story format to tell a purely fantastic fairy tale within a relatively more realistic frame-story. The film version ofThe Wizard of Ozdoes the same thing by making its inner story into a dream. Lewis Carroll's celebratedAlicebooks use the same device of a dream as an excuse for fantasy, while Carroll's less well-knownSylvie and Brunosubverts the trope by allowing the dream figures to enter and interact with the "real" world. In each episode ofMister Rogers' Neighborhood, the main story was realistic fiction, with live action human characters, while an inner story took place in theNeighborhood of Make-Believe, in which most characters were puppets, except Lady Aberlin and occasionally Mr. McFeely, played byBetty AberlinandDavid Newellin both realms.
Some stories feature what might be called a literary version of theDroste effect, where an image contains a smaller version of itself (also a common feature in manyfractals). An early version is found in an ancient Chinese proverb, in which an old monk situated in a temple found on a high mountain recursively tells the same story to a younger monk about an old monk who tells a younger monk a story regarding an old monk sitting in a temple located on a high mountain, and so on.[26]The same concept is at the heart ofMichael Ende's classic children's novelThe Neverending Story, which prominently features a book of the same title. This is later revealed to be the same book the audience is reading, when it begins to be retold again from the beginning, thus creating an infinite regression that features as a plot element. Another story that includes versions of itself isNeil Gaiman'sThe Sandman: Worlds' Endwhich contains several instances of multiple storytelling levels, includingCerements(issue #55) where one of the inmost levels corresponds to one of the outer levels, turning the story-within-a-story structure into an infinite regression. Jesse Ball'sThe Way Through Doorsfeatures a deeply nested set of stories within stories, most of which explore alternate versions of the main characters. The frame device is that the main character is telling stories to a woman in a coma (similar to Almodóvar'sTalk to Her, mentioned above).
Richard Adams' classic Watership Down includes several memorable tales about the legendary prince of rabbits, El-Ahraira, as told by master storyteller, Dandelion.
Samuel Delany's great surrealist sci-fi classicDhalgrenfeatures the main character discovering a diary apparently written by a version of himself, with incidents that usually reflect, but sometimes contrast with the main narrative. The last section of the book is taken up entirely by journal entries, about which readers must choose whether to take as completing the narrator's own story. Similarly, inKiese Laymon'sLong Division, the main character discovers a book, also calledLong Division, featuring what appears to be himself, except as living twenty years earlier. The title book in Charles Yu'sHow to Live Safely in a Science Fictional Universeexists within itself as a stable creation of a closed loop in time. Likewise, in the Will Ferrell comedyStranger than Fictionthe main character discovers he is a character in a book that (along with its author) also exists in the same universe. The 1979 bookGödel, Escher, BachbyDouglas Hofstadterincludes a narrative betweenAchilles and the Tortoise(characters borrowed fromLewis Carroll, who in turn borrowed them fromZeno), and within this story they find the book "Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe", which they begin to read, the Tortoise taking the part of the Tortoise, and Achilles taking the part of Achilles. Within this self-referential narrative, the two characters find the book "Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe", which they begin to read, this time each taking the other's part. The 1979 experimental novelIf on a winter's night a travelerbyItalo Calvinofollows a reader, addressed in the second person, trying to read the very same book, but being interrupted by ten other recursively nested incomplete stories.
Robert Altman's satirical Hollywood noirThe Playerends with theantiherobeing pitched a movie version of his own story, complete with an unlikely happy ending. The long-running musicalA Chorus Linedramatizes its own creation, and the life stories of its own original cast members. The famous final number does double duty as the showstopper for both the musical the audience is watching and the one the characters are appearing in.Austin Powers in Goldmemberbegins with an action film opening, which turns out to be a sequence being filmed bySteven Spielberg. Near the ending, the events of the film itself are revealed to be a movie being enjoyed by the characters. Jim Henson'sThe Muppet Movieis framed as a screening of the movie itself, and the screenplay for the movie is present inside the movie, which ends with an abstracted, abbreviated re-staging of its own events. The 1985 Tim Burton filmPee-Wee's Big Adventureends with the main characters watching a film version of their own adventures, but as reimagined as a Hollywood blockbuster action film, withJames Brolinas a more stereotypically manly version of thePaul Reubenstitle character. Episode 14 of theanimeseriesMartian Successor Nadesicois essentially a clip show, but has several newly animated segments based onGekigangar III, an anime that exists within its universe and that many characters are fans of, that involves the characters of that show watching Nadesico. The episode ends with the crew of the Nadesico watching the very same episode of Gekigangar, causing aparadox.Mel Brooks's 1974 comedyBlazing Saddlesleaves its Western setting when the climactic fight scene breaks out, revealing the setting to have been a set in theWarner Bros.studio lot; the fight spills out onto an adjacent musical set, then into the studio canteen, and finally onto the streets. The two protagonists arrive atGrauman's Chinese Theatre, which is showing the "premiere" ofBlazing Saddles; they enter the cinema to watch the conclusion of their own film. Brooks recycled the gag in his 1987Star Warsparody,Spaceballs, where the villains are able to locate the heroes by watching a copy of the movie they are in on VHS video tape (a comic exaggeration of the phenomenon of films being available on video before their theatrical release). Brooks also made the 1976 parodySilent Movieabout a buffoonish team of filmmakers trying to make the first Hollywood silent film in forty years—which is essentially that film itself (another forty years later, life imitated art imitating art, when an actual modern silent movie became a hit, the Oscar winnerThe Artist).
The film-within-a-film format is used in theScreamhorror series. InScream 2, the opening scene takes place in a movie theater where a screening ofStabis played which depicts the events fromthe first film. In between the events ofScream 2andScream 3, a second film was released calledStab 2.Scream 3is about the actors filming a fictional third installment in the Stab series. The actors playing the trilogy's characters end up getting killed, much in the same way as the characters they are playing on screen and in the same order. In between the events ofScream 3andScream 4, four other Stab films are released. In the opening sequence ofScream 4two characters are watchingStab 7before they get killed. There's also a party in which all seven Stab movies were going to be shown. References are also made toStab 5involvingtime travelas a plot device. In the fifth installment of the series, also namedScream, an eighth Stab film is mentioned having been released before the film takes place. The characters in the film, several of which are fans of the series, heavily criticize the film, similar to howScream 4was criticized. Additionally, late in the film, Mindy watches the first Stab by herself. During the depiction of Ghostface sneaking up behind Randy on the couch from the first film in Stab, Ghostface sneaks up on Mindy and attacks and stabs her.
DirectorSpike Jonze'sAdaptationis a fictionalized version of screenwriterCharlie Kaufman's struggles to adapt the non-cinematic bookThe Orchid Thiefinto a Hollywood blockbuster. As his onscreen self succumbs to the temptation to commercialize the narrative, Kaufman incorporates those techniques into the script, including tropes such as an invented romance, a car chase, a drug-running sequence, and an imaginary identical twin for the protagonist. (The movie also features scenes about the making ofBeing John Malkovich, previously written by Kaufman and directed by Jonze.) Similarly, in Kaufman's self-directed 2008 filmSynecdoche, New York, the main character Caden Cotard is a skilled director of plays who receives a grant, and ends up creating a remarkable theater piece intended as a carbon copy of the outside world. The layers of copies of the world ends up several layers deep. The same conceit was previously used by frequent Kaufman collaboratorMichel Gondryin his music video for theBjörksong "Bachelorette", which features a musical that is about, in part, the creation of that musical. A mini-theater and small audience appear on stage to watch the musical-within-a-musical, and at some point, within that second musical a yet-smaller theater and audience appear.
Fractal fiction is sometimes utilized invideo gamesto play with the concept of player choice: In the first chapter ofStories Untold, the player is required to play atext adventure, which eventually becomes apparent to be happening in the same environment the player is in; inSuperhotthe narrative itself is constructed around the player playing a game called Superhot.
Occasionally, a story within a story becomes such a popular element that the producer(s) decide to develop it autonomously as a separate and distinct work. This is an example of aspin-off. Such spin-offs may be produced as a way of providing additional information on the fictional world for fans.
InHomestuckbyAndrew Hussie, there is a comic calledSweet Bro and Hella Jeff, created by one of the characters, Dave Strider. It was later adapted to its own ongoing series.
In theToy Storyfilm universe,Buzz Lightyearis an animated toy action figure, which was based on a fictitious cartoon series,Buzz Lightyear of Star Command, which did not exist in the real world except for snippets seen withinToy Story. Later,Buzz Lightyear of Star Commandwas produced in the real world and was itself later joined byLightyear, a film described as the source material for the toy and cartoon series.
Kujibiki Unbalance, a series in theGenshikenuniverse, has spawned merchandise of its own, and been remade into a series on its own.
The popularDog Manseries of children's graphic novels is presented as a creation of the main characters of authorDav Pilkey's earlier series,Captain Underpants.
In the animated online franchiseHomestar Runner, many of the best-known features were spun off from each other. The best known was "Strong Bad Emails", which depicted the villain of the original story giving snarky answers to fan emails, but that in turn spawned several other long-running features which started out as figments of Strong Bad's imagination, including the teen-oriented cartoon parody "Teen Girl Squad" and the anime parody "20X6".
In theHarry Potterseries, three such supplemental books have been produced:Fantastic Beasts and Where to Find Them, a guidebook used by the characters;Quidditch Through the Ages, a book from the school library; andThe Tales of Beedle the Bard, presenting fairy tales told to children of the wizarding world.
In the works ofKurt Vonnegut,Kilgore Trouthas written a novel calledVenus on the Half-Shell. In 1975 real-world authorPhilip José Farmerwrote a science-fiction novel calledVenus on the Half-Shell, published under the name Kilgore Trout.
Captain Proton: Defender of the Earth, a story byDean Wesley Smith, was adapted from the holonovelCaptain Protonin theStar Trekuniverse.
One unique example is theTyler Perrycomedy/horror hitBoo! A Madea Halloween, which originated as a parody of Tyler Perry films in theChris RockfilmTop 5.
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Tamanna(Urdu:تمنا;transl.Desire) is2014British-Pakistani filmdistributed by Royal Palm Group t/a Summit Entertainment (Pak) and Super Cinema andARY Filmsand produced by Concordia Productions. A drama in theneo-noirgenre[3]the film is directed by aBritishdirector Steven Moore and produced by Pakistani producerSarah Tareen. The film starsOmair Rana,Salman Shahid,Mehreen RaheelandFaryal Gohar.[4]
Prior to release the film won an award at theLondon Asian Film Festival[5]for the first released song byRahat Fateh Ali Khan. Other songs included in the film are sung byAli AzmatandAmanat Ali. The original score for the film was written by Arthur Rathbone Pullen son ofBooker Prizenominee and best selling English authorJulian Rathbone.
The film incorporates elements of dark humour, melodrama, crime, passion and revenge and is based onAnthony Shaffer’s play,Sleuth. The film's hero is Rizwan Ahmed (Omair Rana) a struggling actor who meets Mian Tariq Ali (Salman Shahid), a relic of the once thriving film industry. The struggling actor is there to convince Ali to divorce his wife. A contest of male dominance between the two men ensues, starting quite reasonably, playfully even, but eventually turning angry and violent.[6]
Whilst some of the interactions between the two men are similar to the playSleuth, the film has roles for not just the Wyke character's wife, but also his second, younger wife, who is the protagonist's object of desire. The milieu is Pakistan's film industry,Lollywoodin its dying days. The outcome for the characters is dark, with more emphasis on being sacrificed than self-sacrifice, and is used an allegory of wider issues. The dialogue, in Urdu, and the scenario are adapted in numerous ways for Pakistani culture.
BothSalman ShahidandFaryal Goharwere cast first around 3 years before the film was eventually made, as early as 2009. They appear together in the video for the songKoi Dil Meintogether not long after that in 2010.Omair Ranawas not part of the film initially, his role as Riz theprotagonistwas given toHameed Sheikh, who is famous for his role of Sher Shah inShoaib Mansoor’sKhuda Kay Liyeand as Omar Boloch inKandahar Break.Mehreen Raheelwas brought in very late, just before principal photography which took place in October 2012. Amongst scenes that were cut in the edit was Omair Rana withRasheed NazinWazir Khan Mosquein theWalled City of Lahore, but does appear briefly on a TV set in the background at one point during the film.Sahir Lodhi, a famous Pakistani TV presenter, also makes a cameo voice appearance as a TV interviewer.
The film contains both original score and individual songs. The score of the film is composed by British Composer and musician Arthur Rathbone Pullen. The songs are sung by Pakistan's best known playback singerRahat Fateh Ali KhanandAli Azmatand composed bySahir Ali Bagga.[7]Amanat Ali sung the title track of Tamanna, composed by Afzal Hussain. The video of the titular song was filmed at Lahore's historic Barood Khana Haveli.[2]
The first Look trailer was released in June 2013 and Koi Dil Mein before that as a song with video including some of the film's oldLollywoodrecreation footage from amise en abymetechnique of a film within a film.
The film premiered on 13 June 2014[8]in Lahore Karachi and Islamabad then ran subsequently in cinemas around Pakistan for two weeks.[2][9][10]The film is expected to screen at selected festivals in late 2014/2015 and was already screened at theTricycle Theatreas part of the 16th London Asian Film Festival on 8 June 2014.[11]The film had its worldwide TV premiere on 10 May 2015 onARY Digitaland is scheduled for regular showing by the channel which owns TV distribution rights.
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Inanalytic philosophyandcomputer science,referential transparencyandreferential opacityare properties of linguistic constructions,[a]and by extension of languages. A linguistic construction is calledreferentially transparentwhen for any expression built from it,replacinga subexpression with another one thatdenotesthe same value[b]does not change the value of the expression.[1][2]Otherwise, it is calledreferentially opaque. Each expression built from a referentially opaque linguistic construction states something about a subexpression, whereas each expression built from a referentially transparent linguistic construction states something not about a subexpression, meaning that the subexpressions are ‘transparent’ to the expression, acting merely as ‘references’ to something else.[3]For example, the linguistic construction ‘_ was wise’ is referentially transparent (e.g.,Socrates was wiseis equivalent toThe founder of Western philosophy was wise) but ‘_ said _’ is referentially opaque (e.g.,Xenophon said ‘Socrates was wise’is not equivalent toXenophon said ‘The founder of Western philosophy was wise’).
Referential transparency, in programming languages, depends on semantic equivalences among denotations of expressions, or oncontextual equivalenceof expressions themselves. That is, referential transparency depends on the semantics of the language. So, bothdeclarative languagesandimperative languagescan have referentially transparent positions, referentially opaque positions, or (usually) both, according to the semantics they are given.
The importance of referentially transparent positions is that they allow theprogrammerand thecompilerto reason about program behavior as arewrite systemat those positions. This can help in provingcorrectness, simplifying analgorithm, assisting in modifying code without breaking it, oroptimizingcode by means ofmemoization,common subexpression elimination,lazy evaluation, orparallelization.
The concept originated inAlfred North WhiteheadandBertrand Russell'sPrincipia Mathematica(1910–1913):[3]
A proposition as the vehicle of truth or falsehood is a particular occurrence, while a proposition considered factually is a class of similar occurrences. It is the proposition considered factually that occurs in such statements as “Abelievesp“ and “pis aboutA.”
Of course it is possible to make statements about the particular fact “Socrates is Greek.” We may say how many centimetres long it is; we may say it is black; and so on. But these are not the statements that a philosopher or logician is tempted to make.
When an assertion occurs, it is made by means of a particular fact, which is an instance of the proposition asserted. But this particular fact is, so to speak, “transparent”; nothing is said about it, but by means of it something is said about something else. It is this “transparent” quality that belongs to propositions as they occur in truth-functions. This belongs topwhenpis asserted, but not when we say “pis true.”
It was adopted in analytic philosophy inWillard Van Orman Quine'sWord and Object(1960):[1]
When a singular term is used in a sentence purely to specify its object, and the sentence is true of the object, then certainly the sentence will stay true when any other singular term is substituted that designates the same object. Here we have a criterion for what may be calledpurely referential position: the position must be subject to thesubstitutivity of identity.
[…]
Referential transparency has to do with constructions (§ 11); modes of containment, more specifically, of singular terms or sentences in singular terms or sentences. I call a mode of containmentφreferentially transparent if, whenever an occurrence of a singular termtis purely referential in a term or sentenceψ(t), it is purely referential also in the containing term or sentenceφ(ψ(t)).
The term appeared in its contemporary computer science usage in the discussion ofvariablesinprogramming languagesinChristopher Strachey's seminal set of lecture notesFundamental Concepts in Programming Languages(1967):[2]
One of the most useful properties of expressions is that called by Quine [4]referential transparency. In essence this means that if we wish to find the value of an expression which contains a sub-expression, the only thing we need to know about the sub-expression is its value. Any other features of the sub-expression, such as its internal structure, the number and nature of its components, the order in which they are evaluated or the colour of the ink in which they are written, are irrelevant to the value of the main expression.
There are three fundamental properties concerning substitutivity in formal languages: referential transparency, definiteness, and unfoldability.[4]
Let’s denote syntactic equivalence with ≡ and semantic equivalence with =.
Apositionis defined by a sequence of natural numbers. The empty sequence is denoted by ε and the sequence constructor by ‘.’.
Example.— Position 2.1 in the expression(+ (∗e1e1) (∗e2e2))is the place occupied by the first occurrence ofe2.
Expressionewithexpressione′inserted atpositionpis denoted bye[e′/p]and defined by
Example.— Ife≡ (+ (∗e1e1) (∗e2e2))thene[e3/2.1] ≡ (+ (∗e1e1) (∗e3e2)).
Positionpispurely referentialin expressioneis defined by
In other words, a position is purely referential in an expression if and only if it is subject to the substitutivity of equals.εis purely referential in all expressions.
OperatorΩisreferentially transparentin placeiis defined by
OtherwiseΩisreferentially opaquein placei.
An operator isreferentially transparentis defined by it is referentially transparent in all places. Otherwise it isreferentially opaque.
A formal language isreferentially transparentis defined by all its operators are referentially transparent. Otherwise it isreferentially opaque.
Example.— The ‘_ lives in _’ operator is referentially transparent:
Indeed, the second position is purely referential in the assertion because substitutingThe capital of the United KingdomforLondondoes not change the value of the assertion. The first position is also purely referential for the same substitutivity reason.
Example.— The ‘_ contains _’ and quote operators are referentially opaque:
Indeed, the first position is not purely referential in the statement because substitutingThe capital of the United KingdomforLondonchanges the value of the statement and the quotation. So in the first position, the ‘_ contains _’ and quote operators destroy the relation between an expression and the value that it denotes.
Example.— The ‘_ refers to _’ operator is referentially transparent, despite the referential opacity of the quote operator:
Indeed, the first position is purely referential in the statement, though it is not in the quotation, because substitutingThe capital of the United KingdomforLondondoes not change the value of the statement. So in the first position, the ‘_ refers to _’ operator restores the relation between an expression and the value that it denotes. The second position is also purely referential for the same substitutivity reason.
A formal language isdefiniteis defined by all the occurrences of a variable within its scope denote the same value.
Example.— Mathematics is definite:
Indeed, the two occurrences ofxdenote the same value.
A formal language isunfoldableis defined by all expressions areβ-reducible.
Example.— Thelambda calculusis unfoldable:
Indeed,((λx.x+ 1) 3) = (x+ 1)[3/x].
Referential transparency, definiteness, and unfoldability are independent.
Definiteness implies unfoldability only for deterministic languages.
Non-deterministic languages cannot have definiteness and unfoldability at the same time.
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A Strange Loopis a musical with book, music, and lyrics byMichael R. Jackson, and winner of the 2020Pulitzer Prize for Drama.[1]First producedoff-Broadwayin 2019, then staged inWashington, D.C.in 2021,[2]A Strange Looppremiered on Broadway at theLyceum Theatrein April 2022.[3][4]The show wonBest MusicalandBest Book of a Musicalat the75th Tony Awards.
While working as an usher atThe Lion King,aspiring musical theater writer Usher contemplates the show he is writing, wanting it to represent what it is like to "travel the world in a fat, black, queer body" ("Intermission Song"). He plans to change himself, but his thoughts are too disruptive ("Today"). His mother, who constantly reminds him how hard she and his father worked to raise him, calls with a request that he write aTyler Perry-stylegospelplay ("We Wanna Know").
Usher wishes he could act more like his "inner white girl" but is held back by expectations put on Black boys ("Inner White Girl"). His Thoughts criticize the show, claiming the main character should have more sex appeal and telling him to add certain elements. Usher's father leaves a voicemail saying he foundScott Rudin's number and urges him to leverage their common sexuality to make a connection ("Didn't Want Nothin'").
At a medical checkup, Usher's doctor inquires about his sex life and prescribesTruvada. Usher starts using dating apps but is rejected, causing him to rage against the gay community ("Exile In Gayville"). A stranger flirts with Usher before revealing he is a figment of Usher's imagination and dismissing the war between the "second-wave feminist" and "the dick-sucking Black gay man" within him ("Second Wave").
Usher's agent tells himTyler Perryis seeking a ghostwriter for a gospel play, but Usher has a low opinion of Perry's work. Appearing as famous Black figures, his Thoughts accuse him of being a race traitor and persuade him to take the job ("Tyler Perry Writes Real Life"). Usher writes the play, acting out all the characters as caricatures ("Writing a Gospel Play").
At work, Usher tells a patron he cannot continue the show without confronting his parents with his artistic self. The patron advises him to live without fear ("A Sympathetic Ear"). Usher speaks with his parents over the phone, his father asking if he has contractedHIVand his mother asking about the status of the gospel play. After hooking up with "Inwood Daddy", a white man who fetishizes him and calls him racial slurs, Usher questions his "Boundaries".
On his birthday, Usher's mother leaves a voicemail reminding him that homosexuality is a sin ("Periodically") while his father calls to inform him their church does not approve of his music ("Didn't Want Nothin' [Reprise]"). While arguing with his parents, Usher's mother, horrified by her portrayal in the play, accuses him of hating and disappointing her. Usher recalls how his friend Darnell, thinking he deserved to die, refused HIV medication and concludes living with AIDS is worse than dying from it ("Precious Little Dream/AIDS Is God's Punishment"). His mother tells Usher he is loved but thinks he is struggling because of his homosexuality.
The Thought who is portraying Usher's mother breaks thefourth wall, asking Usher if he wants the show to end with hateful caricatures of his parents. Usher claims that he is depicting life as it was when he was 17, to which the Thoughts remind him he is now 26. Usher then reflects on his childhood, realizing his perceptions must change before he can change ("Memory Song"). With his back to the audience, Usher wonders what will happen when the show ends. He turns around, concluding that he does not need to change because change is an illusion, and he, like everyone else, is in "A Strange Loop".
A Strange Loopbegan previews atoff-BroadwayvenuePlaywrights Horizonson May 24, 2019. It opened on June 17, 2019, with closing scheduled for July 7, 2019,[9]before extending to July 28, 2019.[10]The show featuredLarry Owensas Usher. The creative team credits included Michael R. Jackson as writer of book, music, and lyrics,Stephen Brackettas director,Raja Feather Kellyas choreographer,Charlie Rosenas orchestrator, and Rona Siddiqui as music director.[11]
TheWashington, DCproduction atWoolly Mammoth Theatre Companywas originally scheduled for September 2020, but postponed to December 2021 due to theCOVID-19 pandemic.[12][2]The six-week limited run began previews November 22, 2021, and opened December 3, 2021.[13]The show extended another week, changing its closing date from January 2 to 9, 2022.[14]
The Broadway production ofA Strange Loopwas announced December 20, 2021.[15]Many notable people from the entertainment industry served as the show's producers. They included lead producerBarbara Whitman, as well asBenj Pasek,Justin Paul,Jennifer Hudson,RuPaul Charles,Marc Platt,Megan EllisonofAnnapurna Pictures,Don Cheadle,Frank Marshall,James L. Nederlander,Alan Cumming,Ilana Glazer,Mindy KalingandBilly Porter.[16]Previews were scheduled to begin on April 6, but were postponed to April 14 due toCOVID-19breakouts among the cast.[17]The show officially opened April 26, 2022. In October 2022, it was announced that the show would play its final performance on Broadway on January 15, 2023.[18]
The London production opened at theBarbican Centreon June 17, 2023, for a limited run until September 9. It was produced byHoward Panterfor Trafalgar Theatre Productions, theNational Theatre,Barbara Whitmanand Wessex Grove.[19]Brackett, Kelly, Rosen, Siddiqui, and the rest of the Broadway creative team returned for the London production.[20]Kyle Ramar Freeman, who understudied Usher on Broadway, starred in the production alongside an all-British cast of Thoughts.[21]Freeman played his final performance on August 12 due to prior commitments. Kyle Birch stepped into the role of Usher on August 14 and continued through the show's final performance.[22]
A Strange Loopplayed as a co-production betweenAmerican Conservatory Theater(A.C.T.) in San Francisco and theCenter Theatre Groupin Los Angeles, presented at A.C.T. from April 18 to May 12, 2024, and the Center Theatre Group'sAhmanson Theatrefrom June 5 to June 30, 2024. The production was helmed by the same creative team as the Broadway and London mountings and featured a largely new cast as Usher and the Thoughts, with John-Andrew Morrison reprising his role as Thought 4.[23][24]
Upon opening off-Broadway on June 17, 2019.A Strange Loopreceived critical acclaim. In particular, it was praised for its emotional honesty and meta themes within both the writing and the musical compositions. It was also praised for the performances that the cast gave, calling them “physically exhaustive.” However, it was deemed as unlikely to become a Broadway show due to it potentially getting "too easily lost in a Broadway House."[25]
A Strange Loopopened on Broadway on April 26, 2022, and also received critical acclaim. In particular, it was praised for its themes and tone which were successfully retained from the off-Broadway version and then cleaned up for the larger, moreconsumerbased crowd which would be found on Broadway. There was a light critiquing about how working within the largeinstitutionof Broadway instead of merely peering in has made some of the commentary become shallow.[26][27]
The original off-Broadway cast recording was released on September 27, 2019, on Yellow Sound Label.[28]The album peaked at number 6 on theBillboardCast Albums chart.[29]A Broadway cast album was recorded on April 10, 2022, and released on June 10, 2022, throughSh-K-Boom Records, Yellow Sound Label, Barbara Whitman Productions, andGhostlight Records.[30]It debuted at number two on the Cast Albums chart.[31]
On June 14, 2022,Deadlinereported that the musical filled 98% of its available seats during the week ending June 12. The musical grossed $676,316 for seven performances.[32]The musical also broke theLyceum Theatrebox office house record for a standard 8-performance week, taking $860,496 for the week ending June 26, a $15,183 bump over the previous week.[33]As of September 2022,A Strange Loopgrossed around $14.2 million from 136,777 attendance and 157 performances.[34]
On May 4, 2020, thePulitzer Prize for Dramawas awarded to Jackson for the musical, with the committee citing the show as "a metafictional musical that tracks the creative process of an artist transforming issues of identity, race, and sexuality that once pushed him to the margins of the cultural mainstream into a meditation on universal human fears and insecurities." The show is the tenth musical to win the award, as well as the first musical written by a Black person to win and first musical to win without a Broadway run.[35]The show premiered on Broadway in April 2022 and won theTony Award for Best Musical. As one of its producers,Jennifer Hudsonbecame the second Black woman to receiveall four of the major American entertainment awards (EGOT).[36]
The cast recording received a nomination for theGrammy Award for Best Musical Theater Albumfor the2023 Grammy Awards.
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Broadway theatre,[nb 1]orBroadway, is a theater genre that consists of thetheatrical performancespresented in 41 professionaltheaters, each with 500 or more seats, in theTheater DistrictandLincoln CenteralongBroadway, inMidtown Manhattan, New York City.[1][2]Broadway andLondon'sWest Endtogether represent the highest commercial level of live theater in theEnglish-speaking world.[3]
While theBroadway thoroughfareis eponymous with the district, it is closely identified withTimes Square. Only three theaters are located on Broadway itself: theBroadway Theatre,Palace Theatre, andWinter Garden Theatre. The rest are located on the numbered cross streets, extending from theNederlander Theatreone block south of Times Square on West 41st Street, north along either side of Broadway to53rd Street, andVivian Beaumont Theater, atLincoln Centeron West 65th Street. While exceptions exist, the term "Broadway theatre" is used predominantly to describe venues with seating capacities of at least 500 people. Smaller theaters in New York City are referred to asoff-Broadway, regardless of location, while very small venues with fewer than 100 seats are calledoff-off-Broadway, a term that can also apply to non-commercial,avant-garde, or productions held outside of traditional theater venues.[4]
The Theater District is an internationally prominenttourist attraction in New York City. According toThe Broadway League, shows on Broadway sold approximately US$1.54 billion worth of tickets in both the 2022–2023 and the 2023–2024 seasons. Both seasons featured theater attendance of approximately 12.3 million each.[5]
Most Broadway shows aremusicals. HistorianMartin Shefterargues that "Broadway musicals, culminating in the productions ofRodgers and Hammerstein, became enormously influential forms ofAmerican popular culture" and contributed to making New York City thecultural capital of the world.[6]
New York City's first significant theatre was established in the mid-18th century, around 1750, when actor-managers Walter Murray and Thomas Kean established a resident theatre company at theTheatre on Nassau StreetinLower Manhattan, which held about 280 people. They presentedWilliam Shakespeare's plays andballad operassuch asThe Beggar's Opera.[7]In 1752,William Hallamsent a company of twelve actors from Britain to the colonies with his brotherLewisas their manager. They established a theatre inWilliamsburg, Virginia, and opened withThe Merchant of VeniceandThe Anatomist. The company moved to New York in 1753, performingballad operasand ballad-farces likeDamon and Phillida.
During theRevolutionary War, theatre was suspended in New York City. But after the war's end, theatre resumed in 1798, when the 2,000-seatPark Theatrewas built on Chatham Street on present-dayPark Row.[7]A second major theatre,Bowery Theatre, opened in 1826,[8]followed by others.
By the 1840s,P.T. Barnumwas operating an entertainment complex in Lower Manhattan. In 1829, at Broadway and Prince Street,Niblo's Gardenopened and soon became one of New York's premier nightspots. The 3,000-seat theatre presented all sorts ofmusicaland non-musical entertainments. In 1844,Palmo's Opera Houseopened and presented opera for only four seasons before bankruptcy led to its rebranding as a venue for plays under the name Burton's Theatre. TheAstor Opera Houseopened in 1847. A riot broke out in 1849 when the lower-class patrons of the Bowery Theatre objected to what they perceived as snobbery by the upper-class audiences at Astor Place: "After theAstor Place Riotof 1849,entertainment in New York Citywas divided along class lines: opera was chiefly for the upper-middle and upper classes, minstrel shows and melodramas for the middle-class, variety shows in concert saloons for men of the working class and the slumming middle-class."[9]
The plays ofWilliam Shakespearewere frequently performed on the Broadway stage during the period, most notably by American actorEdwin Boothwho was internationally known for his performance asHamlet. Booth played the role for a famous 100 consecutive performances at theWinter Garden Theatrein 1865 (with the run ending just a few months before Booth's brotherJohn Wilkes BoothassassinatedAbraham Lincoln), and would later revive the role at his ownBooth's Theatre(which was managed for a time by his brotherJunius Brutus Booth Jr.). Other renowned Shakespeareans who appeared in New York in this era wereHenry Irving,Tommaso Salvini,Fanny Davenport, andCharles Fechter.
Theatre in New York moved fromDowntowngradually toMidtown Manhattan, beginning around 1850, seeking less expensive real estate. At the beginning of the nineteenth century, the area that now comprises theTheater Districtwas owned by a handful of families and comprised a few farms. In 1836, MayorCornelius Lawrenceopened42nd Streetand invited Manhattanites to "enjoy the pure clean air."[10]Close to 60 years later, theatrical entrepreneurOscar Hammerstein Ibuilt the iconicVictoria Theateron West 42nd Street.[10]
Broadway's first "long-run" musical was a 50-performance hit calledThe Elvesin 1857. In 1870, the heart of Broadway was inUnion Square, and by the end of the century, many theatres were nearMadison Square. Theatres arrived in theTimes Squarearea in the early 1900s, and the Broadway theatres consolidated there after a large number were built around the square in the 1920s and 1930s. New York runs continued to lag far behind those in London,[11]butLaura Keene's "musical burletta"The Seven Sisters(1860) shattered previous New York records with a run of 253 performances.
The first theatre piece that conforms to the modern conception of a musical, adding dance and original music that helped to tell the story, is considered to beThe Black Crook, which premiered in New York on September 12, 1866. The production was five-and-a-half hours long, but despite its length, it ran for a record-breaking 474 performances. The same year,The Black Domino/Between You, Me and the Postwas the first show to call itself a "musical comedy".[12]
Tony Pastoropened the firstvaudevilletheatre one block east of Union Square in 1881, whereLillian Russellperformed. ComediansEdward HarriganandTony Hartproduced and starred in musicals on Broadway between 1878 (The Mulligan Guard Picnic) and 1890, with book and lyrics by Harrigan and music by his father-in-lawDavid Braham. These musical comedies featured characters and situations taken from the everyday life of New York's lower classes and represented a significant step forward from vaudeville and burlesque, towards a more literate form. They starred high-quality professional singers (Lillian Russell,Vivienne Segal, andFay Templeton), instead of the amateurs, often sex workers, who had starred in earlier musical forms.
As transportation improved, poverty in New York diminished, and street lighting made for safer travel at night, the number of potential patrons for the growing number of theatres increased enormously. Plays could run longer and still draw in the audiences, leading to better profits and improved production values. As in England, during the latter half of the century, the theatre began to be cleaned up, with lessprostitutionhindering the attendance of the theatre by women.Gilbert and Sullivan's family-friendlycomic operahits, beginning withH.M.S. Pinaforein 1878, were imported to New York (by the authors and also in numerous unlicensed productions). They were imitated in New York by American productions such asReginald Dekoven'sRobin Hood(1891) andJohn Philip Sousa'sEl Capitan(1896), along with operas, ballets, and other British and European hits.
Charles H. Hoyt'sA Trip toChinatown(1891) became Broadway's long-run champion when it surpassedAdonisand its 603 total performances in 1893, holding the stage for 657 performances.Chinatownitself was surpassed by the musicalIrene(1919) in 1921 as the longest-running Broadway musical, and even earlier, in March 1920, byLightnin'(1918) as the longest-running Broadway show. In 1896, theatre ownersMarc KlawandA. L. Erlangerformed theTheatrical Syndicate, which controlled almost every legitimate theatre in the U.S. for the next sixteen years.[13]However, smaller vaudeville and variety houses proliferated, andOff-Broadwaywas well established by the end of the nineteenth century.
A Trip to Coontown(1898) was the first musical comedy entirely produced and performed byAfrican Americansin a Broadway theatre (inspired largely by the routines of theminstrel shows), followed by theragtime-tingedClorindy: The Origin of the Cakewalk(1898), and the highly successfulIn Dahomey(1902). Hundreds of musical comedies were staged on Broadway in the 1890s and early 1900s made up of songs written in New York'sTin Pan Alleyinvolving composers such asGus Edwards,John Walter Bratton, andGeorge M. Cohan(Little Johnny Jones(1904),45 Minutes From Broadway(1906), andGeorge Washington Jr.(1906)). Still, New York runs continued to be relatively short, with a few exceptions, compared with London runs, untilWorld War I.[11]A few very successful British musicals continued to achieve great success in New York, includingFlorodorain 1900–01.
In the early years of the twentieth century, translations of popular late-nineteenth century continental operettas were joined by the "Princess Theatre" shows of the 1910s, by writers such asP. G. Wodehouse,Guy Bolton, andHarry B. Smith.Victor Herbert, whose work included some intimate musical plays with modern settings as well as his string of famous operettas (The Fortune Teller(1898),Babes in Toyland(1903),Mlle. Modiste(1905),The Red Mill(1906), andNaughty Marietta(1910)).[14]
Beginning withThe Red Mill, Broadway shows installed electric signs outside the theatres. Since colored bulbs burned out too quickly, white lights were used, and Broadway was nicknamed "The Great White Way". In August 1919, theActors' Equity Associationdemanded a standard contract for all professional productions. After a strike shut down all the theatres, the producers were forced to agree. By the 1920s, theShubert Brothershad risen to take over the majority of the theatres from the Erlanger syndicate.[15]
During this time, the playLightnin'byWinchell SmithandFrank Baconbecame the first Broadway show to reach 700 performances. From then, it would go on to become the first show to reach 1,000 performances.Lightnin'was the longest-running Broadway show until being overtaken in performance totals byAbie's Irish Rosein 1925.
The motion picture mounted a challenge to the stage. At first, films weresilentand presented only limited competition. By the end of the 1920s, films likeThe Jazz Singerwere presented with synchronized sound, and critics wondered if cinema would replace live theatre altogether. While live vaudeville could not compete with these inexpensive films that featured vaudeville stars and major comedians of the day, other theatres survived. The musicals of theRoaring Twenties, borrowing from vaudeville,music hall, and other light entertainment, tended to ignore plot in favor of emphasizing star actors and actresses, big dance routines, and popular songs.
Florenz Ziegfeldproduced annual spectacular song-and-dance revues on Broadway featuring extravagant sets and elaborate costumes, but there was little to tie the various numbers together. Typical of the 1920s were lighthearted productions such asSally;Lady Be Good;Sunny;No, No, Nanette;Harlem;Oh, Kay!; andFunny Face. Their books may have been forgettable, but they produced enduring standards fromGeorge Gershwin,Cole Porter,Jerome Kern,Vincent Youmans, andRodgers and Hart, among others, andNoël Coward,Sigmund Romberg, andRudolf Frimlcontinued in the vein of Victor Herbert. Live theatre has survived the invention of cinema.
Leaving these comparatively frivolous entertainments behind and taking the drama a step forward,Show Boatpremiered on December 27, 1927, at theZiegfeld Theatre. It represented a complete integration of book and score, with dramatic themes, as told through the music, dialogue, setting, and movement, woven together more seamlessly than in previous musicals. It ran for 572 performances.[16]
The 1920s also spawned a new age of American playwright with the emergence ofEugene O'Neill, whose playsBeyond the Horizon,Anna Christie,The Hairy Ape,Strange Interlude, andMourning Becomes Electraproved that there was an audience for serious drama on Broadway, and O'Neill's success paved the way for major dramatists likeElmer Rice,Maxwell Anderson,Robert E. Sherwood,Clifford Odets,Tennessee Williams, andArthur Miller, as well as writers of comedy likeGeorge S. KaufmanandMoss Hart. Classical revivals also proved popular with Broadway theatre-goers, notablyJohn BarrymoreinHamletandRichard III,John GielgudinHamlet,The Importance of Being EarnestandMuch Ado About Nothing,Walter HampdenandJosé FerrerinCyrano de Bergerac,Paul Robesonand Ferrer inOthello,Maurice EvansinRichard IIand the plays ofGeorge Bernard Shaw, andKatharine Cornellin such plays asRomeo and Juliet,Antony and Cleopatra, andCandida.
In 1930,Theatre Guild's production ofRoar, China!was Broadway's first play with a majority Asian cast.[17]
AsWorld War IIapproached, a dozen Broadway dramas addressed the rise of Nazism in Europe and the issue of American non-intervention. The most successful wasLillian Hellman'sWatch on the Rhine, which opened in April 1941.[18]
After the lean years of theGreat Depression, Broadway theatre had entered a golden age with the blockbuster hitOklahoma!, in 1943, which ran for 2,212 performances. According toJohn Kenrick's writings on Broadway musicals, "Every season saw new stage musicals send songs to the top of the charts. Public demand, a booming economy and abundant creative talent kept Broadway hopping. To this day, the shows of the 1950s form the core of the musical theatre repertory."[19]
Kenrick notes that "the late 1960s marked a time of cultural upheaval. All those changes would prove painful for many, including those behind the scenes, as well as those in the audience."[20]Of the 1970s, Kenrick writes: "Just when it seemed that traditional book musicals were back in style, the decade ended with critics and audiences giving mixed signals."[21]
Ken Bloomobserved that "The 1960s and 1970s saw a worsening of the area [Times Square] and a drop in the number of legitimate shows produced on Broadway."[22]By way of comparison, in the 1950 to 1951 season (May to May), 94 productions opened on Broadway; in the 1969 to 1970 season (June to May), there were 59 productions (fifteen were revivals).[23][24]In the 1920s, there were 70–80 theaters; however, by 1969, there were 36 left.[25]
During this time, many Broadway productions struggled due to low attendance rates, which resulted in perceived mediocrity among such plays. For this reason, theTheatre Development Fundwas created with the purpose of assisting productions with high cultural value that likely would struggle without subsidization, byoffering tickets to those playsto consumers at reduced prices.[26]
In early 1982,Joe Papp, the theatrical producer and director who establishedThe Public Theater, led the "Save the Theatres" campaign.[27]It was a not-for-profit group supported by theActors Equityunion to save the theater buildings in the neighborhood from demolition by monied Manhattan development interests.[28][29][30][31]Papp provided resources, recruited a publicist and celebrated actors, and provided audio, lighting, and technical crews for the effort.[29]
At Papp's behest, in July 1982, a bill was introduced in the97th Congress, entitled "H.R.6885, A bill to designate the Broadway/Times Square Theatre District in the City of New York as a national historic site".[32]The legislation would have provided certain U.S. government resources and assistance to help the city preserve the district.[32]Faced with strong opposition and lobbying byMayor Ed Koch's Administrationand corporate Manhattan development interests, the bill was not passed. The Save the Theatres campaign then turned their efforts to supporting the establishment of the Theater District as a registeredhistoric district.[33][34]In December 1983, Save the Theatres prepared "The Broadway Theater District, a Preservation Development and Management Plan", and demanded that each theater in the district receive landmark designation.[34]MayorEd Kochultimately reacted by creating a Theater Advisory Council, which included Papp.[29]
Due to theCOVID-19 pandemic in New York City, Broadway theaters closed on March 12, 2020, shuttering 16 shows that were playing or were in the process of opening. The Broadway League shutdown was extended first to April, then to May, then June, then September 2020 and January 2021,[35]and later to June 1, 2021.[36]Then-governorAndrew Cuomoannounced that most sectors ofNew Yorkwould have their restrictions lifted on May 19, 2021, but he stated that Broadway theatres would not be able to immediately resume performances on this date due to logistical reasons. In May 2021, Cuomo announced that Broadway theaters would be allowed to reopen on September 14, and the League confirmed that performances would begin to resume in the fall season.[37]
Springsteen on Broadwaybecame the first full-length show to resume performances, opening on June 26, 2021, to 1,721 vaccinated patrons at theSt. James Theatre.[38]Pass Overthen had its first preview on August 4, and opened on August 22, 2021, becoming the first new play to open.[39][40]HadestownandWaitresswere the first musicals to resume performances on September 2, 2021.[41]The74th Tony Awardswere also postponed; the Tony nominations were announced on October 15, 2020,[42]and took place on September 26, 2021.[43]On July 30, 2021, it was announced that all Broadway theaters required attendees to provide proof of fullCOVID-19 vaccination. The rule applied to guests ages 12+. Those under age 12 were required to provide a negative COVID-19 test (PCR within 72 hours or antigen within six hours of the performance start time). Beginning November 8, those ages 5–11 also had the option to provide proof of at least one vaccination shot. Effective December 14, in accordance with NYC's vaccination mandate, guests ages 5–11 were required to have at least one vaccination shot until January 29, 2022, where they had to be fully vaccinated.[44]The vaccine mandate lasted until April 30,[45][46]and attendees were also required to wearface masksuntil July 1.[47]
During the COVID-19 shutdown, the Shubert Organization, the Nederlander Organization, and Jujamcyn had pledged to increase racial and cultural diversity in their theaters, including naming at least one theater for a Black theatrical personality.[48]TheAugust Wilson Theatre, owned by Jujamcyn, had been renamed after Black playwrightAugust Wilsonin 2005.[49]The Shuberts announced in March 2022 that theCort Theatre, which was under renovation at the time, would be renamed after actorJames Earl Jones.[50][51]In June 2022, the Nederlanders announced that theBrooks Atkinson Theatrewould be renamed afterLena Horne,[52][53][49]The James Earl Jones Theatre was rededicated in September 2022,[54]while the Lena Horne Theatre was rededicated that November.[55]Pre-show announcements talking about masks being encouraged wasn't concluded until June 2023.
Although there are some exceptions, shows with open-ended runs generally have evening performances Tuesday through Saturday, with a 7:00 p.m. or 8:00 p.m. "curtain". The afternoon "matinée" performances are at 2:00 p.m. on Wednesdays and Saturdays and at 3:00 p.m. on Sundays. This makes for an eight-performance week. On this schedule, most shows do not play on Monday and the shows and theatres are said to be "dark" on that day.[56][57]The actors and the crew in these shows tend to regard Sunday evening through Monday evening as their weekend. The Tony award presentation ceremony is usually held on a Sunday evening in June to fit this schedule.
In recent years, some shows have moved their Tuesday show time an hour earlier to 7:00 pm.[56]The rationale for this move was that since fewer tourists take in shows midweek, Tuesday attendance depends more on local patrons. The earlier curtain makes it possible for suburban patrons to get home by a reasonable hour after the show. Some shows, especially thoseDisneyproduces, change their performance schedules fairly frequently depending on the season. This is done in order to maximize access to their target audience.
Most Broadway producers and theatre owners are members ofThe Broadway League(formerly "The League of American Theatres and Producers"), a trade organization that promotes Broadway theatre as a whole, negotiates contracts with the various theatrical unions and agreements with the guilds, and co-administers theTony Awardswith theAmerican Theatre Wing, a service organization. While the League and the theatrical unions are sometimes at loggerheads during those periods when new contracts are being negotiated, they also cooperate on many projects and events designed to promote professional theatre in New York.
Of the four non-profit theatre companies with Broadway theatres, all four (Lincoln Center Theater,Manhattan Theatre Club,Roundabout Theatre Company, andSecond Stage Theatre) belong to theLeague of Resident Theatresand have contracts with the theatrical unions which are negotiated separately from the other Broadway theatre and producers. (Disneyalso negotiates apart from the League, as didLiventbefore it closed down its operations.)
The majority of Broadway theatres are owned or managed by three organizations: theShubert Organization, a for-profit arm of the non-profit Shubert Foundation, which owns seventeen theatres; theNederlander Organization, which controls nine theatres; andATG Entertainment, which owns seven Broadway houses.
Both musicals and straight plays on Broadway often rely on casting well-known performers in leading roles to draw larger audiences or bring in new audience members to the theatre. Actors from film and television are frequently cast for the revivals of Broadway shows or are used to replace actors leaving a cast. There are still, however, performers who are primarily stage actors, spending most of their time "on the boards", and appearing in screen roles only secondarily. As Patrick Healy ofThe New York Timesnoted:
Broadway once had many homegrown stars who committed to working on a show for a year, asNathan Lanehas forThe Addams Family. In 2010, some theater heavyweights like Mr. Lane were not even nominated; instead, several Tony Awards were given for productions that were always intended to be short-timers on Broadway, given that many of their film-star performers had to move on to other commitments.[58]
According to Mark Shenton, "One of the biggest changes to the commercial theatrical landscape—on both sides of the Atlantic—over the past decade or so is that sightings of big star names turning out to do plays has [sic] gone up; but the runs they are prepared to commit to has gone down. Time was that a producer would require a minimum commitment from his star of six months, and perhaps a year; now, the 13-week run is the norm."[59]
The minimum size of the Broadway orchestra is governed by an agreement with the musicians' union (Local 802, American Federation of Musicians) and The Broadway League. For example, the agreement specifies the minimum size of the orchestra at the Minskoff Theatre to be eighteen, while at the Music Box Theatre it is nine.[60]
Most Broadway shows are commercial productions intended to make a profit for the producers and investors ("backers" or "angels"), and therefore have open-ended runs (duration that the production plays), meaning that the length of their presentation is not set beforehand, but depends on critical response, word of mouth, and the effectiveness of the show's advertising, all of which determine ticket sales. Investing in a commercial production carries a varied degree of financial risk. Shows need not make a profit immediately; should they make their "nut" (weekly operating expenses), or lose money at a rate acceptable to the producers, they may continue to run in the expectation that, eventually, they will pay back their initial costs and become profitable. In some borderline situations, producers may ask that royalties be temporarily reduced or waived, or even that performers—with the permission of their unions—take reduced salaries, to prevent a show from closing. Theatre owners, who are not generally profit participants in most productions, may waive or reduce rents, or even lend money to a show to keep it running.
Some Broadway shows are produced by non-commercial organizations as part of a regular subscription season—Lincoln Center Theatre,Roundabout Theatre Company,Manhattan Theatre Club, andSecond Stage Theaterare the four non-profit theatre companies that currently have permanent Broadway venues. Some other productions are produced on Broadway with "limited engagement runs" for several reasons, including financial issues, prior engagements of the performers, or temporary availability of a theatre between the end of one production and the beginning of another. However, some shows with planned limited engagement runs may, after critical acclaim or box office success, extend their engagements or convert to open-ended runs. This was the case with 2007'sAugust: Osage County, 2009'sGod of Carnage, 2012'sNewsies, and 2022'sTake Me Out.[61]
Historically, musicals on Broadway tend to have longer runs than "straight" (i.e., non-musical) plays. On January 9, 2006,The Phantom of the Operaat theMajestic Theatrebecame the longest-running Broadway musical, with 7,486 performances, overtakingCats.[62]The Phantom of the Operaclosed on Broadway on April 16, 2023, soon after celebrating its 35th anniversary, after a total of 13,981 performances.[63][64]
Attending a Broadway show is a commontouristactivity in New York. TheTKTSbooths sell same-day tickets (and in certain cases, next-day matinee tickets) for many Broadway andOff-Broadwayshows at a discount of 20 to 50%.[65]The TKTS booths are located inTimes Square, inLower Manhattan, and atLincoln Center. This service is run byTheatre Development Fund. Many Broadway theatres also offer special student rates, same-day "rush" or "lottery" tickets, or standing-room tickets to help ensure that their theatres are as full—and their grosses as high—as possible.[66]
According toThe Broadway League, total Broadway attendance was 14.77 million in 2018–2019, compared to 13.79 million in 2017–2018.[67]The average age of the Broadway audience in the 2017–18 theater season was 40, the lowest it had been in nearly two decades.[68]By 2018, about 20% of Broadway tickets were sold to international visitors, although many visitors reported not being able to use their tickets.[69]In 2022–2023, the first full season since the COVID-19 pandemic, Broadway theaters sold 12.3 million tickets, of which 35% were to local residents and 17% to international visitors. At the time, the average age of theatergoers was 40.4; nearly two-thirds of the audience were women; and 29% identified as a racial minority.[70]
The classification of theatres is governed by language inActors' Equity Associationcontracts. To be eligible for a Tony, a production must be in a house with 500 seats or more and in the Theater District, which are the criteria that define Broadway theatre.Off-Broadwayandoff-off-Broadwayshows often provide a more experimental, challenging, and intimate performance than is possible in the larger Broadway theatres. Some Broadway shows, however, such as the musicalsHair,Little Shop of Horrors,Spring Awakening,Next to Normal,Rent,Avenue Q,In the Heights,Fun Home,A Chorus Line,Dear Evan Hansen, andHamilton, began their runs Off-Broadway and later transferred to Broadway, seeking to replicate their intimate experience in a larger theatre. Other productions are first developed throughworkshopsand then out-of-towntryoutsbefore transferring to Broadway.Merrily We Roll Alongfamously skipped an out-of-town tryout and attempted to do an in-town tryout—actuallypreview performances—on Broadway before its official opening, with disastrous results.[71][72]
After, or even during, successful runs in Broadway theatres, producers often remount their productions with new casts and crew for the Broadway national tour, which travels to theatres in major cities across the country. Sometimes when a show closes on Broadway, the entire production, with most if not all of the original cast intact, is relaunched as a touring company, hence the name "Broadway national tour". Some shows may even have several touring companies out at a time, whether the show is still running in New York or not, with many companies "sitting down" in other major cities for their own extended runs. For Broadway national tours of top-tier cities, the entire Broadway production is transplanted almost entirely intact and may run for many months (or years) at each stop. For example, the first U.S. tour ofThe Phantom of the Operarequired 26 53-foot-long (16.1 m)semi-trailersto transport all its sets, equipment, and costumes, and it took almost 10 days to properly unload all those trucks and install everything into a theater.[73]
Second-tier and smaller cities can also attract national tours, but these are more likely to be "bus and truck" tours.[73]These are scaled-down versions of the larger, national touring productions, historically acquiring their name because the casts generally traveled by bus instead of by air, while the sets and equipment traveled by truck. Tours of this type often run for weeks rather than months, and frequently feature a reduced physical production to accommodate smaller venues and tighter schedules, and to fit into fewer trucks.[73]A typical second-tier city can usually sell only up to about eight shows (one week) of tickets.[73]For cities smaller than that, a touring production might move twice a week ("split weeks") or every day ("one-nighters").[73]For "bus and truck" tours, the production values are usually less lavish than the typical Broadway national tour or national touring production, and the actors, while still members of the actors' union, are compensated under a different, less lucrative union contract. TheTouring Broadway Awards, presented byThe Broadway League, honored excellence in touring Broadway.
Broadway productions and artists are honored by the annual Antoinette Perry Awards (commonly called the "Tony Awards", or "Tonys"), given by theAmerican Theatre WingandThe Broadway League, and that were first presented in 1947.[74]The Tony is Broadway's most prestigious award, comparable to theAcademy AwardsforHollywoodfilm productions. Their importance has increased since 1967 when the awards presentation show began to be broadcast on national television. In a strategy to improve the television ratings, celebrities are often chosen to host the show, some with scant connection to the theatre.[75]The most recent Tony Awards ceremony was held onJune 16, 2024. Other awards given to Broadway productions include theDrama Desk Award, presented since 1955, theNew York Drama Critics' Circle Awards, first given in 1936, and theOuter Critics Circle Award, initially presented in 1950.
The following shows are confirmed as future Broadway productions. The theatre in which they will run is either not yet known or currently occupied by another show.
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https://en.wikipedia.org/wiki/Broadway_theatre
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Musical theatreis a form oftheatricalperformance that combines songs, spokendialogue, acting and dance. The story and emotional content of a musical – humor,pathos, love, anger – are communicated through words, music, movement and technical aspects of the entertainment as an integrated whole. Although musical theatre overlaps with other theatrical forms like opera and dance, it may be distinguished by the equal importance given to the music as compared with the dialogue, movement and other elements. Since the early 20th century, musical theatre stage works have generally been called, simply,musicals.
Although music has been a part of dramatic presentations since ancient times, modern Western musical theatre emerged during the 19th century, with many structural elements established by thelight operaworks ofJacques Offenbachin France,Gilbert and Sullivanin Britain and the works ofHarriganandHartin America. These were followed byEdwardian musical comedies, which emerged in Britain, and the musical theatre works of American creators likeGeorge M. Cohanat the turn of the 20th century. ThePrincess Theatremusicals (1915–1918) were artistic steps forward beyond therevuesand other frothy entertainments of the early 20th century and led to such groundbreaking works asShow Boat(1927),Of Thee I Sing(1931) andOklahoma!(1943). Some of the best-known musicals through the decades that followed includeMy Fair Lady(1956),The Fantasticks(1960),Hair(1967),A Chorus Line(1975),Les Misérables(1985),The Phantom of the Opera(1986),Rent(1996),Wicked(2003) andHamilton(2015).
Musicals are performed around the world. They may be presented in large venues, such as big-budgetBroadwayorWest Endproductions in New York City or London. Alternatively, musicals may be staged in smaller venues, such asoff-Broadway,off-off-Broadway,regional theatre,fringe theatre, orcommunity theatreproductions, oron tour. Musicals are often presented byamateur and school groupsin churches, schools and other performance spaces. In addition to the United States and Britain, there are vibrant musical theatre scenes in continental Europe, Asia, Australasia, Canada and Latin America.
Since the 20th century, the "book musical" has been defined as a musical play where songs and dances are fully integrated into a well-made story with serious dramatic goals and which is able to evoke genuine emotions other than laughter.[2][3]The three main components of a book musical are itsmusic,lyricsandbook. The book orscriptof a musical refers to the story, character development and dramatic structure, including the spoken dialogue and stage directions, but it can also refer to the dialogue and lyrics together, which are sometimes referred to as thelibretto(Italian for "small book"). The music and lyrics together form thescoreof a musical and include songs,incidental musicand musical scenes, which are "theatrical sequence[s] set to music, often combining song with spoken dialogue."[4]The interpretation of a musical is the responsibility of its creative team, which includes a director, a musical director, usually a choreographer and sometimes anorchestrator. A musical's production is also creatively characterized by technical aspects, such asset design,costumes,stage properties (props),lightingandsound. The creative team, designs and interpretations generally change from the original production to succeeding productions. Some production elements, however, may be retained from the original production, for example,Bob Fosse's choreography inChicago.
There is no fixed length for a musical. While it can range from a short one-act entertainment to severalactsand several hours in length (or even a multi-evening presentation), most musicals range from one and a half to three hours. Musicals are usually presented in two acts, with one shortintermission, and the first act is frequently longer than the second. The first act generally introduces nearly all of the characters and most of the music and often ends with the introduction of a dramatic conflict or plot complication while the second act may introduce a few new songs but usually contains reprises of important musical themes and resolves the conflict or complication. A book musical is usually built around four to six main theme tunes that are reprised later in the show, although it sometimes consists of a series of songs not directly musically related. Spoken dialogue is generally interspersed between musical numbers, although "sung dialogue" orrecitativemay be used, especially in so-called "sung-through" musicals such asJesus Christ Superstar,Falsettos,Les Misérables,EvitaandHamilton. Several shorter musicals on Broadway and in the West End in the 21st century have been presented in one act.
Moments of greatest dramatic intensity in a book musical are often performed in song. Proverbially, "when the emotion becomes too strong for speech, you sing; when it becomes too strong for song, you dance."[5]In a book musical, a song is ideally crafted to suit the character (or characters) and their situation within the story; although there have been times in the history of the musical (e.g. from the 1890s to the 1920s) when this integration between music and story has been tenuous. AsThe New York TimescriticBen Brantleydescribed the ideal of song in theatre when reviewing the 2008 revival ofGypsy: "There is no separation at all between song and character, which is what happens in those uncommon moments when musicals reach upward to achieve their ideal reasons to be."[6]Typically, many fewer words are sung in a five-minute song than are spoken in a five-minute block of dialogue. Therefore, there is less time to develop drama in a musical than in a straight play of equivalent length, since a musical usually devotes more time to music than to dialogue. Within the compressed nature of a musical, the writers must develop the characters and the plot.
The material presented in a musical may be original, or it may be adapted from novels (WickedandMan of La Mancha), plays (Hello, Dolly!andCarousel), classic legends (Camelot), historical events (EvitaandHamilton) or films (The ProducersandBilly Elliot). On the other hand, many successful musical theatre works have been adapted formusical films, such asWest Side Story,My Fair Lady,The Sound of Music,Oliver!andChicago.
Musical theatre is closely related to the theatrical form of opera, but the two are usually distinguished by weighing a number of factors. First, musicals generally have a greater focus on spoken dialogue.[7]Some musicals, however, are entirely accompanied and sung-through, while some operas, such asDie Zauberflöte, and mostoperettas, have some unaccompanied dialogue.[7]Second, musicals usually include more dancing as an essential part of the storytelling, particularly by the principal performers as well as the chorus. Third, musicals often use various genres ofpopular musicor at least popular singing and musical styles.[8]
Finally, musicals usually avoid certain operatic conventions. In particular, a musical is almost always performed in the language of its audience. Musicals produced on Broadway or in the West End, for instance, are invariably sung in English, even if they were originally written in another language. While an opera singer is primarily a singer and only secondarily an actor (and rarely needs to dance), a musical theatre performer is often an actor first but must also be a singer and dancer. Someone who is equally accomplished at all three is referred to as a "triple threat". Composers of music for musicals often consider the vocal demands of roles with musical theatre performers in mind. Today, large theatres that stage musicals generally usemicrophonesandamplificationof the actors' singing voices in a way that would generally be disapproved of in an operatic context.[9]
Some works, including those byGeorge Gershwin,Leonard BernsteinandStephen Sondheim, have been made into both musical theatre and operatic productions.[10][11]Similarly, some older operettas or light operas (such asThe Pirates of PenzancebyGilbert and Sullivan) have been produced in modern adaptations that treat them as musicals. For some works, production styles are almost as important as the work's musical or dramatic content in defining into which art form the piece falls.[12]Sondheim said, "I really think that when something plays Broadway it's a musical, and when it plays in an opera house it's opera. That's it. It's the terrain, the countryside, the expectations of the audience that make it one thing or another."[13]There remains an overlap in form between lighter operatic forms and more musically complex or ambitious musicals. In practice, it is often difficult to distinguish among the various kinds of musical theatre, including "musical play", "musical comedy", "operetta" and "light opera".[14]
Like opera, the singing in musical theatre is generally accompanied by an instrumental ensemble called apit orchestra, located in a lowered area in front of the stage. While opera typically uses a conventionalsymphony orchestra, musicals are generally orchestrated for ensembles ranging from27 players down to only a few players.Rock musicalsusually employ a small group of mostly rock instruments,[15]and some musicals may call for only a piano or two instruments.[16]The music in musicals uses a range of "styles and influences includingoperetta, classical techniques,folk music,jazz[and] local or historical styles [that] are appropriate to the setting."[4]Musicals may begin with anovertureplayed by the orchestra that "weav[es] together excerpts of the score's famous melodies."[17]
There are variousEasterntraditions of theatre that include music, such asChinese opera,Taiwanese opera, JapaneseNohandIndian musical theatre, includingSanskrit drama,Indian classical dance,Parsi theatreandYakshagana.[18]India has, since the 20th century, produced numerous musical films, referred to as "Bollywood" musicals, and in Japan a series of2.5D musicalsbased on popularanimeandmangacomics has developed in recent decades.
Shorter or simplified "junior" versions of many musicals are available for schools and youth groups, and very short works created or adapted for performance by children are sometimes calledminimusicals.[19][20]
The antecedents of musical theatre in Europe can be traced back to thetheatre of ancient Greece, where music and dance were included in stage comedies and tragedies during the 5th century BCE.[21][22]The music from the ancient forms is lost, however, and they had little influence on later development of musical theatre.[23]In the 12th and 13th centuries, religious dramas taught theliturgy. Groups of actors would use outdoorPageant wagons(stages on wheels) to tell each part of the story. Poetic forms sometimes alternated with the prose dialogues, and liturgical chants gave way to new melodies.[24]
The EuropeanRenaissancesaw older forms evolve into two antecedents of musical theatre:commedia dell'arte, where raucous clowns improvised familiar stories, and later,opera buffa. In England, Elizabethan and Jacobean plays frequently included music,[25]and short musical plays began to be included in an evenings' dramatic entertainments.[26]Courtmasquesdeveloped during theTudor periodthat involved music, dancing, singing and acting, often with expensive costumes and a complexstage design.[27][28]These developed into sung plays that are recognizable as English operas, the first usually being thought of asThe Siege of Rhodes(1656).[29]In France, meanwhile,Molièreturned several of his farcical comedies into musical entertainments with songs (music provided byJean-Baptiste Lully) and dance in the late 17th century. These influenced a brief period ofEnglish opera[30]by composers such asJohn Blow[31]andHenry Purcell.[29]
From the 18th century, the most popular forms of musical theatre in Britain wereballad operas, likeJohn Gay'sThe Beggar's Opera, that included lyrics written to the tunes of popular songs of the day (often spoofing opera), and laterpantomime, which developed from commedia dell'arte, andcomic operawith mostly romantic plot lines, likeMichael Balfe'sThe Bohemian Girl(1845). Meanwhile, on the continent,singspiel,comédie en vaudeville,opéra comique,zarzuelaand other forms of light musical entertainment were emerging.The Beggar's Operawas the first recorded long-running play of any kind, running for 62 successive performances in 1728. It would take almost a century afterwards before any play broke 100 performances,[32]but the record soon reached 150 in the late 1820s.[33][34]Other musical theatre forms developed in England by the 19th century, such asmusic hall,melodramaandburletta, which were popularized partly because most London theatres were licensed only as music halls and not allowed to present plays without music.
Colonial America did not have a significant theatre presence until 1752, when London entrepreneur William Hallam sent a company of actors to the colonies managed by his brotherLewis.[35]In New York in the summer of 1753, they performed ballad-operas, such asThe Beggar's Opera, and ballad-farces.[35]By the 1840s,P. T. Barnumwas operating an entertainment complex in lower Manhattan.[36]Other early musical theatre in America consisted of British forms, such as burletta and pantomime,[23]but what a piece was called did not necessarily define what it was. The 1852 BroadwayextravaganzaThe Magic Deeradvertised itself as "A Serio Comico Tragico Operatical Historical Extravaganzical Burletical Tale of Enchantment."[37]Theatre in New York moved from downtown gradually to midtown from around 1850 and did not arrive in the Times Square area until the 1920s and 1930s. New York runs lagged far behind those in London, butLaura Keene's "musical burletta"Seven Sisters(1860) shattered previous New York musical theatre record, with a run of 253 performances.[38]
Around 1850, the French composerHervéwas experimenting with a form of comic musical theatre he calledopérette.[39]The best known composers ofoperettawereJacques Offenbachfrom the 1850s to the 1870s andJohann Strauss IIin the 1870s and 1880s.[23]Offenbach's fertile melodies, combined with his librettists' witty satire, formed a model for the musical theatre that followed.[39]Adaptations of the French operettas (played in mostly bad, risqué translations),musical burlesques, music hall, pantomime and burletta dominated the London musical stage into the 1870s.[40]
In America, mid-19th century musical theatre entertainments included crudevariety revue, which eventually developed intovaudeville,minstrel shows, which soon crossed the Atlantic to Britain, and Victorian burlesque, first popularized in the US by British troupes.[23]Kurt GänzlconsidersThe Doctor of Alcantara(1862), with music composed byJulius Eichbergand a book and lyrics by Benjamin E Woolf, to be the "first American musical",[41]though he also points to even earlier works.[42]A hugely successful musical entertainment that premiered in New York in 1866,The Black Crook, combined dance and some original music that helped to tell the story. The spectacular production, famous for its skimpy costumes, ran for a record-breaking 474 performances.[43]The same year,The Black Domino/Between You, Me and the Postwas the first show to call itself a "musical comedy". In 1874,Evangeline or The Belle of Arcadia, byEdward E. RiceandJ. Cheever Goodwin, based loosely onLongfellow’sEvangeline, with an original American story and music, opened successfully in New York and was revived in Boston, New York, and in repeated tours.[44]ComediansEdward HarriganandTony Hartproduced and starred in musicals on Broadway between 1878 (The Mulligan Guard Picnic) and 1885. These musical comedies featured characters and situations taken from the everyday life of New York's lower classes. They starred high quality singers (Lillian Russell,Vivienne SegalandFay Templeton) instead of the ladies of questionable repute who had starred in earlier musical forms. In 1879,The Brookby Nate Salsbury was another national success with contemporary American dance styles and an American story about "members of an acting company taking a trip down a river ... with lots of obstacles and mishaps along the way".[44]
As transportation improved, poverty in London and New York diminished, and street lighting made for safer travel at night, the number of patrons for the growing number of theatres increased enormously. Plays ran longer, leading to better profits and improved production values, and men began to bring their families to the theatre. The first musical theatre piece to exceed 500 consecutive performances was the French operettaThe Chimes of Normandyin 1878 (705 performances).[33][45]Englishcomic operaadopted many of the successful ideas of European operetta, none more successfully than the series of more than a dozen long-runningGilbert and Sullivancomic operas, includingH.M.S. Pinafore(1878) andThe Mikado(1885).[39]These were sensations on both sides of the Atlantic and in Australia and helped to raise the standard for what was considered a successful show.[46]These shows were designed for family audiences, a marked contrast from the risqué burlesques, bawdy music hall shows and French operettas that sometimes drew a crowd seeking less wholesome entertainment.[40]Only a few 19th-century musical pieces exceeded the run ofThe Mikado, such asDorothy, which opened in 1886 and set a new record with a run of 931 performances. Gilbert and Sullivan's influence on later musical theatre was profound, creating examples of how to "integrate" musicals so that the lyrics and dialogue advanced a coherent story.[47][48]Their works wereadmired and copiedby early authors and composers of musicals in Britain[49][50]and America.[46][51]
A Trip to Chinatown(1891) was Broadway's long-run champion (untilIrenein 1919), running for 657 performances, but New York runs continued to be relatively short, with a few exceptions, compared with London runs, until the 1920s.[33]Gilbert and Sullivan were widely pirated and also were imitated in New York by productions such asReginald De Koven'sRobin Hood(1891) andJohn Philip Sousa'sEl Capitan(1896).A Trip to Coontown(1898) was the first musical comedy entirely produced and performed by African Americans on Broadway (largely inspired by the routines of theminstrel shows), followed byragtime-tinged shows. Hundreds of musical comedies were staged on Broadway in the 1890s and early 20th century, composed of songs written in New York'sTin Pan Alley, including those byGeorge M. Cohan, who worked to create an American style distinct from the Gilbert and Sullivan works. The most successful New York shows were often followed by extensive national tours.[52]
Meanwhile, musicals took over the London stage in theGay Nineties, led by producerGeorge Edwardes, who perceived that audiences wanted a new alternative to theSavoy-style comic operas and their intellectual, political, absurdist satire. He experimented with a modern-dress, family-friendly musical theatre style, with breezy, popular songs, snappy, romantic banter, and stylish spectacle at theGaietyand his other theatres. These drew on the traditions of comic opera and used elements of burlesque and of the Harrigan and Hart pieces. He replaced the bawdy women of burlesque with his "respectable" corps ofGaiety Girlsto complete the musical and visual fun. The success of the first of these,In Town(1892) andA Gaiety Girl(1893) set the style for the next three decades. The plots were generally light, romantic "poor maiden loves aristocrat and wins him against all odds" shows, with music byIvan Caryll,Sidney JonesandLionel Monckton. These shows were immediately widely copied in America, andEdwardian musical comedyswept away the earlier musical forms of comic opera and operetta.The Geisha(1896) was one of the most successful in the 1890s, running for more than two years and achieving great international success.
The Belle of New York(1898) became the first American musical to run for over a year in London. The British musical comedyFlorodora(1899) was a popular success on both sides of the Atlantic, as wasA Chinese Honeymoon(1901), which ran for a record-setting 1,074 performances in London and 376 in New York.[34]After the turn of the 20th century,Seymour Hicksjoined forces with Edwardes and American producerCharles Frohmanto create another decade of popular shows. Other enduring Edwardian musical comedy hits includedThe Arcadians(1909) andThe Quaker Girl(1910).[53]
Virtually eliminated from the English-speaking stage by competition from the ubiquitous Edwardian musical comedies, operettas returned to London and Broadway in 1907 withThe Merry Widow, and adaptations of continental operettas became direct competitors with musicals.Franz LehárandOscar Strauscomposed new operettas that were popular in English until World War I.[54]In America,Victor Herbertproduced a string of enduring operettas includingThe Fortune Teller(1898),Babes in Toyland(1903),Mlle. Modiste(1905),The Red Mill(1906) andNaughty Marietta(1910).
In the 1910s, the team ofP. G. Wodehouse,Guy BoltonandJerome Kern, following in the footsteps ofGilbert and Sullivan, created the "Princess Theatreshows" and paved the way for Kern's later work by showing that a musical could combine light, popular entertainment with continuity between its story and songs.[47]HistorianGerald Bordmanwrote:
These shows built and polished the mold from which almost all later major musical comedies evolved. ... The characters and situations were, within the limitations of musical comedy license, believable and the humor came from the situations or the nature of the characters. Kern's exquisitely flowing melodies were employed to further the action or develop characterization. ... [Edwardian] musical comedy was often guilty of inserting songs in a hit-or-miss fashion. The Princess Theatre musicals brought about a change in approach. P. G. Wodehouse, the most observant, literate and witty lyricist of his day, and the team of Bolton, Wodehouse and Kern had an influence felt to this day.[55]
The theatre-going public needed escapist entertainment during the dark times ofWorld War I, and they flocked to the theatre. The 1919 hit musicalIreneran for 670 performances, a Broadway record that held until 1938.[56]The British theatre public supported far longer runs like that ofThe Maid of the Mountains(1,352 performances) and especiallyChu Chin Chow. Its run of 2,238 performances was more than twice as long as any previous musical, setting a record that stood for nearly forty years.[57]Even a revival ofThe Beggar's Operaheld the stage for 1,463 performances.[58]Revues likeThe Bing Boys Are Herein Britain, and those ofFlorenz Ziegfeldand his imitators in America, were also extraordinarily popular.[37]
The musicals of theRoaring Twenties, borrowing from vaudeville,music halland other light entertainments, tended to emphasize big dance routines and popular songs at the expense of plot. Typical of the decade were lighthearted productions likeSally;Lady, Be Good;No, No, Nanette;Oh, Kay!; andFunny Face. Despite forgettable stories, these musicals featured stars such asMarilyn MillerandFred Astaireand produced dozens of enduring popular songs by Kern,GeorgeandIra Gershwin,Irving Berlin,Cole PorterandRodgers and Hart. Popular music was dominated by musical theatre standards, such as "Fascinating Rhythm", "Tea for Two" and "Someone to Watch Over Me". Many shows wererevues, series of sketches and songs with little or no connection between them. The best-known of these were the annualZiegfeld Follies, spectacular song-and-dance revues on Broadway featuring extravagant sets, elaborate costumes and beautiful chorus girls.[23]These spectacles also raised production values, and mounting a musical generally became more expensive.[37]Shuffle Along(1921), an all-African American show, was a hit on Broadway.[59]A new generation of composers of operettas also emerged in the 1920s, such asRudolf FrimlandSigmund Romberg, to create a series of popular Broadway hits.[60]
In London, writer-stars such asIvor NovelloandNoël Cowardbecame popular, but the primacy of British musical theatre from the 19th century through 1920 was gradually replaced by American innovation, especially after World War I, as Kern and otherTin Pan Alleycomposers began to bring new musical styles such asragtimeandjazzto the theatres, and theShubert Brotherstook control of the Broadway theatres. Musical theatre writerAndrew Lambnotes, "The operatic and theatrical styles of nineteenth-century social structures were replaced by a musical style more aptly suited to twentieth-century society and its vernacular idiom. It was from America that the more direct style emerged, and in America that it was able to flourish in a developing society less hidebound by nineteenth-century tradition."[61]In France,comédie musicalewas written between in the early decades of the century for such stars asYvonne Printemps.[62]
Progressing far beyond the comparatively frivolous musicals and sentimental operettas of the decade, Broadway'sShow Boat(1927) represented an even more complete integration of book and score than the Princess Theatre musicals, with dramatic themes told through the music, dialogue, setting and movement. This was accomplished by combining the lyricism of Kern's music with the skillful libretto ofOscar Hammerstein II. One historian wrote, "Here we come to a completely new genre – the musical play as distinguished from musical comedy. Now ... everything else was subservient to that play. Now ... came complete integration of song, humor and production numbers into a single and inextricable artistic entity."[63]
As theGreat Depressionset in during the post-Broadway national tour ofShow Boat, the public turned back to mostly light, escapist song-and-dance entertainment.[55]Audiences on both sides of the Atlantic had little money to spend on entertainment, and only a few stage shows anywhere exceeded a run of 500 performances during the decade. The revueThe Band Wagon(1931) starred dancing partners Fred Astaire and his sisterAdele, while Porter'sAnything Goes(1934) confirmedEthel Merman's position as the First Lady of musical theatre, a title she maintained for many years. Coward and Novello continued to deliver old fashioned, sentimental musicals, such asThe Dancing Years, while Rodgers and Hart returned from Hollywood to create a series of successful Broadway shows, includingOn Your Toes(1936, withRay Bolger, the first Broadway musical to make dramatic use of classical dance),Babes in Arms(1937) andThe Boys from Syracuse(1938). Porter addedDu Barry Was a Lady(1939). The longest-running piece of musical theatre of the 1930s in the US wasHellzapoppin(1938), a revue with audience participation, which played for 1,404 performances, setting a new Broadway record.[56]In Britain,Me and My Girlran for 1,646 performances.[58]
Still, a few creative teams began to build onShow Boat's innovations.Of Thee I Sing(1931), a political satire by the Gershwins, was the first musical awarded thePulitzer Prize.[23][64]As Thousands Cheer(1933), a revue byIrving BerlinandMoss Hartin which each song or sketch was based on a newspaper headline, marked the first Broadway show in which an African-American,Ethel Waters, starred alongside white actors. Waters' numbers included "Supper Time", a woman's lament for her husband who has been lynched.[65]The Gershwins'Porgy and Bess(1935) featured an all African-American cast and blended operatic, folk and jazz idioms.The Cradle Will Rock(1937), directed byOrson Welles, was a highly political pro-unionpiece that, despite the controversy surrounding it, ran for 108 performances.[37]Rodgers and Hart'sI'd Rather Be Right(1937) was a political satire withGeorge M. Cohanas PresidentFranklin D. Roosevelt, andKurt Weill'sKnickerbocker Holidaydepicted New York City's early history while good-naturedly satirizing Roosevelt's good intentions.
The motion picture mounted a challenge to the stage. Silent films had presented only limited competition, but by the end of the 1920s, films likeThe Jazz Singercould be presented with synchronized sound."Talkie"films at low prices effectively killed offvaudevilleby the early 1930s.[66]Despite the economic woes of the 1930s and the competition from film, the musical survived. In fact, it continued to evolve thematically beyond the gags and showgirls musicals of theGay NinetiesandRoaring Twentiesand the sentimental romance of operetta, adding technical expertise and the fast-paced staging and naturalistic dialogue style led by directorGeorge Abbott.[23]
The 1940s began with more hits from Porter,Irving Berlin, Rodgers and Hart, Weill and Gershwin, some with runs over 500 performances as the economy rebounded, but artistic change was in the air.Rodgers and Hammerstein'sOklahoma!(1943) completed the revolution begun byShow Boat, by tightly integrating all the aspects of musical theatre, with a cohesive plot, songs that furthered the action of the story, and featured dream ballets and other dances that advanced the plot and developed the characters, rather than using dance as an excuse to parade scantily clad women across the stage.[3]Rodgers and Hammerstein hired ballet choreographerAgnes de Mille, who used everyday motions to help the characters express their ideas. It defied musical conventions by raising its first act curtain not on a bevy of chorus girls, but rather on a woman churning butter, with an off-stage voice singing the opening lines ofOh, What a Beautiful Mornin'unaccompanied. It drew rave reviews, set off a box-office frenzy and received aPulitzer Prize.[67]Brooks Atkinsonwrote inThe New York Timesthat the show's opening number changed the history of musical theatre: "After a verse like that, sung to a buoyant melody, the banalities of the old musical stage became intolerable."[68]It was the first "blockbuster" Broadway show, running a total of 2,212 performances, and was made into a hit film. It remains one of the most frequently produced of the team's projects. William A. Everett andPaul R. Lairdwrote that this was a "show, that, likeShow Boat, became a milestone, so that later historians writing about important moments in twentieth-century theatre would begin to identify eras according to their relationship toOklahoma!".[69]
"AfterOklahoma!, Rodgers and Hammerstein were the most important contributors to the musical-play form... The examples they set in creating vital plays, often rich with social thought, provided the necessary encouragement for other gifted writers to create musical plays of their own".[63]The two collaborators created an extraordinary collection of some of musical theatre's best loved and most enduring classics, includingCarousel(1945),South Pacific(1949),The King and I(1951) andThe Sound of Music(1959). Some of these musicals treat more serious subject matter than most earlier shows: the villain inOklahoma!is a suspected murderer and psychopath;Carouseldeals with spousal abuse, thievery, suicide and the afterlife;South Pacificexplores miscegenation even more thoroughly thanShow Boat; the hero ofThe King and Idies onstage; and the backdrop ofThe Sound of Musicis theannexation of Austria by Nazi Germany in 1938.
The show's creativity stimulated Rodgers and Hammerstein's contemporaries and ushered in the "Golden Age" of American musical theatre.[68]Americana was displayed on Broadway during the "Golden Age", as the wartime cycle of shows began to arrive. An example of this isOn the Town(1944), written byBetty ComdenandAdolph Green, composed byLeonard Bernsteinand choreographed byJerome Robbins. The story is set during wartime and concerns three sailors who are on a 24-hour shore leave in New York City, during which each falls in love. The show also gives the impression of a country with an uncertain future, as the sailors and their women also have.Irving Berlinused sharpshooterAnnie Oakley's career as a basis for hisAnnie Get Your Gun(1946, 1,147 performances);Burton Lane,E. Y. HarburgandFred Saidycombined political satire with Irish whimsy for their fantasyFinian's Rainbow(1947, 725 performances); and Cole Porter found inspiration inWilliam Shakespeare'sThe Taming of the ShrewforKiss Me, Kate(1948, 1,077 performances). The American musicals overwhelmed the old-fashioned British Coward/Novello-style shows, one of the last big successes of which was Novello'sPerchance to Dream(1945, 1,021 performances).[58]The formula for the Golden Age musicals reflected one or more of four widely held perceptions of the "American dream": That stability and worth derives from a love relationship sanctioned and restricted by Protestant ideals of marriage; that a married couple should make a moral home with children away from the city in a suburb or small town; that the woman's function was as homemaker and mother; and that Americans incorporate an independent and pioneering spirit or that their success is self-made.[70]
The 1950s were crucial to the development of the American musical.[71]Damon Runyon's eclectic characters were at the core ofFrank Loesser's andAbe Burrows'Guys and Dolls, (1950, 1,200 performances); and theGold Rushwas the setting forAlan Jay LernerandFrederick Loewe'sPaint Your Wagon(1951). The relatively brief seven-month run of that show did not discourageLerner and Loewefrom collaborating again, this time onMy Fair Lady(1956), an adaptation ofGeorge Bernard Shaw'sPygmalionstarringRex HarrisonandJulie Andrews, which at 2,717 performances held the long-run record for many years. Popular Hollywood films were made of all of these musicals. Two hits by British creators in this decade wereThe Boy Friend(1954), which ran for 2,078 performances in London and marked Andrews' American debut, andSalad Days(1954), which broke the British long-run record with a run of 2,283 performances.[58][57]
Another record was set byThe Threepenny Opera, which ran for 2,707 performances, becoming the longest-running off-Broadway musical untilThe Fantasticks. The production also broke ground by showing that musicals could be profitable off-Broadway in a small-scale, small orchestra format. This was confirmed in 1959 when a revival ofJerome KernandP. G. Wodehouse'sLeave It to Janeran for more than two years. The 1959–1960off-Broadwayseason included a dozen musicals and revues includingLittle Mary Sunshine,The FantasticksandErnest in Love, a musical adaptation ofOscar Wilde's 1895 hitThe Importance of Being Earnest.[72]
West Side Story(1957) transportedRomeo and Julietto modern day New York City and converted the feuding Montague and Capulet families into opposing ethnic gangs, the Jets and the Sharks. The book was adapted byArthur Laurents, with music byLeonard Bernsteinand lyrics by newcomerStephen Sondheim. It was praised by critics for its innovations in music and choreography[73][74]but was less commercially successful than the same year'sThe Music Man, written and composed byMeredith Willson, which won theTony Award for Best Musicalthat year.[75]West Side Storywould get afilm adaptationin 1961, which proved successful both critically and commercially.[76][77]Laurents and Sondheim teamed up again forGypsy(1959), withJule Styneproviding the music for a story aboutRose Thompson Hovick, the mother of the titular stripperGypsy Rose Lee.
Although directors and choreographers have had a major influence on musical theatre style since at least the 19th century,[78]George Abbott and his collaborators and successors took a central role in integrating movement and dance fully into musical theatre productions in the Golden Age.[79]Abbott introduced ballet as a story-telling device inOn Your Toesin 1936, which was followed byAgnes de Mille's ballet and choreography inOklahoma!.[80]After Abbott collaborated with Jerome Robbins inOn the Townand other shows, Robbins combined the roles of director and choreographer, emphasizing the story-telling power of dance inWest Side Story,A Funny Thing Happened on the Way to the Forum(1962) andFiddler on the Roof(1964).Bob Fossechoreographed for Abbott inThe Pajama Game(1956) andDamn Yankees(1957), injecting playful sexuality into those hits. He was later the director-choreographer forSweet Charity(1968),Pippin(1972) andChicago(1975). Other notable director-choreographers have includedGower Champion,Tommy Tune,Michael Bennett,Gillian LynneandSusan Stroman. Prominent directors have includedHal Prince, who also got his start with Abbott,[79]andTrevor Nunn.[81]
During the Golden Age, automotive companies and other large corporations began to hire Broadway talent to writecorporate musicals, private shows only seen by their employees or customers.[82][83]The 1950s ended withRodgers and Hammerstein's last hit,The Sound of Music, which also became another hit for Mary Martin. It ran for 1,443 performances and shared the Tony Award for Best Musical. Together with its extremely successful1965 film version, it has become one of the most popular musicals in history.
In 1960,The Fantastickswas first produced off-Broadway. This intimate allegorical show would quietly run for over 40 years at the Sullivan Street Theatre inGreenwich Village, becoming by far the longest-running musical in history. Its authors produced other innovative works in the 1960s, such asCelebrationandI Do! I Do!, the first two-character Broadway musical. The 1960s would see a number of blockbusters, likeFiddler on the Roof(1964; 3,242 performances),Hello, Dolly!(1964; 2,844 performances),Funny Girl(1964; 1,348 performances) andMan of La Mancha(1965; 2,328 performances), and some more risqué pieces likeCabaret, before ending with the emergence of therock musical. In Britain,Oliver!(1960) ran for 2,618 performances, but the long-run champion of the decade wasThe Black and White Minstrel Show(1962), which played for 4,344 performances.[58]Two men had considerable impact on musical theatre history beginning in this decade:Stephen SondheimandJerry Herman.
The first project for which Sondheim wrote both music and lyrics wasA Funny Thing Happened on the Way to the Forum(1962, 964 performances), with a book based on the works ofPlautusbyBurt SheveloveandLarry Gelbart, starringZero Mostel. Sondheim moved the musical beyond its concentration on the romantic plots typical of earlier eras; his work tended to be darker, exploring the grittier sides of life both present and past. Other early Sondheim works includeAnyone Can Whistle(1964, which ran only nine performances, despite having starsLee RemickandAngela Lansbury), and the successfulCompany(1970),Follies(1971) andA Little Night Music(1973). Later, Sondheim found inspiration in unlikely sources: the opening of Japan to Western trade forPacific Overtures(1976), a legendary murderous barber seeking revenge in theIndustrial Ageof London forSweeney Todd(1979), the paintings ofGeorges SeuratforSunday in the Park with George(1984), fairy tales forInto the Woods(1987), and a collection of presidential assassins inAssassins(1990).
While some critics have argued that some of Sondheim's musicals lack commercial appeal, others have praised their lyrical sophistication and musical complexity, as well as the interplay of lyrics and music in his shows. Some of Sondheim's notable innovations include a show presented in reverse (Merrily We Roll Along) and the above-mentionedAnyone Can Whistle, in which the first act ends with the cast informing the audience that they are mad.
Jerry Herman played a significant role in American musical theatre, beginning with his first Broadway production,Milk and Honey(1961, 563 performances), about the founding of the state of Israel, and continuing with the blockbuster hitsHello, Dolly!(1964, 2,844 performances),Mame(1966, 1,508 performances), andLa Cage aux Folles(1983, 1,761 performances). Even his less successful shows likeDear World(1969) andMack and Mabel(1974) have had memorable scores (Mack and Mabelwas later reworked into a London hit). Writing both words and music, many of Herman'sshow tuneshave become popular standards, including "Hello, Dolly!", "We Need a Little Christmas", "I Am What I Am", "Mame", "The Best of Times", "Before the Parade Passes By", "Put On Your Sunday Clothes", "It Only Takes a Moment", "Bosom Buddies" and "I Won't Send Roses", recorded by such artists asLouis Armstrong,Eydie Gormé,Barbra Streisand,Petula Clarkand Bernadette Peters. Herman's songbook has been the subject of two popular musical revues,Jerry's Girls(Broadway, 1985) andShowtune(off-Broadway, 2003).
The musical started to diverge from the relatively narrow confines of the 1950s. Rock music would be used in several Broadway musicals, beginning withHair, which featured not only rock music but also nudity and controversial opinions about theVietnam War, race relations and other social issues.[84]
AfterShow BoatandPorgy and Bess, and as the struggle in America and elsewhere for minorities'civil rightsprogressed, Hammerstein,Harold Arlen,Yip Harburgand others were emboldened to write more musicals and operas that aimed to normalize societal toleration of minorities and urged racial harmony. Early Golden Age works that focused on racial tolerance includedFinian's RainbowandSouth Pacific. Towards the end of the Golden Age, several shows tackled Jewish subjects and issues, such asFiddler on the Roof,Milk and Honey,Blitz!and laterRags. The original concept that becameWest Side Storywas set in theLower East Sideduring Easter-Passover celebrations; the rival gangs were to be Jewish andItalianCatholic. The creative team later decided that the Polish (white) vs.Puerto Ricanconflict was fresher.[85]
Tolerance as an important theme in musicals has continued in recent decades. The final expression ofWest Side Storyleft a message of racial tolerance. By the end of the 1960s, musicals became racially integrated, with black and white cast members even covering each other's roles, as they did inHair.[86]Homosexuality has also been explored in musicals, starting withHair, and even more overtly inLa Cage aux Folles,Falsettos,Rent,Hedwig and the Angry Inchand other shows in recent decades.Paradeis a sensitive exploration of bothanti-Semitismand historical American racism, andRagtimesimilarly explores the experience of immigrants and minorities in America.
After the success ofHair,rock musicalsflourished in the 1970s, withJesus Christ Superstar,Godspell,The Rocky Horror Show,EvitaandTwo Gentlemen of Verona. Some of those began as "concept albums" which were then adapted to the stage, most notablyJesus Christ SuperstarandEvita. Others had no dialogue or were otherwise reminiscent of opera, with dramatic, emotional themes; these sometimes started as concept albums and were referred to asrock operas. Shows likeRaisin,Dreamgirls,PurlieandThe Wizbrought a significant African-American influence to Broadway. More varied musical genres and styles were incorporated into musicals both on and especially off-Broadway. At the same time, Stephen Sondheim found success with some of his musicals, as mentioned above.
In 1975, the dance musicalA Chorus Lineemerged from recorded group therapy-style sessionsMichael Bennettconducted with "gypsies" – those who sing and dance in support of the leading players – from the Broadway community. From hundreds of hours of tapes,James Kirkwood Jr.andNick Dantefashioned a book about an audition for a musical, incorporating many real-life stories from the sessions; some who attended the sessions eventually played variations of themselves or each other in the show. With music byMarvin Hamlischand lyrics byEdward Kleban,A Chorus Linefirst opened atJoseph Papp'sPublic Theaterin lower Manhattan. What initially had been planned as a limited engagement eventually moved to theShubert Theatreon Broadway[87]for a run of 6,137 performances, becoming the longest-running production in Broadway history up to that time. The show swept the Tony Awards and won thePulitzer Prize, and its hit song,What I Did for Love, became a standard.[88]
Broadway audiences welcomed musicals that varied from the golden age style and substance.John KanderandFred Ebbexplored the rise ofNazismin Germany inCabaret, and murder and the media inProhibition-eraChicago, which relied on oldvaudevilletechniques.Pippin, byStephen Schwartz, was set in the days ofCharlemagne.Federico Fellini's autobiographical film8½becameMaury Yeston'sNine. At the end of the decade,EvitaandSweeney Toddwere precursors of the darker, big budget musicals of the 1980s that depended on dramatic stories, sweeping scores and spectacular effects. At the same time, old-fashioned values were still embraced in such hits asAnnie,42nd Street,My One and Only, and popular revivals ofNo, No, NanetteandIrene. Although many film versions of musicals were made in the 1970s, few were critical or box office successes, with the notable exceptions ofFiddler on the Roof,CabaretandGrease.[89]
The 1980s saw the influence of European "megamusicals" on Broadway, in the West End and elsewhere. These typically feature a pop-influenced score, large casts and spectacular sets and special effects – a fallingchandelier(inThe Phantom of the Opera); a helicopter landing on stage (inMiss Saigon) – and big budgets. Some were based on novels or other works of literature. The British team of composerAndrew Lloyd Webberand producerCameron Mackintoshstarted the megamusical phenomenon with their 1981 musicalCats, based on the poems ofT. S. Eliot, which overtookA Chorus Lineto become the longest-running Broadway show. Lloyd Webber followed up withStarlight Express(1984), performed on roller skates;The Phantom of the Opera(1986; also with Mackintosh), derived from thenovel of the same name; andSunset Boulevard(1993), from the 1950film of the same name.Phantomwould surpassCatsto become the longest-running show in Broadway history, a record it still holds.[90][91]The French team ofClaude-Michel SchönbergandAlain BoublilwroteLes Misérables, based on thenovel of the same name, whose 1985 London production was produced by Mackintosh and became, and still is, thelongest-running musical in West End and Broadway history. The team produced another hit withMiss Saigon(1989), which was inspired by the Puccini operaMadama Butterfly.[90][91]
The megamusicals' huge budgets redefined expectations for financial success on Broadway and in the West End. In earlier years, it was possible for a show to be considered a hit after a run of several hundred performances, but with multimillion-dollar production costs, a show must run for years simply to turn a profit. Megamusicals were also reproduced in productions around the world, multiplying their profit potential while expanding the global audience for musical theatre.[91]
In the 1990s, a new generation of theatrical composers emerged, includingJason Robert BrownandMichael John LaChiusa, who began with productions off-Broadway. The most conspicuous success of these artists wasJonathan Larson's showRent(1996), a rock musical (based on the operaLa bohème) about a struggling community of artists in Manhattan. While the cost of tickets to Broadway and West End musicals was escalating beyond the budget of many theatregoers,Rentwas marketed to increase the popularity of musicals among a younger audience. It featured a young cast and a heavily rock-influenced score; the musical became a hit. Its young fans, many of them students, calling themselves RENTheads, camped out at theNederlander Theatrein hopes of winning the lottery for $20 front row tickets, and some saw the show dozens of times. Other shows on Broadway followedRent's lead by offering heavily discounted day-of-performance or standing-room tickets, although often the discounts are offered only to students.[92]
The 1990s also saw the influence of large corporations on the production of musicals. The most important has beenDisney Theatrical Productions, which began adapting some ofDisney'sanimated film musicals for the stage, starting withBeauty and the Beast(1994),The Lion King(1997) andAida(2000), the latter two with music byElton John.The Lion Kingis thehighest-grossing musicalin Broadway history.[93]The Who's Tommy(1993), a theatrical adaptation of the rock operaTommy, achieved a healthy run of 899 performances but was criticized for sanitizing the story and "musical theatre-izing" the rock music.[94]
Despite the growing number of large-scale musicals in the 1980s and 1990s, a number of lower-budget, smaller-scale musicals managed to find critical and financial success, such asFalsettoland,Little Shop of Horrors,Bat Boy: The MusicalandBlood Brothers, which ran for 10,013 performances.[95]The topics of these pieces vary widely, and the music ranges from rock to pop, but they often are produced off-Broadway, or for smaller London theatres, and some of these stagings have been regarded as imaginative and innovative.[96]
In the new century, familiarity has been embraced by producers and investors anxious to guarantee that they recoup their considerable investments. Some took (usually modest-budget) chances on new and creative material, such asUrinetown(2001),Avenue Q(2003),The Light in the Piazza(2005),Spring Awakening(2006),In the Heights(2008),Next to Normal(2009),American Idiot(2010) andThe Book of Mormon(2011).Hamilton(2015), transformed "under-dramatized American history" into an unusual hip-hop inflected hit.[97]In 2011, Sondheim argued that of all forms of "contemporary pop music",rapwas "the closest to traditional musical theatre" and was "one pathway to the future."[98]
However, most major-market 21st-century productions have taken a safe route, with revivals of familiar fare, such asFiddler on the Roof,A Chorus Line,South Pacific,Gypsy,Hair,West Side StoryandGrease, or with adaptations of other proven material, such as literature (The Scarlet Pimpernel,WickedandFun Home), hoping that the shows would have a built-in audience as a result. This trend is especially persistent with film adaptations, includingThe Producers,Spamalot,Hairspray,Legally Blonde,The Color Purple,Xanadu,Billy Elliot,Shrek,WaitressandGroundhog Day.[99]Some critics have argued that the reuse of film plots, especially those from Disney (such asMary PoppinsandThe Little Mermaid), equate the Broadway and West End musical to a tourist attraction, rather than a creative outlet.[37]
Today, it is less likely that a sole producer, such asDavid MerrickorCameron Mackintosh, backs a production. Corporate sponsors dominate Broadway, and often alliances are formed to stage musicals, which require an investment of $10 million or more. In 2002, the credits forThoroughly Modern Millielisted ten producers, and among those names were entities composed of several individuals.[100]Typically, off-Broadway and regional theatres tend to produce smaller and therefore less expensive musicals, and development of new musicals has increasingly taken place outside of New York and London or in smaller venues. For example,Spring Awakening,Fun HomeandHamiltonwere developed off-Broadway before being launched on Broadway.
Several musicals returned to the spectacle format that was so successful in the 1980s, recallingextravaganzasthat have been presented at times, throughout theatre history, since the ancient Romans staged mock sea battles. Examples include the musical adaptations ofLord of the Rings(2007),Gone with the Wind(2008) andSpider-Man: Turn Off the Dark(2011). These musicals involved songwriters with little theatrical experience, and the expensive productions generally lost money. Conversely,The Drowsy Chaperone,Avenue Q,The 25th Annual Putnam County Spelling Bee,XanaduandFun Home, among others, have been presented in smaller-scale productions, mostly uninterrupted by an intermission, with short running times, and enjoyed financial success. In 2013,Timemagazine reported that a trend off-Broadway has been "immersive" theatre, citing shows such asNatasha, Pierre & The Great Comet of 1812(2012) andHere Lies Love(2013) in which the staging takes place around and within the audience.[101]The shows set a joint record, each receiving 11 nominations forLucille Lortel Awards,[102]and feature contemporary scores.[103][104]
In 2013,Cyndi Lauperwas the "first female composer to win the [Tony for] Best Score without a male collaborator" for writing the music and lyrics forKinky Boots. In 2015, for the first time, anall-female writing team,Lisa KronandJeanine Tesori, won theTony Award for Best Original Score(andBest Bookfor Kron) forFun Home,[105]although work by male songwriters continues to be produced more often.[106]
Another trend has been to create a minimal plot to fit a collection of songs that have already been hits. Following the earlier success ofBuddy – The Buddy Holly Story, these have includedMovin' Out(2002, based on the tunes ofBilly Joel),Jersey Boys(2006,The Four Seasons),Rock of Ages(2009, featuring classic rock of the 1980s),Thriller – Live(2009,Michael Jackson), and many others. This style is often referred to as the "jukebox musical".[107]Similar but more plot-driven musicals have been built around the canon of a particular pop group includingMamma Mia!(1999, based on the songs ofABBA),Our House(2002, based on the songs ofMadness) andWe Will Rock You(2002, based on the songs ofQueen).
Live-action film musicals were nearly dead in the 1980s and early 1990s, with exceptions ofVictor/Victoria,Little Shop of Horrorsandthe 1996 film ofEvita.[108]In the new century,Baz Luhrmannbegan a revival of the film musical withMoulin Rouge!(2001). This was followed byChicago(2002);Phantom of the Opera(2004);Rent(2005);Dreamgirls(2006);Hairspray,EnchantedandSweeney Todd(all in 2007);Mamma Mia!(2008);Nine(2009);Les MisérablesandPitch Perfect(both in 2012),Into The Woods,The Last Five Years(2014),La La Land(2016),The Greatest Showman(2017),A Star Is BornandMary Poppins Returns(both 2018),Rocketman(2019) andIn the HeightsandSteven Spielberg's version ofWest Side Story(both in 2021), among others.Dr. Seuss'sHow the Grinch Stole Christmas!(2000) andThe Cat in the Hat(2003), turned children's books into live-action film musicals. After the immense success of Disney and other houses with animated film musicals beginning withThe Little Mermaidin 1989 and running throughout the 1990s (including some more adult-themed films, likeSouth Park: Bigger, Longer & Uncut(1999)), fewer animated film musicals were released in the first decade of the 21st century.[108]The genre made a comeback beginning in 2010 withTangled(2010),Rio(2011) andFrozen(2013). In Asia, India continues to produce numerous "Bollywood" film musicals, and Japan produces "Anime" and "Manga" film musicals.
Made for TV musical films were popular in the 1990s, such asGypsy(1993),Cinderella(1997) andAnnie(1999). Several made for TV musicals in the first decade of the 21st century were adaptations of the stage version, such asSouth Pacific(2001),The Music Man(2003) andOnce Upon a Mattress(2005), and a televised version of the stage musicalLegally Blondein 2007. Additionally, several musicals were filmed on stage and broadcast on Public Television, for exampleContactin 2002 andKiss Me, KateandOklahoma!in 2003. The made-for-TV musicalHigh School Musical(2006), and its several sequels, enjoyed particular success and were adapted for stage musicals and other media.
In 2013,NBCbegan a series of live television broadcasts of musicals withThe Sound of Music Live![109]Although the production received mixed reviews, it was a ratings success.[110]Further broadcasts have includedPeter Pan Live!(NBC 2014),The Wiz Live!(NBC 2015),[111]a UK broadcast,The Sound of Music Live(ITV2015)[112]Grease: Live(Fox2016),[113][114]Hairspray Live!(NBC, 2016),A Christmas Story Live!(Fox, 2017),[115]andRent: Live(Fox 2019).[116]
Some television shows have set episodes as a musical. Examples include episodes ofAlly McBeal,Xena: Warrior Princess("The Bitter Suite" and "Lyre, Lyre, Heart's On Fire"),Psych("Psych: The Musical"),Buffy the Vampire Slayer("Once More, with Feeling"),That's So Raven,Daria,Dexter's Laboratory,The Powerpuff Girls,The Flash,Once Upon a Time,Oz,Scrubs(one episode was written by the creators ofAvenue Q),Batman: The Brave and the Bold("Mayhem of the Music Meister") andThat '70s Show(the 100th episode, "That '70s Musical"). Others have included scenes where characters suddenly begin singing and dancing in a musical-theatre style during an episode, such as in several episodes ofThe Simpsons,30 Rock,Hannah Montana,South Park,Bob's BurgersandFamily Guy.[117]Television series that have extensively used the musical format have includedCop Rock,Flight of the Conchords,Glee,SmashandCrazy Ex-Girlfriend.
There have also been musicals made for the internet, includingDr. Horrible's Sing-Along Blog,about a low-rent super-villain played byNeil Patrick Harris. It was written during theWGA writer's strike.[118]Since 2006, reality TV shows have been used to help market musical revivals by holding a talent competition to cast (usually female) leads. Examples of these areHow Do You Solve a Problem like Maria?,Grease: You're the One That I Want!,Any Dream Will Do,Legally Blonde: The Musical – The Search for Elle Woods,I'd Do AnythingandOver the Rainbow.In 2021,Schmigadoon!was a parody of, and homage to, Golden Age musicals of the 1940s and 1950s.[119]
TheCOVID-19 pandemiccaused theclosure of theatres and theatre festivals around the worldin early 2020, including all Broadway[120]and West End theatres.[121]Many performing arts institutions attempted to adapt, or reduce their losses, by offering new (or expanded) digital services. In particular this resulted in theonline streamingof previously recorded performances of many companies,[122][123][124]as well as bespoke crowdsourcing projects.[125][126]For example, TheSydney Theatre Companycommissioned actors to film themselves at home discussing, then performing, a monologue from one of the characters they had previously played on stage.[127]The casts of musicals, such asHamiltonandMamma Mia!united on Zoom calls to entertain individuals and the public.[128][129]Some performances were streamed live, or presented outdoors or in other "socially distanced" ways, sometimes allowing audience members to interact with the cast.[130]Radio theatre festivals were broadcast.[131]Virtual, and even crowd-sourced musicals were created, such asRatatouille the Musical.[132][133]Filmed versions of major musicals, likeHamilton, were released on streaming platforms.[134]Andrew Lloyd Webber released recordings of his musicals on YouTube.[135]
Due to the closures and loss of ticket sales, many theatre companies were placed in financial peril. Some governments offered emergency aid to the arts.[136][137][138]Some musical theatre markets began to reopen in fits and starts by early 2021,[139]with West End theatres postponing their reopening from June to July,[140]and Broadway starting in September.[141]Throughout 2021, however, spikes in the pandemic have caused some closures even after markets reopened.[142][143]
The U.S. and Britain were the most active sources of book musicals from the 19th century through much of the 20th century (although Europe produced various forms of popularlight operaand operetta, for example SpanishZarzuela, during that period and even earlier). However, the light musical stage in other countries has become more active in recent decades.
Musicals from other English-speaking countries (notably Australia and Canada) often do well locally and occasionally even reach Broadway or the West End (e.g.,The Boy from OzandThe Drowsy Chaperone). South Africa has an active musical theatre scene, with revues likeAfrican FootprintandUmojaand book musicals, such asKat and the KingsandSarafina!touring internationally. Locally, musicals likeVere,Love and Green Onions,Over the Rainbow: the all-new all-gay... extravaganzaandBangbroek MountainandIn Briefs – a queer little Musicalhave been produced successfully.
Successful musicals from continental Europe include shows from (among other countries) Germany (ElixierandLudwig II), Austria (Tanz der Vampire,Elisabeth,Mozart!andRebecca), Czech Republic (Dracula), France (Starmania,Notre-Dame de Paris,Les Misérables,Roméo et JulietteandMozart, l'opéra rock) and Spain (Hoy no me puedo levantarandThe Musical Sancho Panza).
Japan has recently seen the growth of an indigenous form of musical theatre, both animated and live action, mostly based onAnimeandManga, such asKiki's Delivery ServiceandTenimyu. The popularSailor Moonmetaseries has had twenty-nineSailor Moon musicals, spanning thirteen years. Beginning in 1914, a series of popularrevueshave been performed by the all-femaleTakarazuka Revue, which currently fields five performing troupes. Elsewhere in Asia, the IndianBollywoodmusical, mostly in the form of motion pictures, is tremendously successful.[144]
Beginning with a 2002 tour ofLes Misérables, various Western musicals have been imported to mainland China and staged in English.[145]Attempts at localizing Western productions in China began in 2008 whenFamewas produced in Mandarin with a full Chinese cast at theCentral Academy of Dramain Beijing.[146]Since then, other western productions have been staged in China in Mandarin with a Chinese cast. The first Chinese production in the style of Western musical theatre wasThe Gold Sandin 2005.[145]In addition, Li Dun, a well-known Chinese producer, producedButterflies, based on a classic Chinese love tragedy, in 2007 as well asLove U Teresain 2011.[145]
Musicals are often presented byamateurand school groups in churches, schools and other performance spaces.[147][148]Although amateur theatre has existed for centuries, even in the New World,[149]François Cellierand Cunningham Bridgeman wrote, in 1914, that prior to the late 19th century, amateur actors were treated with contempt by professionals. After the formation of amateurGilbert and Sullivancompanies licensed to perform theSavoy operas, professionals recognized that the amateur societies "support the culture of music and the drama. They are now accepted as useful training schools for the legitimate stage, and from the volunteer ranks have sprung many present-day favourites."[150]TheNational Operatic and Dramatic Associationwas founded in the UK in 1899. It reported, in 1914, that nearly 200 amateur dramatic societies were producing Gilbert and Sullivan works in Britain that year.[150]Similarly, more than 100 amateur theatres were founded in the US in the early 20th century. This number has grown to an estimated 18,000 in the US.[149]The Educational Theater Association in the US has nearly 5,000 member schools.[151]
The Broadway Leagueannounced that in the 2007–08 season, 12.27 million tickets were purchased for Broadway shows for a gross sale amount of almost a billion dollars.[152]The League further reported that during the 2006–07 season, approximately 65% of Broadway tickets were purchased by tourists, and that foreign tourists were 16% of attendees.[153]The Society of London Theatre reported that 2007 set a record for attendance in London. Total attendees in the major commercial and grant-aided theatres in Central London were 13.6 million, and total ticket revenues were £469.7 million.[154]The international musicals scene has been increasingly active in recent decades. Nevertheless, Stephen Sondheim commented in the year 2000:
You have two kinds of shows on Broadway – revivals and the same kind of musicals over and over again, all spectacles. You get your tickets forThe Lion Kinga year in advance, and essentially a family ... pass on to their children the idea that that's what the theater is – a spectacular musical you see once a year, a stage version of a movie. It has nothing to do with theater at all. It has to do with seeing what is familiar. ... I don't think the theatre will dieper se, but it's never going to be what it was. ... It's a tourist attraction."[155]
However, noting the success in recent decades of original material, and creative re-imaginings of film, plays and literature, theatre historianJohn Kenrickcountered:
Is the Musical dead? ... Absolutely not! Changing? Always! The musical has been changing ever sinceOffenbachdid his first rewrite in the 1850s. And change is the clearest sign that the musical is still a living, growing genre. Will we ever return to the so-called 'golden age', with musicals at the center of popular culture? Probably not. Public taste has undergone fundamental changes, and the commercial arts can only flow where the paying public allows.[37]
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Michael R. Jackson(born 1981)[1]is an American playwright, composer, and lyricist, best known for his musicalA Strange Loop, which won the 2020Pulitzer Prize for Dramaand the 2022Tony Award for Best Musical. He is originally from Detroit.
Jackson interned for a time at ABC, working on daytime programming, specificallyAll My Children. Jackson has described himself as "a huge soap person," and wanted to originally write for soaps.[2]
Jackson wrote the book and lyrics forOnly Childrenwith composer Rachel Peters, which was presented atNYU's Frederick Loewe Theatre.[3]
He also wrote lyrics and co-wrote the book, with Anna K. Jacobs, for the musical adaptation of the 2007 indie filmTeeth.[4]He sang "Lonesome of the Road" on a tribute album forElizabeth Swados.[5][6]
In 2019, his song cycle,The Kids on the Lawn, was published inThe New York Times Magazine'sculture issue. The issue, organized around the theme "America 2024", imagines what America will be like five years into the future.[7]
Jackson's musical,A Strange Loop, received its world premiere atPlaywrights Horizonsin New York City in 2019.[8]After a six-week run at theWoolly Mammoth Theatre Companyin Washington, D.C., in 2021,A Strange Loopopened on Broadway at theLyceum Theatrein April 2022.[9][10]
His musicalWhite Girl in Dangerbegan previews at theTony Kiser Theateron March 15, 2023, and opened on April 10, 2023. The musical explores the intersections of race, class, and identity in daytime soap operas.[11]
In 2017, Jackson received a Jonathan Larson Grant from the American Theatre Wing[12]and was one of 11 winners of the 2017 Lincoln Center Emerging Artist Award.[13]He was also a Sundance Theatre Institute Composer Fellow and a 2016–2017 Dramatist Guild Fellow.[14]
Jackson was named one of the "Black Male Writers for our Time" byThe New York Timesin 2018.[15]In 2019, he received aWhiting Awardfor drama and a Helen Merrill Award for Playwriting.[16][17]In 2020, Jackson was awarded thePulitzer Prize for DramaforA Strange Loop, becoming the first black musical theatre writer to win the award.[18]He was also the winner of theLambda Literary Award for Drama[19]and aFred EbbAward for aspiring musical theatre songwriters.[20]Additionally, Jackson received twoDrama Desk Awards, twoObie Awards, twoOuter Critics Circle AwardHonors, and anAntonyo Awardfor Best Book forA Strange Loop.[21]
In June 2020, in honor of the 50th anniversary of the first LGBTQPride parade,Queertynamed him among the fifty heroes "leading the nation toward equality, acceptance, and dignity for all people".[22][23]In 2022, Jackson was featured in the book50 Key Figures in Queer US Theatre, with a profile written by theatre scholar Aviva Helena Neff.[24]
In March 2021, Jackson was awarded theWindham–Campbell Literature Prizefor drama.[25]
At the75th Tony Awards, Jackson's musicalA Strange Loopwas nominated for 11 awards, winningBest MusicalandBest Book of a Musical.[26]
Jackson studied atCass Technical High Schooland attended New York University.[27]He is openly gay.[28]
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Absurdismis thephilosophicaltheory that the universe isirrationaland meaningless. It states that trying to find meaning leads people into conflict with a seemingly meaningless world. This conflict can be betweenrationalman and an irrational universe, betweenintentionand outcome, or betweensubjectiveassessment and objective worth, but the precise definition of the term is disputed. Absurdism claims that, due to one or more of these conflicts, existenceas a wholeisabsurd. It differs in this regard from the less global thesis that someparticularsituations, persons, or phases in life are absurd.
Various components of the absurd are discussed in the academic literature, and different theorists frequently concentrate their definition and research on different components. On the practical level, the conflict underlying the absurd is characterized by the individual's struggle to find meaning in a meaningless world. The theoretical component, on the other hand, emphasizes more theepistemicinability of reason to penetrate and understandreality. Traditionally, the conflict is characterized as a collision between an internal component of human nature, and an external component of the universe. However, some later theorists have suggested that both components may be internal: the capacity to see through the arbitrariness of any ultimate purpose, on the one hand, and the incapacity to stop caring about such purposes, on the other hand. Certain accounts also involve ametacognitivecomponent by holding that anawarenessof the conflict is necessary for the absurd to arise.
Some arguments in favor of absurdism focus on the human insignificance in the universe, on the role ofdeath, or on the implausibility or irrationality of positing an ultimate purpose. Objections to absurdism often contend that life is in fact meaningful or point out certain problematic consequences or inconsistencies of absurdism. Defenders of absurdism often complain that it does not receive the attention of professional philosophers it merits in virtue of the topic's importance and its potential psychological impact on the affected individuals in the form ofexistential crises. Various possible responses to deal with absurdism and its impact have been suggested. The three responses discussed in the traditional absurdist literature aresuicide,religious beliefin a higher purpose, and rebellion against the absurd. Of these, rebellion is usually presented as the recommended response since, unlike the other two responses, it does not escape the absurd and instead recognizes it for what it is. Later theorists have suggested additional responses, like usingironyto take life less seriously or remaining ignorant of the responsible conflict. Some absurdists argue that whether and how one responds is insignificant. This is based on the idea that if nothing really matters then the human response toward this fact does not matter either.
The term "absurdism" is most closely associated with the philosophy ofAlbert Camus. However, important precursors and discussions of the absurd are also found in the works ofSøren Kierkegaard. Absurdism is intimately related to various other concepts and theories. Its basic outlook is inspired byexistentialistphilosophy. However, existentialism includes additional theoretical commitments and often takes a more optimistic attitude toward the possibility of finding or creating meaning in one's life. Absurdism andnihilismshare the belief that life is meaningless, but absurdists do not treat this as an isolated fact and are instead interested in the conflict between the humandesirefor meaning and the world's lack thereof. Being confronted with this conflict may trigger an existential crisis, in which unpleasantexperienceslikeanxietyordepressionmay push the affected to find a response for dealing with the conflict. Recognizing the absence of objective meaning, however, does not preclude the conscious thinker from finding subjective meaning.
Absurdism is thephilosophicalthesis that life, or the world in general, is absurd. There is wide agreement that the term "absurd" implies a lack ofmeaningor purpose but there is also significant dispute concerning its exact definition and various versions have been suggested.[1][2][3][4][5]The choice of one's definition has important implications for whether the thesis of absurdism is correct and for the arguments cited for and against it: it may be true on one definition and false on another.[6]
In a general sense, the absurd is that which lacks a sense, often because it involves some form ofcontradiction. The absurd is paradoxical in the sense that it cannot be grasped byreason.[7][8][9]But in the context of absurdism, the term is usually used in a more specific sense. According to most definitions, it involves a conflict, discrepancy, or collision between two things. Opinions differ on what these two things are.[1][2][3][4]For example, it is traditionally identified as the confrontation ofrationalman with an irrational world or as the attempt to grasp something based on reasons even though it is beyond the limits of rationality.[10][11]Similar definitions see the discrepancy betweenintentionand outcome, between aspiration andreality, or between subjective assessment and objective worth as the source of absurdity.[1][3]Other definitions locate both conflicting sides within man: the ability to apprehend the arbitrariness of final ends and the inability to let go of commitments to them.[4]In regard to the conflict, absurdism differs fromnihilismsince it is not just the thesis that nothing matters. Instead, it includes the component that things seem to matter to us nonetheless and that this impression cannot be shaken off. This difference is expressed in the relational aspect of the absurd in that it constitutes a conflict between two sides.[4][1][2]
Various components of the absurd have been suggested and different researchers often focus their definition and inquiry on one of these components. Some accounts emphasize the practical components concerned with the individual seeking meaning while others stress the theoretical components about being unable toknowthe world or to rationally grasp it. A different disagreement concerns whether the conflict exists only internal to the individual or is between the individual's expectations and theexternal world. Some theorists also include themetacognitivecomponent that the absurd entails that the individual is aware of this conflict.[2][3][12][4]
An important aspect of absurdism is that the absurd is not limited to particular situations but encompasses life as a whole.[2][1][13]There is a general agreement that people are often confronted with absurd situations in everyday life.[7]They often arise when there is a serious mismatch between one's intentions and reality.[2]For example, a person struggling to break down a heavy front door is absurd if the house they are trying to break into lacks a back wall and could easily be entered on this route.[1]But the philosophical thesis of absurdism is much more wide-reaching since it is not restricted to individual situations, persons, or phases in life. Instead, it asserts that life, or the world as a whole, is absurd. The claim that the absurd has such a global extension is controversial, in contrast to the weaker claim that some situations are absurd.[2][1][13]
The perspective of absurdism usually comes into view when the agent takes a step back from their individual everyday engagements with the world to assess their importance from a bigger context.[4][2][14]Such an assessment can result in the insight that the day-to-day engagements matter a lot to us despite the fact that they lack real meaning when evaluated from a wider perspective. This assessment reveals the conflict between the significance seen from the internal perspective and the arbitrariness revealed through the external perspective.[4]The absurd becomes a problem since there is a strongdesirefor meaning and purpose even though they seem to be absent.[7]In this sense, the conflict responsible for the absurd often either constitutes or is accompanied by anexistential crisis.[15][14]
An important component of the absurd on the practical level concerns the seriousness people bring toward life. This seriousness is reflected in many different attitudes and areas, for example, concerning fame,pleasure,justice, knowledge, or survival, both in regard to ourselves as well as in regard to others.[2][8][14]But there seems to be a discrepancy between how seriously we take our lives and the lives of others on the one hand, and how arbitrary they and the world at large seem to be on the other hand. This can be understood in terms ofimportanceand caring: it is absurd that people continue to care about these matters even though they seem to lack importance on an objective level.[16][17]The collision between these two sides can be defined as the absurd. This is perhaps best exemplified when the agent is seriously engaged in choosing between arbitrary options, none of which truly matters.[2][3]
Some theorists characterize theethicalsides of absurdism and nihilism in the same way as the view that it does not matter how we act or that "everything is permitted."[8]On this view, an important aspect of the absurd is that whatever higher end or purpose we choose to pursue, it can also be put into doubt since, in the last step, it always lacks a higher-order justification.[2][1]But usually, a distinction between absurdism and nihilism is made since absurdism involves the additional component that there is a conflict between man's desire for meaning and the absence of meaning.[18][14]
On a more theoretical view, absurdism is thebeliefthat the world is, at its core, indifferent and impenetrable toward human attempts to uncover its deeper reason or that it cannot be known.[12][10]According to this theoretical component, it involves theepistemologicalproblem of the human limitations of knowing the world.[12]This includes the thesis that the world is in critical ways ungraspable to humans, both in relation to what to believe and how to act.[12][10]This is reflected in the chaos and irrationality of the universe, which acts according to its own laws in a manner indifferent to human concerns and aspirations. It is closely related to the idea that the world remains silent when we ask why things are the way they are. This silence arises from the impression that, on the most fundamental level, all things exist without a reason: they are simply there.[12][19][20]An important aspect of these limitations to knowing the world is that they are essential tohuman cognition, i.e. they are not due to following false principles or accidental weaknesses but are inherent in the human cognitive faculties themselves.[12]
Some theorists also link this problem to thecircularity of human reason, which is very skilled at producing chains of justification linking one thing to another while trying and failing to do the same for the chain of justification as a whole when taking a reflective step backward.[2][14]This implies that human reason is not just too limited to grasp life as a whole but that, if one seriously tried to do so anyway, its ungrounded circularity might collapse and lead to madness.[2]
An important disagreement within the academic literature about the nature of absurdism and the absurd focuses specifically on whether the components responsible for the conflict are internal or external.[1][2][3][4]According to the traditional position, the absurd has both internal and external components: it is due to the discrepancy between man's internal desire to lead ameaningful lifeand the external meaninglessness of the world. In this view, humans have, among their desires, some transcendent aspirations that seek a higher form of meaning in life. The absurd arises since these aspirations are ignored by the world, which is indifferent to our "need for validation of the importance of our concerns."[1][3]This implies that the absurd "is not in man ... nor in the world, but in their presence together. " This position has been rejected by some later theorists, who hold that the absurd is purely internal because it "derives not from a collision between our expectations and the world, but from a collision within ourselves".[1][2][4][6]
The distinction is important since, on the latter view, the absurd is built into human nature and would prevail no matter what the world was like. So, it is not just that absurdism is true in the actual world. Instead, anypossible world, even one that was designed by a divine god and guided by them according to their higher purpose, would still be equally absurd to man. In this sense, absurdity is the product of the power of ourconsciousnessto take a step back from whatever it is considering and reflect on the reason of its object. When this process is applied to the world as a whole including God, it is bound to fail its search for a reason or an explanation, no matter what the world is like.[1][2][14]In this sense, absurdity arises from the conflict between features of ourselves: "our capacity to recognize the arbitrariness of our ultimate concerns and our simultaneous incapacity to relinquish our commitment to them".[4]This view has the side-effect that the absurd depends on the fact that the affected person recognizes it. For example, people who fail to apprehend the arbitrariness or the conflict would not be affected.[1][2][14]
According to some researchers, a central aspect of the absurd is that the agent isawareof the existence of the corresponding conflict. This means that the person is conscious both of the seriousness they invest and of how it seems misplaced in an arbitrary world.[2][14]It also implies that other entities that lack this form of consciousness, like non-organic matter or lower life forms, are not absurd and are not faced with this particular problem.[2]Some theorists also emphasize that the conflict remains despite the individual's awareness of it, i.e. that the individual continues to care about their everyday concerns despite their impression that, on the large scale, these concerns are meaningless.[4]Defenders of themetacognitivecomponent have argued that it manages to explain why absurdity is primarily ascribed to human aspirations but not to lower animals: because they lack this metacognitive awareness. However, other researchers reject the metacognitive requirement based on the fact that it would severely limit the scope of the absurd to only those possibly few individuals who clearly recognize the contradiction while sparing the rest. Thus, opponents have argued that not recognizing the conflict is just as absurd as consciously living through it.[1][2][14]
Various popular arguments are often cited in favor of absurdism. Some focus on the future by pointing out that nothing we do today will matter in a million years.[2][14]A similar line of argument points to the fact that our lives are insignificant because of how small they are in relation to the universe as a whole, both concerning their spatial and their temporal dimensions. The thesis of absurdism is also sometimes based on the problem ofdeath, i.e. that there is no final end for us to pursue since we are all going to die.[2][20]In this sense, death is said to destroy all our hard-earned achievements like career, wealth, or knowledge. This argument is mitigated to some extent by the fact that we may have positive or negative effects on the lives of other people as well. But this does not fully solve the issue since the same problem, i.e. the lack of an ultimate end, applies to their lives as well.[2]Thomas Nagelhas objected to these lines of argument based on the claim that they are circular: they assume rather than establish that life is absurd. For example, the claim that our actions today will not matter in a million years does not directly imply that they do not matter today. And similarly, the fact that a process does not reach a meaningful ultimate goal does not entail that the process as a whole is worthless since some parts of the process may contain their justification without depending on a justification external to them.[2][14]
Another argument proceeds indirectly by pointing out how various great thinkers have obvious irrational elements in their systems of thought. These purported mistakes of reason are then taken as signs of absurdism that were meant to hide or avoid it.[12][21]From this perspective, the tendency to posit the existence of a benevolent God may be seen as a form ofdefense mechanismorwishful thinkingto avoid an unsettling and inconvenient truth.[12]This is closely related to the idea that humans have an inborn desire for meaning and purpose, which is dwarfed by a meaningless and indifferent universe.[22][23][24]For example,René Descartesaims to build a philosophical system based on the absolute certainty of the "I think, therefore I am" just to introduce without a proper justification the existence of a benevolent and non-deceiving God in a later step in order to ensure that we can know about the external world.[12][25]A similar problematic step is taken byJohn Locke, who accepts the existence of a God beyondsensory experience, despite his strictempiricism, which demands that all knowledge be based on sensory experience.[12][26]
Other theorists argue in favor of absurdism based on the claim that meaning isrelational. In this sense, for something to be meaningful, it has to stand in relation to something else that is meaningful.[4][21]For example, a word is meaningful because of its relation to a language or someone's life could be meaningful because this person dedicates their efforts to a higher meaningful project, like serving God or fighting poverty. An important consequence of this characterization of meaning is that it threatens to lead to aninfinite regress:[4][21]at each step, something is meaningful because something else is meaningful, which in its turn has meaning only because it is related to yet another meaningful thing, and so on.[27][28]This infinite chain and the corresponding absurdity could be avoided if some things had intrinsic or ultimate meaning, i.e. if their meaning did not depend on the meaning of something else.[4][21]For example, if things on the large scale, like God or fighting poverty, had meaning, then our everyday engagements could be meaningful by standing in the right relation to them. However, if these wider contexts themselves lack meaning then they are unable to act as sources of meaning for other things. This would lead to the absurd when understood as the conflict between the impression that our everyday engagements are meaningful even though they lack meaning because they do not stand in a relation to something else that is meaningful.[4]
Another argument for absurdism is based on the attempt of assessing standards of what matters and why it matters. It has been argued that the only way to answer such a question is in reference to these standards themselves. This means that, in the end, it depends only on us, that "what seems to us important or serious or valuable would not seem so if we were differently constituted". The circularity and groundlessness of these standards themselves are then used to argue for absurdism.[2][14]
The most common criticism of absurdism is to argue that life in fact has meaning.Supernaturalistarguments to this effect are based on the claim that God exists and acts as the source of meaning. Naturalist arguments, on the other hand, contend that various sources of meaning can be found in the natural world without recourse to a supernatural realm. Some of them hold that meaning is subjective. On this view, whether a given thing is meaningful varies from person to person based on their subjective attitude toward this thing. Others find meaning in external values, for example, inmorality, knowledge, orbeauty. All these different positions have in common that they affirm the existence of meaning, in contrast to absurdism.[29][30][21]
Another criticism of absurdism focuses on its negative attitude toward moral values. In the absurdist literature, the moral dimension is sometimes outright denied, for example, by holding that value judgments are to be discarded or that the rejection of God implies the rejection of moral values.[3]On this view, absurdism brings with it a highly controversial form ofmoral nihilism. This means that there is a lack, not just of a higher purpose in life, but also of moral values. These two sides can be linked by the idea that without a higher purpose, nothing is worth pursuing that could give one's life meaning. This worthlessness seems to apply to morally relevant actions equally as to other issues.[3][8]In this sense, "[b]elief in the meaning of life always implies a scale of values" while "[b]elief in the absurd ... teaches the contrary".[31]Various objections to such a position have been presented, for example, that it violatescommon senseor that it leads to numerous radical consequences, like that no one is ever guilty of any blameworthy behavior or that there are no ethical rules.[3][32]
But this negative attitude toward moral values is not always consistently maintained by absurdists and some of the suggested responses on how to deal with the absurd seem to explicitly defend the existence of moral values.[3][20][33]Due to this ambiguity, other critics of absurdism have objected to it based on its inconsistency.[3]The moral values defended by absurdists often overlap with the ethical outlook ofexistentialismand include traits likesincerity,authenticity, andcourageasvirtues.[34][35]In this sense, absurdists often argue that it matters how the agent faces the absurdity of their situation and that the response should exemplify these virtues. This aspect is particularly prominent in the idea that the agent should rebel against the absurd and live their life authentically as a form of passionate revolt.[3][12][10]
Some see the latter position as inconsistent with the idea that there is no meaning in life: if nothing matters then it should also not matter how we respond to this fact.[3][2][1][4]So absurdists seem to be committed both to the claim that moral values exist and that they do not exist. Defenders of absurdism have tried to resist this line of argument by contending that, in contrast to other responses, it remains true to the basic insight of absurdism and the "logic of the absurd" by acknowledging the existence of the absurd instead of denying it.[3][36]But this defense is not always accepted. One of its shortcomings seems to be that it commits theis-ought fallacy: absurdism presents itself as a descriptive claim about the existence and nature of the absurd but then goes on to posit various normative claims.[3][37]Another defense of absurdism consists in weakening the claims about how one should respond to the absurd and which virtues such a response should exemplify. On this view, absurdism may be understood as a form ofself-helpthat merely provides prudential advice. Such prudential advice may be helpful to certain people without pretending to have the status of universally valid moral values or categorical normative judgments. So the value of the prudential advice may merely be relative to the interests of some people but not valuable in a more general sense. This way, absurdists have tried to resolve the apparent inconsistency in their position.[3]
According to absurdism, life in general is absurd: the absurd is not just limited to a few specific cases. Nonetheless, some cases are more paradigmatic examples than others.The Myth of Sisyphusis often treated as a key example of the absurd.[10][3]In it,ZeuspunishesKing Sisyphusby compelling him to roll a massive boulder up a hill. Whenever the boulder reaches the top, it rolls down again, thereby forcing Sisyphus to repeat the same task all over again throughout eternity. This story may be seen as an absurdistparablefor the hopelessness and futility of human life in general: just like Sisyphus, humans in general are condemned to toil day in and day out in the attempt to fulfill pointless tasks, which will be replaced by new pointless tasks once they are completed. It has been argued that a central aspect of Sisyphus' situation is not just the futility of his labor but also his awareness of the futility.[10][38][3]
Another example of the absurdist aspect of the human condition is given inFranz Kafka'sThe Trial.[39][40]In it, the protagonist Josef K. is arrested and prosecuted by an inaccessible authority even though he is convinced that he has done nothing wrong. Throughout the story, he desperately tries to discover what crimes he is accused of and how to defend himself. But in the end, he lets go of his futile attempts and submits to his execution without ever finding out what he was accused of. The absurd nature of the world is exemplified by the mysterious and impenetrable functioning of the judicial system, which seems indifferent to Josef K. and resists all of his attempts of making sense of it.[41][39][40]
Philosophers of absurdism often complain that the topic of the absurd does not receive the attention of professional philosophers it merits, especially when compared to other perennial philosophical areas of inquiry. It has been argued, for example, that this can be seen in the tendency of various philosophers throughout the ages to include the epistemically dubitable existence of God in their philosophical systems as a source of ultimate explanation of the mysteries of existence. In that regard, this tendency may be seen as a form of defense mechanism or wishful thinking constituting a side-effect of the unacknowledged and ignored importance of the absurd.[12][21]While some discussions of absurdism happen explicitly in the philosophical literature, it is often presented in a less explicit manner in the form of novels or plays. These presentations usually happen by telling stories that exemplify some of the key aspects of absurdism even though they may not explicitly discuss the topic.[10][3]
It has been argued that acknowledging the existence of the absurd has important consequences for epistemology, especially in relation to philosophy but also when applied more widely to other fields.[12][10]The reason for this is that acknowledging the absurd includes becoming aware of human cognitive limitations and may lead to a form of epistemic humbleness.[12]
The impression that life is absurd may in some cases have serious psychological consequences like triggering an existential crisis. In this regard, an awareness both of absurdism itself and the possible responses to it can be central to avoiding or resolving such consequences.[3][15][14]
... in spite of or in defiance of the whole of existence he wills to be himself with it, to take it along, almost defying his torment. For to hope in the possibility of help, not to speak of help by virtue of the absurd, that for God all things are possible—no, that he will not do. And as for seeking help from any other—no, that he will not do for all the world; rather than seek help he would prefer to be himself—with all the tortures of hell, if so it must be.
Most researchers argue that the basic conflict posed by the absurd cannot be truly resolved. This means that any attempt to do so is bound to fail even though their protagonists may not be aware of their failure. On this view, there are still several possible responses, some better than others, but none able to solve the fundamental conflict. Traditional absurdism, as exemplified byAlbert Camus, holds that there are three possible responses to absurdism:suicide,religious belief, or revolting against the absurd.[10][3]Later researchers have suggested more ways of responding to absurdism.[2][4][14]
A very blunt and simple response, though quite radical, is to commit suicide.[13]According to Camus, for example, the problem of suicide is the only "really serious philosophical problem". It consists in seeking an answer to the question "Should I kill myself?".[20]This response is motivated by the insight that, no matter how hard the agent tries, they may never reach their goal of leading a meaningful life, which can then justify the rejection of continuing to live at all.[3]Most researchers acknowledge that this is one form of response to the absurd but reject it due to its radical and irreversible nature and argue instead for a different approach.[13][20]
One such alternative response to the apparent absurdity of life is to assume that there is some higher ultimate purpose in which the individual may participate, like service to society,progressof history, or God's glory.[2][3][13]While the individual may only play a small part in the realization of this overarching purpose, it may still act as a source of meaning. This way, the individual may find meaning and thereby escape the absurd. One serious issue with this approach is that the problem of absurdity applies to this alleged higher purpose as well. So just like the aims of a single individual life can be put into doubt, this applies equally to a larger purpose shared by many.[4][21]And if this purpose is itself absurd, it fails to act as a source of meaning for the individual participating in it. Camus identifies this response as a form of suicide as well, pertaining not to the physical but to the philosophical level. It is a philosophical suicide in the sense that the individual just assumes that the chosen higher purpose is meaningful and thereby fails to reflect on its absurdity.[2][3]
Traditional absurdists usually reject both physical and philosophical suicide as the recommended response to the absurd, usually with the argument that both these responses constitute some form of escape that fails to face the absurd for what it is. Despite the gravity and inevitability of the absurd, they recommend that we should face it directly, i.e. not escape from it by retreating into the illusion of false hope or by ending one's life.[12][10][1]In this sense, accepting the reality of the absurd means rejecting any hopes for a happyafterlifefree of those contradictions.[10][2]Instead, the individual should acknowledge the absurd and engage in a rebellion against it.[12][10][1]Such a revolt usually exemplifies certain virtues closely related toexistentialism, like the affirmation of one'sfreedomin the face of adversity as well as acceptingresponsibilityand defining one's ownessence.[12][3]An important aspect of this lifestyle is that life is lived passionately and intensely by inviting and seeking newexperiences. Such a lifestyle might be exemplified by anactor, a conqueror, or aseduction artistwho is constantly on the lookout for new roles, conquests, or attractive people despite their awareness of the absurdity of these enterprises.[10][43]Another aspect lies increativity, i.e. that the agent sees themselves as and acts as the creator of their own works and paths in life. This constitutes a form of rebellion in the sense that the agent remains aware of the absurdity of the world and their part in it but keeps on opposing it instead of resigning and admitting defeat.[10]But this response does not solve the problem of the absurd at its core: even a life dedicated to the rebellion against the absurd is itself still absurd.[2][1]Defenders of the rebellious response to absurdism have pointed out that, despite its possible shortcomings, it has one important advantage over many of its alternatives: it manages to accept the absurd for what it is without denying it by rejecting that it exists or by stopping one's own existence. Some even hold that it is the only philosophically coherent response to the absurd.[3]
While these three responses are the most prominent ones in the traditional absurdist literature, various other responses have also been suggested. Instead of rebellion, for example, absurdism may also lead to a form ofirony. This irony is not sufficient to escape the absurdity of life altogether, but it may mitigate it to some extent by distancing oneself to some degree from the seriousness of life.[2][1][4][14]According toThomas Nagel, there may be, at least theoretically, two responses to actually resolving the problem of the absurd. This is based on the idea that the absurd arises from the consciousness of a conflict between two aspects of human life: that humans care about various things and that the world seems arbitrary and does not merit this concern.[4][2][14]The absurd would not arise if either of the conflicting elements would cease to exist, i.e. if the individual would stop caring about things, as someEastern religionsseem to suggest, or if one could find something that possesses a non-arbitrary meaning that merits the concern. For theorists who give importance to theconsciousnessof this conflict for the absurd, a further option presents itself: to remain ignorant of it to the extent that this is possible.[4][2][14]
Other theorists hold that a proper response to the absurd may neither be possible nor necessary, that it just remains one of the basic aspects of life no matter how it is confronted. This lack of response may be justified through the thesis of absurdism itself: if nothing really matters on the grand scale, then this applies equally to human responses toward this fact. From this perspective, the passionate rebellion against an apparently trivial or unimportant state of affairs seems less like a heroic quest and more like afool's errand.[2][1][4]Jeffrey Gordon has objected to this criticism based on the claim that there is a difference between absurdity and lack of importance. So even if life as a whole is absurd, some facts about life may still be more important than others and the fact that life as a whole is absurd would be a good candidate for the more important facts.[1]
Absurdism has its origins in the work of the 19th-centuryDanishphilosopherSøren Kierkegaard, who chose to confront the crisis that humans face with the Absurd by developing his ownexistentialist philosophy.[44]Absurdism as a belief system was born of the European existentialist movement that ensued, specifically when Camus rejected certain aspects of that philosophical line of thought[45]and published his essayThe Myth of Sisyphus. The aftermath ofWorld War IIprovided the social environment that stimulated absurdist views and allowed for their popular development, especially in the devastated country ofFrance.FoucaultviewedShakespearean theateras a precursor of absurdism.[46]
An idea very close to the concept of the absurd is due toImmanuel Kant, who distinguishes betweenphenomenaandnoumena.[12]This distinction refers to the gap between how things appear to us and what they are like in themselves. For example, according to Kant, space and times are dimensions belonging to the realm of phenomena since this is how sensory impressions are organized by themind, but may not be found on the level of noumena.[47][48]The concept of the absurd corresponds to the thesis that there is such a gap and human limitations may limit the mind from ever truly grasping reality, i.e. that reality in this sense remains absurd to the mind.[12]
A century beforeCamus, the 19th-century Danish philosopherSøren Kierkegaardwrote extensively about the absurdity of the world. In his journals, Kierkegaard writes about the absurd:
What is the Absurd? It is, as may quite easily be seen, that I, a rational being, must act in a case where my reason, my powers of reflection, tell me: you can just as well do the one thing as the other, that is to say where my reason and reflection say: you cannot act and yet here is where I have to act... The Absurd, or to act by virtue of the absurd, is to act upon faith ... I must act, but reflection has closed the road so I take one of the possibilities and say: This is what I do, I cannot do otherwise because I am brought to a standstill by my powers of reflection.[50]
Here is another example of the Absurd from his writings:
What, then, is the absurd? The absurd is that the eternal truth has come into existence in time, that God has come into existence, has been born, has grown up. etc., has come into existence exactly as an individual human being, indistinguishable from any other human being, in as much as all immediate recognizability is pre-Socratic paganism and from the Jewish point of view is idolatry.
How can this absurdity be held or believed? Kierkegaard says:
I gladly undertake, by way of brief repetition, to emphasize what other pseudonyms have emphasized. The absurd is not the absurd or absurdities without any distinction (wherefore Johannes de Silentio: "How many of our age understand what the absurd is?"). The absurd is a category, and the most developed thought is required to define the Christian absurd accurately and with conceptual correctness. The absurd is a category, the negative criterion, of the divine or of the relationship to the divine. When the believer has faith, the absurd is not the absurd—faith transforms it, but in every weak moment it is again more or less absurd to him. The passion of faith is the only thing which masters the absurd—if not, then faith is not faith in the strictest sense, but a kind of knowledge. The absurd terminates negatively before the sphere of faith, which is a sphere by itself. To a third person the believer relates himself by virtue of the absurd; so must a third person judge, for a third person does not have the passion of faith. Johannes de Silentio has never claimed to be a believer; just the opposite, he has explained that he is not a believer—in order to illuminate faith negatively.
Kierkegaard provides an example inFear and Trembling(1843), which was published under the pseudonymJohannes de Silentio. In the story ofAbrahamin theBook of Genesis, Abraham is told byGodtokill his sonIsaac. Just as Abraham is about to kill Isaac, an angel stops Abraham from doing so. Kierkegaard believes that through virtue of the absurd, Abraham, defying all reason and ethical duties ("you cannot act"), got back his son and reaffirmed his faith ("where I have to act").[52]
Another instance of absurdist themes in Kierkegaard's work appears inThe Sickness Unto Death, which Kierkegaard signed with pseudonymAnti-Climacus. Exploring the forms of despair, Kierkegaard examines the type of despair known as defiance.[53]In the opening quotation reproduced at the beginning of the article, Kierkegaard describes how such a man would endure such a defiance and identifies the three major traits of the Absurd Man, later discussed by Albert Camus: a rejection of escaping existence (suicide), a rejection of help from a higher power and acceptance of his absurd (and despairing) condition.
According to Kierkegaard in his autobiographyThe Point of View of My Work as an Author, most of his pseudonymous writings are not necessarily reflective of his own opinions. Nevertheless, his work anticipated many absurdist themes and provided its theoretical background.
The philosophy of Albert Camus, or more precisely the “camusian absurd” (French:l'absurde camusien), refers with absurdism to the work and philosophical thought of the French writerAlbert Camus. This philosophy is influenced by the author's political,libertarian, social and ecological ideas; and is inspired by previous philosophical trends, such asGreek philosophy,nihilism, theNietzschean thoughtorexistentialism. It revolves around three major cycles: “the absurd (l'absurde)”, “the revolt (la révolte)” and “love (l'amour)”. Each cycle is linked to a Greek myth (Sisyphus,Prometheus,Nemesis) and explores specific themes and objects; the common thread remaining the solitude and despair of the human, constantly driven by the tireless search for the meaning of the world and of life.
I had a precise plan when I started my work: I wanted to first express negation. In three forms. Romanesque: it wasThe Stranger. Drama:CaligulaandThe Misunderstanding. Ideological:The Myth of Sisyphus. I wouldn't have been able to talk about it if I hadn't experienced it; I have no imagination. But for me it was, if you like,the methodical doubtof Descartes. I knew that we cannot live in negation and I announced it in the preface to the Myth of Sisyphus; I anticipated the positive in all three forms again. Romance:The Plague. Drama:The State of SiegeandThe Righteous. Ideological:The Rebel. I already saw a third layer around the theme of love. These are the projects I have in progress
The cycle of the absurd, ornegation, primarily addressessuicideand thehuman condition. It is expressed through four of Camus's works: thenovelThe Strangerand theessayThe Myth of Sisyphus(1942), then the playsCaligulaandThe Misunderstanding(1944). By refusing the refuge of belief, Human becomes aware that his existence revolves around repetitive and meaningless acts. The certainty of death only reinforces, according to the writer, the feeling of uselessness of all existence. The absurd is therefore the feeling that man feels when confronted with the absence of meaning in the face of the Universe, the painful realization of his separation from the world. The question then arises of the legitimacy of suicide.
The cycle of revolt, calledthe positive, is a direct response to the absurd and is also expressed by four of his works: the novelThe Plague(1947), the playsThe State of Siege(1948) andThe Just Assassins(1949), then the essayThe Rebel(1951). Positive concept of affirmation of the individual, where only action and commitment count in the face of the tragedy of the world,revoltis for the writer the way of experiencing the absurd, knowing our fatal destiny and nevertheless facing it : “Man refuses the world as it is, without agreeing to escape it.” It is intelligence grappling with the “unreasonable silence of the world”. Depriving ourselves of eternal life frees us from the constraints imposed by an improbable future; Man gains freedom of action, lucidity and dignity.
The philosophy of Camus therefore has as its finitude a singularhumanism. Advancing a message of lucidity, resilience and emancipation in the face of the absurdity of life, it encourages people to create their own meanings through personal choices and commitments, and to embrace their freedom to the fullest. Because he affirms that, even in the absurd, there is room for passion and rebellion; and although the Universe may be indifferent to our search for meaning, this search isin itselfmeaningful. InThe Myth of Sisyphus, despite his absurd destiny, Sisyphus finds a form of liberation in his incessant work: “one must imagine Sisyphus happy”. With the cycle of love and the “midday thought” (French:la pensée de midi), the philosophy of the absurd is completed by a principle of measurement and pleasure, close toEpicureanism.
Though the notion of the 'absurd' pervades allAlbert Camus's writing,The Myth of Sisyphusis his chief work on the subject. In it, Camus considers absurdity as a confrontation, an opposition, a conflict or a "divorce" between two ideals. Specifically, he defines the human condition as absurd, as the confrontation between man's desire for significance, meaning and clarity on the one hand—and the silent, cold universe on the other. He continues that there are specific human experiences evoking notions of absurdity. Such a realization or encounter with the absurd leaves the individual with a choice:suicide, aleap of faith, or recognition. He concludes that recognition is the only defensible option.[57]
For Camus, suicide is a "confession" that life is not worth living; it is a choice that implicitly declares that life is "too much." Suicide offers the most basic "way out" of absurdity: the immediate termination of the self and its place in the universe.
The absurd encounter can also arouse a "leap of faith," a term derived from one of Kierkegaard's early pseudonyms,Johannes de Silentio(although the term was not used by Kierkegaard himself),[58]where one believes that there is more than the rational life (aesthetic or ethical). To take a "leap of faith," one must act with the "virtue of the absurd" (asJohannes de Silentioput it), where a suspension of the ethical may need to exist. This faith has no expectations, but is a flexible power initiated by a recognition of the absurd. Camus states that because the leap of faith escapes rationality and defers to abstraction over personal experience, the leap of faith is not absurd. Camus considers the leap of faith as "philosophical suicide," rejecting both this and physical suicide.[58][59]
Lastly, a person can choose to embrace the absurd condition. According to Camus, one's freedom—and the opportunity to give life meaning—lies in the recognition of absurdity. If the absurd experience is truly the realization that the universe is fundamentally devoid of absolutes, then we as individuals are truly free. "To live without appeal,"[60]as he puts it, is a philosophical move to define absolutes and universals subjectively, rather than objectively. The freedom of man is thus established in one's natural ability and opportunity to create their own meaning and purpose; to decide (or think) for oneself. The individual becomes the most precious unit of existence, representing a set of unique ideals that can be characterized as an entire universe in its own right. In acknowledging the absurdity of seeking any inherent meaning, but continuing this search regardless, one can be happy, gradually developing meaning from the search alone."Happiness and the absurd are two sons of the same earth. They are inseparable."[61]
Camus states inThe Myth of Sisyphus: "Thus I draw from the absurd three consequences, which are my revolt, my freedom, and my passion. By the mere activity of consciousness I transform into a rule of life what was an invitation to death, and I refuse suicide."[62]"Revolt" here refers to the refusal of suicide and search for meaning despite the revelation of the Absurd; "Freedom" refers to the lack of imprisonment by religious devotion or others' moral codes; "Passion" refers to the most wholehearted experiencing of life, since hope has been rejected, and so he concludes that every moment must be lived fully.
Absurdism originated from (as well as alongside) the 20th-century strains ofexistentialismandnihilism; it shares some prominent starting points with both, though also entails conclusions that are uniquely distinct from these other schools of thought. All three arose from the human experience of anguish and confusion stemming from existence: the apparent meaninglessness of a world in which humans, nevertheless, are compelled to find or create meaning.[63]The three schools of thought diverge from there. Existentialists have generally advocated the individual's construction of their own meaning in life as well as thefree willof the individual. Nihilists, on the contrary, contend that "it is futile to seek or to affirm meaning where none can be found."[64]Absurdists, following Camus' formulation, hesitantly allow the possibility for some meaning or value in life, but are neither as certain as existentialists are about the value of one's own constructed meaning nor as nihilists are about the total inability to create meaning. Absurdists following Camus also devalue or outright reject free will, encouraging merely that the individual live defiantly and authenticallyin spite ofthe psychological tension of the Absurd.[65]
Camus himself passionately worked to counternihilism, as he explained in his essay "The Rebel", while he also categorically rejected the label of "existentialist" in his essay "Enigma" and in the compilationThe Lyrical and Critical Essays of Albert Camus, though he was, and still is, often broadly characterized by others as an existentialist.[66]Both existentialism and absurdism entail consideration of the practical applications of becoming conscious of the truth ofexistential nihilism: i.e., how a driven seeker of meaning should act when suddenly confronted with the seeming concealment, or downright absence, of meaning in the universe.
While absurdism can be seen as a kind of response to existentialism, it can be debated exactly how substantively the two positions differ from each other. The existentialist, after all, does not deny the reality of death. But the absurdist seems to reaffirm the way in which death ultimately nullifies our meaning-making activities, a conclusion the existentialists seem to resist through various notions of posterity or, inSartre's case, participation in a grand humanist project.[67]
The basic problem of absurdism is usually not encountered through a dispassionate philosophical inquiry but as the manifestation of anexistential crisis.[15][3][14]Existential crises are inner conflicts in which the individual wrestles with the impression that life lacksmeaning. They are accompanied by various negativeexperiences, such asstress,anxiety, despair, anddepression, which can disturb the individual's normal functioning in everyday life.[22][23][24]In this sense, the conflict underlying the absurdist perspective poses a psychological challenge to the affected. This challenge is due to the impression that the agent's vigorous daily engagement stands in incongruity with its apparent insignificance encountered through philosophical reflection.[15]Realizing this incongruity is usually not a pleasant occurrence and may lead to estrangement, alienation, and hopelessness.[68][14]The intimate relation to psychological crises is also manifested in the problem of finding the right response to this unwelcome conflict, for example, by denying it, by taking life less seriously, or by revolting against the absurd.[15]But accepting the position of absurdism may also have certain positive psychological effects. In this sense, it can help the individual achieve a certain psychological distance from unexamined dogmas and thus help them evaluate their situation from a more encompassing and objective perspective. However, it brings with it the danger of leveling all significant differences and thereby making it difficult for the individual to decide what to do or how to live their life.[8]
It has been argued that absurdism in the practical domain resemblesepistemological skepticismin the theoretical domain.[2][12]In the case of epistemology, we usually take for granted our knowledge of the world around us even though, whenmethodological doubtis applied, it turns out that this knowledge is not as unshakable as initially assumed.[69]For example, the agent may decide to trust their perception that the sun is shining but its reliability depends on the assumption that the agent is not dreaming, which they would not know even if they were dreaming. In a similar sense in the practical domain, the agent may decide to take aspirin in order to avoid a headache even though they may be unable to give a reason for why they should be concerned with their ownwellbeingat all.[2]In both cases, the agent goes ahead with a form of unsupported natural confidence and takes life largely for granted despite the fact that their power to justify is only limited to a rather small range and fails when applied to the larger context, on which the small range depends.[2][14]
It has been argued that absurdism is opposed to various fundamental principles and assumptions guidingeducation, like the importance oftruthand of fostering rationality in the students.[8]
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The termautopoiesis(fromGreekαὐτo-(auto)'self'andποίησις(poiesis)'creation, production'), one of several current theories of life, refers to asystemcapable of producing and maintaining itself by creating its own parts.[1]The term was introduced in the 1972 publicationAutopoiesis and Cognition: The Realization of the Livingby Chilean biologistsHumberto MaturanaandFrancisco Varelato define the self-maintainingchemistryof livingcells.[2]
The concept has since been applied to the fields ofcognition,neurobiology,systems theory,architectureandsociology.Niklas Luhmannbriefly introduced the concept of autopoiesis toorganizational theory.[3]
In their 1972 bookAutopoiesis and Cognition, Chilean biologists Maturana and Varela described how they invented the word autopoiesis.[4]: 89: 16
"It was in these circumstances ... in which he analyzed Don Quixote's dilemma of whether to follow the path of arms (praxis, action) or the path of letters (poiesis, creation, production), I understood for the first time the power of the word "poiesis" and invented the word that we needed:autopoiesis. This was a word without a history, a word that could directly mean what takes place in the dynamics of the autonomy proper to living systems."
They explained that,[4]: 78
"An autopoieticmachineis a machine organized (defined as a unity) as a network of processes of production (transformation and destruction) of components which: (i) through their interactions and transformations continuously regenerate and realize the network of processes (relations) that produced them; and (ii) constitute it (the machine) as a concrete unity in space in which they (the components) exist by specifying the topological domain of its realization as such a network."
They described the "space defined by an autopoietic system" as "self-contained", a space that "cannot be described by using dimensions that define another space. When we refer to our interactions with a concrete autopoietic system, however, we project this system on the space of our manipulations and make a description of this projection."[4]: 89
Autopoiesis was originally presented as a system description that was said to define and explain the nature ofliving systems. A canonical example of an autopoietic system is thebiological cell. Theeukaryoticcell, for example, is made of variousbiochemicalcomponents such asnucleic acidsandproteins, and is organized into bounded structures such as thecell nucleus, variousorganelles, acell membraneandcytoskeleton. These structures, based on an internal flow of molecules and energy,producethe components which, in turn, continue to maintain the organized bounded structure that gives rise to these components.
An autopoietic system is to be contrasted with anallopoieticsystem, such as a car factory, which uses raw materials (components) to generate a car (an organized structure) which is somethingotherthan itself (the factory). However, if the system is extended from the factory to include components in the factory's "environment", such as supply chains, plant / equipment, workers, dealerships, customers, contracts, competitors, cars, spare parts, and so on, then as a total viable system it could be considered to be autopoietic.[5]
Of course, cells also require raw materials (nutrients), and produce numerous products -waste products, the extracellular matrix, intracellular messaging molecules, etc.
Autopoiesis in biological systems can be viewed as a network of constraints that work to maintain themselves. This concept has been called organizational closure[6]or constraint closure[7]and is closely related to the study ofautocatalytic chemical networkswhere constraints are reactions required to sustain life.
Though others have often used the term as a synonym forself-organization, Maturana himself stated he would "[n]ever use the notion of self-organization ... Operationally it is impossible. That is, if the organization of a thing changes, the thing changes".[8]Moreover, an autopoietic system is autonomous and operationally closed, in the sense that there are sufficient processes within it to maintain the whole. Autopoietic systems are "structurally coupled" with their medium, embedded in a dynamic of changes that can be recalled assensory-motor coupling.[9]This continuous dynamic is considered as a rudimentary form ofknowledgeorcognitionand can be observed throughout life-forms.
An application of the concept of autopoiesis tosociologycan be found in Niklas Luhmann'sSystems Theory, which was subsequently adapted byBob Jessopin his studies of the capitalist state system.Marjatta Maulaadapted the concept of autopoiesis in a business context.[10]The theory of autopoiesis has also been applied in the context of legal systems by not only Niklas Luhmann, but also Gunther Teubner.[11][12]Patrik Schumacherhas applied the term to refer to the 'discursive self-referential making of architecture.'[13][14]Varela eventually further applied autopoesis to develop models of mind, brain, and behavior called non-representationalist,enactive,embodied cognitive neuroscience, culminating inneurophenomenology.
In the context of textual studies,Jerome McGannargues that texts are "autopoietic mechanisms operating as self-generating feedback systems that cannot be separated from those who manipulate and use them".[15]Citing Maturana and Varela, he defines an autopoietic system as "a closed topological space that 'continuously generates and specifies its own organization through its operation as a system of production of its own components, and does this in an endless turnover of components'", concluding that "Autopoietic systems are thus distinguished from allopoietic systems, which are Cartesian and which 'have as the product of their functioning something different from themselves'". Coding and markup appearallopoietic", McGann argues, but are generative parts of the system they serve to maintain, and thus language and print or electronic technology are autopoietic systems.[16]: 200–1
The philosopherSlavoj Žižek, in his discussion ofHegel, argues:
"Hegel is – to use today's terms – the ultimate thinker of autopoiesis, of the process of the emergence of necessary features out of chaotic contingency, the thinker of contingency's gradual self-organisation, of the gradual rise of order out of chaos."[17]
Autopoiesis can be defined as the ratio between the complexity of a system and the complexity of its environment.[18]
This generalized view of autopoiesis considers systems as self-producing not in terms of their physical components, but in terms of its organization, which can be measured in terms of information and complexity. In other words, we can describe autopoietic systems as those producing more of their own complexity than the one produced by their environment.
Autopoiesis has been proposed as a potential mechanism ofabiogenesis, by which molecules evolved into more complex cells that could support the development of life.[20]
Autopoiesis is just one of several current theories of life, including thechemoton[21]ofTibor Gánti, thehypercycleofManfred EigenandPeter Schuster,[22][23][24]the(M,R) systems[25][26]ofRobert Rosen, and theautocatalytic sets[27]ofStuart Kauffman, similar to an earlier proposal byFreeman Dyson.[28]All of these (including autopoiesis) found their original inspiration in Erwin Schrödinger's bookWhat is Life?[29]but at first they appear to have little in common with one another, largely because the authors did not communicate with one another, and none of them made any reference in their principal publications to any of the other theories. Nonetheless, there are more similarities than may be obvious at first sight, for example between Gánti and Rosen.[30]Until recently[31][32][33]there have been almost no attempts to compare the different theories and discuss them together.
An extensive discussion of the connection of autopoiesis tocognitionis provided by Evan Thompson in his 2007 publication,Mind in Life.[34]The basic notion of autopoiesis as involving constructive interaction with the environment is extended to include cognition. Initially, Maturana defined cognition as behavior of an organism "with relevance to the maintenance of itself".[35]: 13However, computer models that are self-maintaining but non-cognitive have been devised, so some additional restrictions are needed, and the suggestion is that the maintenance process, to be cognitive, involves readjustment of the internal workings of the system in somemetabolic process. On this basis it is claimed that autopoiesis is a necessary but not a sufficient condition for cognition.[36]Thompson wrote that this distinction may or may not be fruitful, but what matters is that living systems involve autopoiesis and (if it is necessary to add this point) cognition as well.[37]: 127It can be noted that this definition of 'cognition' is restricted, and does not necessarily entail any awareness orconsciousnessby the living system. With the publication of The Embodied Mind in 1991, Varela, Thompson and Rosch applied autopoesis to make non-representationalist, andenactive models of mind, brain and behavior, which further developedembodied cognitive neuroscience, later culminating inneurophenomenology.
The connection of autopoiesis to cognition, or if necessary, of living systems to cognition, is an objective assessment ascertainable by observation of a living system.
One question that arises is about the connection between cognition seen in this manner and consciousness. The separation of cognition and consciousness recognizes that the organism may be unaware of the substratum where decisions are made. What is the connection between these realms? Thompson refers to this issue as the "explanatory gap", and one aspect of it is thehard problem of consciousness, how and why we havequalia.[38]
A second question is whether autopoiesis can provide a bridge between these concepts. Thompson discusses this issue from the standpoint ofenactivism. An autopoietic cell actively relates to its environment. Its sensory responses trigger motor behavior governed by autopoiesis, and this behavior (it is claimed) is a simplified version of a nervous system behavior. The further claim is that real-time interactions like this require attention, and an implication of attention is awareness.[39]
There are multiple criticisms of the use of the term in both its original context, as an attempt to define and explain the living, and its various expanded usages, such as applying it to self-organizing systems in general or social systems in particular.[40]Critics have argued that the concept and its theory fail to define or explain living systems and that, because of the extreme language ofself-referentialityit uses without any external reference, it is really an attempt to give substantiation to Maturana's radicalconstructivistorsolipsisticepistemology,[41]or whatDanilo Zolo[42][43]has called instead a "desolate theology". An example is the assertion by Maturana and Varela that "We do not see what we do not see and what we do not see does not exist".[44]
According to Razeto-Barry, the influence ofAutopoiesis and Cognition: The Realization of the Livingin mainstream biology has proven to be limited. Razeto-Barry believes that autopoiesis is not commonly used as the criterion for life.[45]
Zoologist and philosopherDonna Harawayalso criticizes the usage of the term, arguing that "nothing makes itself; nothing is really autopoietic or self-organizing",[46]and suggests the use ofsympoiesis, meaning "making-with", instead.
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Atemporal paradox,time paradox, ortime travel paradox, is aparadox, an apparent contradiction, or logical contradiction associated with the idea oftime travelor other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistictime dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.
Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox.[1]Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps."[2][3]Consistency paradoxes, on the other hand, are those where future events influence the past to cause an apparent contradiction, exemplified by thegrandfather paradox, where a person travels to the past to prevent the conception of one of their ancestors, thus eliminating all the ancestor's descendants.[4]Newcomb's paradoxstems from the apparent contradictions that stem from the assumptions of bothfree willand foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop,[2]but although they appear similar, causal loops are unchanging and self-originating, whereas time loops are constantly resetting.[5]
A bootstrap paradox, also known as aninformation loop, aninformation paradox,[6]anontological paradox,[7]or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of eitherretrocausalityortime travel.[8][9][10][11]
Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere".[8]Such causally looped events then exist inspacetime, but their origin cannot be determined.[8][9]The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical.[9][6][12]A notable example occurs in the 1958science fictionshort story"—All You Zombies—", byRobert A. Heinlein, wherein the main character, anintersexindividual, becomes both their own mother and father; the 2014 filmPredestinationis based on the story. Allen Everett gives the movieSomewhere in Timeas an example involving an object with no origin: an old woman gives a watch to a playwright who later travels back in time and meets the same woman when she was young, and shows her the watch that she will later give to him.[6]An example of information which "came from nowhere" is in the movieStar Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula fortransparent aluminumto the 20th-century engineer who supposedly invented it.
Smeenk uses the term "predestination paradox" to refer specifically to situations in which a time traveler goes back in time to try to prevent some event in the past.[7]
The "predestination paradox" is a concept in time travel and temporal mechanics, often explored in science fiction. It occurs when a future event is the cause of a past event, which in turn becomes the cause of the future event, forming a self-sustaining loop in time. This paradox challenges conventional understandings of cause and effect, as the events involved are both the origin and the result of each other. A notable example is found in the TV seriesDoctor Who, where acharacter saves her father in the past, fulfilling a memory he had shared with her as a child about a strange woman having saved his life. The predestination paradox raises philosophical questions about free will, determinism, and the nature of time itself. It is commonly used as a narrative device in fiction to highlight the interconnectedness of events and the inevitability of certain outcomes.
The consistency paradox or grandfather paradox occurs when the past is changed in any way that directly negates the conditions required for the time travel to occur in the first place, thus creating a contradiction. A common example given is traveling to the past and preventing the conception of one's ancestors (such as causing the death of the ancestor's parent beforehand), thus preventing the conception of oneself. If the traveler were not born, then it would not be possible to undertake such an act in the first place; therefore, the ancestor proceeds to beget the traveler's next-generation ancestor and secure the line to the traveler. There is no predicted outcome to this scenario.[8]Consistency paradoxes occur whenever changing the past is possible.[9]A possible resolution is that a time travellercando anything thatdidhappen, butcannotdo anything thatdid nothappen. Doing something that did not happen results in a contradiction.[8]This is referred to as theNovikov self-consistency principle.
The grandfather paradox encompasses any change to the past,[13]and it is presented in many variations, including killing one's past self.[14][15]Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written intoAmazing Storiesin the 1920s.[16]Another variant of the grandfather paradox is the "Hitler paradox" or "Hitler's murder paradox", in which the protagonist travels back in time to murderAdolf Hitlerbefore he can rise to power in Germany, thus preventingWorld War IIand theHolocaust. Rather than necessarily physically preventing time travel, the action removes anyreasonfor the travel, along with any knowledge that the reason ever existed.[17]
Physicist John Garrison et al. give a variation of the paradox of an electronic circuit that sends a signal through a time machine to shut itself off, and receives the signal before it sends it.[18][19]
Newcomb's paradox is athought experimentshowing an apparent contradiction between theexpected utilityprinciple and thestrategic dominanceprinciple.[20]The thought experiment is often extended to explorecausalityand free will by allowing for "perfect predictors": if perfect predictors of the future exist, for example if time travel exists as a mechanism for making perfect predictions[how?], then perfect predictions appear to contradict free will because decisions apparently made with free will are already known to the perfect predictor[clarification needed].[21][22]Predestinationdoes not necessarily involve asupernaturalpower, and could be the result of other "infallible foreknowledge" mechanisms.[23]Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.[24]
Even without knowing whether time travel to the past is physically possible, it is possible to show usingmodal logicthat changing the past results in a logical contradiction. If it is necessarily true that the past happened in a certain way, then it is false and impossible for the past to have occurred in any other way. A time traveler would not be able to change the past from the way itis,but would only act in a way that is already consistent with whatnecessarilyhappened.[25][26]
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible. For example, the philosopherBradley Dowdenmade this sort of argument in the textbookLogical Reasoning, arguing that the possibility of creating a contradiction rules out time travel to the past entirely. However, some philosophers and scientists believe that time travel into the past need not be logically impossible provided that there is no possibility of changing the past,[13]as suggested, for example, by theNovikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopherNorman Swartz.[27]
A recent proposed resolution argues that if time is not an inherent property of the universe but is insteademergentfrom the laws ofentropy, as some modern theories suggest,[28][29]then it presents a natural solution to the Grandfather Paradox.[30]In this framework, "time travel" is reinterpreted not as movement along a linear continuum but as a reconfiguration of the present state of the universe to match a prior entropic configuration. Because the original chronological sequence—including events like the time traveler’s birth—remains preserved in the universe’s irreversible entropic progression, actions within the reconfigured state cannot alter the causal history that produced the traveler. This avoids paradoxes by treating time as a thermodynamic artifact rather than a mutable dimension.
Consideration of the possibility of backward time travel in a hypothetical universe described by aGödel metricled famed logicianKurt Gödelto assert that time might itself be a sort of illusion.[31][32]He suggests something along the lines of theblock timeview, in which time is just another dimension like space, with all events at all times being fixed within this four-dimensional "block".[citation needed]
Sergey Krasnikovwrites that these bootstrap paradoxes – information or an object looping through time – are the same; the primary apparent paradox is a physical system evolving into a state in a way that is not governed by its laws.[33]: 4He does not find these paradoxical and attributes problems regarding the validity of time travel to other factors in the interpretation of general relativity.[33]: 14–16
A 1992 paper by physicists Andrei Lossev andIgor Novikovlabeled such items without origin asJinn, with the singular termJinnee.[34]: 2311–2312This terminology was inspired by theJinnof theQuran, which are described as leaving no trace when they disappear.[35]: 200–203Lossev and Novikov allowed the term "Jinn" to cover both objects and information with the reflexive origin; they called the former "Jinn of the first kind", and the latter "Jinn of the second kind".[6][34]: 2315–2317[35]: 208They point out that an object making circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency; thesecond law of thermodynamicsseems to require that the object tends to a lower energy state throughout its history, and such objects that are identical in repeating points in their history seem to contradict this, but Lossev and Novikov argued that since the second law only requires entropy to increase inclosedsystems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy.[6][35]: 200–203They emphasize that there is no "strict difference" between Jinn of the first and second kind.[34]: 2320Krasnikov equivocates between "Jinn", "self-sufficient loops", and "self-existing objects", calling them "lions" or "looping or intruding objects", and asserts that they are no less physical than conventional objects, "which, after all, also could appear only from either infinity or a singularity."[33]: 8–9
The self-consistency principle developed byIgor Dmitriyevich Novikov[36]: p. 42 note 10expresses one view as to how backwardtime travelwould be possible without the generation of paradoxes. According to this hypothesis, even thoughgeneral relativitypermits someexact solutionsthat allow fortime travel[37]that containclosed timelike curvesthat lead back to the same point in spacetime,[38]physics in or nearclosed timelike curves(time machines) can only be consistent with the universal laws of physics, and thus only self-consistent events can occur. Anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from happening, since this would represent an inconsistency. The authors concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.[39]
PhysicistJoseph Polchinskiconsidered a potentially paradoxical situation involving abilliard ballthat is fired into awormholeat just the right angle such that it will be sent back in time and collides with its earlier self, knocking it off course, which would stop it from entering the wormhole in the first place.Kip Thornereferred to this problem as "Polchinski's paradox".[39]Thorne and two of his students at Caltech, Fernando Echeverria and Gunnar Klinkhammer, went on to find a solution that avoided any inconsistencies, and found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case.[40]Later analysis by Thorne andRobert Forwardshowed that for certain initial trajectories of the billiard ball, there could be an infinite number of self-consistent solutions.[39]It is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.[41]: 184The lack of constraints on initial conditions only applies to spacetime outside of thechronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.[41]: 187–188
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as anad hocsolution, and supposes that there are far more damaging implications of time travel.[42]Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity.[33]: 14–16Another conjecture, thecosmic censorship hypothesis, suggests that every closed timelike curve passes through anevent horizon, which prevents such causal loops from being observed.[43]
The interacting-multiple-universes approach is a variation of themany-worlds interpretationof quantum mechanics that involves time travelers arriving in a different universe than the one from which they came; it has been argued that, since travelers arrive in a different universe's history and not their history, this is not "genuine" time travel.[44]Stephen Hawking has argued for thechronology protection conjecture, that even if the MWI is correct, we should expect each time traveler to experience a single self-consistent history so that time travelers remain within their world rather than traveling to a different one.[45]
David Deutschhas proposed thatquantum computationwith a negative delay—backward time travel—produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning.[46]However Deutsch's self-consistency condition has been demonstrated as capable of being fulfilled to arbitrary precision by any system subject to the laws of classicalstatistical mechanics, even if it is not built up by quantum systems.[47]Allen Everett has also argued that even if Deutsch's approach is correct, it would imply that any macroscopic object composed of multiple particles would be split apart when traveling back in time, with different particles emerging in different worlds.[48]
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Adilemma(fromAncient Greekδίλημμα(dílēmma)'doubleproposition') is aproblemoffering two possibilities, neither of which is unambiguously acceptable or preferable. The possibilities are termed thehornsof the dilemma, aclichédusage, but distinguishing the dilemma from other kinds of predicament as a matter of usage.[1]
The termdilemmais attributed byGabriel NuchelmanstoLorenzo Vallain the 15th century, in later versions of his logic text traditionally calledDialectica. Valla claimed that it was the appropriate Latin equivalent of theGreekdilemmaton. Nuchelmans argued that his probable source was a logic text ofc.1433ofGeorge of Trebizond.[2]He also concluded that Valla had reintroduced to the Latin West a type of argument that had fallen into disuse.[3]
Valla'sneologismdid not immediately take hold, preference being given to the established Latin termcomplexio, used byCicero, withconversioapplied to the upsetting of dilemmatic reasoning. With the support ofJuan Luis Vives, however,dilemmawas widely applied by the end of the 16th century.[4]
A dilemma is often phrased as "you must accept either A, or B", where A and B are propositions each leading to some further conclusion. In the case where this is true, it can be called a "dichotomy", but when it is not true, the dilemma constitutes afalse dichotomy, which is a logicalfallacy. Traditional usage distinguished the dilemma as a "hornedsyllogism" from thesophismthat attracted the Latin namecornutus.[5]The original use of the wordhornsin English has been attributed toNicholas Udallin his 1548 bookParaphrases, translating from the Latin termcornuta interrogatio.[6]
The dilemma is sometimes used as arhetorical device. Its isolation as textbook material has been attributed toHermogenes of Tarsusin his workOn Invention.[7]C. S. Peircegave a definition ofdilemmatic argumentas any argument relying onexcluded middle.[8]
Inpropositional logic,dilemmais applied to a group ofrules of inference, which are in themselves valid rather than fallacious. They each have three premises, and include theconstructive dilemmaanddestructive dilemma.[9]Such arguments can be refuted by showing that the disjunctive premise – the "horns of the dilemma" – does not in fact hold, because it presents a false dichotomy. You are asked to accept "A or B", but counter by showing that is not all. Successfully undermining that premise is called "escaping through the horns of the dilemma".[10]
Dilemmatic reasoning has been attributed toMelissus of Samos, aPresocraticphilosopher whose works survive in fragmentary form, making the origins of the technique in philosophy imponderable.[11]It was established withDiodorus Cronus(diedc.284BCE).[12]The paradoxes ofZeno of Eleawere reported byAristotlein dilemma form, but that may have been to conform with whatPlatosaid about Zeno's style.[13]
In cases where twomoral principlesappear to be inconsistent, an actor confronts a dilemma in terms of which principle to follow. This kind of moral case study is attributed toCicero, in book III of hisDe Officiis.[14]In the Christian tradition ofcasuistry, an approach to abstract ranking of principles introduced byBartolomé de Medinain the 16th century became tainted with the accusation oflaxism, as did casuistry itself.[15]Another approach, with legal roots, is to lay emphasis on particular features present in a given case: in other words, the exact framing of the dilemma.[16]
In law, Valentin Jeutner has argued that the term "legal dilemma" could be used as aterm-of-art, to describe a situation where a legal subject is confronted with two or more legal norms that the legal subject cannot simultaneously comply with.[17]
Examples include contradictory contracts where one clause directly negates another clause, or conflicts between fundamental (e.g.constitutional) legal norms.Leibniz's1666 doctoral dissertationDe casibus perplexis(Perplexing Cases) is an early study of contradictory legal conditions.[18]In domestic law, it has been argued that theGerman Constitutional Courtconfronted a legal dilemma when determining, in connection with proceedings relating to the GermanAviation Security Act, whether a government official could intentionally kill innocent civilians by shooting down a hijacked airplane that would otherwise have crashed into a football stadium, killing tens of thousands.[19]
In international law, it has been suggested that theInternational Court of Justiceconfronted a legal dilemma in its 1996Nuclear Weapons Advisory Opinion. It was faced with the question whether, in an extreme circumstance of self-defence, it is astate's right to self-defenceor international law's general prohibition ofnuclear weaponsthat should take priority.[20]
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TheEuthyphro dilemmais found inPlato's dialogueEuthyphro, in whichSocratesasksEuthyphro, "Is thepious(τὸ ὅσιον) loved by thegodsbecause it is pious, or is it pious because it is loved by the gods?" (10a)
Although it was originally applied to the ancientGreek pantheon, the dilemma has implications for modernmonotheistic religions.Gottfried Leibnizasked whether the good and just "is good and just because God wills it or whether God wills it because it is good and just".[1]Ever since Plato's original discussion, this question has presented a problem for some theists, though others have thought it afalse dilemma, and it continues to be an object of theological and philosophical discussion today.
Socrates and Euthyphro discuss the nature of piety in Plato'sEuthyphro. Euthyphro proposes (6e) that the pious (τὸ ὅσιον) is the same thing as that which is loved by the gods (τὸ θεοφιλές), but Socrates finds a problem with this proposal: the gods may disagree among themselves (7e). Euthyphro then revises his definition, so that piety is only that which is loved by all of the gods unanimously (9e).
At this point thedilemmasurfaces. Socrates asks whether the gods love the pious because it is the pious, or whether the pious is pious only because it is loved by the gods (10a). Socrates and Euthyphro both contemplate the first option: surely the gods love the pious because it is the pious. But this means, Socrates argues, that we are forced to reject the second option: the fact that the gods love something cannot explain why the pious is the pious (10d). Socrates points out that if both options were true, they would yield a vicious circle, with the gods loving the pious because it is the pious, and the pious being the pious because the gods love it. And this, in turn, means Socrates argues, that the pious is not the same as the god-beloved, for what makes the pious the pious is not what makes the god-beloved the god-beloved. After all, what makes the god-beloved the god-beloved is that the gods love it, whereas what makes the pious the pious is something else (9d-11a). Thus Euthyphro's theory does not give us the verynatureof the pious, but at most aqualityof the pious (11ab).
The dilemma can be modified to apply to philosophical theism, where it is still the object of theological and philosophical discussion, largely within the Christian, Jewish, and Islamic traditions. AsGermanphilosopherandmathematicianGottfried Leibnizpresented this version of the dilemma: "It is generally agreed that whatever God wills is good and just. But there remains the question whether it is good and just because God wills it or whether God wills it because it is good and just; in other words, whether justice and goodness are arbitrary or whether they belong to the necessary and eternal truths about the nature of things."[2]Many philosophers and theologians have addressed the Euthyphro dilemma since the time of Plato, though not always with reference to the Platonic dialogue. According to scholarTerence Irwin, the issue and its connection with Plato was revived by Ralph Cudworth and Samuel Clarke in the 17th and 18th centuries.[3]More recently, it has received a great deal of attention from contemporary philosophers working inmetaethicsand thephilosophy of religion. Philosophers and theologians aiming to defend theism against the threat of the dilemma have developed a variety of responses.
The first horn of the dilemma (i.e. that which is right is commanded by Godbecause it is right) goes by a variety of names, includingintellectualism,rationalism,realism,naturalism, andobjectivism. Roughly, it is the view that there are independent moral standards: some actions are right or wrong in themselves, independent of God's commands. This is the view accepted by Socrates and Euthyphro in Plato's dialogue. TheMu'tazilahschool ofIslamic theologyalso defended the view (with, for example,Nazzammaintaining that God is powerless to engage in injustice or lying),[4]as did theIslamic philosopherAverroes.[5]Thomas Aquinasnever explicitly addresses the Euthyphro dilemma, but Aquinas scholars often put him on this side of the issue.[6][7]Aquinas draws a distinction between what is good or evil in itself and what is good or evil because of God's commands,[8]with unchangeable moral standards forming the bulk ofnatural law.[9]Thus he contends that not even God can change theTen Commandments(adding, however, that Godcanchange what individuals deserve in particular cases, in what might look like special dispensations to murder or stealing).[10]Among laterScholastics,Gabriel Vásquezis particularly clear-cut about obligations existing prior to anyone's will, even God's.[11][12]Modernnatural law theory sawGrotiusandLeibnizalso putting morality prior toGod's will, comparing moral truths to unchangeable mathematical truths, and engagingvoluntaristslikePufendorfin philosophical controversy.[13]Cambridge PlatonistslikeBenjamin WhichcoteandRalph Cudworthmounted seminal attacks on voluntarist theories, paving the way for the later rationalistmetaethicsofSamuel ClarkeandRichard Price;[14][15][16]what emerged was a view on which eternal moral standards, though dependent on God in some way, exist independently of God's will and prior to God's commands. Contemporaryphilosophers of religionwho embrace this horn of the Euthyphro dilemma includeRichard Swinburne[17][18]andT. J. Mawson[19](though see below for complications).
Contemporary philosophers Joshua Hoffman and Gary S. Rosenkrantz take the first horn of the dilemma, branding divine command theory a "subjective theory of value" that makes morality arbitrary.[30]They accept a theory of morality on which, "right and wrong, good and bad, are in a sense independent of whatanyonebelieves, wants, or prefers."[31]They do not address the problems mentioned above with the first horn, but do consider a related problem concerning God's omnipotence: namely, that it might be handicapped by his inability to bring about what is independently evil. To this they reply that God is omnipotent, even though there are states of affairs he cannot bring about: omnipotence is a matter of "maximal power", not an ability to bring about all possible states of affairs. And supposing that it is impossible for God not to exist, then since there cannot be more than one omnipotent being, it is therefore impossible for any being to have more power than God (e.g., a being who is omnipotent but notomnibenevolent). Thus God's omnipotence remains intact.[32]
Richard SwinburneandT. J. Mawsonhave a slightly more complicated view. They both take the first horn of the dilemma when it comes tonecessarymoral truths. But divine commands are not totally irrelevant, for God and his will can still effectcontingentmoral truths.[33][34][18][19]On the one hand, the most fundamental moral truths hold true regardless of whether God exists or what God has commanded: "Genocide and torturing children are wrong and would remain so whatever commands any person issued."[24]This is because, according to Swinburne, such truths are true as a matter oflogical necessity: like the laws of logic, one cannot deny them without contradiction.[35]This parallel offers a solution to the aforementioned problems of God's sovereignty, omnipotence, and freedom: namely, that these necessary truths of morality pose no more of a threat than the laws of logic.[36][37][38]On the other hand, there is still an important role for God's will. First, there are some divine commands that can directly create moral obligations: e.g., the command to worship on Sundays instead of on Tuesdays.[39]Notably, not even these commands, for which Swinburne and Mawson take the second horn of the dilemma, have ultimate, underived authority. Rather, they create obligations only because of God's role as creator and sustainer and indeed owner of the universe, together with the necessary moral truth that we owe some limited consideration to benefactors and owners.[40][41]Second, God can make anindirectmoral difference by deciding what sort of universe to create. For example, whether a public policy is morally good might indirectly depend on God's creative acts: the policy's goodness or badness might depend on its effects, and those effects would in turn depend on the sort of universe God has decided to create.[42][43]
The second horn of the dilemma (i.e. that which is right is rightbecause it is commanded by God) is sometimes known asdivine command theoryorvoluntarism. Roughly, it is the view that there are no moral standards other than God's will: without God's commands, nothing would be right or wrong. This view was partially defended byDuns Scotus, who argued that not allTen Commandmentsbelong to theNatural Lawin the strictest sense.[44]Scotus held that while our duties to God (the first three commandments, traditionally thought of as the First Tablet) areself-evident,true by definition, and unchangeable even by God, our duties to others (found on the second tablet) were arbitrarily willed by God and are within his power to revoke and replace (although, the third commandment, to honour the Sabbath and keep it holy, has a little of both, as we are absolutely obliged to render worship to God, but there is no obligation in natural law to do it on this day or that). Scotus does note, however that the last seven commandments "are highly consonant with [the natural law], though they do not follow necessarily from first practical principles that are known in virtue of their terms and are necessarily known by any intellect [that understands their terms. And it is certain that all the precepts of the second table belong to the natural law in this second way, since their rectitude is highly consonant with first practical principles that are known necessarily".[45][46][47][48]Scotus justifies this position with the example of a peaceful society, noting that the possession of private property is not necessary to have a peaceful society, but that "those of weak character" would be more easily made peaceful with private property than without.
William of Ockhamwent further, contending that (since there is no contradiction in it) God could command us not to love God[49]and even tohateGod.[50]LaterScholasticslikePierre D'Aillyand his studentJean de Gersonexplicitly confronted the Euthyphro dilemma, taking the voluntarist position that God does not "command good actions because they are good or prohibit evil ones because they are evil; but... these are therefore good because they are commanded and evil because prohibited."[51]ProtestantreformersMartin LutherandJohn Calvinboth stressed the absolute sovereignty of God's will, with Luther writing that "for [God's] will there is no cause or reason that can be laid down as a rule or measure for it",[52]and Calvin writing that "everything which [God] wills must be held to be righteous by the mere fact of his willing it."[53]The voluntarist emphasis on God's absolute power was carried further byDescartes, who notoriously held that God had freely created the eternal truths oflogicandmathematics, and that God was therefore capable of givingcirclesunequalradii,[54]givingtrianglesother than 180 internal degrees, and even makingcontradictionstrue.[55]Descartes explicitly seconded Ockham: "why should [God] not have been able to give this command [i.e., the command to hate God] to one of his creatures?"[56]Thomas Hobbesnotoriously reduced the justice of God to "irresistible power"[57](drawing the complaint ofBishop Bramhallthat this "overturns... all law").[58]AndWilliam Paleyheld that all moral obligations bottom out in the self-interested "urge" to avoidHelland enterHeavenby acting in accord with God's commands.[59]Islam'sAsh'arite theologians,al-Ghazaliforemost among them, embraced voluntarism: scholar George Hourani writes that the view "was probably more prominent and widespread in Islam than in any other civilization."[60][61]Wittgensteinsaid that of "the two interpretations of the Essence of the Good", that which holds that "the Good is good, in virtue of the fact that God wills it" is "the deeper", while that which holds that "God wills the good, because it is good" is "the shallow, rationalistic one, in that it behaves 'as though' that which is good could be given some further foundation".[62]Today, divine command theory is defended by many philosophers of religion, though typically in a restricted form (seebelow).
This horn of the dilemma also faces several problems:
One common response to the Euthyphro dilemma centers on a distinction betweenvalueandobligation. Obligation, which concerns rightness and wrongness (or what is required, forbidden, or permissible), is given a voluntarist treatment. But value, which concerns goodness and badness, is treated as independent of divine commands. The result is arestricteddivine command theory that applies only to a specific region of morality: thedeonticregion of obligation. This response is found inFrancisco Suárez's discussion of natural law and voluntarism inDe legibus[85]and has been prominent in contemporary philosophy of religion, appearing in the work of Robert M. Adams,[86]Philip L. Quinn,[87]and William P. Alston.[88]
A significant attraction of such a view is that, since it allows for a non-voluntarist treatment of goodness and badness, and therefore of God's own moral attributes, some of the aforementioned problems with voluntarism can perhaps be answered. God's commands are not arbitrary: there are reasons which guide his commands based ultimately on this goodness and badness.[89]God could not issue horrible commands: God's own essential goodness[81][90][91]or loving character[92]would keep him from issuing any unsuitable commands. Our obligation to obey God's commands does not result incircular reasoning; it might instead be based on a gratitude whose appropriateness is itself independent of divine commands.[93]These proposed solutions are controversial,[94]and some steer the view back into problems associated with the first horn.[95]
One problem remains for such views: if God's own essential goodness does not depend on divine commands, then the question regards what itdoesdepend on. Perhaps something other than God. Here the restricted divine command theory is commonly combined with a view reminiscent of Plato: God is identical to the ultimate standard for goodness.[96]Alston offers the analogy ofthe standard meter bar in France. Something is a meter long inasmuch as it is the same length as the standard meter bar, and likewise, something is good inasmuch as it approximates God. If one asks whyGodis identified as the ultimate standard for goodness, Alston replies that this is "the end of the line," with no further explanation available, but adds that this is no more arbitrary than a view that invokes a fundamental moral standard.[97]On this view, then, even though goodness is independent of God'swill, it still depends onGod, and thus God's sovereignty remains intact.
This solution has been criticized byWes Morriston. If we identify the ultimate standard for goodness with God's nature, then it seems we are identifying it with certain properties of God (e.g., being loving, being just). If so, then the dilemma resurfaces: God is either good because he has those properties, or those properties are good because God has them.[98]Nevertheless, Morriston concludes that the appeal to God's essential goodness is the divine-command theorist's best bet. To produce a satisfying result, however, it would have to give an account of God's goodness that does not trivialize it and does not make God subject to an independent standard of goodness.[99]
Moral philosopherPeter Singer, disputing the perspective that "God is good" and could never advocate something like torture, states that those who propose this are "caught in a trap of their own making, for what can they possibly mean by the assertion that God is good? That God is approved of by God?"[100]
Augustine,Anselm, and Aquinas all wrote about the problems raised by the Euthyphro dilemma, although, likeWilliam James[101]and Wittgenstein[62]later, they did not mention it by name. As philosopher and Anselm scholar Katherin A. Rogers observes, many contemporary philosophers of religion suppose that there are true propositions which exist as platonicabstractaindependently of God.[102]Among these are propositions constituting a moral order, to which God must conform in order to be good.[103]ClassicalJudaeo-Christiantheism, however, rejects such a view as inconsistent with God's omnipotence, which requires that God and what he has made is all that there is.[102]"The classical tradition," Rogers notes, "also steers clear of the other horn of the Euthyphro dilemma, divine command theory."[104]From a classical theistic perspective, therefore, the Euthyphro dilemma is false. As Rogers puts it, "Anselm, like Augustine before him and Aquinas later, rejects both horns of the Euthyphro dilemma. God neither conforms to nor invents the moral order. Rather His very nature is the standard for value."[102]Another criticism raised byPeter Geachis that the dilemma implies you must search for a definition that fits piety rather than work backwards by deciding pious acts (i.e. you must know what piety is before you can list acts which are pious).[105]It also implies something can not be pious if it is only intended to serve the Gods without actually fulfilling any useful purpose.
The basis of the false dilemma response—God's nature is the standard for value—predates the dilemma itself, appearing first in the thought of the eighth-century BCHebrewprophets,Amos,Hosea,MicahandIsaiah. (Amos lived some three centuries before Socrates and two beforeThales, traditionally regarded as the first Greek philosopher.) "Their message," writes British scholarNorman H. Snaith, "is recognized by all as marking a considerable advance on all previous ideas,"[106]not least in its "special consideration for the poor and down-trodden."[107]As Snaith observes,tsedeq, the Hebrew word forrighteousness, "actually stands for the establishment of God's will in the land." This includes justice, but goes beyond it, "because God's will is wider than justice. He has a particular regard for the helpless ones on earth."[108]Tsedeq"is the norm by which all must be judged" and it "depends entirely upon the Nature of God."[109]
Hebrew has fewabstract nouns. What the Greeks thought of as ideas or abstractions, the Hebrews thought of as activities.[110]In contrast to the Greekdikaiosune(justice) of the philosophers,tsedeqis not an idea abstracted from this world of affairs. As Snaith writes:
Tsedeqis something that happens here, and can be seen, and recognized, and known. It follows, therefore, that when the Hebrew thought oftsedeq(righteousness), he did not think of Righteousness in general, or of Righteousness as an Idea. On the contrary, he thought of a particular righteous act, an action, concrete, capable of exact description, fixed in time and space.... If the word had anything like a general meaning for him, then it was as it was represented by a whole series of events, the sum-total of a number of particular happenings.[109]
The Hebrew stance on what came to be called theproblem of universals, as on much else, was very different from that of Plato and precluded anything like the Euthyphro dilemma.[111]This has not changed. In 2005,Jonathan Sackswrote, "In Judaism, the Euthyphro dilemma does not exist."[112]Jewish philosophers Avi Sagi and Daniel Statman criticized the Euthyphro dilemma as "misleading" because "it is not exhaustive": it leaves out a third option, namely that God "acts only out of His nature."[113]
In Aquinas' view, to speak of abstractions not only as existent, but as more perfect exemplars than fully designated particulars, is to put a premium on generality and vagueness.[114]On this analysis, the abstract "good" in the first horn of the Euthyphro dilemma is an unnecessary obfuscation. Aquinas frequently quoted with approval Aristotle's definition, "Good is what all desire."[115][116]As he clarified, "When we say that good is what all desire, it is not to be understood that every kind of good thing is desired by all, but that whatever is desired has the nature of good."[117]In other words, even those who desire evil desire it "only under the aspect of good," i.e., of what is desirable.[118]The difference between desiring good and desiring evil is that in the former, will and reason are in harmony, whereas in the latter, they are in discord.[119]
Aquinas's discussion ofsinprovides a good point of entry to his philosophical explanation of why the nature of God is the standard for value. "Every sin," he writes, "consists in the longing for a passing [i.e., ultimately unreal or false] good."[120]Thus, "in a certain sense it is true what Socrates says, namely that no one sins with full knowledge."[121]"No sin in the will happens without an ignorance of the understanding."[122]God, however, has full knowledge (omniscience) and therefore by definition (that of Socrates, Plato, and Aristotle as well as Aquinas) can never will anything other than what is good. It has been claimed – for instance, byNicolai Hartmann, who wrote: "There is no freedom for the good that would not be at the same time freedom for evil"[123]– that this would limit God's freedom, and therefore his omnipotence.Josef Pieper, however, replies that such arguments rest upon an impermissiblyanthropomorphicconception of God.[124]In the case of humans, as Aquinas says, to be able to sin is indeed a consequence,[125]or even a sign, of freedom (quodam libertatis signum).[126]Humans, in other words, are not puppets manipulated by God so that they always do what is right. However, "it does not belong to theessenceof the free will to be able to decide for evil."[127]"To will evil is neither freedom nor a part of freedom."[126]It is precisely humans' creatureliness – that is, their not being God and therefore omniscient – that makes them capable of sinning.[128]Consequently, writes Pieper, "the inability to sin should be looked on as the very signature of a higher freedom – contrary to the usual way of conceiving the issue."[124]Pieper concludes: "Onlythewill [i.e., God's] can be the right standard of its own willing and must will what is right necessarily, from within itself, and always. A deviation from the norm would not even be thinkable. And obviously only the absolute divine will is the right standard of its own act"[129][130]– and consequently of all human acts. Thus the second horn of the Euthyphro dilemma, divine command theory, is also disposed of.
Thomist philosopherEdward Feserwrites, "Divine simplicity [entails] that God's will just is God's goodness which just is His immutable and necessary existence. That means that what is objectively good and what God wills for us as morally obligatory are really the same thing considered under different descriptions, and that neither could have been other than they are. There can be no question then, either of God's having arbitrarily commanded something different for us (torturing babies for fun, or whatever) or of there being a standard of goodness apart from Him. Again, the Euthyphro dilemma is a false one; the third option that it fails to consider is that what is morally obligatory is what God commands in accordance with a non-arbitrary and unchanging standard of goodness that is not independent of Him... He is notunderthe moral law precisely because Heisthe moral law."[131]
William James, in his essay "The Moral Philosopher and the Moral Life", dismisses the first horn of the Euthyphro dilemma and stays clear of the second. He writes: "Our ordinary attitude of regarding ourselves as subject to an overarching system of moral relations, true 'in themselves,' is ... either an out-and-out superstition, or else it must be treated as a merely provisional abstraction from that real Thinker ... to whom the existence of the universe is due."[132]Moral obligations are created by "personal demands," whether these demands[133]come from the weakest creatures, from the most insignificant persons, or from God. It follows that "ethics have as genuine a foothold in a universe where the highest consciousness is human, as in a universe where there is a God as well." However, whether "the purely human system" works "as well as the other is a different question."[132]
For James, the deepest practical difference in the moral life is between what he calls "the easy-going and the strenuous mood."[134]In a purely human moral system, it is hard to rise above the easy-going mood, since the thinker's "various ideals, known to him to be mere preferences of his own, are too nearly of the same denominational value;[135]he can play fast and loose with them at will. This too is why, in a merely human world without a God, the appeal to our moral energy falls short of its maximum stimulating power." Our attitude is "entirely different" in a world where there are none but "finite demanders" from that in a world where there is also "an infinite demander." This is because "the stable and systematic moral universe for which the ethical philosopher asks is fully possible only in a world where there is a divine thinker with all-enveloping demands", for in that case, "actualized in his thought already must be that ethical philosophy which we seek as the pattern which our own must evermore approach." Even though "exactly what the thought of this infinite thinker may be is hidden from us", our postulation of him serves "to let loose in us the strenuous mood"[134]and confront us with anexistential[136]"challenge" in which "our total character and personal genius ... are on trial; and if we invoke any so-called philosophy, our choice and use of that also are but revelations of our personal aptitude or incapacity for moral life. From this unsparing practical ordeal no professor's lectures and no array of books can save us."[134]In the words ofRichard M. Gale, "God inspires us to lead the morally strenuous life in virtue of our conceiving of him as unsurpassablygood. This supplies James with an adequate answer to the underlying question of theEuthyphro."[137]
Alexander Rosenberguses a version of the Euthyphro dilemma to argue that objective morality cannot exist and hence an acceptance ofmoral nihilismis warranted.[138]He asks, is objective morality correct because evolution discovered it or did evolution discover objective morality because it is correct? If the first horn of the dilemma is true then our current morality cannot be objectively correct by accident because if evolution had given us another type of morality then that would have been objectively correct. If the second horn of dilemma is true then one must account for how the random process of evolution managed to only select for objectively correct moral traits while ignoring the wrong moral traits. Given the knowledge that evolution has given us tendencies to be xenophobic and sexist it is mistaken to claim that evolution has only selected for objective morality as evidently it did not. Because both horns of the dilemma do not give an adequate account for how the evolutionary process instantiated objective morality in humans, a position ofMoral nihilismis warranted.
Yale Law SchoolProfessorMyres S. McDougal, formerly a classicist, later a scholar of property law, posed the question, "Do we protect it because it's a property right, or is it a property right because we protect it?"[139]The dilemma has also been restated in legal terms byGeoffrey Hodgson, who asked: "Does a state make a law because it is a customary rule, or does law become a customary rule because it is approved by the state?"[140]
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Atemporal paradox,time paradox, ortime travel paradox, is aparadox, an apparent contradiction, or logical contradiction associated with the idea oftime travelor other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistictime dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.
Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox.[1]Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps."[2][3]Consistency paradoxes, on the other hand, are those where future events influence the past to cause an apparent contradiction, exemplified by thegrandfather paradox, where a person travels to the past to prevent the conception of one of their ancestors, thus eliminating all the ancestor's descendants.[4]Newcomb's paradoxstems from the apparent contradictions that stem from the assumptions of bothfree willand foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop,[2]but although they appear similar, causal loops are unchanging and self-originating, whereas time loops are constantly resetting.[5]
A bootstrap paradox, also known as aninformation loop, aninformation paradox,[6]anontological paradox,[7]or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of eitherretrocausalityortime travel.[8][9][10][11]
Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere".[8]Such causally looped events then exist inspacetime, but their origin cannot be determined.[8][9]The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical.[9][6][12]A notable example occurs in the 1958science fictionshort story"—All You Zombies—", byRobert A. Heinlein, wherein the main character, anintersexindividual, becomes both their own mother and father; the 2014 filmPredestinationis based on the story. Allen Everett gives the movieSomewhere in Timeas an example involving an object with no origin: an old woman gives a watch to a playwright who later travels back in time and meets the same woman when she was young, and shows her the watch that she will later give to him.[6]An example of information which "came from nowhere" is in the movieStar Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula fortransparent aluminumto the 20th-century engineer who supposedly invented it.
Smeenk uses the term "predestination paradox" to refer specifically to situations in which a time traveler goes back in time to try to prevent some event in the past.[7]
The "predestination paradox" is a concept in time travel and temporal mechanics, often explored in science fiction. It occurs when a future event is the cause of a past event, which in turn becomes the cause of the future event, forming a self-sustaining loop in time. This paradox challenges conventional understandings of cause and effect, as the events involved are both the origin and the result of each other. A notable example is found in the TV seriesDoctor Who, where acharacter saves her father in the past, fulfilling a memory he had shared with her as a child about a strange woman having saved his life. The predestination paradox raises philosophical questions about free will, determinism, and the nature of time itself. It is commonly used as a narrative device in fiction to highlight the interconnectedness of events and the inevitability of certain outcomes.
The consistency paradox or grandfather paradox occurs when the past is changed in any way that directly negates the conditions required for the time travel to occur in the first place, thus creating a contradiction. A common example given is traveling to the past and preventing the conception of one's ancestors (such as causing the death of the ancestor's parent beforehand), thus preventing the conception of oneself. If the traveler were not born, then it would not be possible to undertake such an act in the first place; therefore, the ancestor proceeds to beget the traveler's next-generation ancestor and secure the line to the traveler. There is no predicted outcome to this scenario.[8]Consistency paradoxes occur whenever changing the past is possible.[9]A possible resolution is that a time travellercando anything thatdidhappen, butcannotdo anything thatdid nothappen. Doing something that did not happen results in a contradiction.[8]This is referred to as theNovikov self-consistency principle.
The grandfather paradox encompasses any change to the past,[13]and it is presented in many variations, including killing one's past self.[14][15]Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written intoAmazing Storiesin the 1920s.[16]Another variant of the grandfather paradox is the "Hitler paradox" or "Hitler's murder paradox", in which the protagonist travels back in time to murderAdolf Hitlerbefore he can rise to power in Germany, thus preventingWorld War IIand theHolocaust. Rather than necessarily physically preventing time travel, the action removes anyreasonfor the travel, along with any knowledge that the reason ever existed.[17]
Physicist John Garrison et al. give a variation of the paradox of an electronic circuit that sends a signal through a time machine to shut itself off, and receives the signal before it sends it.[18][19]
Newcomb's paradox is athought experimentshowing an apparent contradiction between theexpected utilityprinciple and thestrategic dominanceprinciple.[20]The thought experiment is often extended to explorecausalityand free will by allowing for "perfect predictors": if perfect predictors of the future exist, for example if time travel exists as a mechanism for making perfect predictions[how?], then perfect predictions appear to contradict free will because decisions apparently made with free will are already known to the perfect predictor[clarification needed].[21][22]Predestinationdoes not necessarily involve asupernaturalpower, and could be the result of other "infallible foreknowledge" mechanisms.[23]Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.[24]
Even without knowing whether time travel to the past is physically possible, it is possible to show usingmodal logicthat changing the past results in a logical contradiction. If it is necessarily true that the past happened in a certain way, then it is false and impossible for the past to have occurred in any other way. A time traveler would not be able to change the past from the way itis,but would only act in a way that is already consistent with whatnecessarilyhappened.[25][26]
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible. For example, the philosopherBradley Dowdenmade this sort of argument in the textbookLogical Reasoning, arguing that the possibility of creating a contradiction rules out time travel to the past entirely. However, some philosophers and scientists believe that time travel into the past need not be logically impossible provided that there is no possibility of changing the past,[13]as suggested, for example, by theNovikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopherNorman Swartz.[27]
A recent proposed resolution argues that if time is not an inherent property of the universe but is insteademergentfrom the laws ofentropy, as some modern theories suggest,[28][29]then it presents a natural solution to the Grandfather Paradox.[30]In this framework, "time travel" is reinterpreted not as movement along a linear continuum but as a reconfiguration of the present state of the universe to match a prior entropic configuration. Because the original chronological sequence—including events like the time traveler’s birth—remains preserved in the universe’s irreversible entropic progression, actions within the reconfigured state cannot alter the causal history that produced the traveler. This avoids paradoxes by treating time as a thermodynamic artifact rather than a mutable dimension.
Consideration of the possibility of backward time travel in a hypothetical universe described by aGödel metricled famed logicianKurt Gödelto assert that time might itself be a sort of illusion.[31][32]He suggests something along the lines of theblock timeview, in which time is just another dimension like space, with all events at all times being fixed within this four-dimensional "block".[citation needed]
Sergey Krasnikovwrites that these bootstrap paradoxes – information or an object looping through time – are the same; the primary apparent paradox is a physical system evolving into a state in a way that is not governed by its laws.[33]: 4He does not find these paradoxical and attributes problems regarding the validity of time travel to other factors in the interpretation of general relativity.[33]: 14–16
A 1992 paper by physicists Andrei Lossev andIgor Novikovlabeled such items without origin asJinn, with the singular termJinnee.[34]: 2311–2312This terminology was inspired by theJinnof theQuran, which are described as leaving no trace when they disappear.[35]: 200–203Lossev and Novikov allowed the term "Jinn" to cover both objects and information with the reflexive origin; they called the former "Jinn of the first kind", and the latter "Jinn of the second kind".[6][34]: 2315–2317[35]: 208They point out that an object making circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency; thesecond law of thermodynamicsseems to require that the object tends to a lower energy state throughout its history, and such objects that are identical in repeating points in their history seem to contradict this, but Lossev and Novikov argued that since the second law only requires entropy to increase inclosedsystems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy.[6][35]: 200–203They emphasize that there is no "strict difference" between Jinn of the first and second kind.[34]: 2320Krasnikov equivocates between "Jinn", "self-sufficient loops", and "self-existing objects", calling them "lions" or "looping or intruding objects", and asserts that they are no less physical than conventional objects, "which, after all, also could appear only from either infinity or a singularity."[33]: 8–9
The self-consistency principle developed byIgor Dmitriyevich Novikov[36]: p. 42 note 10expresses one view as to how backwardtime travelwould be possible without the generation of paradoxes. According to this hypothesis, even thoughgeneral relativitypermits someexact solutionsthat allow fortime travel[37]that containclosed timelike curvesthat lead back to the same point in spacetime,[38]physics in or nearclosed timelike curves(time machines) can only be consistent with the universal laws of physics, and thus only self-consistent events can occur. Anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from happening, since this would represent an inconsistency. The authors concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.[39]
PhysicistJoseph Polchinskiconsidered a potentially paradoxical situation involving abilliard ballthat is fired into awormholeat just the right angle such that it will be sent back in time and collides with its earlier self, knocking it off course, which would stop it from entering the wormhole in the first place.Kip Thornereferred to this problem as "Polchinski's paradox".[39]Thorne and two of his students at Caltech, Fernando Echeverria and Gunnar Klinkhammer, went on to find a solution that avoided any inconsistencies, and found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case.[40]Later analysis by Thorne andRobert Forwardshowed that for certain initial trajectories of the billiard ball, there could be an infinite number of self-consistent solutions.[39]It is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.[41]: 184The lack of constraints on initial conditions only applies to spacetime outside of thechronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.[41]: 187–188
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as anad hocsolution, and supposes that there are far more damaging implications of time travel.[42]Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity.[33]: 14–16Another conjecture, thecosmic censorship hypothesis, suggests that every closed timelike curve passes through anevent horizon, which prevents such causal loops from being observed.[43]
The interacting-multiple-universes approach is a variation of themany-worlds interpretationof quantum mechanics that involves time travelers arriving in a different universe than the one from which they came; it has been argued that, since travelers arrive in a different universe's history and not their history, this is not "genuine" time travel.[44]Stephen Hawking has argued for thechronology protection conjecture, that even if the MWI is correct, we should expect each time traveler to experience a single self-consistent history so that time travelers remain within their world rather than traveling to a different one.[45]
David Deutschhas proposed thatquantum computationwith a negative delay—backward time travel—produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning.[46]However Deutsch's self-consistency condition has been demonstrated as capable of being fulfilled to arbitrary precision by any system subject to the laws of classicalstatistical mechanics, even if it is not built up by quantum systems.[47]Allen Everett has also argued that even if Deutsch's approach is correct, it would imply that any macroscopic object composed of multiple particles would be split apart when traveling back in time, with different particles emerging in different worlds.[48]
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Thehysteron proteron(from theGreek:ὕστερον πρότερον,hýsteron próteron, "later earlier") is arhetoricaldevice. It occurs when the first key word of the idea refers to something that happens temporally later than the second key word. The goal is to call attention to the more important idea by placing it first.[1]
The standard example comes from theAeneidofVirgil: "Moriamur, et in media arma ruamus" ("Let us die, and charge into the thick of the fight"; ii. 353).[2]An example of hysteron proteron encountered in everyday life is the common reference to putting on one's "shoes and socks", rather than "socks and shoes".
By this deliberate reversal, hysteron proteron draws attention to the important point, so giving it primacy.
Hysteron proteron is a form ofhyperbaton, which describes general rearrangements of the sentence.[3]
It can also be defined as a figure of speech consisting of the reversal of a natural or rational order (as in "then came the thunder and the lightning").[4]
An example from theQuranthat demonstrates hysteron proteron, verse (aya) number 89–90 fromSuraNumber 21 says that God grantedZechariah'sprayer for a son even though Zechariah was very old and his wife was sterile:
We granted his prayer and gave himJohn, and we made his wife fertile for him.
A more conventional phrasing would be: "We granted his prayer; we made his wife fertile for him; and [having done so] we gave him John." The reversal of the expected sequence (hysteron proteron) in the verse suggests immediacy: Zechariah's prayer was granted without any delay at all, so much so that the detail itself,"We made his wife fertile for him,"was not allowed to intervene between the prayer and its acceptance.[5]
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Inmathematics, theKlein bottle(/ˈklaɪn/) is an example of anon-orientablesurface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is atwo-dimensionalmanifoldon which one cannot define anormal vectorat each point that variescontinuouslyover the whole manifold. Other related non-orientable surfaces include theMöbius stripand thereal projective plane. While a Möbius strip is a surface with aboundary, a Klein bottle has no boundary. For comparison, asphereis an orientable surface with no boundary.
The Klein bottle was first described in 1882 by the mathematicianFelix Klein.[1]
The following square is afundamental polygonof the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing, in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.[2]
To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus animmersionof the Klein bottle in thethree-dimensional space.
This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has noboundary, where the surface stops abruptly, and it isnon-orientable, as reflected in the one-sidedness of the immersion.
The common physical model of a Klein bottle is a similar construction. TheScience Museum in Londonhas a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles were made for the museum by Alan Bennett in 1995.[3]
The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane.[4]
Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed inxyzt-space. The accompanying illustration ("Time evolution...") shows one useful evolution of the figure. Att= 0the wall sprouts from a bud somewhere near the "intersection" point. After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like theCheshire Catbut leaving its ever-expanding smile behind. By the time the growth front gets to where the bud had been, there is nothing there to intersect and the growth completes without piercing existing structure. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space.[4]
More formally, the Klein bottle is thequotient spacedescribed as thesquare[0,1] × [0,1] with sides identified by the relations(0,y) ~ (1,y)for0 ≤y≤ 1and(x, 0) ~ (1 −x, 1)for0 ≤x≤ 1.
Like theMöbius strip, the Klein bottle is a two-dimensionalmanifoldwhich is notorientable. Unlike the Möbius strip, it is aclosedmanifold, meaning it is acompactmanifold without boundary. While the Möbius strip can be embedded in three-dimensionalEuclidean spaceR3, the Klein bottle cannot. It can be embedded inR4, however.[4]
Continuing this sequence, for example creating a 3-manifold which cannot be embedded inR4but can be inR5, is possible; in this case, connecting two ends of aspherinderto each other in the same manner as the two ends of a cylinder for a Klein bottle, creates a figure, referred to as a "spherinder Klein bottle", that cannot fully be embedded inR4.[5]
The Klein bottle can be seen as afiber bundleover thecircleS1, with fibreS1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to beE, the total space, while the base spaceBis given by the unit interval iny, modulo1~0. The projection π:E→Bis then given byπ([x,y]) = [y].
The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips, as described in the followinglimerickbyLeo Moser:[6]
A mathematician namedKleinThought the Möbius band was divine.Said he: "If you glueThe edges of two,You'll get a weird bottle like mine."
The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given aCW complexstructure with one 0-cellP, two 1-cellsC1,C2and one 2-cellD. ItsEuler characteristicis therefore1 − 2 + 1 = 0. The boundary homomorphism is given by∂D= 2C1and∂C1= ∂C2= 0, yielding thehomology groupsof the Klein bottleKto beH0(K,Z) =Z,H1(K,Z) =Z×(Z/2Z)andHn(K,Z) = 0forn> 1.
There is a 2-1covering mapfrom thetorusto the Klein bottle, because two copies of thefundamental regionof the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. Theuniversal coverof both the torus and the Klein bottle is the planeR2.
Thefundamental groupof the Klein bottle can be determined as thegroup of deck transformationsof the universal cover and has thepresentation⟨a,b|ab=b−1a⟩. It follows that it is isomorphic toZ⋊Z{\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }, the only nontrivial semidirect product of the additive group of integersZ{\displaystyle \mathbb {Z} }with itself.
Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to theHeawood conjecture, a generalization of thefour color theorem, which would require seven.
A Klein bottle is homeomorphic to theconnected sumof twoprojective planes.[7]It is also homeomorphic to a sphere plus twocross-caps.
When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.[2]
Dissecting a Klein bottle into halves along itsplane of symmetryresults in two mirror imageMöbius strips, i.e. one with a left-handed half-twist and the other with a right-handed half-twist (one of these is pictured on the right). Remember that the intersection pictured is not really there.[8]
One description of the types of simple-closed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients. This group is isomorphic toZ×Z2. Up to reversal of orientation, the only homology classes which contain simple-closed curves are as follows: (0,0), (1,0), (1,1), (2,0), (0,1). Up to reversal of the orientation of a simple closed curve, if it lies within one of the two cross-caps that make up the Klein bottle, then it is in homology class (1,0) or (1,1); if it cuts the Klein bottle into two Möbius strips, then it is in homology class (2,0); if it cuts the Klein bottle into an annulus, then it is in homology class (0,1); and if bounds a disk, then it is in homology class (0,0).[4]
To make the "figure 8" or "bagel"immersionof the Klein bottle, one can start with aMöbius stripand curl it to bring the edge to the midline; since there is only one edge, it will meet itself there, passing through the midline. It has a particularly simple parametrization as a "figure-8" torus with a half-twist:[4]
for 0 ≤θ< 2π, 0 ≤v< 2π andr> 2.
In this immersion, the self-intersection circle (where sin(v) is zero) is a geometriccirclein thexyplane. The positive constantris the radius of this circle. The parameterθgives the angle in thexyplane as well as the rotation of the figure 8, andvspecifies the position around the 8-shaped cross section. With the above parametrization the cross section is a 2:1Lissajous curve.
A non-intersecting 4-D parametrization can be modeled after that of theflat torus:
whereRandPare constants that determine aspect ratio,θandvare similar to as defined above.vdetermines the position around the figure-8 as well as the position in the x-y plane.θdetermines the rotational angle of the figure-8 as well and the position around the z-w plane.εis any small constant andεsinvis a smallvdependent bump inz-wspace to avoid self intersection. Thevbump causes the self intersecting 2-D/planar figure-8 to spread out into a 3-D stylized "potato chip" or saddle shape in the x-y-w and x-y-z space viewed edge on. Whenε=0the self intersection is a circle in the z-w plane <0, 0, cosθ, sinθ>.[4]
The pinched torus is perhaps the simplest parametrization of the Klein bottle in both three and four dimensions. It can be viewed as a variant of a torus that, in three dimensions, flattens and passes through itself on one side. Unfortunately, in three dimensions this parametrization has twopinch points, which makes it undesirable for some applications. In four dimensions thezamplitude rotates into thewamplitude and there are no self intersections or pinch points.[4]
One can view this as a tube or cylinder that wraps around, as in a torus, but its circular cross section flips over in four dimensions, presenting its "backside" as it reconnects, just as a Möbius strip cross section rotates before it reconnects. The 3D orthogonal projection of this is the pinched torus shown above. Just as a Möbius strip is a subset of a solid torus, the Möbius tube is a subset of a toroidally closedspherinder(solidspheritorus).
The following parametrization of the usual 3-dimensional immersion of the bottle itself is much more complicated.
for 0 ≤u< π and 0 ≤v< 2π.[4]
Regular 3D immersions of the Klein bottle fall into threeregular homotopyclasses.[9]The three are represented by:
The traditional Klein bottle immersion isachiral. The figure-8 immersion is chiral. (The pinched torus immersion above is not regular, as it has pinch points, so it is not relevant to this section.)
If the traditional Klein bottle is cut in its plane of symmetry it breaks into two Möbius strips of opposite chirality. A figure-8 Klein bottle can be cut into two Möbius strips of thesamechirality, and cannot be regularly deformed into its mirror image.[4]
The generalization of the Klein bottle to highergenusis given in the article on thefundamental polygon.[10]
In another order of ideas, constructing3-manifolds, it is known that asolid Klein bottleishomeomorphicto theCartesian productof aMöbius stripand a closed interval. Thesolid Klein bottleis the non-orientable version of thesolid torus, equivalent toD2×S1.{\displaystyle D^{2}\times S^{1}.}
AKlein surfaceis, as forRiemann surfaces, a surface with an atlas allowing thetransition mapsto be composed usingcomplex conjugation. One can obtain the so-calleddianalytic structureof the space, and it has only one side.[11]
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Meno(/ˈmiːnoʊ/;Ancient Greek:Μένων,Ménōn) is aSocratic dialoguewritten byPlatoaround 385 BC., but set at an earlier date around 402 BC.[1]Menobegins the dialogue by asking Socrates whethervirtue(inAncient Greek:ἀρετή,aretē) can be taught, acquired by practice, or comes bynature.[2]In order to determine whether virtue is teachable or not,Socratestells Meno that they first need to determine what virtue is.[3]When the characters speak of virtue, oraretē, they refer to virtue in general, rather than particular virtues, such as justice or temperance. The first part of the work showcasesSocratic dialectical style; Meno, unable to adequately define virtue, is reduced to confusion oraporia.[4]Socrates suggests that they seek an adequate definition for virtue together. In response, Meno suggests that it is impossible to seek what one does not know, because one will be unable to determine whether one has found it.[5]
Socrates challenges Meno's argument, often called "Meno's Paradox", "Learner's Paradox", or the "Arabic Paradox",[citation needed]by introducing the theory of knowledge as recollection (anamnesis). As presented in the dialogue, the theory proposes that souls are immortal and know all things in a disembodied state; learning in the embodied is actually a process of recollecting that which the soul knew before it came into a body.[6]Socrates demonstrates recollection in action by posing a mathematical puzzle to one of Meno's slaves.[7]Subsequently, Socrates and Meno return to the question of whether virtue is teachable, employing the method of hypothesis. Near the end of the dialogue, Meno poses another famous puzzle, called "the Meno problem" or "the Value Problem for Knowledge", which questions why knowledge is valued more highly than true belief.[8]In response, Socrates provides a famous and somewhat enigmatic distinction between knowledge and true belief.[9]
Plato'sMenois a Socratic dialogue in which the two main speakers,SocratesandMeno(alsotransliteratedas "Menon"), discuss human virtue: what it is, and whether or not it can be taught. Meno is visiting Athens fromThessalywith a large entourage of slaves attending him. Young, good-looking and well-born, he is a student ofGorgias, a prominentsophistwhose views on virtue clearly influence that of Meno's. Early in the dialogue, Meno claims that he has held forth many times on the subject of virtue, and in front of large audiences.[10]
One of Meno'sslavesalso has a speaking role, as one of the features of the dialogue is Socrates' engagement with the slave to demonstrate his idea ofanamnesis: certain knowledge is innate and "recollected" by the soul through proper inquiry.
Another participant later in the dialogue is Athenian politicianAnytus,[11]later one of theprosecutors of Socrates,[12]with whom Meno is friendly.
The dialogue begins with Meno asking Socrates to tell him whether virtue can be taught. Socrates says that he does not know what virtue is, and neither does anyone else he knows.[3]Meno responds that, according to Gorgias, his Sophist mentor, virtue means different for different people, that what is virtuous for a man is to conduct himself in the city so that he helps his friends, injures his enemies, and takes care all the while that he personally comes to no harm. Virtue is different for a woman, he says. Her domain is the management of the household, and she is supposed to obey her husband. He says that children (male and female) have their own proper virtue, and so do old men—free orslaves.[13]Socrates objects: there must be some virtue common to all human beings.
Socrates rejects the idea that human virtue depends on a person's sex or age. He leads Meno towards the idea that virtues are common to all people, thatsophrosunê('temperance', i.e. exercise ofself-control) anddikê(akadikaiosunê; 'justice', i.e. refrain from harming others) are virtues even in children and old men.[14]Meno proposes to Socrates that the "capacity to govern men" may be a virtue common to all people. Socrates points out to the slaveholder that "governing well" cannot be a virtue of a slave, because then he would not be a slave.[15]
One of the errors that Socrates points out is that Meno lists many particular virtues without defining a common feature inherent to virtues which makes them thus. Socrates remarks that Meno makes many out of one, like somebody who breaks a plate.[16]
Meno proposes that virtue is the desire for good things and the power to get them. Socrates points out that this raises a second problem—many people do not recognize evil.[17]The discussion then turns to the question of accounting for the fact that so many people are mistaken about good and evil and take one for the other. Socrates asks Meno to consider whether good things must be acquired virtuously in order to be really good.[18]Socrates leads onto the question of whether virtue is one thing or many.
No satisfactory definition of virtue emerges in theMeno. Socrates' comments, however, show that he considers a successful definition to be unitary, rather than a list of varieties of virtue, that it must contain all and only those terms which are genuine instances of virtue, and must not becircular.[19]
Meno asks Socrates:[20][21]
And how will you enquire, Socrates, into that which you do not know? What will you put forth as the subject of enquiry? And if you find what you want, how will you ever know that this is the thing which you did not know?
Socrates rephrases the question, which has come to be the canonical statement of Meno's paradox or the paradox of inquiry:[20][22]
[A] man cannot enquire either about that which he knows, or about that which he does not know; for if he knows, he has no need to enquire; and if not, he cannot; for he does not know the very subject about which he is to enquire.
Socrates responds to thissophisticalparadox with amythos('narrative' or 'fiction') according to which souls are immortal and have learned everything prior totransmigratinginto the human body. Since the soul has had contact with real things prior to birth, we have only to 'recollect' them when alive. Such recollection requiresSocratic questioning, which according to Socrates is not teaching. Socrates demonstrates his method of questioning and recollection by interrogating a slave who is ignorant of geometry.
Socrates begins one of the most influential dialogues of Western philosophy regarding the argument forinborn knowledge. By drawing geometric figures in the ground Socrates demonstrates that the slave is initially unaware of the length that a side must be in order to double the area of a square with 2-foot sides. The slave guesses first that the original side must be doubled in length (4 feet), and when this proves too much, that it must be 3 feet. This is still too much, and the slave is at a loss.
Socrates claims that before he got hold of him the slave (who has been picked at random from Meno's entourage) might have thought he could speak "well and fluently" on the subject of a square double the size of a given square.[23]Socrates comments that this "numbing" he caused in the slave has done him no harm and has even benefited him.[24]
Socrates then adds three more squares to the original square, to form a larger square four times the size. He draws four diagonal lines which bisect each of the smaller squares. Through questioning, Socrates leads the slave to the discovery that the square formed by these diagonals has an area of eight square feet, double that of the original. He says that the slave has "spontaneously recovered" knowledge which he knew from before he was born,[25]without having been taught. Socrates is satisfied that new beliefs were "newly aroused" in the slave.
After witnessing the example with the slave boy, Meno tells Socrates that he thinks that Socrates is correct in his theory of recollection, to which Socrates agrees:[20][26]
Some things I have said of which I am not altogether confident. But that we shall be better and braver and less helpless if we think that we ought to enquire, than we should have been if we indulged in the idle fancy that there was no knowing and no use in seeking to know what we do not know; that is a theme upon which I am ready to fight, in word and deed, to the utmost of my power.
Meno now beseeches Socrates to return to the original question, how virtue is acquired, and in particular, whether or not it is acquired by teaching or through life experience. Socrates proceeds on the hypothesis that virtue is knowledge, and it is quickly agreed that, if this is true, virtue is teachable. They turn to the question of whether virtue is indeed knowledge. Socrates is hesitant, because, if virtue were knowledge, there should be teachers and learners of it, but there are none.
CoincidentallyAnytusappears, whom Socrates praises as the son ofAnthemion, who earned his fortune with intelligence and hard work. He says that Anthemion had his son well-educated and so Anytus is well-suited to join the investigation. Socrates suggests that the sophists are teachers of virtue. Anytus is horrified, saying that he neither knows any, nor cares to know any. Socrates then questions why it is that men do not always produce sons of the same virtue as themselves. He alludes to other notable male figures, such asThemistocles,Aristides,PericlesandThucydides, and casts doubt on whether these men produced sons as capable of virtue as themselves. Anytus becomes offended and accuses Socrates ofslander, warning him to be careful expressing such opinions. (The historical Anytus was one of Socrates' accusers in histrial.) Socrates suggests that Anytus does not realize what slander is, and continues his dialogue with Meno as to the definition of virtue.
After the discussion with Anytus, Socrates returns to quizzing Meno for his own thoughts on whether the sophists are teachers of virtue and whether virtue can be taught. Meno is again at a loss, and Socrates suggests that they have made a mistake in agreeing that knowledge is required for virtue. He points out the similarities and differences between "true belief" and "knowledge". True beliefs are as useful to us as knowledge, but they often fail to "stay in their place" and must be "tethered" by what he callsaitiaslogismos('calculation of reason' or 'reasoned explanation'), immediately adding that this isanamnesis, or recollection.[28]
Whether Plato intends that the tethering of true beliefs with reasoned explanations must always involveanamnesisis explored in later interpretations of the text.[29][30]Socrates' distinction between "true belief" and "knowledge" forms the basis of the philosophicaldefinition of knowledgeas "justified true belief".Myles Burnyeatand others, however, have argued that the phraseaitias logismosrefers to a practical working out of a solution, rather than a justification.[31]
Socrates concludes that, in the virtuous people of the present and the past, at least, virtue has been the result of divine inspiration, akin to the inspiration of the poets, whereas a knowledge of it will require answering the basic question,what is virtue?. In most modern readings these closing remarks are "evidently ironic",[32]but Socrates' invocation of the gods may be sincere, albeit "highly tentative".[33]
This passage in the Meno is often seen as the first statement of the problem of the value of knowledge or "the Meno problem":how is knowledge more valuable than mere true belief?[34]The nature of knowledge and belief is also discussed in theThaetetus.
Meno'stheme is also dealt with in the dialogueProtagoras, where Plato ultimately has Socrates arrive at the opposite conclusion: virtuecanbe taught. Likewise, while inProtagorasknowledge is uncompromisingly this-worldly, inMenothe theory of recollection points to a link between knowledge and eternal truths.[19]
"[A] man cannot search either for what he knows or for what he does not know; He cannot search for what he knows--since he knows it, there is no need to search--nor for what he does not know, for he does not know what to look for."
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Metamorphic codeis code that when run outputs alogically equivalentversion of its own code under someinterpretation. This is similar to aquine, except that a quine'ssource codeis exactly equivalent to its own output. Metamorphic code also usually outputsmachine codeand not its own source code.
Metamorphic code is used bycomputer virusesto avoid thepattern recognitionofanti-virus software. Metamorphic viruses often translate their own binary code into a temporary representation, editing the temporary representation of themselves and then translate the edited form back to machine code again.[1]This procedure is done with the virus itself, and thus also the metamorphic engine itself undergoes changes, which means that no part of the virus stays the same. This differs frompolymorphic code, where the polymorphic engine can not rewrite its own code.
Metamorphic code is used by some viruses when they are about to infect new files, and the result is that the next generation will never look like current generation. The mutated code will do exactly the same thing (under theinterpretationused), but the child's binary representation will typically be completely different from the parent's. Mutation can be achieved using techniques like insertingNOPinstructions (brute force), changing whatregistersto use, changing flow control with jumps, changing machine instructions to equivalent ones or reordering independent instructions.
Metamorphism does not protect a virus againstheuristic analysis.[citation needed]
Metamorphic code can also mean that a virus is capable of infecting executables from two or more differentoperating systems(such asWindowsandLinux) or even differentcomputer architectures. Often, the virus does this by carrying several viruses within itself. The beginning of the virus is then coded so that it translates to correct machine-code for all of the platforms that it is supposed to execute in.[2]This is used primarily inremote exploit injection codewhere the target platform is unknown.
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Inmathematics, aMöbius strip,Möbius band, orMöbius loop[a]is asurfacethat can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered byJohann Benedict ListingandAugust Ferdinand Möbiusin 1858, but it had already appeared inRomanmosaics from the third centuryCE. The Möbius strip is anon-orientablesurface, meaning that within it one cannot consistently distinguishclockwisefrom counterclockwise turns. Every non-orientable surface contains a Möbius strip.
As an abstracttopological space, the Möbius strip can be embedded into three-dimensionalEuclidean spacein many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with aknottedcenterline. Any two embeddings with the same knot for the centerline and the same number and direction of twists aretopologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a singleboundary curve.
Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as aruled surfaceby a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as adevelopable surfaceor befolded flat; the flattened Möbius strips include thetrihexaflexagon. The Sudanese Möbius strip is aminimal surfacein ahypersphere, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can formsurfaces of constant curvature. Certain highly symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip.
The many applications of Möbius strips includemechanical beltsthat wear evenly on both sides, dual-trackroller coasterswhose carriages alternate between the two tracks, andworld mapsprinted so thatantipodesappear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results insocial choice theory. In popular culture, Möbius strips appear in artworks byM. C. Escher,Max Bill, and others, and in the design of therecycling symbol. Many architectural concepts have been inspired by the Möbius strip, including the building design for theNASCAR Hall of Fame. Performers includingHarry Blackstone Sr.andThomas Nelson Downshave based stage magic tricks on the properties of the Möbius strip. ThecanonsofJ. S. Bachhave been analyzed using Möbius strips. Many works ofspeculative fictionfeature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.
The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematiciansJohann Benedict ListingandAugust Ferdinand Möbiusin1858.[2]However, it had been known long before, both as a physical object and in artistic depictions; in particular, it can be seen in several Roman mosaics from thethird century CE.[3][4]In many cases these merely depict coiled ribbons as boundaries. When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent tountwisted rings. Therefore, whether the ribbon is a Möbius strip may be coincidental, rather than a deliberate choice. In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did notmatch up.[3]Another mosaic from the town ofSentinum(depicted) shows thezodiac, held by the godAion, as a band with only a single twist. There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip. Some other ancient depictions of theourobourosor offigure-eight-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type isunclear.[4]
Independently of the mathematical tradition, machinists have long known thatmechanical beltswear half as quickly when they form Möbius strips, because they use the entire surface of the belt rather than only the inner surface of an untwisted belt.[3]Additionally, such a belt may be less prone to curling from side to side. An early written description of this technique dates to 1871, which is after the first mathematical publications regarding the Möbius strip. Much earlier, an image of achain pumpin a work ofIsmail al-Jazarifrom 1206 depicts a Möbius strip configuration for its drivechain.[4]Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto agarment.[3]
The Möbius strip has several curious properties. It is anon-orientable surface: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image. In particular, a curved arrow pointing clockwise (↻) would return as an arrow pointing counterclockwise (↺), implying that, within the Möbius strip, it is impossible to consistently define what it means to be clockwise or counterclockwise. It is the simplest non-orientable surface: any other surface is non-orientable if and only if it has a Möbius strip as asubset.[5]Relatedly, when embedded intoEuclidean space, the Möbius strip has only one side. A three-dimensional object that slides one time around the surface of the strip is not mirrored, but instead returns to the same point of the strip on what appears locally to be its other side, showing that both positions are really part of a single side. This behavior is different from familiarorientable surfacesin three dimensions such as those modeled by flat sheets of paper, cylindrical drinking straws, or hollow balls, for which one side of the surface is not connected to the other.[6]However, this is a property of its embedding into space rather than an intrinsic property of the Möbius strip itself: there exist other topological spaces in which the Möbius strip can be embedded so that it has twosides.[7]For instance, if the front and back faces of a cube are glued to each other with a left-right mirror reflection, the result is a three-dimensional topological space (theCartesian productof a Möbius strip with an interval) in which the top and bottom halves of the cube can be separated from each other by a two-sided Möbiusstrip.[b]In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed anuncountable setofdisjointcopies into three-dimensional space, only a countable number of Möbius strips can be simultaneouslyembedded.[9][10][11]
A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, includes all boundary points of the Möbius strip in a single continuous curve. For a Möbius strip formed by gluing and twisting a rectangle, it has twice the length of the centerline of the strip. In this sense, the Möbius strip is different from an untwisted ring and like a circular disk in having only oneboundary.[6]A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is achiralobject with right- orleft-handedness.[12]Möbius strips with odd numbers of half-twists greater than one, or that are knotted before gluing, are distinct as embedded subsets of three-dimensional space, even though they are all equivalent as two-dimensional topologicalsurfaces.[13]More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot and they have the same number of twists as eachother.[14]With an even number of twists, however, one obtains a different topological surface, called theannulus.[15]
The Möbius strip can be continuously transformed into its centerline, by making it narrower while fixing the points on the centerline. This transformation is an example of adeformation retraction, and its existence means that the Möbius strip has many of the same properties as its centerline, which is topologically a circle. In particular, itsfundamental groupis the same as the fundamental group of a circle, aninfinite cyclic group. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up tohomotopy) only by the number of times they loop around the strip.[16]
Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with four half-twists in it (relative to an untwisted annulus or cylinder) rather than two separate strips. Two of the half-twists come from the fact that this thinner strip goes two times through the half-twist in the original Möbius strip, and the other two come from the way the two halves of the thinner strip wrap around each other. The result is not a Möbius strip, but instead is topologically equivalent to a cylinder. Cutting this double-twisted strip again along its centerline produces two linked double-twisted strips. If, instead, a Möbius strip is cut lengthwise, a third of the way across its width, it produces two linked strips. One of the two is a central, thinner, Möbius strip, while the other has twohalf-twists.[6]These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes calledparadromicrings.[17][18]
The Möbius strip can be cut into six mutually adjacent regions, showing that maps on the surface of the Möbius strip can sometimes require six colors, in contrast to thefour color theoremfor theplane.[19]Six colors are always enough. This result is part of theRingel–Youngs theorem, which states how many colors each topological surfaceneeds.[20]The edges and vertices of these six regions formTietze's graph, which is adual graphon this surface for the six-vertexcomplete graphbut cannot bedrawn without crossings on a plane. Another family of graphs that can beembeddedon the Möbius strip, but not on the plane, are theMöbius ladders, the boundaries of subdivisions of the Möbius strip into rectangles meetingend-to-end.[21]These include the utility graph, a six-vertexcomplete bipartite graphwhose embedding into the Möbius strip shows that, unlike in the plane, thethree utilities problemcan be solved on a transparent Möbiusstrip.[22]TheEuler characteristicof the Möbius strip iszero, meaning that for any subdivision of the strip by vertices and edges into regions, the numbersV{\displaystyle V},E{\displaystyle E}, andF{\displaystyle F}of vertices, edges, and regions satisfyV−E+F=0{\displaystyle V-E+F=0}. For instance, Tietze's graph has12{\displaystyle 12}vertices,18{\displaystyle 18}edges, and6{\displaystyle 6}regions;12−18+6=0{\displaystyle 12-18+6=0}.[19]
There are many different ways of defining geometric surfaces with the topology of the Möbius strip, yielding realizations with additional geometric properties.
One way to embed the Möbius strip in three-dimensional Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of itslines.[23]For the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity of the plane's rotation. This can be described as aparametric surfacedefined by equations for theCartesian coordinatesof its points,x(u,v)=(1+v2cosu2)cosuy(u,v)=(1+v2cosu2)sinuz(u,v)=v2sinu2{\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}}for0≤u<2π{\displaystyle 0\leq u<2\pi }and−1≤v≤1{\displaystyle -1\leq v\leq 1},where one parameteru{\displaystyle u}describes the rotation angle of the plane around its central axis and the other parameterv{\displaystyle v}describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in thexy{\displaystyle xy}-plane and is centered at(0,0,0){\displaystyle (0,0,0)}.[24]The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in the plane that it rotates within, and the Möbius strip that it generates forms a slice through thesolid torusswept out by this disk. Because of the one-sidedness of this slice, the sliced torus remainsconnected.[25]
A line or line segment swept in a different motion, rotating in a horizontal plane around the origin as it moves up and down, formsPlücker's conoidor cylindroid, an algebraicruled surfacein the form of a self-crossing Möbiusstrip.[26]It has applications in the design ofgears.[27]
A strip of paper can form aflattenedMöbius strip in the plane by folding it at60∘{\displaystyle 60^{\circ }}angles so that its center line lies along anequilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral triangles, folded at the edges where two triangles meet. Itsaspect ratio– the ratio of the strip's length[c]to its width – is3≈1.73{\displaystyle {\sqrt {3}}\approx 1.73},and the same folding method works for any larger aspectratio.[28][29]For a strip of nine equilateral triangles, the result is atrihexaflexagon, which can be flexed to reveal different parts of itssurface.[30]For strips too short to apply this method directly, one can first "accordion fold" the strip in its wide direction back and forth using an even number of folds. With two folds, for example, a1×1{\displaystyle 1\times 1}strip would become a1×13{\displaystyle 1\times {\tfrac {1}{3}}}folded strip whosecross sectionis in the shape of an 'N' and would remain an 'N' after a half-twist. The narrower accordion-folded strip can then be folded and joined in the same way that a longer stripwould be.[28][29]
The Möbius strip can also be embedded as apolyhedral surfacein space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than thecylinder, which requires six triangles and six vertices, even when represented more abstractly as asimplicial complex.[31][d]A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of afour-dimensional regular simplex. This four-dimensional polyhedral Möbius strip is the onlytightMöbius strip, one that is fully four-dimensional and for which all cuts byhyperplanesseparate it into two parts that are topologically equivalent to disks orcircles.[32]
Other polyhedral embeddings of Möbius strips include one with four convexquadrilateralsas faces, another with three non-convex quadrilateralfaces,[33]and one using the vertices and center point of a regularoctahedron, with a triangularboundary.[34]Every abstract triangulation of theprojective planecan be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of itsfaces;[35]an example is the six-vertex projective plane obtained by adding one vertex to the five-vertex Möbius strip, connected by triangles to each of its boundaryedges.[31]However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedralsurface.[36]To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in thetriangulation.[37]
A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than3≈1.73{\displaystyle {\sqrt {3}}\approx 1.73},the same ratio as for the flat-folded equilateral-triangle version of the Möbiusstrip.[38]This flat triangular embedding can lift to a smooth[e]embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of theplanes.[38]Mathematically, a smoothly embedded sheet of paper can be modeled as adevelopable surface, that can bend but cannotstretch.[39][40]As its aspect ratio decreases toward3{\displaystyle {\sqrt {3}}}, all smooth embeddings seem to approach the same triangularform.[41]
The lengthwise folds of an accordion-folded flat Möbius strip prevent it from forming a three-dimensional embedding in which the layers are separated from each other and bend smoothly without crumpling or stretching away from thefolds.[29]Instead, unlike in the flat-folded case, there is a lower limit to the aspect ratio of smooth rectangular Möbius strips. Their aspect ratio cannot be less thanπ/2≈1.57{\displaystyle \pi /2\approx 1.57},even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above thisbound.[29][42]Without self-intersections, the aspect ratio must be atleast[43]233+23≈1.695.{\displaystyle {\frac {2}{3}}{\sqrt {3+2{\sqrt {3}}}}\approx 1.695.}
For aspect ratios between this boundand3{\displaystyle {\sqrt {3}}},it has been an open problem whether smooth embeddings, without self-intersection,exist.[29][42][43]In 2023,Richard Schwartzannounced a proof that they do not exist, but this result still awaits peer review and publication.[44][45]If the requirement of smoothness is relaxed to allowcontinuously differentiablesurfaces, theNash–Kuiper theoremimplies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratiobecomes.[g]The limiting case, a surface obtained from an infinite strip of the plane between two parallel lines, glued with the opposite orientation to each other, is called theunbounded Möbius stripor the realtautological line bundle.[46]Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclideanspace.[47]
The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have a known analytic description, but can be calculated numerically, and has been the subject of much study inplate theorysince the initial work on this subject in 1930 byMichael Sadowsky.[39][40]It is also possible to findalgebraic surfacesthat contain rectangular developable Möbiusstrips.[48][49]
The edge, orboundary, of a Möbius strip istopologically equivalentto acircle. In common forms of the Möbius strip, it has a different shape from a circle, but it isunknotted, and therefore the whole strip can be stretched without crossing itself to make the edge perfectlycircular.[50]One such example is based on the topology of theKlein bottle, a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can beimmersed(allowing the surface to cross itself in certain restricted ways). A Klein bottle is the surface that results when two Möbius strips are glued together edge-to-edge, and – reversing that process – a Klein bottle can be sliced along a carefully chosen cut to produce two Möbiusstrips.[51]For a form of the Klein bottle known as Lawson's Klein bottle, the curve along which it is sliced can be made circular, resulting in Möbius strips with circularedges.[52]
Lawson's Klein bottle is a self-crossingminimal surfacein theunit hypersphereof 4-dimensional space, the set of points of the form(cosθcosϕ,sinθcosϕ,cos2θsinϕ,sin2θsinϕ){\displaystyle (\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )}for0≤θ<π,0≤ϕ<2π{\displaystyle 0\leq \theta <\pi ,0\leq \phi <2\pi }.[53]Half of this Klein bottle, the subset with0≤ϕ<π{\displaystyle 0\leq \phi <\pi }, gives a Möbius strip embedded in the hypersphere as a minimal surface with agreat circleas itsboundary.[54]This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the1970s.[55]Geometrically Lawson's Klein bottle can be constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the Sudanese Möbius strip is obtained by sweeping a semicircle instead of a circle, or equivalently by slicing the Klein bottle along a circle that is perpendicular to all of the sweptcircles.[52][56]Stereographic projectiontransforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of itsboundary.[52]The most symmetric projection is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles, but produces an unbounded embedding with the projection point removed from itscenterline.[54]Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to theorthogonal groupO(2){\displaystyle \mathrm {O} (2)},the group of symmetries of acircle.[53]
The Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself. Another form of the Möbius strip, called thecross-caporcrosscap, also has a circular boundary, but otherwise stays on only one side of the plane of thiscircle,[57]making it more convenient for attaching onto circular holes in other surfaces. In order to do so, it crosses itself. It can be formed by removing aquadrilateralfrom the top of a hemisphere, orienting the edges of the quadrilateral in alternating directions, and then gluing opposite pairs of these edges consistently with thisorientation.[58]The two parts of the surface formed by the two glued pairs of edges cross each other with apinch pointlike that of aWhitney umbrellaat each end of the crossingsegment,[59]the same topological structure seen in Plücker'sconoid.[26]
The open Möbius strip is therelative interiorof a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given aRiemannian geometryof constant positive, negative, or zeroGaussian curvature. The cases of negative and zero curvature form geodesically complete surfaces, which means that allgeodesics("straight lines" on the surface) may be extended indefinitely in either direction.
Theminimal surfacesare described as having constant zeromean curvatureinstead of constant Gaussian curvature. The Sudanese Möbius strip was constructed as a minimal surface bounded by a great circle in a 3-sphere, but there is also a unique complete (boundaryless) minimal surface immersed in Euclidean space that has the topology of an open Möbius strip. It is called the Meeks Möbiusstrip,[64]after its 1982 description byWilliam Hamilton Meeks, III.[65]Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimalsurfaces.[66]Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to theBjörling problem, which defines a minimal surface uniquely from its boundary curve and tangent planes along thiscurve.[67]
The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines istopologically equivalentto the open Möbiusstrip.[68]One way to see this is to extend the Euclidean plane to thereal projective planeby adding one more line, theline at infinity. Byprojective dualitythe space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of Euclidean lines, punctures this space of projectivelines.[69]Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbiusstrip.[63]The space of lines in thehyperbolic planecan be parameterized byunordered pairsof distinct points on a circle, the pairs of points at infinity of each line. This space, again, has the topology of an open Möbiusstrip.[70]
These spaces of lines are highly symmetric. The symmetries of Euclidean lines include theaffine transformations, and the symmetries of hyperbolic lines include theMöbius transformations.[71]The affine transformations and Möbius transformations both form6-dimensionalLie groups, topological spaces having a compatiblealgebraic structuredescribing the composition ofsymmetries.[72][73]Because every line in the plane is symmetric to every other line, the open Möbius strip is ahomogeneous space, a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are calledsolvmanifolds, and the Möbius strip can be used as acounterexample, showing that not every solvmanifold is anilmanifold, and that not every solvmanifold can be factored into adirect productof acompactsolvmanifoldwithRn{\displaystyle \mathbb {R} ^{n}}.These symmetries also provide another way to construct the Möbius strip itself, as agroup modelof these Lie groups. A group model consists of a Lie group and astabilizer subgroupof its action; contracting thecosetsof the subgroup to points produces a space with the same topology as the underlying homogenous space. In the case of the symmetries of Euclidean lines, the stabilizer of thex{\displaystyle x}-axisconsists of all symmetries that take the axis to itself. Each lineℓ{\displaystyle \ell }corresponds to a coset, the set of symmetries that mapℓ{\displaystyle \ell }to thex{\displaystyle x}-axis.Therefore, thequotient space, a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbiusstrip.[74]
Beyond the already-discussed applications of Möbius strips to the design of mechanical belts that wear evenly on their entire surface, and of the Plücker conoid to the design of gears, other applications of Möbius strips include:
Scientists have also studied the energetics ofsoap filmsshaped as Möbius strips,[88][89]thechemical synthesisofmoleculeswith a Möbius strip shape,[90][91]and the formation of largernanoscaleMöbius strips usingDNA origami.[92]
Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting byCorrado Cagli(memorialized in a poem byCharles Olson),[93][94]and two prints byM. C. Escher:Möbius Band I(1961), depicting three foldedflatfishbiting each others' tails; andMöbius Band II(1963), depicting ants crawling around alemniscate-shaped Möbius strip.[95][96]It is also a popular subject ofmathematical sculpture, including works byMax Bill(Endless Ribbon, 1953),José de Rivera(Infinity, 1967), andSebastián.[93]Atrefoil-knottedMöbius strip was used inJohn Robinson'sImmortality(1982).[97]Charles O. Perry'sContinuum(1976) is one of several pieces by Perry exploring variations of the Möbius strip.[98]
Because of their easily recognized form, Möbius strips are a common element ofgraphic design.[97]The familiarthree-arrow logoforrecycling, designed in 1970, is based on the smooth triangular form of the Möbiusstrip,[99]as was the logo for the environmentally-themedExpo '74.[100]Some variations of the recycling symbol use a different embedding with three half-twists instead ofone,[99]and the original version of theGoogle Drivelogo used a flat-folded three-twist Möbius strip, as have other similar designs.[101]The BrazilianInstituto Nacional de Matemática Pura e Aplicada(IMPA) uses a stylized smooth Möbius strip as their logo, and has a matching large sculpture of a Möbius strip on display in their building.[102]The Möbius strip has also featured in the artwork forpostage stampsfrom countries including Brazil, Belgium, the Netherlands, andSwitzerland.[103][104]
Möbius strips have been a frequent inspiration for the architectural design of buildings and bridges. However, many of these are projects or conceptual designs rather than constructed objects, or stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture.[105][106]An example is theNational Library of Kazakhstan, for which a building was planned in the shape of a thickened Möbius strip but refinished with a different design after the original architects pulled out of the project.[107]One notable building incorporating a Möbius strip is theNASCAR Hall of Fame, which is surrounded by a large twisted ribbon of stainless steel acting as a façade and canopy, and evoking the curved shapes of racing tracks.[108]On a smaller scale,Moebius Chair(2006) byPedro Reyesis acourting benchwhose base and sides have the form of a Möbius strip.[109]As a form ofmathematics and fiber arts,scarveshave beenknitinto Möbius strips since the work ofElizabeth Zimmermannin the early 1980s.[110]Infood styling, Möbius strips have been used for slicingbagels,[111]making loops out ofbacon,[112]and creating new shapes forpasta.[113]
Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked inspeculative fictionas the basis for atime loopinto which unwary victims may become trapped. Examples of this trope includeMartin Gardner's "No-Sided Professor" (1946),Armin Joseph Deutsch's "A Subway Named Mobius" (1950) and the filmMoebius(1996) based on it. An entire world shaped like a Möbius strip is the setting ofArthur C. Clarke's "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever inventions in multiple stories ofWilliam Hazlett Upsonfrom the 1940s.[114]Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with a twist; these includeMarcel Proust'sIn Search of Lost Time(1913–1927),Luigi Pirandello'sSix Characters in Search of an Author(1921),Frank Capra'sIt's a Wonderful Life(1946),John Barth'sLost in the Funhouse(1968),Samuel R. Delany'sDhalgren(1975) and the filmDonnie Darko(2001).[115]
One of themusical canonsbyJ. S. Bach, the fifth of 14 canons (BWV 1087) discovered in 1974 in Bach's copy of theGoldberg Variations, features aglide-reflectsymmetry in which each voice in the canon repeats, withinverted notes, the same motif from two measures earlier. Because of this symmetry, this canon can be thought of as having its score written on a Möbius strip.[116][h]Inmusic theory, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, thechromatic circle. Because the Möbius strip is theconfiguration spaceof two unordered points on a circle, the space of alltwo-note chordstakes the shape of a Möbius strip. This conception, and generalizations to more points, is a significantapplication of orbifolds to music theory.[117][118]Modern musical groups taking their name from the Möbius strip include American electronic rock trioMobius Band[119]and Norwegian progressive rock bandRing Van Möbius.[120]
Möbius strips and their properties have been used in the design ofstage magic. One such trick, known as the Afghan bands, uses the fact that the Möbius strip remains in one piece as a single strip when cut lengthwise. It originated in the 1880s, and was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such asHarry Blackstone Sr.andThomas Nelson Downs.[121][122]
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Atemporal paradox,time paradox, ortime travel paradox, is aparadox, an apparent contradiction, or logical contradiction associated with the idea oftime travelor other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistictime dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.
Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox.[1]Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps."[2][3]Consistency paradoxes, on the other hand, are those where future events influence the past to cause an apparent contradiction, exemplified by thegrandfather paradox, where a person travels to the past to prevent the conception of one of their ancestors, thus eliminating all the ancestor's descendants.[4]Newcomb's paradoxstems from the apparent contradictions that stem from the assumptions of bothfree willand foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop,[2]but although they appear similar, causal loops are unchanging and self-originating, whereas time loops are constantly resetting.[5]
A bootstrap paradox, also known as aninformation loop, aninformation paradox,[6]anontological paradox,[7]or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of eitherretrocausalityortime travel.[8][9][10][11]
Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere".[8]Such causally looped events then exist inspacetime, but their origin cannot be determined.[8][9]The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical.[9][6][12]A notable example occurs in the 1958science fictionshort story"—All You Zombies—", byRobert A. Heinlein, wherein the main character, anintersexindividual, becomes both their own mother and father; the 2014 filmPredestinationis based on the story. Allen Everett gives the movieSomewhere in Timeas an example involving an object with no origin: an old woman gives a watch to a playwright who later travels back in time and meets the same woman when she was young, and shows her the watch that she will later give to him.[6]An example of information which "came from nowhere" is in the movieStar Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula fortransparent aluminumto the 20th-century engineer who supposedly invented it.
Smeenk uses the term "predestination paradox" to refer specifically to situations in which a time traveler goes back in time to try to prevent some event in the past.[7]
The "predestination paradox" is a concept in time travel and temporal mechanics, often explored in science fiction. It occurs when a future event is the cause of a past event, which in turn becomes the cause of the future event, forming a self-sustaining loop in time. This paradox challenges conventional understandings of cause and effect, as the events involved are both the origin and the result of each other. A notable example is found in the TV seriesDoctor Who, where acharacter saves her father in the past, fulfilling a memory he had shared with her as a child about a strange woman having saved his life. The predestination paradox raises philosophical questions about free will, determinism, and the nature of time itself. It is commonly used as a narrative device in fiction to highlight the interconnectedness of events and the inevitability of certain outcomes.
The consistency paradox or grandfather paradox occurs when the past is changed in any way that directly negates the conditions required for the time travel to occur in the first place, thus creating a contradiction. A common example given is traveling to the past and preventing the conception of one's ancestors (such as causing the death of the ancestor's parent beforehand), thus preventing the conception of oneself. If the traveler were not born, then it would not be possible to undertake such an act in the first place; therefore, the ancestor proceeds to beget the traveler's next-generation ancestor and secure the line to the traveler. There is no predicted outcome to this scenario.[8]Consistency paradoxes occur whenever changing the past is possible.[9]A possible resolution is that a time travellercando anything thatdidhappen, butcannotdo anything thatdid nothappen. Doing something that did not happen results in a contradiction.[8]This is referred to as theNovikov self-consistency principle.
The grandfather paradox encompasses any change to the past,[13]and it is presented in many variations, including killing one's past self.[14][15]Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written intoAmazing Storiesin the 1920s.[16]Another variant of the grandfather paradox is the "Hitler paradox" or "Hitler's murder paradox", in which the protagonist travels back in time to murderAdolf Hitlerbefore he can rise to power in Germany, thus preventingWorld War IIand theHolocaust. Rather than necessarily physically preventing time travel, the action removes anyreasonfor the travel, along with any knowledge that the reason ever existed.[17]
Physicist John Garrison et al. give a variation of the paradox of an electronic circuit that sends a signal through a time machine to shut itself off, and receives the signal before it sends it.[18][19]
Newcomb's paradox is athought experimentshowing an apparent contradiction between theexpected utilityprinciple and thestrategic dominanceprinciple.[20]The thought experiment is often extended to explorecausalityand free will by allowing for "perfect predictors": if perfect predictors of the future exist, for example if time travel exists as a mechanism for making perfect predictions[how?], then perfect predictions appear to contradict free will because decisions apparently made with free will are already known to the perfect predictor[clarification needed].[21][22]Predestinationdoes not necessarily involve asupernaturalpower, and could be the result of other "infallible foreknowledge" mechanisms.[23]Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.[24]
Even without knowing whether time travel to the past is physically possible, it is possible to show usingmodal logicthat changing the past results in a logical contradiction. If it is necessarily true that the past happened in a certain way, then it is false and impossible for the past to have occurred in any other way. A time traveler would not be able to change the past from the way itis,but would only act in a way that is already consistent with whatnecessarilyhappened.[25][26]
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible. For example, the philosopherBradley Dowdenmade this sort of argument in the textbookLogical Reasoning, arguing that the possibility of creating a contradiction rules out time travel to the past entirely. However, some philosophers and scientists believe that time travel into the past need not be logically impossible provided that there is no possibility of changing the past,[13]as suggested, for example, by theNovikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopherNorman Swartz.[27]
A recent proposed resolution argues that if time is not an inherent property of the universe but is insteademergentfrom the laws ofentropy, as some modern theories suggest,[28][29]then it presents a natural solution to the Grandfather Paradox.[30]In this framework, "time travel" is reinterpreted not as movement along a linear continuum but as a reconfiguration of the present state of the universe to match a prior entropic configuration. Because the original chronological sequence—including events like the time traveler’s birth—remains preserved in the universe’s irreversible entropic progression, actions within the reconfigured state cannot alter the causal history that produced the traveler. This avoids paradoxes by treating time as a thermodynamic artifact rather than a mutable dimension.
Consideration of the possibility of backward time travel in a hypothetical universe described by aGödel metricled famed logicianKurt Gödelto assert that time might itself be a sort of illusion.[31][32]He suggests something along the lines of theblock timeview, in which time is just another dimension like space, with all events at all times being fixed within this four-dimensional "block".[citation needed]
Sergey Krasnikovwrites that these bootstrap paradoxes – information or an object looping through time – are the same; the primary apparent paradox is a physical system evolving into a state in a way that is not governed by its laws.[33]: 4He does not find these paradoxical and attributes problems regarding the validity of time travel to other factors in the interpretation of general relativity.[33]: 14–16
A 1992 paper by physicists Andrei Lossev andIgor Novikovlabeled such items without origin asJinn, with the singular termJinnee.[34]: 2311–2312This terminology was inspired by theJinnof theQuran, which are described as leaving no trace when they disappear.[35]: 200–203Lossev and Novikov allowed the term "Jinn" to cover both objects and information with the reflexive origin; they called the former "Jinn of the first kind", and the latter "Jinn of the second kind".[6][34]: 2315–2317[35]: 208They point out that an object making circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency; thesecond law of thermodynamicsseems to require that the object tends to a lower energy state throughout its history, and such objects that are identical in repeating points in their history seem to contradict this, but Lossev and Novikov argued that since the second law only requires entropy to increase inclosedsystems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy.[6][35]: 200–203They emphasize that there is no "strict difference" between Jinn of the first and second kind.[34]: 2320Krasnikov equivocates between "Jinn", "self-sufficient loops", and "self-existing objects", calling them "lions" or "looping or intruding objects", and asserts that they are no less physical than conventional objects, "which, after all, also could appear only from either infinity or a singularity."[33]: 8–9
The self-consistency principle developed byIgor Dmitriyevich Novikov[36]: p. 42 note 10expresses one view as to how backwardtime travelwould be possible without the generation of paradoxes. According to this hypothesis, even thoughgeneral relativitypermits someexact solutionsthat allow fortime travel[37]that containclosed timelike curvesthat lead back to the same point in spacetime,[38]physics in or nearclosed timelike curves(time machines) can only be consistent with the universal laws of physics, and thus only self-consistent events can occur. Anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from happening, since this would represent an inconsistency. The authors concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.[39]
PhysicistJoseph Polchinskiconsidered a potentially paradoxical situation involving abilliard ballthat is fired into awormholeat just the right angle such that it will be sent back in time and collides with its earlier self, knocking it off course, which would stop it from entering the wormhole in the first place.Kip Thornereferred to this problem as "Polchinski's paradox".[39]Thorne and two of his students at Caltech, Fernando Echeverria and Gunnar Klinkhammer, went on to find a solution that avoided any inconsistencies, and found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case.[40]Later analysis by Thorne andRobert Forwardshowed that for certain initial trajectories of the billiard ball, there could be an infinite number of self-consistent solutions.[39]It is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.[41]: 184The lack of constraints on initial conditions only applies to spacetime outside of thechronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.[41]: 187–188
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as anad hocsolution, and supposes that there are far more damaging implications of time travel.[42]Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity.[33]: 14–16Another conjecture, thecosmic censorship hypothesis, suggests that every closed timelike curve passes through anevent horizon, which prevents such causal loops from being observed.[43]
The interacting-multiple-universes approach is a variation of themany-worlds interpretationof quantum mechanics that involves time travelers arriving in a different universe than the one from which they came; it has been argued that, since travelers arrive in a different universe's history and not their history, this is not "genuine" time travel.[44]Stephen Hawking has argued for thechronology protection conjecture, that even if the MWI is correct, we should expect each time traveler to experience a single self-consistent history so that time travelers remain within their world rather than traveling to a different one.[45]
David Deutschhas proposed thatquantum computationwith a negative delay—backward time travel—produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning.[46]However Deutsch's self-consistency condition has been demonstrated as capable of being fulfilled to arbitrary precision by any system subject to the laws of classicalstatistical mechanics, even if it is not built up by quantum systems.[47]Allen Everett has also argued that even if Deutsch's approach is correct, it would imply that any macroscopic object composed of multiple particles would be split apart when traveling back in time, with different particles emerging in different worlds.[48]
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Theouroborosoruroboros(/ˌjʊərəˈbɒrəs/;[2]/ˌʊərəˈbɒrəs/[3]) is an ancientsymboldepicting asnakeordragon[4]eating its own tail. The ouroboros entered Western tradition viaancient Egyptian iconographyand theGreek magical tradition. It was adopted as a symbol inGnosticismandHermeticismand, most notably, inalchemy. Some snakes, such asrat snakes, have been known to consume themselves.[5]
The term derives fromAncient Greekοὐροβόρος,[6]fromοὐράoura'tail' plus-βορός-boros'-eating'.[7][8]
Theouroborosis often interpreted as a symbol for eternal cyclic renewal or acycle of life, death and rebirth; the snake'sskin-sloughingsymbolises thetransmigration of souls. The snake biting its own tail is a fertility symbol in some religions: the tail is aphallic symboland the mouth is ayonicor womb-like symbol.[9]
One of the earliest known ouroborosmotifsis found in theEnigmatic Book of the Netherworld, anancient Egyptian funerary textinKV62, the tomb ofTutankhamun, in the 14th century BCE. The text concerns the actions ofRaand his union withOsirisin theunderworld. The ouroboros is depicted twice on the figure: holding their tails in their mouths, one encircling the head and upper chest, the other surrounding the feet of a large figure, which may represent the unified Ra-Osiris (Osirisborn again asRa). Both serpents are manifestations of the deityMehen, who in other funerary texts protects Ra in his underworld journey. The whole divine figure represents the beginning and the end of time.[10]
The ouroboros appears elsewhere in Egyptian sources, where, like many Egyptian serpent deities, it represents the formless disorder that surrounds the orderly world and is involved in that world's periodic renewal.[11]The symbol persisted from Egyptian intoRoman times, when it frequently appeared on magicaltalismans, sometimes in combination with other magical emblems.[12]The 4th-century CE Latin commentatorServiuswas aware of the Egyptian use of the symbol, noting that the image of a snake biting its tail represents the cyclical nature of the year.[13]
InGnosticism, a serpent biting its tail symbolised eternity and the soul of the world.[14]The GnosticPistis Sophia(c. 400 CE) describes the ouroboros as a twelve-part dragon surrounding the world with its tail in its mouth.[15]
The famous ouroboros drawing from the earlyalchemicaltext,TheChrysopoeiaof Cleopatra(Κλεοπάτρας χρυσοποιία), probably originally dating to the 3rd centuryAlexandria, but first known in a 10th-century copy, encloses the wordshen to pan(ἓν τὸ πᾶν), "the all isone". Its black and white halves may perhaps represent aGnosticdualityof existence, analogous to theTaoistyin and yangsymbol.[16]Thechrysopoeiaouroboros ofCleopatra the Alchemistis one of the oldest images of the ouroboros to be linked with the legendaryopusof the alchemists, thephilosopher's stone.[citation needed]
A 15th-century alchemical manuscript,The Aurora Consurgens, features the ouroboros, where it is used among symbols of the sun, moon, and mercury.[17]
InNorse mythology, the ouroboros appears as the serpentJörmungandr, one of the three children ofLokiandAngrboda, which grew so large that it could encircle the world and grasp its tail in its teeth. In the legends ofRagnar Lodbrok, such asRagnarssona þáttr, the Geatish kingHerraudgives a smalllindwormas a gift to his daughterÞóra Town-Hartafter which it grows into a large serpent which encircles the girl'sbowerand bites itself in the tail. The serpent is slain by Ragnar Lodbrok who marries Þóra. Ragnar later has a son with another woman namedKrákaand this son is born with the image of a white snake in one eye. This snake encircled the iris and bit itself in the tail, and the son was namedSigurd Snake-in-the-Eye.[19]
It is a common belief amongindigenous peopleof the tropical lowlands of South America that waters at the edge of the world-disc are encircled by a snake, often an anaconda, biting its own tail.[20]
The ouroboros has certain features in common with the BiblicalLeviathan. According to theZohar, the Leviathan is a singular creature with no mate, "its tail is placed in its mouth", whileRashionBaba Batra74b describes it as "twisting around and encompassing the entire world". The identification appears to go back as far as the poems ofKalirin the 6th–7th centuries.[citation needed]
In theAitareya Brahmana, aVedictext of the early 1st millennium BCE, the nature of theVedic ritualsis compared to "a snake biting its own tail."[21]
Ouroboros symbolism has been used to describe theKundalini.[22]According to the medievalYoga-kundalini Upanishad: "The divine power, Kundalini, shines like the stem of a young lotus; like a snake, coiled round upon herself she holds her tail in her mouth and lies resting half asleep as the base of the body" (1.82).[23]
Storl (2004) also refers to the ouroboros image in reference to the "cycle ofsamsara".[24]
Swiss psychiatristCarl Jungsaw the ouroboros as anarchetypeand the basicmandalaof alchemy. Jung also defined the relationship of the ouroboros to alchemy: Carl Jung,Collected Works, Vol. 14 para. 513.
The alchemists, who in their own way knew more about the nature of theindividuationprocess than we moderns do, expressed this paradox through the symbol of the Ouroboros, the snake that eats its own tail. The Ouroboros has been said to have a meaning of infinity or wholeness. In the age-old image of the Ouroboros lies the thought of devouring oneself and turning oneself into a circulatory process, for it was clear to the more astute alchemists that theprima materiaof the art was man himself. The Ouroboros is a dramatic symbol for the integration and assimilation of the opposite, i.e. of the shadow. This 'feedback' process is at the same time a symbol of immortality since it is said of the Ouroboros that he slays himself and brings himself to life, fertilizes himself, and gives birth to himself. He symbolizes the One, who proceeds from the clash of opposites, and he, therefore, constitutes the secret of theprima materiawhich ... unquestionably stems from man's unconscious.
The Jungian psychologistErich Neumannwrites of it as a representation of the pre-ego "dawn state", depicting the undifferentiated infancy experience of both humankind and the individual child.[25]
The German organic chemistAugust Kekulédescribed theeureka momentwhen he realised the structure ofbenzene, after he saw a vision of Ouroboros:[26]
I was sitting, writing at my text-book; but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by the repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together; all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.
Martin Reesused the ouroboros to illustrate the various scales of the universe, ranging from 10−20cm (subatomic) at the tail, up to 1025cm (supragalactic) at the head.[27]Rees stressed "the intimate links between the microworld and the cosmos, symbolised by theouraborus", as tail and head meet to complete the circle.[28]
W. Ross Ashbyapplied ideas from biology to his own work as a psychiatrist in "Design for a Brain" (1952): that living things maintain essential variables of the body within critical limits with the brain as a regulator of the necessary feedback loops. Parmar contextualises his practices as an artist in applying the cybernetic Ouroboros principle to musical improvisation.[29]
Hence the snake eating its tail is an accepted image or metaphor in the autopoietic calculus for self-reference,[30]or self-indication, the logical processual notation for analysing and explaining self-producing autonomous systems and "the riddle of the living", developed byFrancisco Varela. Reichel describes this as:
an abstract concept of a system whose structure is maintained through the self-production of and through that structure. In the words of
Kauffman, is "the ancient mythological symbol of the worm ouroboros embedded in a mathematical, non-numerical calculus".[31][32]
The calculus derives from the confluence of the cybernetic logic of feedback, the sub-disciplines ofautopoiesisdeveloped by Varela andHumberto Maturana, and calculus of indications ofGeorge Spencer Brown. In another related biological application:
It is remarkable, that Rosen's insight, that metabolism is just a mapping ..., which may be too cursory for a biologist, turns out to show us the way to constructrecursively, by a limiting process, solutions of the self-referential Ouroborus equation f(f) = f, for an unknown function f, a way that mathematicians had not imagined before Rosen.[33][34]
Second-order cybernetics, or the cybernetics of cybernetics, applies the principle of self-referentiality, or the participation of the observer in the observed, to explore observer involvement.[35]including D. J. Stewart's domain of "observer valued imparities".[36]
The genus of thearmadillo girdled lizard,Ouroborus cataphractus, takes its name from the animal's defensive posture: curling into a ball and holding its own tail in its mouth.[37]
A medium-sizedEuropean hake, known in Spanish aspescadillaand in Portuguese aspescada, is often presented with its mouth biting its tail. In Spanish it receives the name ofpescadilla de rosca("torushake").[38]Both expressionsUma pescadinha de rabo na boca"tail-in mouth little hake" andLa pescadilla que se muerde la cola, "the hake that bites its tail", are proverbial Portuguese and Spanish expressions forcircular reasoningandvicious circles.[39]
TheKobe, Japan-basedDragon GatePro-Wrestling promotion used a stylised ouroboros as their logo for the first 20 years of the company's existence. The logo is a silhouetted dragon twisted into the shape of an infinity symbol, devouring its own tail. In 2019, the promotion dropped the infinity dragon logo in favour of a shield logo.
A variation of the Ouroboros motif is an important symbol in the fantasy novelThe Neverending StorybyMichael Ende: featuring two snakes, one black and one white, biting the other's tail, this symbol represents the powerfulAURYNand the infinite nature of the story. The symbol is also featured prominently on the cover of both the fictional book and the novel.
The Worm Ouroborosis a high-fantasy novel written byE. R. Eddison. Much like the cyclical symbol of the ouroboros eating its own tail, the novel ends as it begins. The main villain has a ring in the form of Ouroboros.
InMexican Gothicthe symbol is used throughout the story, portraying the immortality of the home and the family, as well as the persistence of outdated ideologies.[40]
InThe Wheel of Timeand its2021 television adaption, the Aes Sedai wear a "Great Serpent" ring, described as a snake consuming its own tail.[41]
In the science fiction short story "All You Zombies" (1958) by American writerRobert A. Heinlein, the character Jane wears an Ouroboros ring, "the worm Ouroboros, the world snake".[42]The short story later inspired the moviePredestination(2014).
In theSCP Foundationuniverse, the proposal tale "The Ouroboros Cycle"[43]spans the story of the SCP Foundation from its creation to its ending.
In theA Discovery of Witchesnovels andtelevisionadaptation, the crest of the de Clermont family is an ouroboros. The symbol plays a significant role in thealchemicalplot of the story.
InThe Witcher, the Ouroboros and the "snake biting its own tail" is a recurring theme.
The Ouroboros is the adopted symbol of theEnd Times-obsessedMillennium Groupin the TV seriesMillennium.[44]It also briefly appears whenDana Scullygets a tattoo of it inThe X-FilesSeason 4 episode "Never Again" (1997).[45]
"Ouroboros" is an episode of the British science-fiction sitcomRed Dwarf, in whichDave Listerlearns that he is his own father through time travel.[46]
In Season 1 (2012) ofNinjagotitled "Ninjago: Rise of the Snakes", the Lost City of Ouroboros (also referred to as the Ancient City of Ouroboros) serves as a pivotal location in the Serpentine's plan for vengeance against Ninjago. Once a massive Serpentine city, Ouroboros was buried beneath the Sea of Sand after the Serpentine War. The city was key to Pythor and the Serpentine's efforts to awaken the Great Devourer, which had been imprisoned beneath the city. After retrieving the four Fangblades, Pythor returned to Ouroboros and successfully released the Great Devourer, causing significant damage to the city. Despite the destruction, the Serpentine continued to use the city as a temporary base before abandoning it to journey to the tomb of the Stone Army.
InHemlock Grove(2013-2015), the ouroboros plays an important part throughout the series.
In Season 3 (2014),Ninjago: Rebooted, during the Nindroid crisis, Pythor once again used Ouroboros as a base of operations. Here, he led an army of Nindroids and launched a giant rocket into space in search of the comet that held the remnants of the Golden Weapons.
In Season 1 (2018) of thecyberpunkNetflix seriesAltered Carbon, the protagonist Takeshi Kovacs gets an ouroboros tattoo in shape of aninfinity symbol, and it features in the show's title sequence, tying in to the themes of rebirth and the twisting of the natural cycle of life and death.[47]
In the season 2 premiere of the television seriesLoki, a character named Ouroboros (played byKe Huy Quan) is introduced. He is an employee of the Time Variance Authority. In the fourth episode, he also references a snake biting its own tail.[48]
In the animeFullmetal Alchemist: Brotherhood, members of thehomunculirace are identified by having the symbol carved/tattooed/branded/marked on them.[49]
The Abiranariba inThe Dark Crystal: Age of Resistanceis based on the ouroboros.
Splatoon 3has a serpent-like Salmonid creature named after it, the Horrorboros.[50]
Ace Combat 3: Electrosphere's main antagonist group is a terrorist organization called Ouroboros, whose intention is to cripple Strangereal's megacorporations in the continent of Usea.
The Legend of Heroes: Trailsfeatures the enigmatic Society of Ouroboros, whose members serve as recurring antagonists in the series.
InXenoblade Chronicles 3, the player's party wields a power named after Ouroboros, which is subversively used toopposethe world's cycle of death and rebirth, rather than representing it.
InThe Witcher 3: Wild Hunt, an ouroboros spins on loading screens as an indiciation for the game loading.
A three-headed ouroboros is the logo ofElder Scrolls Online, with a lion, a dragon, and an eagle that represent the three main factions of the game.
InInscryption, Ouroboros is a playable card that has the ability to return to the player's hand as a stronger version of itself after it has been killed.
InKing Woman's albumCelestial Blues(2021), Ourobouros is alluded to in the song "Golgotha": "The snake eats its tail, we return again to this hell".[51]
Ouroboros, a large public sculpture by Australian artistLindy Leeat theNational Gallery of Australiaforecourt.[52]Members of the public are free to enter its 4m "mouth".[53]
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ThePenrose stairsorPenrose steps, also dubbed theimpossible staircase, is animpossible objectcreated byOscar Reutersvärdin 1937[1][2][3][4]and later independently discovered and made popular byLionel Penroseand his sonRoger Penrose.[5]A variation on thePenrose triangle, it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three-dimensionalEuclidean geometrybut possible in somenon-Euclidean geometrylike innil geometry.[6]
The "continuous staircase" was first presented in an article that the Penroses wrote in 1959, based on the so-called "triangle of Penrose" published by Roger Penrose in theBritish Journal of Psychologyin 1958.[5]M.C. Escherthen discovered the Penrose stairs in the following year and made his now famous lithographKlimmen en dalen(Ascending and Descending) in March 1960. Penrose and Escher were informed of each other's work that same year.[7]Escher developed the theme further in his printWaterval(Waterfall), which appeared in 1961.
In their original article the Penroses noted that "each part of the structure is acceptable as representing a flight of steps but the connections are such that the picture, as a whole, is inconsistent: the steps continually descend in a clockwise direction."[8]
Escher, in the 1950s, had not yet drawn any impossible stairs and was not aware of their existence. Roger Penrose had been introduced to Escher's work at theInternational Congress of Mathematiciansin Amsterdam in 1954. He was "absolutely spellbound" by Escher's work, and on his journey back to England he decided to produce something "impossible" on his own. After experimenting with various designs of bars overlying each other he finally arrived at the impossible triangle. Roger showed his drawings to his father, who immediately produced several variants, including the impossible flight of stairs. They wanted to publish their findings but did not know in what field the subject belonged. Because Lionel Penrose knew the editor of theBritish Journal of Psychologyand convinced him to publish their short manuscript, the finding was finally presented as apsychologicalsubject. After the publication in 1958 the Penroses sent a copy of the article to Escher as a token of their esteem.[9]
While the Penroses credited Escher in their article, Escher noted in a letter to his son in January 1960 that he was:
working on the design of a new picture, which featured a flight of stairs which only ever ascended or descended, depending on how you saw it. [The stairs] form a closed, circular construction, rather like a snake biting its own tail. And yet they can be drawn in correct perspective: each step higher (or lower) than the previous one. [...] I discovered the principle in an article which was sent to me, and in which I myself was named as the maker of various 'impossible objects'. But I was not familiar with the continuous steps of which the author had included a clear, if perfunctory, sketch, although I was employing some of his other examples.[10]
Escher was captivated by the endless stairs and subsequently wrote a letter to the Penroses in April 1960:
A few months ago, a friend of mine sent me a photocopy of your article... Your figures 3 and 4, the 'continuous flight of steps', were entirely new to me, and I was so taken by the idea that they recently inspired me to produce a new picture, which I would like to send to you as a token of my esteem. Should you have published other articles on impossible objects or related topics, or should you know of any such articles, I would be most grateful if you could send me further details.[10]
At a conference in Rome in 1985, Roger Penrose said that he had been greatly inspired by Escher's work when he and his father discovered both the Penrose tribar structure (that is, the Penrose triangle) and the continuous steps.
The staircase design had been discovered previously by the Swedish artistOscar Reutersvärd, but neither Penrose nor Escher was aware of his designs.[4]Inspired by a radio programme onMozart's method of composition—described as "creative automatism"; that is, each creative idea written down inspired a new idea—Reutersvärd started to draw a series of impossible objects on a journey from Stockholm to Paris in 1950 in the same "unconscious, automatic" way. He did not realize that his figure was a continuous flight of stairs while drawing, but the process enabled him to trace his increasingly complex designs step by step. When M.C. Escher'sAscending and Descendingwas sent to Reutersvärd in 1961, he was impressed but didn't like the irregularities of the stairs (2 × 15 + 2 × 9). Throughout the 1960s, Reutersvärd sent several letters to Escher to express his admiration for his work, but the Dutch artist failed to respond.[11]Roger Penrose only discovered Reutersvärd's work in 1984.[9]
The Escherian Stairwellis aviral videobased on the Penrose stairs illusion. The video, filmed atRochester Institute of Technologyby Michael Lacanilao, was edited to create a seemingly cyclic stairwell such that if someone walks in either direction, they will end up where they started.[12]The video claims that the stairwell, whose name evokes M.C. Escher's impossible objects, was built in the 1960s by the fictitious architect Rafael Nelson Aboganda. The video was revealed to be anInternet hoax, as individuals have travelled to Rochester Institute of Technology to view the staircase.[13]
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Perpetual motionis the motion of bodies that continues forever in an unperturbed system. Aperpetual motion machineis a hypothetical machine that can do work indefinitely without an externalenergysource. This kind of machine is impossible, since its existence would violate thefirstand/orsecondlaws of thermodynamics.[2][3][4][5]Theselaws of thermodynamicsapply regardless of the size of the system. Thus, machines that extract energy from finite sources cannot operate indefinitely because they are driven by the energy stored in the source, which will eventually be exhausted. A common example is devices powered by ocean currents, whose energy is ultimately derived from the Sun, which itself will eventuallyburn out.
In 2016,[6]new states of matter,time crystals, were discovered in which, on a microscopic scale, the component atoms are in continual repetitive motion, thus satisfying the literal definition of "perpetual motion".[7][8][9][10]However, these do not constitute perpetual motion machines in the traditional sense, or violate thermodynamic laws, because they are in their quantumground state, so no energy can be extracted from them; they exhibit motion without energy.
The history of perpetual motion machines dates back to the Middle Ages.[11]For millennia, it was not clear whether perpetual motion devices were possible or not, until the development of modern theories of thermodynamics showed that they were impossible. Despite this, many attempts have been made to create such machines, continuing into modern times.[12]Modern designers and proponents often use other terms, such as "over unity",[13]to describe their inventions.
Oh ye seekers after perpetual motion, how many vain chimeras have you pursued? Go and take your place with the alchemists.
There is ascientific consensusthat perpetual motion in anisolated systemviolates either thefirst law of thermodynamics, thesecond law of thermodynamics, or both. The first law of thermodynamics is a version of the law ofconservation of energy. The second law can be phrased in several different ways, the most intuitive of which is thatheatflows spontaneously from hotter to colder places; relevant here is that the law observes that in every macroscopic process, there is friction or something close to it; another statement is that noheat engine(an engine which produces work while moving heat from a high temperature to a low temperature) can be more efficient than aCarnot heat engineoperating between the same two temperatures.
In other words:
Statements 2 and 3 apply to heat engines. Other types of engines that convert e.g. mechanical into electromagnetic energy, cannot operate with 100% efficiency, because it is impossible to design any system that is free of energy dissipation.
Machines that comply with both laws of thermodynamics by accessing energy from unconventional sources are sometimes referred to as perpetual motion machines, although they do not meet the standard criteria for the name. By way of example, clocks and other low-power machines, such asCox's timepiece, have been designed to run on the differences in barometric pressure or temperature between night and day. These machines have a source of energy, albeit one which is not readily apparent, so that they only seem to violate the laws of thermodynamics.
Even machines that extract energy from long-lived sources - such as ocean currents - will run down when their energy sources inevitably do. They are not perpetual motion machines because they are consuming energy from an external source and are not isolated systems.
One classification of perpetual motion machines refers to the particular law of thermodynamics the machines purport to violate:[16]
"Epistemic impossibility" describes things which absolutely cannot occur within ourcurrentformulation of the physical laws. This interpretation of the word "impossible" is what is intended in discussions of the impossibility of perpetual motion in a closed system.[19]
The conservation laws are particularly robust from a mathematical perspective.Noether's theorem, which wasproven mathematicallyin 1915, states that any conservation law can be derived from a corresponding continuous symmetry of theactionof a physical system.[20]The symmetry which is equivalent to conservation of energy is thetime invarianceof physical laws. Therefore, if the laws of physics do not change with time, then the conservation of energy follows. For energy conservation to be violated to allow perpetual motion would require that the foundations of physics would change.[21]
Scientific investigations as to whether the laws of physics are invariant over time use telescopes to examine the universe in the distant past to discover, to the limits of our measurements, whether ancient stars were identical to stars today. Combining different measurements such asspectroscopy, direct measurement of thespeed of light in the pastand similar measurements demonstrates that physics has remained substantially the same, if not identical, for all of observable time spanningbillionsof years.[22]
The principles of thermodynamics are so well established, both theoretically and experimentally, that proposals for perpetual motion machines are universally dismissed by physicists. Any proposed perpetual motion design offers a potentially instructive challenge to physicists: one is certain that it cannot work, so one must explainhowit fails to work. The difficulty (and the value) of such an exercise depends on the subtlety of the proposal; the best ones tend to arise from physicists' ownthought experimentsand often shed light upon certain aspects of physics. So, for example, the thought experiment of aBrownian ratchetas a perpetual motion machine was first discussed byGabriel Lippmannin 1900 but it was not until 1912 thatMarian Smoluchowskigave an adequate explanation for why it cannot work.[23]However, during that twelve-year period scientists did not believe that the machine was possible. They were merely unaware of the exact mechanism by which it would inevitably fail.
The law that entropy always increases – the second law of thermodynamics – holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
In the mid-19th-centuryHenry Dircksinvestigated the history of perpetual motion experiments, writing a vitriolic attack on those who continued to attempt what he believed to be impossible:
There is something lamentable, degrading, and almost insane in pursuing the visionary schemes of past ages with dogged determination, in paths of learning which have been investigated by superior minds, and with which such adventurous persons are totally unacquainted. The history of Perpetual Motion is a history of the fool-hardiness of either half-learned, or totally ignorant persons.[24]
One day man will connect his apparatus to the very wheelwork of the universe [...] and the very forces that motivate the planets in their orbits and cause them to rotate will rotate his own machinery.
Some common ideas recur repeatedly in perpetual motion machine designs. Many ideas that continue to appear today were stated as early as 1670 byJohn Wilkins,Bishop of Chesterand an official of theRoyal Society. He outlined three potential sources of power for a perpetual motion machine, "Chymical [sic] Extractions", "Magnetical Virtues" and "the Natural Affection of Gravity".[1]
The seemingly mysterious ability ofmagnetsto influence motion at a distance without any apparent energy source has long appealed to inventors. One of the earliest examples of amagnetic motorwas proposed by Wilkins and has been widely copied since: it consists of a ramp with a magnet at the top, which pulled a metal ball up the ramp. Near the magnet was a small hole that was supposed to allow the ball to drop under the ramp and return to the bottom, where a flap allowed it to return to the top again. However, if the magnet is to be strong enough to pull the ball up the ramp, it cannot then be weak enough to allow gravity to pull it through the hole. Faced with this problem, more modern versions typically use a series of ramps and magnets, positioned so the ball is to be handed off from one magnet to another as it moves. The problem remains the same.
Gravityalso acts at a distance, without an apparent energy source, but to get energy out of a gravitational field (for instance, by dropping a heavy object, producing kinetic energy as it falls) one has to put energy in (for instance, by lifting the object up), and some energy is always dissipated in the process. A typical application of gravity in a perpetual motion machine isBhaskara's wheel in the 12th century, whose key idea is itself a recurring theme, often called the overbalanced wheel: moving weights are attached to a wheel in such a way that they fall to a position further from the wheel's center for one half of the wheel's rotation, and closer to the center for the other half. Since weights further from the center apply a greatertorque, it was thought that the wheel would rotate forever. However, since the side with weights further from the center has fewer weights than the other side, at that moment, the torque is balanced and perpetual movement is not achieved.[25]The moving weights may be hammers on pivoted arms, or rolling balls, or mercury in tubes; the principle is the same.
Another theoretical machine involves a frictionless environment for motion. This involves the use ofdiamagneticorelectromagnetic levitationto float an object. This is done in avacuumto eliminate air friction and friction from an axle. The levitated object is then free to rotate around its center of gravity without interference. However, this machine has no practical purpose because the rotated object cannot do any work as work requires the levitated object to cause motion in other objects, bringing friction into the problem. Furthermore, aperfectvacuum is an unattainable goal since both the container and the object itself would slowlyvaporize, thereby degrading the vacuum.
To extract work from heat, thus producing a perpetual motion machine of the second kind, the most common approach (dating back at least toMaxwell's demon) isunidirectionality. Only molecules moving fast enough and in the right direction are allowed through the demon's trap door. In aBrownian ratchet, forces tending to turn the ratchet one way are able to do so while forces in the other direction are not. A diode in a heat bath allows through currents in one direction and not the other. These schemes typically fail in two ways: either maintaining the unidirectionality costs energy (requiring Maxwell's demon to perform more thermodynamic work to gauge the speed of the molecules than the amount of energy gained by the difference of temperature caused) or the unidirectionality is an illusion and occasional big violations make up for the frequent small non-violations (the Brownian ratchet will be subject to internal Brownian forces and therefore will sometimes turn the wrong way).
Buoyancyis another frequently misunderstood phenomenon. Some proposed perpetual-motion machines miss the fact that to push a volume of air down in a fluid takes the same work as to raise a corresponding volume of fluid up against gravity. These types of machines may involve two chambers with pistons, and a mechanism to squeeze the air out of the top chamber into the bottom one, which then becomes buoyant and floats to the top. The squeezing mechanism in these designs would not be able to do enough work to move the air down, or would leave no excess work available to be extracted.
Proposals for such inoperable machines have become so common that theUnited States Patent and Trademark Office(USPTO) has made an official policy of refusing to grantpatentsfor perpetual motion machines without a working model. The USPTO Manual of Patent Examining Practice states:
With the exception of cases involving perpetual motion, a model is not ordinarily required by the Office to demonstrate the operability of a device. If operability of a device is questioned, the applicant must establish it to the satisfaction of theexaminer, but he or she may choose his or her own way of so doing.[26]
And, further, that:
A rejection [of a patent application] on the ground of lack of utility includes the more specific grounds of inoperativeness, involving perpetual motion. A rejection under 35 U.S.C. 101 for lack of utility should not be based on grounds that the invention is frivolous, fraudulent or against public policy.[27]
The filing of a patent application is a clerical task, and the USPTO will not refuse filings for perpetual motion machines; the application will be filed and then most probably rejected by the patent examiner, after he has done a formal examination.[28]Even if a patent is granted, it does not mean that the invention actually works, it just means that the examiner believes that it works, or was unable to figure out why it would not work.[28]
TheUnited Kingdom Patent Officehas a specific practice on perpetual motion; Section 4.05 of the UKPO Manual of Patent Practice states:
Processes or articles alleged to operate in a manner which is clearly contrary to well-established physical laws, such as perpetual motion machines, are regarded as not having industrial application.[29]
Examples of decisions by the UK Patent Office to refuse patent applications for perpetual motion machines include:[30][self-published source]
TheEuropean Patent Classification(ECLA) has classes including patent applications on perpetual motion systems: ECLA classes "F03B17/04: Alleged perpetua mobilia" and "F03B17/00B: [... machines or engines] (with closed loop circulation or similar : ... Installations wherein the liquid circulates in a closed loop; Alleged perpetua mobilia of this or similar kind".[33]
As a perpetual motion machine can only be defined in a finite isolated system with discrete parameters, and since true isolated systems do not exist (among other things, due toquantum uncertainty), "perpetual motion" in the context of this article is better defined as a "perpetual motion machine", since a machine is a "a mechanically, electrically, or electronically operated device for performing a task",[34]whereas "motion" is simply movement (such asBrownian motion). Distinctions aside, on the macro scale, there are concepts and technical drafts that propose "perpetual motion", but on closer analysis it is revealed that they actually "consume" some sort of natural resource or latent energy, such as the phase changes ofwateror other fluids or small natural temperature gradients, or simply cannot sustain indefinite operation. In general, extracting work from these devices is impossible.
Some examples of such devices include:
In some cases athought experimentappears to suggest that perpetual motion may be possible through accepted and understood physical processes. However, in all cases, a flaw has been found when all of the relevant physics is considered. Examples include:
Despite being dismissed aspseudoscientific, perpetual motion machines have become the focus ofconspiracy theories, alleging that they are being hidden from the public by corporations or governments, who would lose economic control if a power source capable of producing energy cheaply was made available.[45][46]
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Thephoenixis alegendaryimmortal bird that cyclically regenerates or is otherwise born again. Originating inGreek mythology, it has analogs in many cultures, such asEgyptianandPersian mythology. Associated with the sun, a phoenix obtains new life by rising from theashesof its predecessor. Some legends say it dies in a show of flames and combustion, while others say that it simply dies and decomposes before being born again.[1]In theMotif-Index of Folk-Literature, a tool used byfolklorists, the phoenix is classified as motif B32.[2]
The origin of the phoenix has been attributed toAncient EgyptbyHerodotusand later 19th-century scholars, but other scholars think the Egyptian texts may have been influenced by classical folklore. Over time, the phoenix motif spread and gained a variety of new associations;Herodotus,Lucan,Pliny the Elder,Pope Clement I,Lactantius,Ovid, andIsidore of Sevilleare among those who have contributed to the retelling and transmission of the phoenix motif. Over time, extending beyond its origins, the phoenix could variously "symbolize renewal in general as well as the sun, time,the Roman Empire,metempsychosis,consecration,resurrection, life in the heavenlyParadise,Christ,Mary,virginity, the exceptional man, and certain aspects of Christian life".[3]Some scholars have claimed that the poemDe ave phoenicemay present the mythological phoenix motif as a symbol ofChrist's resurrection.[4]
The modern English wordphoenixentered theEnglish languagefromLatin, later reinforced byFrench. The word first entered the English language by way of a borrowing of LatinphoenīxintoOld English(fenix). This borrowing was later reinforced by French influence, which had also borrowed the Latin noun. In time, the word developed specialized use in the English language: For example, the term could refer to an "excellent person" (12th century), a variety of heraldic emblem (15th century), and the name of aconstellation(17th century).[5]
The Latin word comes fromGreekφοῖνιξ(phoinix).[6]The Greek word is first attested in theMycenaean Greekpo-ni-ke, which probably meant "griffin", though it might have meant "palm tree". That word is probably a borrowing from aWest Semiticword formadder, a reddyemade fromRubia tinctorum. The wordPhoenicianappears to be from the same root, meaning "those who work with red dyes". Sophoenixalso mean "the Phoenician bird" or "the purplish-red bird".[7]
Apart from theLinear Bmention above fromMycenaean Greece, the earliest clear mention of the phoenix in ancient Greek literature occurs in a fragment of thePrecepts of Chiron, attributed to 8th-century BC Greek poetHesiod. In the fragment, the wisecentaurChirontells a young heroAchillesthe following:[8]
A chattering crow lives now nine generations of aged men,but a stag's life is four time a crow's,and a raven's life makes three stags old,while the phoenix outlives nine ravens,but we, the rich-hairedNymphsdaughters ofZeustheaegis-holder,outlive ten phoenixes.
There by describing the phoenix's lifetime as approximately 972 times the length of a human's.
Classical discourse attributes a potential origin of the phoenix toAncient Egypt.Herodotus, writing in the 5th century BC, provides the following account of the phoenix:[9]
[The Egyptians] have also another sacred bird called the phoenix which I myself have never seen, except in pictures. Indeed it is a great rarity, even in Egypt, only coming there (according to the accounts of the people of Heliopolis) once in five hundred years, when the old phoenix dies. Its size and appearance, if it is like the pictures, are as follow: The plumage is partly red, partly golden, while the general make and size are almost exactly that of theeagle. They tell a story of what this bird does, which does not seem to me to be credible: that he comes all the way fromArabia, and brings the parent bird, all plastered over withmyrrh, to thetemple of the Sun, and there buries the body. In order to bring him, they say, he first forms a ball of myrrh as big as he finds that he can carry; then he hollows out the ball and puts his parent inside, after which he covers over the opening with fresh myrrh, and the ball is then of exactly the same weight as at first; so he brings it to Egypt, plastered over as I have said, and deposits it in the temple of the Sun. Such is the story they tell of the doings of this bird.
In the 19th century, scholastic suspicions appeared to be confirmed by the discovery that Egyptians inHeliopolishad venerated theBennu, a solar bird similar in some respects to the Greek phoenix. However, the Egyptian sources regarding the bennu are often problematic and open to a variety of interpretations. Some of these sources may have actually been influenced by Greek notions of the phoenix, rather than the other way around.[10]
The phoenix is often depicted in ancient and medieval literature and medieval art endowed with ahalo, emphasizing the bird's connection with theSun.[15]The earliest recorded images of the phoenix feature nimbuses that often have seven rays, likeHelios(the Greek personification of the Sun).[16]Pliny the Elder[17]also describes the bird as having a crest of feathers on its head,[15]andEzekiel the Dramatistcompared it to a rooster.[18]
The phoenix came to be associated with specific colors over time. Although the phoenix was generally believed to be colorful and vibrant, sources provide no clear consensus about its exact coloration.Tacitussays that its color made it stand out from all other birds.[19]Some said that the bird had peacock-like coloring, andHerodotus's claim of the Phoenix being red and yellow is popular in many versions of the story on record.[20]Ezekiel the Tragediandeclared that the phoenix had red legs and striking yellow eyes,[18]butLactantiussaid that its eyes were blue like sapphires[21]and that its legs were covered in yellow-gold scales with rose-colored talons.[22]
Herodotus, Pliny,Solinus, andPhilostratusdescribe the phoenix as similar in size to an eagle,[23]but Lactantius and Ezekiel the Dramatist both claim that the phoenix was larger, with Lactantius declaring that it was even larger than anostrich.[24]
According to Pliny'sNatural History,[25]
aquilae narratur magnitudine, auri fulgore circa colla, cetero purpureus, caeruleam roseis caudam pinnis distinguentibus, cristis fauces, caputque plumeo apice honestante.
The story is that it is as large as an eagle, and has a gleam of gold round its neck and all the rest of it is purple, but the tail blue picked out with rosecoloured feathers and the throat picked out with tufts, and a feathered crest adorning its head.
According toClaudian's poem "The Phoenix",[26]
arcanum radiant oculi iubar. igneus oracingit honos. rutilo cognatum vertice sidusattollit cristatus apex tenebrasque serenaluce secat. Tyrio pinguntur crura veneno.antevolant Zephyros pinnae, quas caerulus ambitflore color sparsoque super ditescit in auro.
A mysterious fire flashes from its eye,and a flamingaureoleenriches its head. Its crestshines with the sun's own light and shatters thedarkness with its calm brilliance. Its legs are ofTyrianpurple; swifter than those of theZephyrsare its wingsof flower-like blue dappled with rich gold.
According to Pliny the Elder, a senator Manilius (Marcus Manilius?) had written that the phoenix appeared at the end of eachGreat Year, which he wrote of "in the consulship ofGnaeus CorneliusandPublius Licinius", that is, in 96 BC, that a cycle was 540 years, and that it was 215 into the cycle (i.e. it began in 311 BC).[25]Another of Pliny's sources, Cornelius Valerianus, is cited for an appearance of the phoenix in 36 AD "in the consulship ofQuintus PlautiusandSextus Papinius".[25]Pliny states that a purported phoenix seen in Egypt in 47 AD was brought to the capital and exhibited in theComitiumin time for the 800th anniversary of thefoundation of RomebyRomulus, though he added that "nobody would doubt that this phoenix was a fabrication".[25]
A second recording of the phoenix was made byTacitus, who said that the phoenix had appeared instead in 34 AD "in the consulship ofPaulus FabiusandLucius Vitellius" and that the cycle was either 500 years or 1461 years (which was the Great Year based on the EgyptianSothic cycle), and that it had previously been seen in the reigns first of Sesosis, then of Amasis, and finally of Ptolemy (third of the Macedonian dynasty).[28]A third recording was made byCassius Dio, who also said that the phoenix was seen in the consulship of Quintus Plautus and Sextus Papinius.[29]
In time, the motif and concept of the phoenix extended from its origins in ancient Greek folklore. For example, the classical motif of the phoenix continues into theGnosticmanuscriptOn the Origin of the Worldfrom theNag Hammadi Librarycollection in Egypt, generally dated to the 4th century:[30]
Thus whenSophia Zoesaw that the rulers of darkness had laid a curse upon her counterparts, she was indignant. And coming out of the first heaven with full power, she chased those rulers out of their heavens and cast them into the sinful world, so that there they should dwell, in the form of evil spirits upon the earth.[...], so that in their world it might pass the thousand years in paradise—a soul-endowed living creature called "phoenix". It kills itself and brings itself back to life as a witness to the judgement against them, for they did wrong toAdamand his race, unto the consummation of the age. There are [...] three men, and also his posterities, unto the consummation of the world: the spirit-endowed of eternity, and the soul-endowed, and the earthly. Likewise, there are three phoenixes in paradise—the first is immortal, the second lives 1,000 years; as for the third, it is written in the sacred book that it is consumed. So, too, there are three baptisms—the first is spiritual, the second is by fire, the third is by water. Just as the phoenix appears as a witness concerning theangels, so the case of the waterhydriin Egypt, which has been a witness to those going down into the baptism of a true man. The two bulls in Egypt posses a mystery, the Sun and the Moon, being a witness toSabaoth: namely, that over themSophiareceived the universe; from the day that she made the Sun and Moon, she put a seal upon her heaven, unto eternity. And the worm that has been born out of the phoenix is a human being as well. It is written concerning it, "the just man will blossom like a phoenix". And the phoenix first appears in a living state, and dies, and rises again, being a sign of what has become apparent at the consummation of the age.
The anonymous 10th-century Old EnglishExeter Bookcontains a 677-line 9th-century alliterative poem consisting of a paraphrase and abbreviation of Lactantius, followed by an explication of the Phoenix as anallegoryfor theresurrectionofChrist.[31]
Þisses fugles gecynd fela gelicesbi þam gecornum Cristes þegnum;beacnað in burgum hu hi beorhtne gefeanþurh fæder fultum on þas frecnan tidhealdaþ under heofonum ond him heanna blædin þam uplican eðle gestrynaþ.
This bird's nature is much liketo the chosen servants of Christ;pointeth out to men how they bright joythrough the Father's aid in this perilous timemay under heaven possess, and exalted happinessin the celestial country may gain.
In the 14th century, Italian poetDante Alighierirefers to the phoenix in Canto XXIV of theDivine Comedy'sInferno:
Così per li gran savi si confessache la fenice more e poi rinasce,quando al cinquecentesimo anno appressa;erba né biado in sua vita non pasce,ma sol d'incenso lagrime e d'amomo,e nardo e mirra son l'ultime fasce.
Even thus by the great sages 'tis confessedThe phoenix dies, and then is born again,When it approaches its five-hundredth year;On herb or grain it feeds not in its life,But only on tears ofincenseandamomum,Andnardandmyrrhare its lastwinding-sheet.
In the 17th-century playHenry VIIIby English playwrightsWilliam ShakespeareandJohn Fletcher,Archbishop Cranmersays inAct V, Scene vin reference to Elizabeth (who was to becomeQueen Elizabeth I):
... Nor shall this peace sleep with her; but as whenThe bird of wonder dies, the maiden phoenix,Her ashes new create another heirAs great in admiration as herself;So shall she leave her blessedness to one,When heaven shall call her from this cloud of darkness,Who from the sacred ashes of her honourShall star-like rise as great in fame as she was,And so stand fix'd ...
In the 19th-century novelSartor ResartusbyThomas Carlyle, Diogenes Teufelsdröckh uses the phoenix as a metaphor for thecyclical patternof history, remarking upon the "burning of a World-Phoenix" and the "Palingenesia, or Newbirth of Society" from its ashes:
When the Phoenix is fanning her funeral pyre, will there not be sparks flying! Alas, some millions of men, and among them such as aNapoleon, have already been licked into that high-eddying Flame, and like moths consumed there. Still also have we to fear that incautious beards will get singed.For the rest, in what year of grace such Phoenix-cremation will be completed, you need not ask. The law of Perseverance is among the deepest in man: by nature he hates change; seldom will he quit his old house till it has actually fallen about his ears. Thus have I seen Solemnities linger as Ceremonies, sacred Symbols as idle Pageants, to the extent of three hundred years and more after all life and sacredness had evaporated out of them. And then, finally, what time the Phoenix Death-Birth itself will require, depends on unseen contingencies.—Meanwhile, would Destiny offer Mankind, that after, say two centuries of convulsion and conflagration, more or less vivid, the fire-creation should be accomplished, and we to find ourselves again in a Living Society, and no longer fighting but working,—were it not perhaps prudent in Mankind to strike the bargain?[34]
Phoenixes are present and relatively common in Europeanheraldry, which developed during theHigh Middle Ages. They most often appear ascrests, and more rarely ascharges. The heraldic phoenix is depicted as the head, chest and wings of an eagle rising from a fire; the entire creature is never depicted.[35]
Scholars have observed analogues to the phoenix in a variety of cultures. These analogues include theHindugaruda(गरुड) andbherunda(भेरुण्ड), theSlavicfirebird(жар-птица) andRaróg, thePersiansimorgh(سیمرغ), theGeorgianpaskunji(ფასკუნჯი), theArabiananqa(عنقاء), theTurkishKonrul, also calledZümrüdü Anka("emerald anqa"), theTibetanMe byi karmo, theChineseFenghuang(鳳凰) andZhuque(朱雀).[36]These perceived analogues are sometimes included as part of theMotif-Index of Folk-Literaturephoenixmotif (B32).[2]
There are many works of modern literature make reference to the phoenix. Examples include:
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Pitch circularityis a fixed series oftonesthat are perceived to ascend or descend endlessly inpitch. It's an example of anauditory illusion.
Pitch is often defined as extending along a one-dimensionalcontinuumfrom high to low, as can be experienced by sweeping one’s hand up or down a piano keyboard. This continuum is known as pitch height. However pitch also varies in a circular fashion, known aspitch class: as one plays up a keyboard in semitone steps, C, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯and B sound in succession, followed by C again, but oneoctavehigher. Because the octave is the most consonant interval after theunison, tones that stand in octave relation, and are so of the same pitch class, have a certain perceptual equivalence—all Cs sound more alike to other Cs than to any other pitch class, as do all D♯s, and so on; this creates the auditory equivalent of aBarber's pole, where all tones of the same pitch class are located on the same side of the pole, but at different heights.
Researchers have demonstrated that by creating banks of tones whose note names are clearly defined perceptually but whose perceived heights are ambiguous, one can create scales that appear to ascend or descend endlessly in pitch.Roger Shepardachieved this ambiguity of height by creating banks of complex tones, with each tone composed only of components that stood in octave relationship. In other words, the components of the complex tone C consisted only of Cs, but in different octaves, and the components of the complex tone F♯consisted only of F♯s, but in different octaves.[2]When such complex tones are played in semitone steps the listener perceives a scale that appears to ascend endlessly in pitch.Jean-Claude Rissetachieved the same effect using gliding tones instead, so that a single tone appeared to glide up or down endlessly in pitch.[3]Circularity effects based on this principle have been produced in orchestral music and electronic music, by having multiple instruments playing simultaneously in different octaves.
Normann et al.[4]showed that pitch circularity can be created using a bank of single tones; here the relative amplitudes of the odd and even harmonics of each tone are manipulated so as to create ambiguities of height.
A different algorithm that creates ambiguities of pitch height by manipulating the relative amplitudes of the odd and even harmonics, was developed byDiana Deutschand colleagues.[5]Using this algorithm, gliding tones that appear to ascend or descend endlessly are also produced. This development has led to the intriguing possibility that, using this new algorithm, one might transform banks of natural instrument samples so as to produce tones that sound like those of natural instruments but still have the property of circularity. This development opens up new avenues for music composition and performance.[6]
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Polytely(fromGreekrootspoly-and-tel-meaning "many goals") comprises complexproblem-solvingsituations characterized by the presence of multiple simultaneous goals.[1]These goals may be contradictory or otherwise conflict with one another, requiring prioritisation of desired outcomes.[1]
Polytely is a feature of complex problem-solving that adds difficulty to finding an optimum solution. Funke describes polytely as a feature "not... inherent in a system, but [referring] to certain decisions of the experimenter", especially decisions relating to what goals are to be followed in solving the problem.[2]In the complex problem of nuclear waste disposal, Flüeler cites both trust betweenstates(as a factor innuclear proliferation: "Some states disarm whilst others re-arm – both do it for the sake of our planet's peace"), and safe andsustainabledisposal of nuclear waste as situations where considering in terms of polytely helps elaborate and then balance important but conflicting goals.[3]
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Atemporal paradox,time paradox, ortime travel paradox, is aparadox, an apparent contradiction, or logical contradiction associated with the idea oftime travelor other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistictime dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.
Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox.[1]Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps."[2][3]Consistency paradoxes, on the other hand, are those where future events influence the past to cause an apparent contradiction, exemplified by thegrandfather paradox, where a person travels to the past to prevent the conception of one of their ancestors, thus eliminating all the ancestor's descendants.[4]Newcomb's paradoxstems from the apparent contradictions that stem from the assumptions of bothfree willand foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop,[2]but although they appear similar, causal loops are unchanging and self-originating, whereas time loops are constantly resetting.[5]
A bootstrap paradox, also known as aninformation loop, aninformation paradox,[6]anontological paradox,[7]or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of eitherretrocausalityortime travel.[8][9][10][11]
Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere".[8]Such causally looped events then exist inspacetime, but their origin cannot be determined.[8][9]The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical.[9][6][12]A notable example occurs in the 1958science fictionshort story"—All You Zombies—", byRobert A. Heinlein, wherein the main character, anintersexindividual, becomes both their own mother and father; the 2014 filmPredestinationis based on the story. Allen Everett gives the movieSomewhere in Timeas an example involving an object with no origin: an old woman gives a watch to a playwright who later travels back in time and meets the same woman when she was young, and shows her the watch that she will later give to him.[6]An example of information which "came from nowhere" is in the movieStar Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula fortransparent aluminumto the 20th-century engineer who supposedly invented it.
Smeenk uses the term "predestination paradox" to refer specifically to situations in which a time traveler goes back in time to try to prevent some event in the past.[7]
The "predestination paradox" is a concept in time travel and temporal mechanics, often explored in science fiction. It occurs when a future event is the cause of a past event, which in turn becomes the cause of the future event, forming a self-sustaining loop in time. This paradox challenges conventional understandings of cause and effect, as the events involved are both the origin and the result of each other. A notable example is found in the TV seriesDoctor Who, where acharacter saves her father in the past, fulfilling a memory he had shared with her as a child about a strange woman having saved his life. The predestination paradox raises philosophical questions about free will, determinism, and the nature of time itself. It is commonly used as a narrative device in fiction to highlight the interconnectedness of events and the inevitability of certain outcomes.
The consistency paradox or grandfather paradox occurs when the past is changed in any way that directly negates the conditions required for the time travel to occur in the first place, thus creating a contradiction. A common example given is traveling to the past and preventing the conception of one's ancestors (such as causing the death of the ancestor's parent beforehand), thus preventing the conception of oneself. If the traveler were not born, then it would not be possible to undertake such an act in the first place; therefore, the ancestor proceeds to beget the traveler's next-generation ancestor and secure the line to the traveler. There is no predicted outcome to this scenario.[8]Consistency paradoxes occur whenever changing the past is possible.[9]A possible resolution is that a time travellercando anything thatdidhappen, butcannotdo anything thatdid nothappen. Doing something that did not happen results in a contradiction.[8]This is referred to as theNovikov self-consistency principle.
The grandfather paradox encompasses any change to the past,[13]and it is presented in many variations, including killing one's past self.[14][15]Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written intoAmazing Storiesin the 1920s.[16]Another variant of the grandfather paradox is the "Hitler paradox" or "Hitler's murder paradox", in which the protagonist travels back in time to murderAdolf Hitlerbefore he can rise to power in Germany, thus preventingWorld War IIand theHolocaust. Rather than necessarily physically preventing time travel, the action removes anyreasonfor the travel, along with any knowledge that the reason ever existed.[17]
Physicist John Garrison et al. give a variation of the paradox of an electronic circuit that sends a signal through a time machine to shut itself off, and receives the signal before it sends it.[18][19]
Newcomb's paradox is athought experimentshowing an apparent contradiction between theexpected utilityprinciple and thestrategic dominanceprinciple.[20]The thought experiment is often extended to explorecausalityand free will by allowing for "perfect predictors": if perfect predictors of the future exist, for example if time travel exists as a mechanism for making perfect predictions[how?], then perfect predictions appear to contradict free will because decisions apparently made with free will are already known to the perfect predictor[clarification needed].[21][22]Predestinationdoes not necessarily involve asupernaturalpower, and could be the result of other "infallible foreknowledge" mechanisms.[23]Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.[24]
Even without knowing whether time travel to the past is physically possible, it is possible to show usingmodal logicthat changing the past results in a logical contradiction. If it is necessarily true that the past happened in a certain way, then it is false and impossible for the past to have occurred in any other way. A time traveler would not be able to change the past from the way itis,but would only act in a way that is already consistent with whatnecessarilyhappened.[25][26]
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible. For example, the philosopherBradley Dowdenmade this sort of argument in the textbookLogical Reasoning, arguing that the possibility of creating a contradiction rules out time travel to the past entirely. However, some philosophers and scientists believe that time travel into the past need not be logically impossible provided that there is no possibility of changing the past,[13]as suggested, for example, by theNovikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopherNorman Swartz.[27]
A recent proposed resolution argues that if time is not an inherent property of the universe but is insteademergentfrom the laws ofentropy, as some modern theories suggest,[28][29]then it presents a natural solution to the Grandfather Paradox.[30]In this framework, "time travel" is reinterpreted not as movement along a linear continuum but as a reconfiguration of the present state of the universe to match a prior entropic configuration. Because the original chronological sequence—including events like the time traveler’s birth—remains preserved in the universe’s irreversible entropic progression, actions within the reconfigured state cannot alter the causal history that produced the traveler. This avoids paradoxes by treating time as a thermodynamic artifact rather than a mutable dimension.
Consideration of the possibility of backward time travel in a hypothetical universe described by aGödel metricled famed logicianKurt Gödelto assert that time might itself be a sort of illusion.[31][32]He suggests something along the lines of theblock timeview, in which time is just another dimension like space, with all events at all times being fixed within this four-dimensional "block".[citation needed]
Sergey Krasnikovwrites that these bootstrap paradoxes – information or an object looping through time – are the same; the primary apparent paradox is a physical system evolving into a state in a way that is not governed by its laws.[33]: 4He does not find these paradoxical and attributes problems regarding the validity of time travel to other factors in the interpretation of general relativity.[33]: 14–16
A 1992 paper by physicists Andrei Lossev andIgor Novikovlabeled such items without origin asJinn, with the singular termJinnee.[34]: 2311–2312This terminology was inspired by theJinnof theQuran, which are described as leaving no trace when they disappear.[35]: 200–203Lossev and Novikov allowed the term "Jinn" to cover both objects and information with the reflexive origin; they called the former "Jinn of the first kind", and the latter "Jinn of the second kind".[6][34]: 2315–2317[35]: 208They point out that an object making circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency; thesecond law of thermodynamicsseems to require that the object tends to a lower energy state throughout its history, and such objects that are identical in repeating points in their history seem to contradict this, but Lossev and Novikov argued that since the second law only requires entropy to increase inclosedsystems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy.[6][35]: 200–203They emphasize that there is no "strict difference" between Jinn of the first and second kind.[34]: 2320Krasnikov equivocates between "Jinn", "self-sufficient loops", and "self-existing objects", calling them "lions" or "looping or intruding objects", and asserts that they are no less physical than conventional objects, "which, after all, also could appear only from either infinity or a singularity."[33]: 8–9
The self-consistency principle developed byIgor Dmitriyevich Novikov[36]: p. 42 note 10expresses one view as to how backwardtime travelwould be possible without the generation of paradoxes. According to this hypothesis, even thoughgeneral relativitypermits someexact solutionsthat allow fortime travel[37]that containclosed timelike curvesthat lead back to the same point in spacetime,[38]physics in or nearclosed timelike curves(time machines) can only be consistent with the universal laws of physics, and thus only self-consistent events can occur. Anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from happening, since this would represent an inconsistency. The authors concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.[39]
PhysicistJoseph Polchinskiconsidered a potentially paradoxical situation involving abilliard ballthat is fired into awormholeat just the right angle such that it will be sent back in time and collides with its earlier self, knocking it off course, which would stop it from entering the wormhole in the first place.Kip Thornereferred to this problem as "Polchinski's paradox".[39]Thorne and two of his students at Caltech, Fernando Echeverria and Gunnar Klinkhammer, went on to find a solution that avoided any inconsistencies, and found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case.[40]Later analysis by Thorne andRobert Forwardshowed that for certain initial trajectories of the billiard ball, there could be an infinite number of self-consistent solutions.[39]It is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.[41]: 184The lack of constraints on initial conditions only applies to spacetime outside of thechronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.[41]: 187–188
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as anad hocsolution, and supposes that there are far more damaging implications of time travel.[42]Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity.[33]: 14–16Another conjecture, thecosmic censorship hypothesis, suggests that every closed timelike curve passes through anevent horizon, which prevents such causal loops from being observed.[43]
The interacting-multiple-universes approach is a variation of themany-worlds interpretationof quantum mechanics that involves time travelers arriving in a different universe than the one from which they came; it has been argued that, since travelers arrive in a different universe's history and not their history, this is not "genuine" time travel.[44]Stephen Hawking has argued for thechronology protection conjecture, that even if the MWI is correct, we should expect each time traveler to experience a single self-consistent history so that time travelers remain within their world rather than traveling to a different one.[45]
David Deutschhas proposed thatquantum computationwith a negative delay—backward time travel—produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning.[46]However Deutsch's self-consistency condition has been demonstrated as capable of being fulfilled to arbitrary precision by any system subject to the laws of classicalstatistical mechanics, even if it is not built up by quantum systems.[47]Allen Everett has also argued that even if Deutsch's approach is correct, it would imply that any macroscopic object composed of multiple particles would be split apart when traveling back in time, with different particles emerging in different worlds.[48]
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Inepistemology, and more specifically, thesociology of knowledge,reflexivityrefers to circular relationships betweencause and effect, especially as embedded in human belief structures. A reflexive relationship is multi-directional when the causes and the effects affect the reflexive agent in a layered or complex sociological relationship. The complexity of this relationship can be furthered when epistemology includesreligion.
Withinsociologymore broadly—the field of origin—reflexivitymeans an act ofself-referencewhere existence engenders examination, by which the thinking action "bends back on", refers to, and affects the entity instigating the action or examination. It commonly refers to the capacity of anagentto recognise forces ofsocialisationand alter their place in thesocial structure. A low level of reflexivity would result in individuals shaped largely by their environment (or "society"). A high level of social reflexivity would be defined by individuals shapingtheir ownnorms, tastes, politics, desires, and so on. This is similar to the notion ofautonomy. (See alsostructure and agencyandsocial mobility.)
Withineconomics,reflexivityrefers to the self-reinforcing effect of market sentiment, whereby rising prices attract buyers whose actions drive prices higher still until the process becomes unsustainable. This is an instance of apositive feedbackloop. The same process can operate in reverse leading to a catastrophic collapse in prices.
Insocial theory,reflexivitymay occur when theories in a discipline should apply equally to the discipline itself; for example, in the case that the theories of knowledge construction in the field ofsociology of scientific knowledgeshould apply equally to knowledge construction by sociology of scientific knowledge practitioners, or when the subject matter of a discipline should apply equally to the individual practitioners of that discipline (e.g., when psychological theory should explain the psychological processes of psychologists). More broadly, reflexivity is considered to occur when the observations of observers in the social system affect the very situations they are observing, or when theory being formulated is disseminated to and affects the behaviour of the individuals or systems the theory is meant to be objectively modelling. Thus, for example, an anthropologist living in an isolated village may affect the village and the behaviour of its citizens under study. The observations are not independent of the participation of the observer.
Reflexivity is, therefore, a methodological issue in the social sciences analogous to theobserver effect. Within that part of recentsociology of sciencethat has been called thestrong programme, reflexivity is suggested as a methodological norm or principle, meaning that a full theoretical account of the social construction of, say, scientific, religious or ethical knowledge systems, should itself be explainable by the same principles and methods as used for accounting for these other knowledge systems. This points to a general feature ofnaturalised epistemologies, that such theories of knowledge allow for specific fields of research to elucidate other fields as part of an overall self-reflective process: any particular field of research occupied with aspects of knowledge processes in general (e.g., history of science, cognitive science, sociology of science, psychology of perception, semiotics, logic, neuroscience) may reflexively study other such fields yielding to an overall improved reflection on the conditions for creating knowledge.
Reflexivity includes both a subjective process ofself-consciousnessinquiry and the study ofsocial behaviourwith reference to theories aboutsocial relationships.
The principle of reflexivity was perhaps first enunciated by the sociologistsWilliam I. ThomasandDorothy Swaine Thomas, in their 1928 bookThe child in America: "If men define situations as real, they are real in their consequences".[1]The theory was later termed the "Thomas theorem".
SociologistRobert K. Merton(1948, 1949) built on the Thomas principle to define the notion of aself-fulfilling prophecy: that once a prediction or prophecy is made, actors may accommodate their behaviours and actions so that a statement that would have been false becomes true or, conversely, a statement that would have been true becomes false - as a consequence of the prediction or prophecy being made. The prophecy has a constitutive impact on the outcome or result, changing the outcome from what would otherwise have happened.
Reflexivity was taken up as an issue in science in general byKarl Popper(1957), who in his bookThe Poverty of Historicismhighlighted the influence of a prediction upon the event predicted, calling this the 'Oedipus effect' in reference to the Greek tale in which the sequence of events fulfilling the Oracle's prophecy is greatly influenced by the prophecy itself. Popper initially considered such self-fulfilling prophecy a distinguishing feature of social science, but later came to see that in the natural sciences, particularly biology and even molecular biology, something equivalent to expectation comes into play and can act to bring about that which has been expected.[2]It was also taken up byErnest Nagel(1961). Reflexivity presents a problem for science because if a prediction can lead to changes in the system that the prediction is made in relation to, it becomes difficult to assess scientific hypotheses by comparing the predictions they entail with the events that actually occur. The problem is even more difficult in the social sciences.
Reflexivity has been taken up as the issue of "reflexive prediction" in economic science by Grunberg and Modigliani (1954) andHerbert A. Simon(1954), has been debated as a major issue in relation to theLucas critique, and has been raised as a methodological issue in economic science arising from the issue of reflexivity in thesociology of scientific knowledge(SSK) literature.
Reflexivity has emerged as both an issue and a solution in modern approaches to the problem ofstructure and agency, for example in the work ofAnthony Giddensin hisstructuration theoryandPierre Bourdieuin hisgenetic structuralism.
Giddens, for example, noted that constitutive reflexivity is possible in any social system, and that this presents a distinct methodological problem for the social sciences. Giddens accentuated this theme with his notion of "reflexive modernity" – the argument that, over time, society is becoming increasingly more self-aware, reflective, and hence reflexive.
Bourdieuargued that the social scientist is inherently laden withbiases, and only by becoming reflexively aware of those biases can the social scientists free themselves from them and aspire to the practice of an objective science. For Bourdieu, therefore, reflexivity is part of the solution, not the problem.
Michel Foucault'sThe order of thingscan be said to touch on the issue of Reflexivity. Foucault examines the history of Western thought since the Renaissance and argues that each historical epoch (he identifies three and proposes a fourth) has anepisteme, or "a historicala priori", that structures and organises knowledge. Foucault argues that the concept of man emerged in the early 19th century, what he calls the "Age of Man", with the philosophy ofImmanuel Kant. He finishes the book by posing the problem of the age of man and our pursuit of knowledge- where "man is both knowing subject and the object of his own study"; thus, Foucault argues that the social sciences, far from being objective, produce truth in their own mutually exclusivediscourses.
Economic philosopherGeorge Soros, influenced by ideas put forward by his tutor,Karl Popper(1957),[3]has been an active promoter of the relevance of reflexivity to economics, first propounding it publicly in his 1987 bookThe alchemy of finance.[4]He regards his insights into market behaviour from applying the principle as a major factor in the success of his financial career.
Reflexivity is inconsistent withgeneral equilibrium theory, which stipulates that markets move towards equilibrium and that non-equilibrium fluctuations are merely random noise that will soon be corrected. In equilibrium theory, prices in the long run at equilibrium reflect the underlyingeconomic fundamentals, which are unaffected by prices. Reflexivity asserts that prices do in fact influence the fundamentals and that these newly influenced sets of fundamentals then proceed to change expectations, thus influencing prices; the process continues in a self-reinforcing pattern. Because the pattern is self-reinforcing, markets tend towards disequilibrium. Sooner or later they reach a point where the sentiment is reversed and negative expectations become self-reinforcing in the downward direction, thereby explaining the familiar pattern of boom and bust cycles.[5]An example Soros cites is theprocyclicalnature of lending, that is, the willingness of banks to ease lending standards for real estate loans when prices are rising, then raising standards when real estate prices are falling, reinforcing the boom and bust cycle. He further suggests that property price inflation is essentially a reflexive phenomenon: house prices are influenced by the sums that banks are prepared to advance for their purchase, and these sums are determined by the banks' estimation of the prices that the property would command.
Soros has often claimed that his grasp of the principle of reflexivity is what has given him his "edge" and that it is the major factor contributing to his successes as a trader. For several decades there was little sign of the principle being accepted in mainstream economic circles, but there has been an increase of interest following the crash of 2008, with academic journals, economists, and investors discussing his theories.[6]
Economist and former columnist of theFinancial Times,Anatole Kaletsky, argued that Soros' concept of reflexivity is useful in understanding China's economy and how the Chinese government manages it.[7]
In 2009, Soros funded the launch of theInstitute for New Economic Thinkingwith the hope that it would develop reflexivity further.[8]The Institute works with several types ofheterodox economics, particularly thepost-Keynesianbranch.[9]
Margaret Archerhas written extensively on laypeople's reflexivity. For her, human reflexivity is amediating mechanismbetween structural properties, or the individual's social context, and action, or the individual's ultimate concerns.[10]Reflexive activity, according to Archer, increasingly takes the place of habitual action in late modernity since routine forms prove ineffective in dealing with the complexity of modern life trajectories.[11]
While Archer emphasises the agentic aspect of reflexivity, reflexive orientations can themselves be seen as being "socially and temporally embedded".[12]For example, Elster points out that reflexivity cannot be understood without taking into account the fact that it draws on background configurations (e.g., shared meanings, as well as past social engagement and lived experiences of the social world) to be operative.[12]
In anthropology, reflexivity has come to have two distinct meanings, one that refers to the researcher's awareness of an analytic focus on his or her relationship to the field of study, and the other that attends to the ways thatcultural practicesinvolve consciousness and commentary on themselves.
The first sense of reflexivity in anthropology is part of social science's more general self-critique in the wake of theories byMichel Foucaultand others about the relationship of power and knowledge production. Reflexivity about the research process became an important part of the critique of the colonial roots[13]and scientistic methods of anthropology in the "writing cultures"[14]movement associated withJames CliffordandGeorge Marcus, as well as many other anthropologists. Rooted in literary criticism and philosophical analysis of the relationship among the anthropologists, the people represented in texts, and their textual representations, this approach has fundamentally changed ethical and methodological approaches in anthropology. As with thefeministandanti-colonialcritiques that provide some of reflexive anthropology's inspiration, the reflexive understanding of the academic and political power of representations, analysis of the process of "writing culture" has become a necessary part of understanding the situation of the ethnographer in the fieldwork situation. Objectification of people and cultures and analysis of them only as objects of study has been largely rejected in favor of developing more collaborative approaches that respect local people's values and goals. Nonetheless, many anthropologists have accused the "writing cultures" approach of muddying the scientific aspects of anthropology with too much introspection about fieldwork relationships, and reflexive anthropology have been heavily attacked by more positivist anthropologists.[15]Considerable debate continues in anthropology over the role ofpostmodernismand reflexivity, but most anthropologists accept the value of the critical perspective, and generally only argue about the relevance of critical models that seem to lead anthropology away from its earlier core foci.[16]
The second kind of reflexivity studied by anthropologists involves varieties of self-reference in which people and cultural practices call attention to themselves.[17]One important origin for this approach isRoman Jakobsonin his studies ofdeixisand the poetic function in language, but the work ofMikhail Bakhtinon carnival has also been important. Within anthropology,Gregory Batesondeveloped ideas about meta-messages (subtext) as part of communication, whileClifford Geertz's studies of ritual events such as theBalinese cock-fightpoint to their role as foci for public reflection on the social order. Studies of play and tricksters further expanded ideas about reflexive cultural practices. Reflexivity has been most intensively explored in studies of performance,[18]public events,[19]rituals,[20]and linguistic forms[21]but can be seen any time acts, things, or people are held up and commented upon or otherwise set apart for consideration. In researching cultural practices, reflexivity plays an important role, but because of its complexity and subtlety, it often goes under-investigated or involves highly specialised analyses.[22]
One use of studying reflexivity is in connection toauthenticity. Cultural traditions are often imagined as perpetuated as stable ideals by uncreative actors. Innovation may or may not change tradition, but since reflexivity is intrinsic to many cultural activities, reflexivity is part of tradition and not inauthentic. The study of reflexivity shows that people have both self-awareness and creativity in culture. They can play with, comment upon, debate, modify, and objectify culture through manipulating many different features in recognised ways. This leads to themetacultureof conventions about managing and reflecting upon culture.[23]
Ininternational relations, the question of reflexivity was first raised in the context of the so-called ‘Third Debate’ of the late 1980s. This debate marked a break with the positivist orthodoxy of the discipline. The post-positivist theoretical restructuring was seen to introduce reflexivity as a cornerstone of critical scholarship.[24][25]For Mark Neufeld, reflexivity in International Relations was characterized by 1) self-awareness of underlying premises, 2) an acknowledgment of the political-normative dimension of theoretical paradigms, and 3) the affirmation that judgement about the merits of paradigms is possible despite the impossibility of neutral or apolitical knowledge production.[26]
Since the nineties, reflexivity has become an explicit concern ofconstructivist,poststructuralist,feminist, and other critical approaches to International Relations.[27][25][28][29][30][31]InThe Conduct of Inquiry in International Relations, Patrick Thaddeus Jackson identified reflexivity of one of the four main methodologies into which contemporary International Relations research can be divided, alongside neopositivism, critical realism, and analyticism.[32]
Flanagan has argued that reflexivity complicates all three of the traditional roles that are typically played by a classical science: explanation, prediction and control. The fact that individuals and social collectivities are capable of self-inquiry and adaptation is a key characteristic of real-world social systems, differentiating the social sciences from the physical sciences. Reflexivity, therefore, raises real issues regarding the extent to which the social sciences may ever be viewed as "hard" sciences analogous to classical physics, and raises questions about the nature of the social sciences.[33]
A new generation of scholars has gone beyond (meta-)theoretical discussion to develop concrete research practices for the implementation of reflexivity. These scholars have addressed the ‘how to’ question by turning reflexivity from an informal process into a formal research practice.[34][35][36][37]While most research focuses on how scholars can become more reflexive toward their positionality and situatedness, some have sought to build reflexive methods in relation to other processes of knowledge production, such as the use of language. The latter has been advanced by the work of Professor Audrey Alejandro in a trilogy on reflexive methods. The first article of the trilogy develops what is referred to as Reflexive Discourse Analysis, a critical methodology for the implementation of reflexivity that integrates discourse theory.[31]The second article further expands the methodological tools for practicing reflexivity by introducing a three-stage research method for problematizing linguistic categories.[38]The final piece of the trilogy adds a further method for linguistic reflexivity, namely the Reflexive Review. This method provides four steps that aim to add a linguistic and reflexive dimension to the practice of writing a literature review.[39]
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Rock paper scissors(also known by several other names and word orders, see§ Names) is anintransitivehand game, usually played between two people, in which each player simultaneously forms one of three shapes with an outstretched hand. These shapes are "rock" (a closed fist), "paper" (a flat hand), and "scissors" (a fist with the index finger and middle finger extended, forming a V). The earliest form of "rock paper scissors"-style game originated inChinaand was subsequently imported intoJapan, where it reached its modern standardized form, before being spread throughout the world in the early 20th century.
Asimultaneous,zero-sum game, it has three possible outcomes: a draw, a win, or a loss. A player who decides to play rock will beat another player who chooses scissors ("rock crushes scissors" or "breaks scissors" or sometimes "blunts scissors"[1]), but will lose to one who has played paper ("paper covers rock"); a play of paper will lose to a play of scissors ("scissors cuts paper"). If both players choose the same shape, the game is tied, but is usually replayed until there is a winner.
Rock paper scissors is often used as a fair choosing method between two people, similar tocoin flipping,drawing straws, or throwingdicein order to settle a dispute or make an unbiased group decision. Unlike trulyrandomselection methods, however, rock paper scissors can be played with some degree of skill by recognizing and exploiting non-random behavior in opponents.[2][3]
The name "rock paper scissors" is simply a translation of the Japanese words for the three gestures involved in the game,[4]though the Japanese name for the game is different.
The name Roshambo or Rochambeau has been claimed to refer toCount Rochambeau, who allegedly played the game during theAmerican Revolutionary War. The legend that he played the game is apocryphal, as all evidence points to the game being brought to the United States later than 1910; if this name has anything to do with him it is for some other reason.[5][6]It is unclear why this name became associated with the game, with hypotheses ranging from a slight phonetic similarity with the Japanese namejan-ken-pon,[5]to the presence of a statue of Rochambeau in a neighborhood of Washington, DC.[6]
The modern game is known by several other names such asRochambeau,Roshambo,Ro-sham-bo,Bato Bato Pik, andJak-en-poy.[7][8][9]While the game's name is a list of three items, different countries often have the list in a different order.
In North America and the United Kingdom, it is known as "rock, paper, scissors" or "scissors, paper, stone".[10][11]If this name is chanted while actually playing the game, it might be followed by an exclamation of "shoot" at the moment when the players are to reveal their choice (i.e. "Rock, paper, scissors, shoot!").[12]
In Australia, the most common name is "scissors, paper, rock" (the reverse of the American format).[13]There have been claims that there are regional variations of the name in Australia; one video claimed that it was referred to as "scissors, paper, rock" inNew South Wales, "rock, paper, scissors" inVictoria,South AustraliaandWestern Australia, and "paper, scissors, rock" inQueensland, though this has been disputed.[14]
In New Zealand, the most common name in English is "paper, scissors, rock".[15]InMāori, it is known aspēpa, kutikuti, kōhatu(lit.'paper, scissors, rock').[16]
In France, the game is sometimes called Shifumi (sometimes spelled Chifoumi).[citation needed]
The players may start by counting to three aloud, or by speaking the name of the game (e.g. "Rock! Paper! Scissors!"), raising one hand in a fist and swinging it down with each syllable onto their other hand (or in a less common variant, holding it behind their back). They then "throw" or "shoot" by extending their selected sign towards their opponent on what would have been the fourth count, often saying the word "shoot" while doing so.
Variations include a version where players throw immediately on the third count (thus throwing on the count of "Scissors!"), a version including five counts rather than four ("Rock! Paper! Scissors! Says! Shoot!", almost exclusively localized in the United States toLong Islandand some parts of New York City), a version where players say “Scissors! Paper! Rock!”, and a version where players shake their hands three times before "throwing".[citation needed]
The first known mention of the game was in the bookWuzazu[zh]by theMing-dynastywriterXie Zhaozhe[zh](fl.c.1600), who wrote that the game dated back to the time of theHan dynasty(206 BCE – 220 CE).[17]In the book, the game was calledshoushiling.Li Rihua's bookNote of Liuyanzhaialso mentions this game, calling itshoushiling(t. 手勢令;s. 手势令),huozhitou(t. 豁指頭;s. 豁指头), orhuaquan(划拳).
From China the game was brought to Japan.[18]Throughout Japanese history there are frequent references tosansukumi-ken, meaningken(fist) games "of the three who are afraid of one another" (i.e. A beats B, B beats C, and C beats A).[18]
The earliestsansukumi-kenin Japan was apparentlymushi-ken(虫拳), a version imported directly from China.[18][19]Inmushi-kenthe "frog" (represented by the thumb) triumphs over the "slug" (represented by the little finger), which, in turn prevails over the "snake" (represented by the index finger), which triumphs over the "frog".[18][19](The Chinese and Japanese versions differ in the animals represented; in adopting the game, theChinese charactersfor thevenomouscentipede(蜈蜙) were apparently confused with the characters for the slug (蛞蝓)).[19]
The most popularsansukumi-kengame in Japan[when?]waskitsune-ken(狐拳). In this game, a fox (狐), often attributed supernatural powers in Japanese folklore, defeats the village head, the village head (庄屋) defeats the hunter, and the hunter (猟師) defeats the fox.Kitsune-ken, unlikemushi-kenor rock–paper–scissors, requires gestures with both hands.[20]
Today, the best-knownsansukumi-kenis calledjan-ken(じゃんけん),[19]which is a variation of the Chinese games introduced in the 17th century.[21]Jan-kenuses the rock, paper, and scissors signs[18]and is the direct source of the modern version of rock paper scissors.[19]Hand-games using gestures to represent the three conflicting elements of rock, paper, and scissors have been most common since the modern version of the game was created in the late 19th century, between theEdoandMeijiperiods.[22]
By the early 20th century, rock paper scissors had spread beyond East Asia, especially through increased Japanese contact with the west.[23]Its English-language name is therefore taken from a translation of the names of the three Japanese hand-gestures for rock, paper and scissors;[4]elsewhere in East Asia the open-palm gesture represents "cloth" rather than "paper".[24]The shape of the scissors is also adopted from the Japanese style.[4]
A 1921 article aboutcricketin theSydney Morning Heralddescribed "stone, scissors, and paper" as a "Teutonic method of drawing lots", which the writer "came across when travelling on the Continent once".[25]Another article, from the same year, theWashington Heralddescribed it as a method of "Chinese gambling".[26]In Britain in 1924 it was described in a letter toThe Timesas a hand game, possibly of Mediterranean origin, called "zhot".[27]A reader then wrote in to say that the game "zhot" referred to was evidently Jan-ken-pon, which she had often seen played throughout Japan.[28]Although at this date the game appears to have been new enough to British readers to need explaining, the appearance by 1927 ofGerard Fairlie's popularthrillernovelwith the titleScissors Cut Paper,[29]followed by Fairlie'sStone Blunts Scissors(1929), suggests it quickly became popular.
The game is referred to in two of Hildegard G. Frey's novels in theCampfire Girlsseries:The Campfire Girls Go Motoring(1916)[30]andThe Campfire Girls Larks and Pranks(1917),[31]which suggests that it was known in America at least that early. The first passage where it appears says "In order that no feelings might be involved in any way over which car we other girls traveled in, Nyoda, Solomon-like, proposed that she and Gladys play 'John Kempo' for us. (That isn't spelled right, but no matter.)" There is no explanation in any of the places where it is referenced of what the game actually is. This suggests that the author at least believed that the game was well known enough in America that her readers would understand the reference.
In 1927La Vie au patronage : organe catholique des œuvres de jeunesse, a children's magazine in France, described it in detail,[32]referring to it as a "jeu japonais" ("Japanese game"). Its French name, "Chi-fou-mi", is based on theOld Japanesewords for "one, two, three" ("hi, fu, mi").
A 1932New York Timesarticle on the Tokyo rush hour describes the rules of the game for the benefit of American readers, suggesting it was not at that time widely known in the U.S.[33]Likewise, thetrick-taking card game“Jan-Ken-Po”, first published in 1934, describes the rules of the hand-game without mentioning any American game along the lines of “rock paper scissors”. The 1933 edition of theCompton's Pictured Encyclopediadescribed it as a common method of settling disputes between children in its article on Japan; the name was given as "John Kem Po" and the article pointedly asserted, "This is such a good way of deciding an argument that American boys and girls might like to practice it too."[34]
It is impossible to gain an advantage over an opponent that chooses their moveuniformly at random. However, it is possible to gain a significant advantage over a non-random player by predicting their move, which can be done by exploiting psychological effects or by analyzing statistical patterns of their past behavior.[35][36][37]As a result, there have been programming competitions foralgorithmsthat play rock paper scissors.[35][38][39]
During tournaments, players often prepare their sequence of three gestures prior to the tournament's commencement.[40][41]Some tournament players employ tactics to confuse or trick the other player into making an illegal move, resulting in a loss. One such tactic is to shout the name of one move before throwing another, in order to misdirect and confuse their opponent.[citation needed]
The "rock" move, in particular, is notable in that it is typically represented by a closed fist—often identical to the fist made by players during the initial countdown. If a player is attempting to beat their opponent based on quickly reading their hand gesture as the players are making their moves, it is possible to determine if the opponent is about to throw "rock" based on their lack of hand movement, as both "scissors" and "paper" require the player to reposition their hand. This can likewise be used to deceive an anticipating opponent by keeping one's fist closed until the last possible moment, leading them to believe that one is about to throw "rock".[citation needed]
As a consequence of rock paper scissors programming contests, many strong algorithms have emerged.[35][38][39]For example, Iocaine Powder, which won the First International RoShamBo Programming Competition in 1999,[38]uses a heuristically designed compilation of strategies.[42]For each strategy it employs, it also has six metastrategies which defeat second-guessing, triple-guessing, as well as second-guessing the opponent, and so on. The optimal strategy or metastrategy is chosen based on past performance. The main strategies it employs are history matching, frequency analysis, and random guessing. Its strongest strategy, history matching, searches for a sequence in the past that matches the last few moves in order to predict the next move of the algorithm. In frequency analysis, the program simply identifies the most frequently played move. The random guess is a fallback method that is used to prevent a devastating loss in the event that the other strategies fail. There have since been some innovations, such as using multiple history-matching schemes that each match a different aspect of the history – for example, the opponent's moves, the program's own moves, or a combination of both.[43]There have also been other algorithms based onMarkov chains.[44]
In 2012, researchers from the Ishikawa Watanabe Laboratory at theUniversity of Tokyocreated a robot hand that can play rock paper scissors with a 100% win rate against a human opponent. Using a high-speed camera the robot recognizes within onemillisecondwhich shape the human hand is making, then produces the corresponding winning shape.[45][46]
There exist numerous cultural and personal variations on rock paper scissors. Differences vary from simply playing the same game with different objects to expanding into more weapons and rules.
InKorea, where the standard version of the game is calledgawi-bawi-bo, a more complex version exists by the namemuk-jji-ppa.[47]After showing their hands, the player with the winning throw shouts "muk-jji-ppa!" upon which both players throw again. If they throw differently, whoever wins this second round shouts "muk-jji-ppa!" and thus the play continues until both players throw the same, at which point whoever was the last winner becomes the final winner.
In "rock paper scissors minus one", another popular variant in Korea, both players throw with both hands simultaneously. Each player chooses one hand to remove, and the winner is decided by the remaining hands in play; a tie leads to a replay.[48]This variation was featured in the second season of theNetflixseriesSquid Game.
In Japan, astrip gamevariant of rock paper scissors is known as 野球拳 (Yakyūken). The loser of each round removes an article of clothing. The game is a minor part of porn culture in Japan and other Asian countries after the influence of TV variety shows andSoft On Demand.
In thePhilippines, the game is calledjak-en-poy(from the Japanesejankenpon). In a longer version of the game, a four-line song is sung, with hand gestures displayed at the end of each (or the final) line: "Jack-en-poy! / Hali-hali-hoy! / Sino'ng matalo, / siya'ng unggoy!" ("Jack-en-poy! / Hali-hali-hoy! / Whoever loses is the monkey!") In the former case, the person with the most wins at the end of the song, wins the game. A shorter version of the game uses the chant "Bato-bato-pick" ("Rock-rock-pick [i.e. choose]") instead.[citation needed]
A variation with more players can be played: Players stand in a circle and all throw at once. If rock, paper, and scissors are all thrown, it is a stalemate, and they rethrow. If only two throws are present, all players with the losing throw are eliminated. Play continues until only the winner remains.[49]
InIndonesia, the game is calledsuten,suitor justsut, and the three signs are elephant (slightly raised thumb), human (outstreched index finger) and ant (outstreched pinky finger).[50]Elephant is stronger than human, human is stronger than ant, but elephant is afraid of the ant.
Using the same tripartite division, there is a full-body variation in lieu of the hand signs called "Bear, Hunter, Ninja".[51]In this iteration the participants stand back-to-back and at the count of three (or ro-sham-bo as is traditional) turn around facing each other using their arms evoking one of the totems.[52]The players' choices break down as: Hunter shoots bear; Bear eats ninja; Ninja kills hunter.[53]
Generalized rock-paper-scissors games where the players have a choice of more than three weapons have been studied.[54]Any variation of rock paper scissors is anoriented graph, where thenodesrepresent the symbols (weapons) choosable by the players, and an edge from A to B means that A defeats B. Each oriented graph is a potentially playable rock paper scissors game. According to theoretical calculations, the number of distinguishable (i.e. notisomorphic) oriented graphs grows with the number of weapons = 3, 4, 5, ... as follows:[55][56]
The French gamepierre, papier, ciseaux, puits(stone, paper, scissors,well) is unbalanced; both the stone and scissors fall in the well and lose to it, while paper covers both stone and well. This means two "weapons", well and paper, can defeat two moves, while the other two weapons each defeat only one of the other three choices. The stone has no advantage to well, so optimal strategy is to play each of the other objects (paper, scissors and well) one-third of the time.[57]
Variants in which the number of moves is an odd number and each move defeats exactly half of the other moves while being defeated by the other half are typically considered. Variations with up to 101 different moves have been published.[58]Adding new gestures has the effect of reducing the odds of a tie, while increasing the complexity of the game. The probability of a tie in an odd-number-of-weapons game can be calculated based on the number of weaponsnas 1/n, so the probability of a tie is 1/3 in standard rock paper scissors, but 1/5 in a version that offered five moves instead of three.[59]
One popular five-weapon expansion is "rock paper scissors Spock lizard", invented by Sam Kass and Karen Bryla,[60]which adds "Spock" and "lizard" to the standard three choices. "Spock" is signified with theStar TrekVulcan salute, while "lizard" is shown by forming the hand into a sock-puppet-like mouth. Spock smashes scissors and vaporizes rock; he is poisoned by lizard and disproved by paper. Lizard poisons Spock and eats paper; it is crushed by rock and decapitated by scissors. This variant was mentioned in a 2005 article inThe Timesof London[61]and was later the subject of anepisodeof the American sitcomThe Big Bang Theoryin 2008 (asrock-paper-scissors-lizard-Spock).[62]
Agame-theoretic analysisshowed that 4 variants of 582 possible variations using 5 different weapons have non-trivialmixed strategy equilibria.[56]The most representative game of these 4 is "rock, paper, scissors, fire, water". Rock beats scissors, paper beats rock, scissors beats paper, fire beats everything except water, and water is beaten by everything except it beats fire. The perfect game-theoretic strategy is to use rock, paper, and scissors19{\displaystyle {\frac {1}{9}}}of the time and13{\displaystyle {\frac {1}{3}}}of the time for fire and water. Nevertheless, experiments show that people underuse water and overuse rock, paper, and scissors in this game.[63]
Thecommon side-blotched lizard(Uta stansburiana) exhibits a rock paper scissors pattern in its mating strategies. Of its three throat color types of males, "orange beats blue, blue beats yellow, and yellow beats orange" in competition for females, which is similar to the rules of rock-paper-scissors.[64][65]
Some bacteria also exhibit a rock paper scissors dynamic when they engage inantibioticproduction. The theory for this finding was demonstrated by computer simulation and in the laboratory by Benjamin Kerr, working atStanford UniversitywithBrendan Bohannan.[66]Additionalin vitroresults demonstrate rock paper scissors dynamics in additional species of bacteria.[67]Biologist Benjamin C. Kirkup Jr. demonstrated that these antibiotics,bacteriocins, were active asEscherichia colicompete with each other in the intestines of mice, and that the rock paper scissors dynamics allowed for the continued competition among strains: antibiotic-producers defeat antibiotic-sensitives; antibiotic-resisters multiply and withstand and out-compete the antibiotic-producers, letting antibiotic-sensitives multiply and out-compete others; until antibiotic-producers multiply again.[68]
Rock paper scissors is the subject of continued research in bacterial ecology and evolution. It is considered one of the basic applications ofgame theoryand non-linear dynamics to bacteriology.[69]Models of evolution demonstrate how intragenomic competition can lead to rock paper scissors dynamics from a relatively general evolutionary model.[70]The general nature of this basic non-transitive model is widely applied in theoretical biology to explore bacterial ecology and evolution.[71][72]
In the televisedrobot combatcompetitionBattleBots, relations between "lifters, which had wedged sides and could use forklift-like prongs to flip pure wedges", "spinners, which were smooth, circular wedges with blades on their bottom side for disabling and breaking lifters", and "pure wedges, which could still flip spinners" are analogical to relations in rock paper scissors games and called "robot Darwinism".[73]
In 2006, American federal judgeGregory Presnellfrom theMiddle District of Floridaordered opposing sides in a lengthy court case to settle a trivial (but lengthily debated) point over the appropriate place for adepositionusing the game of rock paper scissors.[74][75]The ruling inAvista Management v. Wausau Underwritersstated:
Upon consideration of the Motion – the latest in a series ofGordian knotsthat the parties have been unable to untangle without enlisting the assistance of the federal courts – it is ORDERED that said Motion is DENIED. Instead, the Court will fashion a new form of alternative dispute resolution, to wit: at 4:00 P.M. on Friday, June 30, 2006, counsel shall convene at a neutral site agreeable to both parties. If counsel cannot agree on a neutral site, they shall meet on the front steps of the Sam M. Gibbons U.S. Courthouse, 801 North Florida Ave., Tampa, Florida 33602. Each lawyer shall be entitled to be accompanied by one paralegal who shall act as an attendant and witness. At that time and location, counsel shall engage in one (1) game of "rock, paper, scissors." The winner of this engagement shall be entitled to select the location for the 30(b)(6) deposition to be held somewhere in Hillsborough County during the period 11–12 July 2006.[76]
In 2005, when Takashi Hashiyama, CEO of Japanese television equipment manufacturerMaspro Denkoh, decided to auction off the collection ofImpressionistpaintings owned by his corporation, including works byPaul Cézanne,Pablo Picasso, andVincent van Gogh, he contacted two leading auction houses,Christie'sInternational andSotheby'sHoldings, seeking their proposals on how they would bring the collection to the market as well as how they would maximize the profits from the sale. Both firms made elaborate proposals, but neither was persuasive enough to earn Hashiyama's approval. Unwilling to split up the collection into separate auctions, Hashiyama asked the firms to decide between themselves who would hold the auction, which included Cézanne'sLarge Trees Under the Jas de Bouffan, estimated to be worth between $12 million to $16 million.
The houses were unable to reach a decision. Hashiyama told the two firms to play rock paper scissors to decide who would get the rights to the auction, explaining that "it probably looks strange to others, but I believe this is the best way to decide between two things which are equally good."
The auction houses had a weekend to come up with a choice of move. Christie's went to the 11-year-old twin daughters of the international director of Christie's Impressionist and Modern Art Department Nicholas Maclean, who suggested "scissors" because "Everybody expects you to choose 'rock'." Sotheby's said that they treated it as agame of chanceand had no particular strategy for the game, but went with "paper".[78]Christie's won the match and sold the $20 million collection, earning millions of dollars of commission for the auction house.
Prior to a 26 October 2018 match in theFA Women's Super League, the referee, upon being without a coin for the pregamecoin toss, had the team captains play rock paper scissors to determine which team wouldkick-off. The referee was subsequently suspended for three weeks byThe Football Association.[79]
In Japan, researchers have taughtchimpanzeesto identify winning hands according to the rules of rock paper scissors.[80]
In many games, it is common for a group of possible choices to interact in a rock paper scissors style, where each selection is strong against a particular choice, but weak against another. Such mechanics can make a game somewhat self-balancing, prevent gameplay from being overwhelmed by a singledominant strategyand single dominant type of unit.[81]
Many card-based video games in Japan use the rock paper scissors system as their core fighting system, with the winner of each round being able to carry out their designated attack. InAlex Kidd in Miracle World, the player has to win games of rock paper scissors against each boss to proceed. Others use simple variants of rock paper scissors as subgames.
ManyNintendorole-playing games prominently feature a rock paper scissors gameplay element. InPokémon, there is a rock paper scissors element in the type effectiveness system. For example, a Grass-typed Pokémon is weak to Fire, Fire is weak to Water, and Water is weak to Grass.[82]In the 3DS remake ofMario & Luigi: Superstar SagaandMario & Luigi: Bowser's Inside Story, the battles in the second mode use a “Power Triangle” system based on the game's three attack types: Melee, Ranged, and Flying. In theFire Emblemseries of strategy role-playing games, the Weapon Triangle and Trinity of Magic influence the hit and damage rates of weapon types based on whether they are at an advantage or a disadvantage in their respective rock paper scissors system. In theSuper Smash Bros.series, the three basic actions used during battles are described in their respective rock paper scissors system: attack, defense, and grab.
The "Card-Jitsu" minigame inClub Penguinis a rock-paper-scissors game using cards that represent the three elements, Fire, Water and Snow. Fire beats snow, snow beats water, water beats fire.
Various competitive rock paper scissors tournaments have been organised by different groups.
Started in 2015, the WRPSA has hosted Professional Rock Paper Scissors Tournaments all around the world.[83][84][85][86][87]
The World Rock Paper Scissors Society hosted Professional Rock Paper Scissors Tournaments from 2002 to 2009. These open, competitive championships were widely attended by players from around the world and attracted widespread international media attention.[88][89][90][91][92]WRPS events were noted for their large cash prizes, elaborate staging, and colorful competitors.[93]In 2004, the championships were broadcast on the U.S. television networkFox Sports Net(later known asBally Sports), with the winner being Lee Rammage, who went on to compete in at least one subsequent championship.[94][95]The 2007 tournament was won by Andrea Farina.[96]The last tournament hosted by the World RPS Society was in Toronto, Canada, on November 14, 2009.[97]
Several RPS events have been organised in the United Kingdom byWacky Nation. The 1st UK Championship took place on 13 July 2007, and were then held annually. The 2019 event was won by Ellie Mac, who went on to pick up the cash prize of £20,000 but was unable to double her earnings in 2020 due to the coronavirus outbreak.[98]
USA Rock Paper Scissors Leagueis sponsored byBud Light. Leo Bryan Pacis was the first commissioner of the USARPS.[citation needed]Cody Louis Brown was elected as the second commissioner of the USARPS in 2014.[citation needed]
In April 2006, the inaugural USARPS Championship was held inLas Vegas. Following months of regional qualifying tournaments held across the US, 257 players were flown to Las Vegas for a single-elimination tournament at theHouse of Blueswhere the winner received $50,000. The tournament was shown on theA&E Networkon 12 June 2006.
The $50,000 2007 USARPS Tournament took place at the Las Vegas Mandalay Bay in May 2007.
In 2008, Sean "Wicked Fingers" Sears beat 300 other contestants and walked out of the Mandalay Bay Hotel and Casino with $50,000 after defeating Julie "Bulldog" Crossley in the finals.
The inaugural Budweiser International Rock, Paper, Scissors Federation Championship was held in Beijing, China after the close of the 2008 Summer Olympics at Club Bud. A Belfast man won the competition.[99]
The XtremeRPS National Competition is a US nationwide RPS competition with Preliminary Qualifying contests that started in January 2007 and ended in May 2008, followed by regional finals in June and July 2008. The national finals were to be held inDes Moines, Iowa, in August 2008, with a chance to win up to $5,000.
The largest rock paper scissors tournament hosted 2,950 players and was achieved by Oomba, Inc. (USA) atGen Con2014 inIndianapolis, Indiana, United States, on 17 August 2014.[100]
FormerCelebrity Poker Showdownhost and USARPS Head Referee[101]Phil Gordonhas hosted an annual $500 World Series of Rock Paper Scissors event in conjunction with theWorld Series of Pokersince 2005.[102]The winner of the WSORPS receives an entry into theWSOP Main Event. The event is an annual fundraiser for the "Cancer Research and Prevention Foundation" via Gordon's charityBad Beat on Cancer. Poker playerAnnie Dukewon the Second Annual World Series of Rock Paper Scissors.[103]The tournament is taped by ESPN and highlights are covered during "The Nuts" section of ESPN's annual WSOP broadcast.[104][105][106]2009 was the fifth year of the tournament.
Jackpot En Poy is a game segment on the Philippines' longest running noontime variety show,Eat Bulaga!. The game is based on the classic children's game rock paper scissors (Jak-en-poy in Filipino, derived from the Japanese Jan-ken-pon) where four players are paired to compete in the three-round segment. In the first round, the first pair plays against each other until one player wins three times. The next pair then plays against each other in the second round. The winners from the first two rounds then compete against each other to finally determine the ultimate winner. The winner of the game then moves on to the final round. In the final round, the player is presented with several Dabarkads, each holding different amounts of cash prize. The player will then pick three Dabarkads who they will play rock paper scissors against. The player plays against them one at a time. If the player wins against any of theEat Bulaga!hosts, they will win the cash prize.[107][108][109]
Notes
Bibliography
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AShepard tone, named afterRoger Shepard, is a sound consisting of asuperpositionofsine wavesseparated byoctaves. When played with the basspitchof the tone moving upward or downward, it is referred to as theShepard scale. This creates theauditory illusionof a tone that seems to continually ascend or descend in pitch, yet which ultimately gets no higher or lower.[1]
Each square in Figure 1 indicates a tone, with any set of squares in vertical alignment together making one Shepard tone. The color of each square indicates theloudnessof the note, with purple being the quietest and green the loudest. Overlapping notes that play at the same time are exactly one octave apart, and each scale fades in and fades out so that hearing the beginning or end of any given scale is impossible.
As a conceptual example of an ascending Shepard scale, the first tone could be an almost inaudible C4(middle C) and a loud C5(an octave higher). The next would be a slightly louder C♯4and a slightly quieter C♯5; the next would be a still louder D4and a still quieter D5. The two frequencies would be equally loud at the middle of the octave (F♯4and F♯5), and the twelfth tone would be a loud B4and an almost inaudible B5with the addition of an almost inaudible B3. The thirteenth tone would then be the same as the first, and the cycle could continue indefinitely. (In other words, each tone consists of two sine waves with frequencies separated by octaves; the intensity of each is e.g. araised cosinefunction of its separation insemitonesfrom a peak frequency, which in the above example would be B4. According to Shepard, "almost any smooth distribution that tapers off to subthreshold levels at low and high frequencies would have done as well as the cosine curve actually employed."[1]
The theory behind the illusion was demonstrated during an episode of the BBC's showBang Goes the Theory, where the effect was described as "a musicalbarber's pole".[2]
The scale as described, with discrete steps between each tone, is known as thediscrete Shepard scale. The illusion is more convincing if there is a short time between successive notes (staccatoormarcatorather thanlegatoorportamento).[citation needed]
Jean-Claude Rissetsubsequently created a version of the scale where the tones glide continuously, and it is appropriately called thecontinuous Risset scaleorShepard–Risset glissando.[3]When done correctly, the tone appears to rise (or fall) continuously in pitch, yet return to its starting note. Risset has also created a similar effect with rhythm in which tempo seems to increase or decrease endlessly.[4]
A sequentially played pair of Shepard tones separated by anintervalof atritone(half an octave) produces thetritone paradox. Shepard had predicted that the two tones would constitute a bistable figure, the auditory equivalent of theNecker cube, that could be heard ascending or descending, but never both at the same time.[1]
In 1986,Diana Deutschdiscovered that the perception of which tone was higher depended on the absolute frequencies involved and that an individual would usually hear the same pitch as the highest (this is determined by the absolute pitch of the notes).[5]Interestingly, different listeners may perceive the same pattern as being either ascending or descending, depending on the language or dialect of the listener (Deutsch, Henthorn, and Dolson found that native speakers ofVietnamese, atonallanguage, heard the tritone paradox differently from Californians who were native speakers of English).[6][7]
Pedro Patricio observed in 2012 that, by using a Shepard tone as a sound source and applying it to a melody, he could reproduce the illusion of a continuously ascending or descending movement characteristic of the Shepard Scale. Regardless of the tempo and theenvelopeof the notes, the auditory illusion is effectively maintained. The uncertainty of the scale the Shepard tones pertain allows composers to experiment with deceiving and disconcerting melodies.[8]
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Thethree hares(orthree rabbits) is a circularmotifappearing insacred sitesfromEast Asia, theMiddle Eastand the churches ofDevon, England (as the "Tinners' Rabbits"),[1]and historical synagogues in Europe.[2][better source needed]It is used as an architecturalornament, a religioussymbol, and in other modernworks of art[3][4]or a logo foradornment(includingtattoos),[5]jewelry, and acoat of armson anescutcheon.[6][7]It is viewed as a puzzle, a visual challenge, and has been rendered as sculpture, drawing, and painting.
The symbol features threeharesorrabbitschasing each other in a circle. Like thetriskelion,[8]thetriquetra, and their antecedents (e.g., thetriple spiral), the symbol of the three hares has a threefoldrotational symmetry. Each of the ears is shared by two hares, so that only three ears are shown. Although its meaning is apparently not explained in contemporary written sources from any of the medieval cultures where it is found, it is thought to have a range of symbolic or mystical associations with fertility and thelunar cycle. When used in Christian churches, it is presumed to be a symbol of theTrinity. Its origins and original significance are uncertain, as are the reasons why it appears in such diverse locations.[1]
The earliest occurrences appear to be in cave temples in China, dated to theSui dynasty(6th to 7th centuries).[9][10]Theiconographyspread along theSilk Road.[11]In other contexts themetaphorhas been given different meaning. For example, Guan Youhui, a retired researcher from theDunhuang Academy, who spent 50 years studying the decorative patterns in theMogao Caves, believes the three rabbits—"like many images inChinese folk artthat carry auspicious symbolism—represent peace and tranquility".[9][10]SeeAurel Stein. The hares have appeared inLotusmotifs.[12]
The three hares appear on 13th centuryMongolmetalwork, and on a copper coin, found inIran, dated to 1281.[13][14][15]
Another appears on an ancient Islamic-madereliquaryfrom southern Russia. Another 13th or early 14th century box, later used as a reliquary, was made inIranunderMongolrule, and is preserved in the treasury of theCathedral of Trierin Germany. On its base, the casket has Islamic designs, and originally featured two images of the three hares. One was lost through damage.[16]
One theory pertaining to the spread of the motif is that it was transported from China across Asia and as far as the south west of England by merchants travelling the Silk Road and that the motif was transported via designs found on expensiveOriental ceramics. This view is supported by the early date of the surviving occurrences in China. However, the majority of representations of the three hares in churches occur in England and northern Germany. This supports a contrary view that the three hares occurred independently as English or early German symbols.[1][9][10][17]
Some claim that the Devon name, Tinners' Rabbits, is related to localtin minersadopting it. The mines generated wealth in the region and funded the building and repair of many local churches, and thus the symbol may have been used as a sign of the miners' patronage.[18]Thearchitectural ornamentof the three hares also occurs in churches that are unrelated to the miners of South West England. Other occurrences in England include floor tiles atChester Cathedral,[19]stained glass atLong Melford, Suffolk[A]and a ceiling inScarborough, Yorkshire.[1]
The motif of the three hares is used in a number of medieval or more recent European churches, particularly in France (e.g., in theBasilica of Notre-Dame de FourvièreinLyon)[20]and Germany. It occurs with the greatest frequency in the churches ofDevon, United Kingdom, where it appears to be a recollection of earlierInsular Celticdesign such as thetriaxially symmetrictriskeleand otherRomano-Britishdesigns which are known from early British 'Celtic' (La Tène) metalwork such as circular enamelled and openwork triskel brooches (fibulae). The motif appears inilluminated manuscriptsamongst similar devices such as the anthropomorphic "beard pullers" seen in manuscripts such as theBook of Kells,[21]architecturalwood carving,stone carving, windowtracery, andstained glass. In South Western England there are over thirty recorded examples of the three hares appearing on 'roof bosses' (carved wooden knobs) on the ceilings inmedievalchurches inDevon, (particularlyDartmoor). There is a good example of a roof boss of the three hares atWidecombe-in-the-Moor,[8]Dartmoor, with another in the town ofTavistockon the edge of the moor. Themotifoccurs with similar central placement in Synagogues.[2]Another occurrence is on theossuarythat by tradition contained the bones ofSt. Lazarus.[22]
Where it occurs in the United Kingdom, the three hares motif usually appears in a prominent place in the church, such as the central rib of thechancelroof, or on a central rib of thenave. This suggests that the symbol held significance to the church, and casts doubt on the theory that they may have been a masons' or carpenters' signature marks.[1]There are two possible and perhaps concurrent reasons why the three hares may have found popularity as a symbol within the church. Firstly, it was widely believed that the hare washermaphroditeand could reproduce without loss ofvirginity.[16]This led to an association with theVirgin Mary, with hares sometimes occurring inilluminated manuscriptsandNorthern Europeanpaintings of the Virgin andChrist Child. The other Christian association may have been with theHoly Trinity,[16][23][unreliable source?]representing the"One in Three and Three in One"of which the triangle or three interlocking shapes such as rings are common symbols. In many locations the three hares are positioned adjacent to theGreen Man, a symbol commonly believed to be associated with the continuance ofAnglo-SaxonorCelticpaganism.[24]These juxtapositions may have been created to imply the contrast of the Divine with man'ssinful, earthly nature.[16]
In Judaism, theshafaninHebrewhas symbolic meaning.[B][C]Rabbits can carry positive symbolic connotations, like lions and eagles. 16th century German scholarRabbiYosef Hayim Yerushalmi, saw the rabbits as a symbol of theJewish diaspora. The replica of theChodorowSynagoguefrom Poland (on display at theMuseum of the Jewish DiasporainTel Aviv) has a ceiling with a large central painting which depicts a double-headed eagle holds two brown rabbits in its claws without harming them. The painting is surrounded by a citation from the end ofDeuteronomy:
כנשר יעיר קינו על גוזליו ירחף. יפרוש כנפיו יקחהו ישאהו על אברתו
This may be translated: "As an eagle that stirreth up her nest, hovereth over her young, spreadeth abroad her wings, taketh them, beareth them on her pinions (...thus isGodto the Jewish people)."[2]
The hare frequently appears in the form of the symbol of the rotating rabbits. An ancient Germanriddledescribes this graphic thus:
There are three hares and only three ears,and yet each hare has two.[26][2]
This curious graphic riddle can be found in all of the famouswooden synagoguesfrom the period of the 17th and 18th century in theAshknazregion (in Germany) that are on museum display inBeth HatefutsothMuseum in Tel Aviv, theJewish Museum Berlinand TheIsrael Museumin Jerusalem. They also appear in the Synagogue fromHorb am Neckar(donated to the Israel Museum). The three animals adorn the wooden panels of the prayer room fromUnterlimpurgnearSchwäbisch Hall, which may be seen in replica in the Jewish Museum Berlin. They also are seen in a main exhibit of the Diaspora Museum in Tel Aviv. Israeli art historian Ida Uberman wrote about this house of worship: "... Here we find depictions of three kinds of animals, all organized in circles: eagles, fishes and hares. These three represent the Kabbalistic elements of the world: earth, water and fire/heavens... The fact that they are always three is important, for that number . . . is important in theKabbalisticcontext".[2]
Not only do they appear among floral and animal ornaments, but they are often in a distinguished location, directly above theTorah ark, the place where theholy scripturesrepose.[2]
They appear onheadstonesinSataniv(Сатанів),Khmelnytsky Oblast, westernUkraine.[27][28]
Jurgis Baltrusaitis's 1955Le Moyen-Âge fantastique: Antiquités et exotismes dans l'art gothique[29]includes a 1576 Dutchengravingwith the puzzle given in Dutch and French around the image. This is the oldest known dated example of the motif as a puzzle, with a caption that translates as:
The secret is not great when one knows it.But it is something to one who does it.Turn and turn again and we will also turn,So that we give pleasure to each of you.And when we have turned, count our ears,It is there, without any disguise, you will find a marvel.[17]
One recent philosophical book poses it as a problem in perception and anoptical illusion—an example ofcontour rivalry. Each rabbit can be individually seen as correct—it is only when you try to see all three at once that you see the problem with defining the hares' ears. This is similar to "The ImpossibleTribar" byRoger Penrose,[17]originated byOscar Reutersvärd. CompareM.C. Escher'simpossible object.
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Inmathematics, theaxiom of regularity(also known as theaxiom of foundation) is an axiom ofZermelo–Fraenkel set theorythat states that everynon-emptysetAcontains an element that isdisjointfromA. Infirst-order logic, the axiom reads:
∀x(x≠∅→(∃y∈x)(y∩x=∅)).{\displaystyle \forall x\,(x\neq \varnothing \rightarrow (\exists y\in x)(y\cap x=\varnothing )).}
The axiom of regularity together with theaxiom of pairingimplies thatno set is an element of itself, and that there is no infinitesequence(an) such thatai+1is an element ofaifor alli. With theaxiom of dependent choice(which is a weakened form of theaxiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
The axiom was originally formulated by von Neumann;[1]it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo.[2]Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity.[3]However, regularity makes some properties ofordinalseasier to prove; and it not only allows induction to be done onwell-ordered setsbut also on proper classes that arewell-founded relational structuressuch as thelexicographical orderingon{(n,α)∣n∈ω∧αis an ordinal}.{\textstyle \{(n,\alpha )\mid n\in \omega \land \alpha {\text{ is an ordinal }}\}\,.}
Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to theaxiom of induction. The axiom of induction tends to be used in place of the axiom of regularity inintuitionistictheories (ones that do not accept thelaw of the excluded middle), where the two axioms are not equivalent.
In addition to omitting the axiom of regularity,non-standard set theorieshave indeed postulated the existence of sets that are elements of themselves.
LetAbe a set, and apply the axiom of regularity to {A}, which is a set by theaxiom of pairing. We see that there must be an element of {A} which is disjoint from {A}. Since the only element of {A} isA, it must be thatAis disjoint from {A}. So, sinceA∩{A}=∅{\textstyle A\cap \{A\}=\varnothing }, we cannot haveAan element ofA(by the definition ofdisjoint).
Suppose, to the contrary, that there is afunction,f, on thenatural numberswithf(n+1) an element off(n) for eachn. DefineS= {f(n):na natural number}, the range off, which can be seen to be a set from theaxiom schema of replacement. Applying the axiom of regularity toS, letBbe an element ofSwhich is disjoint fromS. By the definition ofS,Bmust bef(k) for some natural numberk. However, we are given thatf(k) containsf(k+1) which is also an element ofS. Sof(k+1) is in theintersectionoff(k) andS. This contradicts the fact that they are disjoint sets. Since our supposition led to a contradiction, there must not be any such function,f.
The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.
Notice that this argument only applies to functionsfthat can be represented as sets as opposed to undefinable classes. Thehereditarily finite sets,Vω, satisfy the axiom of regularity (and all other axioms ofZFCexcept theaxiom of infinity). So if one forms a non-trivialultrapowerof Vω, then it will also satisfy the axiom of regularity. The resultingmodelwill contain elements, called non-standard natural numbers, that satisfy the definition of natural numbers in that model but are not really natural numbers.[dubious–discuss]They are "fake" natural numbers which are "larger" than any actual natural number. This model will contain infinite descending sequences of elements.[clarification needed]For example, supposenis a non-standard natural number, then(n−1)∈n{\textstyle (n-1)\in n}and(n−2)∈(n−1){\textstyle (n-2)\in (n-1)}, and so on. For any actual natural numberk,(n−k−1)∈(n−k){\textstyle (n-k-1)\in (n-k)}. This is an unending descending sequence of elements. But this sequence is not definable in the model and thus not a set. So no contradiction to regularity can be proved.
The axiom of regularity enables defining the ordered pair (a,b) as {a,{a,b}}; seeordered pairfor specifics. This definition eliminates one pair of braces from the canonicalKuratowskidefinition (a,b) = {{a},{a,b}}.
This was actually the original form of the axiom in von Neumann's axiomatization.
Supposexis any set. Lettbe thetransitive closureof {x}. Letube the subset oftconsisting of unranked sets. Ifuis empty, thenxis ranked and we are done. Otherwise, apply the axiom of regularity touto get an elementwofuwhich is disjoint fromu. Sincewis inu,wis unranked.wis a subset oftby the definition of transitive closure. Sincewis disjoint fromu, every element ofwis ranked. Applying the axioms of replacement and union to combine the ranks of the elements ofw, we get an ordinal rank forw, to witrank(w)=∪{rank(z)+1∣z∈w}{\textstyle \textstyle \operatorname {rank} (w)=\cup \{\operatorname {rank} (z)+1\mid z\in w\}}. This contradicts the conclusion thatwis unranked. So the assumption thatuwas non-empty must be false andxmust have rank.
LetXandYbe sets. Then apply the axiom of regularity to the set {X,Y} (which exists by the axiom of pairing). We see there must be an element of {X,Y} which is also disjoint from it. It must be eitherXorY. By the definition of disjoint then, we must have eitherYis not an element ofXor vice versa.
Let the non-empty setSbe a counter-example to the axiom of regularity; that is, every element ofShas a non-empty intersection withS. We define a binary relationRonSbyaRb:⇔b∈S∩a{\textstyle aRb:\Leftrightarrow b\in S\cap a}, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) inSsatisfyinganRan+1for allninN. As this is an infinite descending chain, we arrive at a contradiction and so, no suchSexists.
Regularity was shown to be relatively consistent with the rest of ZF by Skolem[4]and von Neumann,[5]meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent.[6]
The axiom of regularity was also shown to beindependentfrom the other axioms of ZFC, assuming they are consistent. The result was announced byPaul Bernaysin 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) Rieger-Bernayspermutation models(or method), which were used for other proofs of independence for non-well-founded systems.[7][8]
Naive set theory(the axiom schema ofunrestricted comprehensionand theaxiom of extensionality) is inconsistent due toRussell's paradox. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weakeraxiom schema of separation. However, this step alone takes one to theories of sets which are considered too weak.[clarification needed][citation needed]So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension.[citation needed][clarification needed]So far, these axioms do not seem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties. These two axioms are known to be relatively consistent.
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is noset of all sets. The axiom of regularity together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.
If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as Russell's paradox) which followed from the original theory would still follow in the extended theory.
The existence ofQuine atoms(sets that satisfy the formula equationx= {x}, i.e. have themselves as their only elements) is consistent with the theory obtained by removing the axiom of regularity from ZFC. Variousnon-wellfounded set theoriesallow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.[9]
In ZF it can be proven that the class⋃αVα{\textstyle \bigcup _{\alpha }V_{\alpha }}, called thevon Neumann universe, is equal to the class of all sets. This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which does not satisfy the axiom of regularity, a model which satisfies it can be constructed by taking only sets in⋃αVα{\textstyle \bigcup _{\alpha }V_{\alpha }}.
Herbert Enderton[10]wrote that "The idea of rank is a descendant of Russell's concept oftype". Comparing ZF withtype theory,Alasdair Urquhartwrote that "Zermelo's system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included.[11][12]
Dana Scott[13]went further and claimed that:
The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of thetheory of types. That was at the basis of both Russell's and Zermelo's intuitions. Indeed the best way to regard Zermelo's theory is as a simplification and extension of Russell's. (We mean Russell'ssimpletheory of types, of course.) The simplification was to make the typescumulative. Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagineextendingthe types into the transfinite—just how far we want to go must necessarily be left open. Now Russell made his typesexplicitin his notation and Zermelo left themimplicit. [emphasis in original]
In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity.[14]
The concept of well-foundedness andrankof a set were both introduced byDmitry Mirimanoff.[15][16]Mirimanoff called a setx"regular" (French:ordinaire) if every descending chainx∋x1∋x2∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets;[17]in later papers Mirimanoff also explored what are now callednon-well-founded sets(extraordinairein Mirimanoff's terminology).[18]
Skolem[4]and von Neumann[1]pointed out that non-well-founded sets are superfluous[19]and in the same publication von Neumann gives an axiom[20]which excludes some, but not all, non-well-founded sets.[21]In a subsequent publication, von Neumann[22]gave an equivalent but more complex version of the axiom of class foundation:[23]
A≠∅→∃x∈A(x∩A=∅).{\displaystyle A\neq \emptyset \rightarrow \exists x\in A\,(x\cap A=\emptyset ).}
The contemporary and final form of the axiom is due to Zermelo.[2]
Urelementsare objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such asZFA, there are. In these theories, the axiom of regularity must be modified. The statement "x≠∅{\textstyle x\neq \emptyset }" needs to be replaced with a statement thatx{\textstyle x}is not empty and is not an urelement. One suitable replacement is(∃y)[y∈x]{\textstyle (\exists y)[y\in x]}, which states thatxisinhabited.
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TheCartesian theateris a term coined by philosopher and cognitive scientistDaniel Dennettto critique a persistent flaw in theories of mind, introduced in his 1991 bookConsciousness Explained.
It mockingly describes the idea of consciousness as a centralized "stage" in the brain where perceptions are presented to an internal observer. Dennett ties this toCartesian materialism, which he considers to be the often unacknowledged residue of René Descartes’dualismin modernmaterialistviews. This model implies aninfinite regress, as each observer would require another to perceive it, a problem Dennett argues misrepresents how consciousness actually emerges.
The phrase echoes earlier skepticism from Dennett’s teacher,Gilbert Ryle, who inThe Concept of Mind(1949) similarly derided Cartesian dualism’s depiction of the mind as a "private theater" or "second theater."[1]
Descartesoriginally claimed thatconsciousnessrequires an immaterial soul, which interacts with the body via thepineal glandof the brain.[2]Dennett says that, when the dualism is removed, what remains of Descartes' original model amounts to imagining a tiny theater in the brain where ahomunculus(small person), now physical, performs the task of observing all the sensory data projected on a screen at a particular instant, making the decisions and sending out commands (seeHomunculus argument).[3]
The term "Cartesian theater" was brought up in the context of themultiple drafts modelthat Dennett posits inConsciousness Explained(1991):
Cartesian materialism is the view that there is a crucial finish line or boundary somewhere in the brain, marking a place where the order of arrival equals the order of "presentation" in experience becausewhat happens thereis what you are conscious of. ... Many theorists would insist that they have explicitly rejected such an obviously bad idea. But ... the persuasive imagery of the Cartesian Theater keeps coming back to haunt us—laypeople and scientists alike—even after its ghostly dualism has been denounced and exorcized.[4]
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Discworldis acomic fantasy[1]book serieswritten by the English author SirTerry Pratchett, set on theDiscworld, aflat planetbalanced on the backs of four elephants which in turn stand on the back of a giant turtle. The series began in 1983 withThe Colour of Magicand continued until the final novelThe Shepherd's Crown, which was published in 2015, following Pratchett's death. The books frequently parody or take inspiration from classic works, usually fantasy or science fiction, as well asmythology,folkloreandfairy tales, and often use them forsatiricalparallels with cultural, political and scientific issues.
Forty-oneDiscworldnovels were published. Apart from the first novel in the series,The Colour of Magic, the original British editions of the first 26 novels, up toThief of Time(2001), had cover art byJosh Kirby. After Kirby's death in 2001, the covers were designed byPaul Kidby. The American editions, published byHarperCollins, used their own cover art. Companion publications include eleven short stories (some only loosely related to the Discworld), four popular science books, and a number of supplementary books and reference guides. The series has been adapted for graphic novels, theatre, computer and board games, and television.
Discworldbooks regularly toppedSunday Timesbest-sellers list, making Pratchett the UK's best-selling author in the 1990s.Discworldnovels have also won awards such as thePrometheus Awardand theCarnegie Medal. In theBBC'sBig Read, fourDiscworldnovels were in the top 100, and a total of fourteen in the top 200. More than 80 millionDiscworldbooks have been sold in 37 languages.[2][3]
Very few of theDiscworldnovels have chapter divisions. Instead, they feature interweaving storylines. Pratchett was quoted as saying that he "just never got into the habit of chapters",[4]later adding that "I have to shove them in the putativeYAbooks because my editor screams until I do".[5]However, the firstDiscworldnovelThe Colour of Magicwas divided into "books", as isPyramids. Additionally,Going PostalandMaking Moneyboth have chapters, a prologue, an epilogue, and brief teasers of what is to come in each chapter, in the style ofA. A. Milne,Jules Verne, andJerome K. Jerome.
TheDiscworldnovels contain common themes and motifs that run through the series. Many of the novels parody fantasy tropes and various subgenres of fantasy, likefairy tales(notablyWitches Abroad) or vampire tales (Carpe Jugulum). Analogies of real-world issues, such as religion (Small Gods), fundamentalism and inner city tension (Thud), business and politics (Making Money), racial prejudice and exploitation (Snuff) recur, as do aspects of culture and entertainment such as opera (Maskerade), rock music (Soul Music), cinema (Moving Pictures), and football (Unseen Academicals). Parodies of non-Discworld fiction also occur frequently, includingShakespeare,Beatrix Potter, and several movies. Major historical events, especially battles, are sometimes the basis for both trivial and key events (Jingo,Eric, andPyramids), as are trends in science, technology, pop culture and modern art (Moving Pictures,Men at Arms,Thud). There are alsohumanistthemes in manyDiscworldnovels, and a focus oncritical thinkingskills in the Witches andTiffany Achingseries.
TheDiscworldnovels and stories are, in principle, stand-alone works. However, a number of novels and stories formnovel sequenceswith distinctstory arcs:
Rincewindwas the first protagonist ofDiscworld. He is a wizard with no skill, no wizardly qualifications, and no interest in heroics. He is extremely cowardly but is constantly thrust into dangerous adventures. He saves Discworld on several occasions, and has an instrumental role in the emergence of life on Roundworld (Science of Discworld).
Other characters in the Rincewind story arc includeCohen the Barbarian, an aging hero of the old fantasy tradition, out of touch with the modern world and still fighting despite his advanced age;Twoflower, a naive tourist from the Agatean Empire (inspired by cultures of East Asia, particularly Japan and China); andThe Luggage, a magical, semi-sentient and aggressive multi-legged travelling accessory. Rincewind appears in eight Discworld novels as well as the fourScience of Discworldsupplementary books.
Death, a seven-footskeleton in a black robewho rides a pale horse named Binky, appears in every novel exceptThe Wee Free MenandSnuff, although sometimes with only a few lines. His dialogue is always depicted inSMALL CAPSwithout quotation marks. Several characters have said that his voice seemed to reach their minds without making a sound.
Death guides souls from this world to the next. Over millennia he has developed a fascination with humanity to a point and feels protective of it. He adopted a human daughter and took on a human apprentice[6]Eventually the daughter and apprentice had a daughter,Susan Sto Helit, a primary character inSoul Music,Hogfather, andThief of Time.
Characters that often appear with Death include his butlerAlbert, his granddaughter Susan Sto Helit, theDeath of Ratsin charge of gathering the souls of rodents,Quoththe raven, and the Auditors of Reality, the closest thing Death has to a nemesis.
Five Discworld novels feature prominently either Death or Susan with Death appearing. He also appears in the short stories
Witchesin Pratchett's universe act asherbalists, nurses, adjudicators and wise women who can usemagicbut generally prefer not to, finding simple but cunningly applied psychology (called "headology") far more effective.
The principal witch,Granny Weatherwax, a taciturn, bitter old crone from the small mountain country ofLancre, largely despises people but acts as their healer and protector because no one else can do this as well as she can. Her closest friend isNanny Ogg, a jolly, personable witch with the "common touch" who enjoys a smoke and a pint of beer, and often sings bawdy folk songs like the notorious "Hedgehog Song". The two take on apprentice witches: firstMagrat Garlick, thenAgnes Nitt, thenTiffany Aching, who become accomplished witches.
Other characters in the Witches series include:
The witches appear in many Discworld books, and are protagonists in seven. They also appeared in the short story "The Sea and Little Fishes". Their stories frequently draw on ancient European folklore and fairy tales, and parody famous works of literature, particularly byShakespeare.
The stories featuring theAnkh-Morpork City Watchareurban fantasy, and frequently depict a traditional, magically-run fantasy world coming into contact with modern technology. They revolve around the growth of theAnkh-MorporkCity Watch from a hopeless gang of three to a fully-equipped and efficient police force. The stories are largelypolice procedurals, featuring crimes with heavy political or societal overtones.
The main characterSam Vimesis a haggard, cynical, working-class street copper. When introduced inGuards! Guards!, he is the alcoholic captain of the three-person Night Watch, which also includes the lazy, cowardly, and none-too-bright sergeantFred Colonand CorporalNobby Nobbs, a petty thief in his own right. ThenCarrot Ironfoundersson, a 6-foot-6-inch-tall (1.98 m) dwarf-by-adoption, joins the Watch.
Other main characters include
Cheery Littlebottom, the Watch'sforensicsexpert and one of the first openly female dwarves, tried to rename herself "Cheri" without success. Constable Visit-the-infidel-with-explanatory-pamphlets appears in some novels, and Sam's wife,Lady Sybil Vimes (née Ramkin)is integral to certain storylines.Inspector A E Pessimalwas recruited by Vimes as his adjutant afterHavelock Vetinari, the Patrician of Ankh-Morpork, sent him as an auditor.
The City Watch feature in eight Discworld stories, and cameoed in a number of others, includingMaking Money, the children's bookWhere's My Cow?,and the short story "Theatre of Cruelty".
Pratchett stated on numerous occasions that the presence of the City Watch makes Ankh-Morpork stories "problematic", as stories set in the city that do not directly involve Vimes and the Watch often require a Watch presence to maintain the story—at which point, it becomes a Watch story by default.[citation needed]
The Wizards ofUnseen University(UU) appear prominently throughout manyDiscworldnovels; the books that centre around them exclusively are The Science of the Discworld series and the novelsUnseen AcademicalsandThe Last Continent. In the early books, the faculty of UU changed frequently; promotion usually involved assassination. However, after the ascension of the bombasticMustrum Ridcullyto the position ofArchchancellor, the hierarchy settled down and characters had the chance to develop. Earlier books featured the wizards in possible invasions of Discworld by creatures from the Dungeon Dimensions, Lovecraftian monsters that hungered for magic.
The wizards of UU employ the traditional "whizz-bang" type of magic seen inDungeons & Dragonsgames, but also investigate the rules and structure of magic in terms highly reminiscent ofparticle physics. Prominent members include
In later novels, Rincewind joins their group, while the Dean leaves to become the Archchancellor of Brazeneck College in the nearby city of Pseudopolis.
The Wizards feature prominently in nineDiscworldbooks and star in TheScience of Discworld seriesand the short story "A Collegiate Casting-Out of Devilish Devices".
Tiffany Achingis a young apprentice witch in a series of Discworld books aimed at young adults. Her stories often parallel mythic heroes' quests, but also deal with Tiffany's difficulties as a young girl maturing into a responsible woman. She is aided in her task by theNac Mac Feegle, a gang of blue-tattooed, 6-inch tall, hard-drinking, loud-mouthedpicts, also called "The Wee Free Men", who serve as her guardians. She is the protagonist of five novels,The Wee Free Men,A Hat Full of Sky,Wintersmith,I Shall Wear Midnight, andThe Shepherd's Crown. Major characters in this series include Miss Tick, a travelling witch who discovers Tiffany; Nac Mac Feegle chieftain Rob Anybody; and the other young witches Annagramma Hawkin and Petulia Gristle. BothGranny WeatherwaxandNanny Oggalso appear in her stories.
Moist von Lipwigis a professional criminal and con man to whom Havelock Vetinari gives a "second chance" after staging his execution, recognising the advantages hisjack-of-all-tradesabilities will give to the development of the city. After putting him in charge of theAnkh-Morpork Post OfficeinGoing Postal, with good results, Vetinari orders him to clear up the city's corrupt financial sector inMaking Money. In a third book,Raising Steam, Vetinari directs Lipwig to oversee the development of a railway network for Dick Simnel's newly invented steam locomotive. Other characters in this series includeAdora Belle Dearheart, Lipwig's acerbic, chain-smoking love interest; Gladys, a golem who develops a strange crush on Lipwig;Stanley Howler, an obsessive young man who was raised by peas and becomes the Discworld's firststamp collector; and the very old Junior Postman Groat, who never got promoted to Senior Postman because there was never a Postmaster alive long enough to promote him.
Several other books can be grouped together as "Other cultures of Discworld" books. They may contain characters or locations from other arcs, typically not as protagonist or antagonist but as a supporting character or even a throwaway reference. These includePyramids(Djelibeybi),Small Gods(Omnia), andMonstrous Regiment(Zlobenia and Borogravia).
Short descriptions of many of the notable characters:
Short stories by Pratchett based in the Discworld, including published miscellanea such as the fictional game origins ofThud, were reprinted in Pratchett's collectionA Blink of the Screen(2012), and elsewhere.
Seven of the short stories or short writings were also collected in a compilation of the majority of Pratchett's known short work namedOnce More* With Footnotes(2004).
Additionally, another short story "Turntables of the Night" (1989) is set in England but featuresDeathas a character; it is available online and in both anthologies.
Five short stories republished inA Stroke of the Pen: The Lost Stories(2023) constitute the first known works by Pratchett that include early versions of places and characters that would later become parts of Discworld. Pratchett authored most of them under a pseudonym that remained unlinked to him for decades, until posthumously discovered in 2022.[29][30][31][32]
Although Terry Pratchett said, "There are no maps. You can't map a sense of humour,"[33]there are four "Mapps":The Streets of Ankh-Morpork(1993),The Discworld Mapp(1995),A Tourist Guide to Lancre(1998), andDeath's Domain(1999). The first two were drawn by Stephen Player, based on plans by Pratchett andStephen Briggs, the third is a collaboration between Briggs andPaul Kidby, and the last is by Kidby. All also contain booklets written by Pratchett and Briggs. Terry later collaborated with the Discworld Emporium to produce two much larger works, each with the associated map with the book in a folder,The Compleat Ankh-Morpork City Guide(2012) andThe Compleat Discworld Atlas(2015).[34]
Death's Domainis a book byTerry PratchettandStephen Briggs,[35]and illustrated byPaul Kidby. It is the fourth in the Mapp series. It was first published inpaperbackbyCorgiin 1999.[36]It was the second in the series to be illustrated by Kidby.[37]As with the other "mapps", the basic design and booklet were compiled by Pratchett and Briggs.
The Mapp shows the parasite universe of Death's Domain. The accompanying booklet provides various details of the Domain, both as portrayed in the Discworld books and newly revealed.
InDeath's Domain, the concept of steam locomotives on Discworld is introduced,[38]which became the main theme of Pratchett's Discworld novelRaising Steamfourteen years later.
In the live-action adaptations ofHogfatherandThe Colour of Magic,Dorney Courtis the real-life location used for the exterior ofMon Repos, Death's house.
Pratchett also collaborated withIan StewartandJack Cohenon four books, using the Discworld to illuminatepopular sciencetopics. Each book alternates chapters of aDiscworldstory and notes on real science related to it. The books are:
David Langfordhas compiled twoDiscworldquizbooks:
Most years see the release of a Discworld Diary and Discworld Calendar, both usually following a particular theme.
The diaries feature background information about their themes. Some topics are later used in the series; the character of Miss Alice Band first appeared in theAssassins' Guild Yearbook, for example.[citation needed]
The Discworld Almanak – The Year of The Prawnhas a similar format and general contents to the diaries.
OtherDiscworldpublications include:
The books take place roughly inreal timeand the characters' ages change to reflect the passing of years. The meetings of various characters from different narrative threads (e.g., Ridcully andGranny WeatherwaxinLords and Ladies, Rincewind and Carrot inThe Last Hero) indicate that all the main storylines take place around the same period (end of the Century of the Fruitbat, beginning of the Century of the Anchovy). The main exception is the stand-alone bookSmall Gods, which appears to take place at some point earlier than most of the other stories, though even this contains cameo appearances by Death and the Librarian.
Some main characters may makecameo appearancesin other books where they are not the primary focus; for example, City Watch membersCarrot IronfounderssonandAnguaappear briefly inGoing Postal,Making Money, andUnseen Academicals(placing those books afterGuards! Guards!andMen at Arms). A number of characters, such as members of staff ofUnseen Universityand Lord Vetinari, appear prominently in many different storylines without having specific storylines of their own. The two most frequently recurring central protagonists, Rincewind andSam Vimes, are very briefly in a room together inThe Last Hero, but they do not interact.
After Terry Pratchett was diagnosed withAlzheimer's disease, he said that he would be happy for his daughterRhiannato continue the series.[44]Pratchett co-founded Narrativia in 2012 along with Rob Wilkins to serve as a production company for adaptations of his works, with Rhianna as a member of its writing team.[44]Rhianna Pratchett said that she would be involved in spin-offs, adaptations and tie-ins, but there would be no more novels.[45]The first such spin-off by Rhianna was the tie-in bookTiffany Aching's Guide to Being a Witch, co-written with children's author Gabrielle Kent.
Most of Pratchett's novels have been released asaudio cassetteand CDaudiobooks.
The Colour of Magic,The Light Fantastic,[50]Mort,[51]Guards! Guards!,[52]andSmall Gods[53]have been adapted intographic novels. Adaptations ofThief of Time,The Wee Free Men, andMonstrous Regimenthave been announced but not yet released.[54]
Due in part to the complexity of the novels,Discworldhas been difficult to adapt to film – Pratchett was fond of an anecdote of a producer attempting to pitch an adaptation ofMortin the early 1990s but was told to "lose the Death angle" by US backers.[55]
Cosgrove Hallproduced several animated adaptations forChannel 4from 1996 to 1997. All three starChristopher Leeas Death. These were made available on DVD and VHS in the US from Acorn Media.
Three television movies were commissioned bySky Onein the late 2000s, each of which were broadcast in two parts. Terry Pratchett cameos as a minor character in all three.
The Amazing Mauriceis a UK-Germany co-productionCGI-animatedfeature film, with a screenplay byTerry Rossioclosely adapting the 28thDiscworldstand-alone novelThe Amazing Maurice and His Educated Rodentsof 2001. The film stars the voices ofHugh Laurie— as the eponymous lead character of the streetwise talking ginger tomcat Maurice, who befriends a group of talking rats and a pet human to run a money-spinning "Pied Piper" scam acrossDiscworld— withEmilia Clarke,Himesh Patel,Gemma Arterton,Ariyon Bakare,David Tennant,Julie Atherton,Joe Sugg,Rob Brydon,Hugh Bonneville,David Thewlis, andPeter Serafinowiczcameoingas Death. The film's musical score was composed byTom Howewith English singer-songwriterGabrielle Aplin. It had its premiere at theManchester Animation Festivalon 13 November 2022 before going on to general release at the end of 2022.[60][61]
The same film production companies are putting together a CGI-animated feature filmsequelto this film due for release in 2027.[62]
There have been severalBBC Radioadaptations of Discworld stories, including:
Other video games are:
Various other types of related merchandise have been produced bycottage industrieswith an interest in the books, includingStephen Briggs,Bernard Pearson,Bonsai Trading,Paul KidbyandClarecraft.
Cripple Mr. Onion was originally a fictionalcard gameplayed by characters in the novelsWyrd Sisters,Reaper Man,Witches Abroad,Men at Arms,WintersmithandLords and Ladies. A game called "Shibo Yangcong-San" (derived fromJapanese死亡shibō, "death;"Chinese洋蔥yángcōng, "onion;" and theJapanese honorificさん-san) appears inInteresting Timesas atile gameplayed in theAgatean Empire. This was used by Dr Andrew Millard andProf. Terry Taoas the basis for an actual card game.[99]
Pratchett co-authored withPhil Masterstworole-playing gamesupplements for Discworld, utilising the third edition of theGURPSsystem:
A revised second edition, theDiscworld Roleplaying Game, was published in 2016. It combined the content of the previous two books with new material, and updated the rules toGURPSFourth Edition.
In August 2023,Royal Mailintroduced a series of eight stamps based on Discworld characters, to mark the 40th anniversary of the first book's publication.[108]
On 5 November 2019, theBBC NewslistedThe Discworld Serieson its list of the100 most influential novels.[111]
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"God of the gaps" is atheologicalconcept that emerged in the19th century, and revolves around the idea that gaps inscientificunderstanding are regarded as indications of theexistence of God.[1][2]This perspective has its origins in the observation that some individuals, often withreligiousinclinations, point to areas where science falls short in explaining natural phenomena as opportunities to insert the presence of adivine creator. The term itself was coined in response to this tendency. This theological view suggests thatGodfills in the gaps left by scientific knowledge, and that these gaps represent moments ofdivine interventionor influence.
This concept has been met with criticism and debate from various quarters. Detractors argue that this perspective is problematic as it seems to rely on gaps in human understanding and ignorance to make its case for the existence of God. As scientific knowledge continues to advance, these gaps tend to shrink, potentially weakening the argument for God's existence. Critics contend that such an approach can undermine religious beliefs by suggesting that God only operates in the unexplained areas of our understanding, leaving little room for divine involvement in a comprehensive and coherent worldview.
The "God of the gaps" perspective has been criticized for its association withlogical fallacies. The "God of the gaps" perspective is also a form ofconfirmation bias, since it involves interpreting ambiguous evidence (or rather no evidence) as supporting one's existing attitudes. This type of reasoning is seen as inherently flawed and does not provide a robust foundation forreligious faith. In this context, some theologians and scientists have proposed that a more satisfactory approach is to view evidence of God's actions within the natural processes themselves, rather than relying on the gaps in scientific understanding to validate religious beliefs.
From the 1880s,Friedrich Nietzsche'sThus Spoke Zarathustra, Part Two, "On Priests", said that "into every gap they put their delusion, their stopgap, which they called God".[3]The concept, although not the exact wording, goes back toHenry Drummond, a 19th-centuryevangelistlecturer, from his 1893 Lowell Lectures onThe Ascent of Man. He chastises thoseChristianswho point to the things that Science has not explained as presence of God — "gaps which they will fill up with God" — and urges them to embrace all nature as God's, as the work of "animmanentGod, which is the God of Evolution, is infinitely grander than the occasional wonder-worker, who is the God of an old theology."[4][5]
In 1933,Ernest Barnes, the Bishop of Birmingham, used the phrase in a discussion ofgeneral relativity's implication of aBig Bang:
Must we then postulate Divine intervention? Are we to bring in God to create the first current of Laplace's nebula or to let off the cosmic firework of Lemaître's imagination? I confess an unwillingness to bring God in this way upon the scene. The circumstances which thus seem to demand his presence are too remote and too obscure to afford me any true satisfaction. Men have thought to find God at the special creation of their own species, or active when mind or life first appeared on earth. They have made him God of the gaps in human knowledge. To me the God of the trigger is as little satisfying as the God of the gaps. It is because throughout the physical Universe I find thought and plan and power that behind it I see God as the creator.[6]
DuringWorld War II, the German theologian and martyrDietrich Bonhoefferexpressed the concept in similar terms in letters he wrote while in a Nazi prison.[7]Bonhoeffer wrote, for example:
how wrong it is to use God as a stop-gap for the incompleteness of our knowledge. If in fact the frontiers of knowledge are being pushed further and further back (and that is bound to be the case), then God is being pushed back with them, and is therefore continually in retreat. We are to find God in what we know, not in what we don't know.[7]
In his 1955 bookScience and Christian BeliefCharles Alfred Coulson(1910−1974) wrote:
There is no 'God of the gaps' to take over at those strategic places where science fails; and the reason is that gaps of this sort have the unpreventable habit of shrinking.[8]
and
Either God is in the whole of Nature, with no gaps, or He's not there at all.[9]
Coulson was a mathematics professor atOxford Universityas well as aMethodistchurch leader, often appearing in the religious programs ofBritish Broadcasting Corporation. His book got national attention,[10]was reissued as a paperback, and was reprinted several times, most recently in 1971.
It is claimed that the actual phrase 'God of the gaps' was invented by Coulson.[11][12]
The term was then used in a 1971 book and a 1978 article, byRichard Bube. He articulated the concept in greater detail inMan come of Age: Bonhoeffer’s Response to the God-of-the-Gaps(1978). Bube attributed modern crises in religious faith in part to the inexorable shrinking of the God-of-the-gaps as scientific knowledge progressed. As humans progressively increased their understanding of nature, the previous "realm" of God seemed to many persons and religions to be getting smaller and smaller by comparison. Bube maintained thatDarwin'sOrigin of Specieswas the "death knell" of the God-of-the-gaps. Bube also maintained that the God-of-the-gaps was not the same as the God of the Bible (that is, he was not making anargumentagainst God per se, but rather asserting there was a fundamental problem with the perception of God as existing in the gaps of present-day knowledge).[13]
The term "God of the gaps" is sometimes used in describing the incremental retreat of religious explanations of physicalphenomenain the face of increasingly comprehensive scientific explanations for those phenomena.[8][13][14]Dorothy Dinnersteinincludes psychological explanations for developmental distortions leading to a person believing in a deity, particularly a male deity.[15][citation needed]
R. Laird Harriswrites of the physical science aspect of this:
The expression, "God of the Gaps," contains a real truth. It is erroneous if it is taken to mean that God is not immanent innatural lawbut is only to be observed in mysteries unexplained by law. No significant Christian group has believed this view. It is true, however, if it be taken to emphasize that God is not only immanent in natural law but also is active in the numerous phenomena associated with the supernatural and the spiritual. There are gaps in a physical-chemical explanation of this world, and there always will be. Because science has learned many marvelous secrets of nature, it cannot be concluded that it can explain all phenomena. Meaning, soul, spirits, and life are subjects incapable of physical-chemical explanation or formation.[16]
The termGod-of-the-gaps fallacycan refer to a position that assumes an act of God as the explanation for an unknown phenomenon, which according to the users of the term, is a variant of anargument from ignorancefallacy.[17][18]Such an argument is sometimes reduced to the following form:
One example of such an argument, which uses God as an explanation of one of the current gaps in biological science, is as follows:
"Because current science can't figure out exactly how life started, it must be God who caused life to start." Critics ofintelligent design creationism, for example, have accused proponents of using this basic type of argument.[19]
God-of-the-gaps arguments have been discouraged by some theologians who assert that such arguments tend to relegate God to the leftovers of science: as scientific knowledge increases, the dominion of God decreases.[7][8][20][21]
The term was invented as a criticism of people who perceive that God only acts in the gaps, and who restrict God's activity to such "gaps".[22]It has also been argued that the God-of-the-gaps view is predicated on the assumption that any event which can be explained by science automatically excludes God; that if God did not do something via direct action, that he had no role in it at all.[23][unreliable source?]
The "God of the gaps" argument, as traditionally advanced by scholarly Christians, was intended as a criticism against weak or tenuous faith, not as a statement against theism or belief in God.[4][7][24][improper synthesis?]
According toJohn HabgoodinThe Westminster Dictionary of Christian Theology, the phrase is generally derogatory, and is inherently a direct criticism of a tendency to postulate acts of God to explain phenomena for which science has not (at least at present) given a satisfactory account.[25]Habgood also states:
It is theologically more satisfactory to look for evidence of God's actions within natural processes rather than apart from them, in much the same way that the meaning of a book transcends, but is not independent of, the paper and ink of which it is comprised.[25]
It has been criticized by both theologians and scientists, who say that it is a logical fallacy to base belief in God on gaps in scientific knowledge. In this vein,Richard Dawkins, an atheist, dedicates a chapter of his bookThe God Delusionto criticism of the God-of-the-gaps argument.[26]He noted that:
Creationists eagerly seek a gap in present-day knowledge or understanding. If an apparent gap is found, it is assumed that God, by default, must fill it. What worries thoughtful theologians such as Bonhoeffer is that gaps shrink as science advances, and God is threatened with eventually having nothing to do and nowhere to hide.[26]
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Kurma(Sanskrit:कूर्म,lit.'Turtle' or 'Tortoise'), is the secondavatarof theHindupreserver deity,Vishnu. Originating inVedicliterature such as theYajurvedaas being synonymous with theSaptarishicalledKashyapa, Kurma is most commonly associated in post-Vedic literature such as thePuranas. He prominently appears in the legend of the churning of theOcean of Milk, referred to as theSamudra Manthana. Along with being synonymous withAkupara, theWorld-Turtlesupporting the Earth, Kurma is listed as the second of theDashavatara, which are the ten principal incarnations of Vishnu.
TheSanskritword 'Kurma' (Devanagari: कूर्म) means 'Tortoise' and 'Turtle'.[1]The tortoise incarnation of Vishnu is also referred to in post-Vedic literature such as theBhagavata Puranaas 'Kacchapam' (कच्छप), 'Kamaṭha' (कमठ), 'Akupara' (अकूपार), and 'Ambucara-Atmana' (अम्बुचर-आत्मना), all of which mean 'tortoise' or 'form of a tortoise'.[2][3][4][5]
Written by thegrammarianYaska, theNiruktais one of the sixVedangasor 'limbs of theVedas', concerned with correctetymologyand interpretation of the Vedas. The entry for the Tortoise states (square brackets '[ ]' are as per the original author):
May we obtain that illimitable gift of thine. The sun is calledakuparaalso, i. e. unlimited, because it is immeasurable. The ocean, too, is called akupara, i. e. unlimited, because it is boundless. A tortoise is also called a-kupa-ara, because it does not move in a well [On account of its shallowness]. Kacchapa (tortoise) is (so called because) it protects (pati) its mouth (kaccham), or it protects itself by means of its shell (kacchena), or it drinks (√pa) by the mouth. Kaccha (mouth or shell of a tortoise) = kha-ccha, i. e. something which covers (chddayatl) space (kham). This other (meaning of) kaccha, 'a bank of a river', is derived from the same (root) also, i.e. water (kam) is covered (chadyate) by it.
As illustrated below,Vedicliterature such as theSamavedaandYajurvedaexplicitly state Akupara/Kurma and the sageKashyapaareSynonymous. Kashyapa - also meaning 'Tortoise' - is considered theProgenitorof all living beings with his thirteen wives, including vegetation, as related by H.R. Zimmer:
Ira [meaning 'fluid']... is known as the queen-consort of still another old creator-god and father of creatures, Kashyapa, the Old Tortoise Man, and as such she is the mother of allvegetablelife.
The legend of the churning of theOcean of Milk(Samudra Manthana) developed in post-Vedic literature is itself inextricably linked with Kurma (as the base of the churning rod) and involves other sons of Kashyapa: thedevas/adityas(born fromAditi) and theasuras/Danavas/Daityas(born fromDanuandDiti) use one of theNaga(born fromKadru) as a churning rope to obtainAmrita.Garuda, the king of birds and mount ofVishnu, is another son of Kashyapa (born fromVinata) often mentioned in this legend. In another, Garuda seeks theAmritaproduced (eating a warringElephant And Tortoisein the process) to free his mother and himself from enslavement from Kadru.[citation needed]
Kurmasana(Tortoise Posture) is aYogaposture. 'Panikacchapika'(Sanskrit पाणिकच्छपिका), meaning 'Hand Tortoise',[8]is a special positioning of the fingers during worship rituals to symbolise Kurma. TheKurmacakrais aYantra, a mystical diagram for worship,[9]in the shape of a tortoise. These are all mentioned in theUpanishadsandPuranas(see below).[citation needed]
The Dashavataras are compared to evolution; Kurma - the amphibian - is regarded the next stage afterMatysa, the fish.[10]
Firmness / Steadiness:W. Caland notes that in relation to 'Akupara Kashyapa' in thePancavimsa Brahmanaand Jaiminiya Brahmana, the tortoise is equal to 'a firm standing... and Kashyapa (The Tortoise) is able to convey (them) across the sea [of material existence]'.[11]P.N. Sinha seems to support this view, adding 'Kurma was a great Avatara as He prepared the way for the spiritual regeneration of the universe, by the Churning of theOcean Of Milk'.[12]
DeityYajna-Purusha:N. Aiyangar states that as the tortoise was 'used as the very basis of the fireAltar, the hidden invisible tortoise, taken together with the altar and the sacred fire, seems to have been regarded as symbolizing the Deity Yajna-Purusha who is an invisible spiritual god extending from the fire altar up to heaven and everywhere... this seems to be the reason why the tortoise is identified with the sun'.[13]
Meditation / Churning the Mind:Aiyangar also surmises that the legend of theSamudra Manthanasymbolises churning the mind throughMeditationto achieve liberation (Moksha). Based on the mention ofVatarasanaḥ('Girdled By The Wind')Munisin theTaittirtya Aranyaka- also referred to asurdhvamanthin, meaning 'those who churn upwards' - and the explanation provided in theShvetashvatara Upanishad, Aiyangar believes this would 'appear to be the hidden pivot on which the gist of the riddle of thePuraniclegend about theChurning For Nectarturns'.[13]R. Jarow seems to agree, stating the churning of the Ocean of Milk represents the 'Churning Of TheDualisticMind'.[14]
AsceticPenance:H.H. Wilsonnotes that 'the account [of the Samudra Manthana] in theHari Vamsa... is explained, by the commentator, as anAllegory, in which the churning of the ocean typifies ascetic penance, and theAmbrosiais finalLiberation' (Linking With The Idea Of 'Steadiness' And 'Firmness'), but personally dismisses this interpretation as 'Mere Mystification' (Note 1, pp. 146).[15]
Astronomy:B.G. Sidharth states that the legend of theSamudra Manthanasymbolises astronomic phenomena, for example that 'Mandara represents the polar regions of Earth [and the] Churning Rope,Vasuki, symbolizes the slow annual motion of Earth...Vishnu, or the Sun himself rests upon a coiled snake... which represents the rotation of the Sun on its own axis'. In regards to the tortoise supporting the Earth, Sidharth adds that the 'Twelve Pillars... are evidently the twelve months of the year, and... The four elephants on which Earth rests are the Dikarin, the sentinels of the four directions.. [Kurma] symbolizes the fact that Earth is supported in space in its annual orbit around the Sun'.[16]
A.A. Macdonell,A.B. Keith, J. Roy,J. Dowson, and W.J. Wilkins all state that the origin of Kurma is in theVedas, specifically theShatapatha Brahmana(related to theYajurVeda), where the name is also synonymous withKashyapa, one of theSaptarishi(seven sages).[17][18][19][20][21]
TheShatapatha Brahmanais the earliest extant text to mention Kurma, the tortoise.[22]TheShatapatha Brahmanaequates the tortoise - Kurma to the creator of all creatures. The god Prajapati assumes the form of Kurma to create all creatures (praja). Since he "made" (kar) all, Prajapati's form was calledKurma. Kurma is equated with Kashyapa (literally "tortoise"), thus all creatures are called "children of Kashyapa". Kurma is also calledSurya(the sun).[23][24]
TheShatapatha Brahmanaalso has the origins ofMatsya, the Fish. Like Kurma, Matsya is also associated as the avatar of Vishnu later in thePuranas.[22]
TheTaittiriya Samhitasuggests a ritual of burying a live tortoise at the base of the sacrificial fire altar (uttar-vedi). By this act, the sacrificer earns the merit of reachingheaven.[23][25]Aiyangar suggests that the tortoise symbolizes Yajna-Purusha, the all-pervading god of Sacrifice.[23]In another instance in theTaittiriya Samhitawhere Prajapati assigns sacrifices for the gods and places the oblation within himself, "the Sacrificial Cake" (Purodasa) is said to become a tortoise.[22][26]
TheTaittiriya Aranyakadescribes a similar practice in a ritual calledArunaketuka-kayanawhere the tortoise is buried under the altar. Here, Prajapati or his "juice" (rasa) the tortoise is called Arunaketu ("one who has red rays"). Prajapati performs austerities (tapas). From hisrasasprings a tortoise swimming in the water. Prajapati declares to the tortoise to be his creation; in response the tortoise says that he has existed from "before" and manifests asPurusha- the primordial being and creates various deities including the sun,Agni(the fire),Indra,Vayu(the wind) and various beings. The tortoise is again treated as the divine Creator of the universe.[27][22]
R.T.H. Griffithstates that tortoises were buried in construction of the Ahavaniya Fire-Altar.[28]In this context, theVajasaneyiSamhitaof the whiteYajurvedadescribes the tortoise as the "lord of the waters".[22][28]The selection of the tortoise may stream from the belief that it supports the world.[28]
Though Kurma is not found in the oldest Hindu scriptureRigveda, the seer Kashyapa (who is equated with Kurma) appears in hymns in the scripture.[29][30]TheAtharvavedaregards Kashyapa, who is mentioned along with or identified with Prajapati, assvayambhu("self-manifested").[22][31]In later Hindu scriptures like the epics and the Puranas, Kashyapa is described as the grandfather ofManu, the progenitor of mankind. Apart from described as one ofSaptarishi(seven great sages), he is described as one of thePrajapatis ("agents of creation") and marries 13 daughters ofDaksha, fathering gods, demons, animals, birds and various living beings.[32]The seer Kashyapa, tortoise, being referred in various later Vedic literature as the progenitor of beings, is inferred by A.A. Macdonell along with other animal-based tribal names in theRigvedato suggesttotemism; howeverE.W. Hopkinsdisagrees.[22]
TheRigvedaalso refers in a hymn that Vayu churned for the sages (munis) andRudradrinks from a cup ofvisha, which can be mean water or poison.John Muirsuggests thatvishain theRigvedarefers to Rudra drinking water, however it may have led to, in the Puranas, the legend of Shiva (who is closely linked to the Vedic Rudra) drinking poison in theSamudra Manthana(churning of the ocean) episode.[33][34]
29. There is the Akupara(Saman). ('The Chant of Akupara').
30. By means of this (Saman), Akupara Kasyapa attained power and greatness. Power and Greatness attains he who in lauding has practised the Akupara(Saman).
The sageKashyapa- stated in theVedas,Itihāsa(Epics), andPuranasto be the progenitor of all living beings (see relevant sections, below) - is also stated to be synonymous withAkupara, the name of the 'World-Turtle' in theMahabharata. Caland explains in his footnote to verse 30 the significance of this name by quoting from theJaiminiya Brahmana:[11]
Akupara Kasyapa descended together with theKalis, into the sea. He sought it in firm standing. He saw this atman and lauded with it. Thereupon, he found a firm standing in the sea, viz., this earth. Since that time, the Kalis sit on his back. This Saman is (Equal To) a firm standing. A firm standing gets he who knows thus. The Chandoma(-Day)s are a sea... and Kasyapa (The Tortoise) is able to convey (Them) across the sea. That there is here this Akupara, is for crossing over the sea.
The Jaiminiya Brahmana explicitly links Akupara, Kashyapa, and the tortoise in regards to providing a 'Firm Standing' to cross over the sea of material existence. As illustrated below, in theYajurveda, Kashyapa is also stated to beSynonymouswithPrajapati(i.e. the Creator-GodBrahma) and with Kurma. In the Puranas, Kashyapa is frequently referred to as 'Prajapati' as well.
Swami Achuthananda states that although varied like other legends, Vishnu's role is "limited" as Kurma, compared his other avatars.[36]
The epics present the earliest known versions of the popular Samudra Manthana narrative.[37]In theAdi ParvaBook of the epicMahabharata, the god Narayana (identified withVishnu) suggests the gods (devas) and the demons (asuras) churn the ocean to obtainamrita(ambrosia) as both of them seek immortality. The gods select Mount Mandara as the churning rod and the serpent Vasuki-Ananta as the rope. Then they approach Kurma-raja, the king of tortoises to support the mount on its back, which it consented. The gods churn from the tail side of the serpent, while asuras on the head side. Various trees and herbs are cast into the ocean. The churned water takes into milk. Ultimately, various precious items like Soma (the moon), the goddess Sri (Lakshmi), Sura (liquor), the white horseUchchaihshravas, the white elephantAiravata, the gemKaustubhaand finally the godDhanvantariwith the vessel ofamritaemerge from the ocean. The poisonkalakutasprings from the ocean and is drunk by Shiva, whose throat becomes blue earning him the epithetNila-kantha(The blue necked). The devas and asuras battle for theamrita. Narayana becomes an enchanting woman (calledMohiniin later scriptures) and snatch the pot ofamritafrom the asuras. Narayana along with Nara battle the asuras, while the enchantress distributes theamritaonly to the gods. Rahu, an asura, disguises himself as a god and tries to drink some Amrita himself. Surya (the sun-god) and Chandra (the moon-god) quickly inform Vishnu, and he uses the Chakra (the divine discus) to decapitate Rahu, leaving the head immortal. Eventually, the gods defeat the asuras with Indra retaining theamritaand appointing Nara as its guardian.[38][39]
In this narrative, Kurma is not related with Vishnu.[40]Though the critical edition of the epic does not refer to Kurma as an avatar of Vishnu,[40][41]some latter insertions in manuscripts of the epic associate Kurma as apradurbhava(manifestation) of Vishnu.[42][43][44]
TheRamayanabriefly mentions the Samudra Manthana episode, however does not mention Kurma in it. The epic mentions the ocean churned being the ocean of milk, theKshirasagara.[45]An passage, generally believed to be interpolated and not part of the critical edition, refers to Kurma as well as the drinking of the poison by Shiva.[46]The mount Mandara sinks to Patala (the underworld) during the churning. On the beseeching of the gods, Vishnu takes the form of the tortoise and raises the mount on his back. Vishnu also supports the mount as holding its peak in a form and another form joins the gods in churning the ocean.[46][47]Later versions of theRamayanalike theAdhyatma Ramayanaassociate Kurma withRama, the male protagonist of theRamayanawho is also regarded as an avatar of Vishnu.[48]
J.W. Wilkins states that the 'probable' origin of Kurma is as an incarnation ofPrajapati(i.e.Brahma) in the Shatapatha Brahmana (7:5:1:5-7), but as 'the worship of Brahma became less popular, whilst that of Vishnu increased in its attraction, the names, attributes, and works of one deity seem to have been transferred to the other'.[21][49]
Kurma as well asVaraha, the boar avatar of Vishnu, was both associated with the Creator Prajapati.Hermann Jacobisuggests that Prajapati may have worshipped in these animal forms.[50]With Vishnu gaining the status of the Supreme God, the actions of Prajapati were transferred to Vishnu.[50]
In post-Vedic literature, including thePuranas, Kurma is inextricably linked with the legend of the churning of theOcean Of Milk, known as theSamudra Manthana. Kurma is also directly linked withAkupara, the so-called 'world-turtle' that supports the Earth, usually withSesa.
The tale of Vishnu appearing as Kurma to support the sinking Mandara mountain is narrated in a chapter in theAgni Puranadedicated to Samudra Manthana. The narrative starts with the curse of sage Durvasa to the gods (devas), who lose to the asuras in battle and seek refuge in Vishnu. The asuras and the devas unite to churn the milk ocean, with Mount Mandara as the churning rod and Vasuki as the rope. Kurma appears to support the mountain. The poison Halahala appears from the ocean, which is drunk by Shiva to save the world. After which, various divine objects emerge from the churning of the ocean, ending with the god Dhanavantri carrying the vessel of Amrita. When the asuras steal the pot, Vishnu assumes the form of the seductress Mohini and grabs it from the asuras and distributes it to the gods. Rahu assumes a form of a deva and drinks the amrita and is decapitated by Vishnu.[51]
A similar narrative is also given in theVishnu Purana; Vishnu is described to participate in the churning in many forms - Kurma as the base of the mount, in one form he sits on top of Mandara and in other forms, helps the gods and the demons pull the serpentine rope.[52][53]TheBrahmanda Puranastates that Vishnu in the form of Brahma supports the mount; while as Narayana invigorated the gods.[54]TheVayu Purana, thePadma Puranahave similar narratives; theBhagavata Puranaalso narrates the tale.[55]
TheBhagavata Puranadescribes the form of Vishnu as Ajita, the son of Vairaja and Sambhuti, who assumed the form of the tortoise to rescue Mandara from drowning.[56]He is further called the first tortoise.[57]In another instance, it states that the ocean tides are a result of the breathing of Kurma, who had become drowsy due to the scratching of Mandara on his back.[58]
Samudra Manthana is alluded briefly in theKurma Purana, theLinga Purana, theBrahma Vaivarta Puranaand theShiva Purana.[55][59]
Variations in these narratives alter the number and order of the divine articles appearing from the churning of the ocean. The number ranges from 9 to the popular list of 14. The common list includes the poison Halahala (Kalakuta), Varuni (Sura) - goddess of liquor, the divine horse Uchhaishravas, the gem Kaustubha, the goddess Lakshmi (Sri), theApsaras, the cow of plentySurabhi, the white elephant Airavata and Dhanavantri with the pot ofamrita(sometimes enumerated as two objects). Other objects include the umbrella ofVaruna, earrings taken by Indra for his motherAditi, the bow of VishnuSharanga, the conch of Vishnu (Shankha), Nidra - the goddess of sloth,Alakshmior Jyestha - the goddess of misfortune and the Tulasi plant.[55]
In theMatsya Purana, Vishnu states that his form theworld turtleKurma, which supports all the worlds on his back, be requested by the gods to aid in the Samudra Manthana. Kurma is placed in Patala as the base of Mount Mandara.[60]TheShiva Puranaexplicitly praises Vishnu as the world turtle who supports the Earth.[61]TheBrahma Vaivarta Puranastates the serpent Shesha who supports the universe over his hoods, sits on Kurma, who lies in the wind or the waters.[62]
TheVishnu Purananarrative of Vishnu's boar avatarVarahaalludes to the Matysa and Kurma avatars, saying that Brahma (identified with Narayana, an epithet transferred to Vishnu) took these forms in previouskalpas.[63]
In the tale of the battle of the demonBhandasuraand the goddessLalitain theBrahmanda Purana, Lalita creates Kurma to shelter her goddess army who was drowning in the ocean, created by a weapon used by the demon.[64]
In theAgni Purana, theShaligramstone for Kurma is described as black in colour with circular lines and an elevated hinder part.[65]
Kurma is invoked in worship of Vishnu in various scriptures.[66][67][68]TheBrahma Puranasalutes Kurma in a hymn as the "great tortoise", who "lifted the Earth and kept the mountain aloft".[69]TheLinga Purana, theGaruda Puranaand theShiva Puranasimilarly praises Vishnu as the one who kept the Mandara mountain aloft or the one who supported Mandara during the churning of the ocean as a tortoise.[70][71][72]
TheAgni Purana, theMarkendeya Purana, theVishnu Puranaand theBrahma Puranastate that Vishnu resides inBharata(theIndian subcontinent) in the form of Kurma.[73][74][75]TheMarkendeya Puranagives a detailed description of various lands of the region and constellations and zodiac stars corresponding to nine parts of the tortoise - mouth, four feet, tail, centre and two sides of its belly.[76][77]TheBhagavata Puranastates Vishnu stays as Kurma in the Himalayan continent (Hiraṇmaya-varsa).[78]
TheKurma Puranais one of four Puranas that bear the names of Vishnu's avatars. The Purana is narrated by Kurma to the kingIndradyumnaand later to the sages and the gods at the time of Samudra Manthana.[79]The detailed tale of the Samudra Manthana is absent from the Purana and alludes to Kurma as the one who supported Mount Mandara.[80]TheKurma Puranais stated to be narrated by Kurma and is prescribed to be gifted with a golden statue of a tortoise in theAgni Purana.[81]
TheAgni Puranaprescribes that Kurma be depicted in zoomorphic form as a tortoise.[82]
In the narrative of the battle between Shiva's manifestationVirabhadraand Vishnu's avatarNarasimhaof theLinga Puranaand theShiva Purana, Virabhadra mocks Narasimha-Vishnu stating that Kurma's skull adorns the necklace of Shiva.[83][84]
In a passing reference in theVishnu Puranaand theMarkendeya Purananarrative ofVaraha, Brahma - identified with Narayana - decides to take the form of the boar Varaha, similar to the forms of the fish (Matsya) and tortoise (Kurma), he took in previouskalpas.[85]
TheLinga Purana, theVaraha Puranaand theShiva Puranamention Kurma as second in its Dashavatara listing.[86][71][87][88]
TheVaraha Puranarecommends avrata(vow) with fasting and worshipping Kurma-Vishnu in a three lunar-day festival culminating on thetwelfth lunar dayin the bright half of thePaushamonth. The first day of thevratais said to be the day when Vishnu assumed the Kurma form in Samundra manthan.[89]
TheBhagavata Puranalists Kurma the eleventh avatar of Vishnu in the list of 22 avatars.[72]TheGaruda Puranalists him as the eleventh of 20 avatars,[90]elsewhere he is mentioned as the second of the Dashavatara.[91]
TheVishnu Sahasranamaversion from theGaruda PuranamentionsKurmaas an epithet of Vishnu.[92]TheGaruda Puranaaddresses Kurma in hymns to Vishnu.[93]He is associated with the south-western direction.[94]
It was this [Mandara] mountain that was formerly lifted up byHari(in the form of [the] Divine Tortoise) and used for churning (the milk ocean) by theDevasandDanavas. Sindhu (the ocean) which extends to six hundred thousand Yojanas is the deep pit made by this mountain. This great mountain was not broken even when it rubbed against the physical body of the Divine Tortoise. O leading king, when it fell into the ocean all the hidden parts of the ocean were exposed by the mountain. O Brahmanas, water gushed out from this mountain [and] went up through the path of the Brahmanda (Cosmic Egg). Great fire was generated by this mountain due to attrition when it came into contact with the bony shell of the (Divine) Tortoise... It was for a great period of time viz. ten thousand years than this mountain ground and rubbed the armlets of the discus-bearing Lord.
In theNarada Purana, a brief synopsis of theSamudra Manthanais given byBrahmatoMohini, as quoted above (Part 4: 8.7-11). There are two other notable mentions of this legend. The first is bySaunakawho said 'When there was an impediment at the time of churning the ocean for the sake of nectar, he [Kurma] held the mount Mandara on his back, for the welfare of the gods. I seek refuge in that Tortoise' (Part 1: 2.37). In the second, it is stated 'it was when the milk-ocean was churning thatKamodawas born among the four jewels of Virgins' (Part 5: Uttara Bhaga: 68.4). Other details include:
Visnuhimself, remaining in the ocean in the form of a tortoise, nourished the gods with unusual lustre... the goddessVarunibecame (manifest), Her eyes were rolling about due to intoxication... [she said:] "I am a goddess giving strength. The demons may take me". Regarding Varuni as impure, the gods let her go. Then the demons took her. She became wine after being taken (by them)... Then the deadly poison (came up). By it all gods and demons with (other) deities were afflicted. Mahadeva [Shiva] took and drank that poison at his will. Due to drinking it Mahadeva had his throat turned dark blue. TheNagasdrank the remaining poison that had come up from the White [Milky] Ocean.
In thePadma Puranathree accounts of theSamudra Manthanaare given, all beginning withIndrabeing cursed byDurvasasfor arrogance. In the first, narrated byPulastya, as a result of the curse the 'three worlds, along with Indra, were void of affluence... [and] theDaityas(sons ofDiti) andDanavas(sons ofDanu) started military operations against [the] gods', forcing them to seek refuge withVishnu.Vasukiis used as a rope to churn the ocean. Notably, during the churning,Varuni(Goddess of Wine) is upon emerging rejected by the gods and accepted by theasuras, the opposite of the account given in theBrahmanda Purana(to explain the meaning of 'Asura'). Unnamed poison also emerges which is drunk byShiva, before the emergence ofDhanvantariwith the nectar of immortality (Amrita) as well asLakshmi. Although the asuras take the nectar, Vishnu assumes the form ofMohinito trick them and give it to the gods. The asuras are destroyed, with the Danavas since then becoming 'eager for (the company of) ladies' (Part 1: 4).
O gods, Indira (i.e.Laksmi), due to whose mere glance the world is endowed with glory, has vanished due to the curse of theBrahmana(viz.Durvasas). Then, O gods, all of you, along with the demons, having uprooted the golden mountain Mandara and making it, with the king of serpents going round it, the churning-rod, churn the milky ocean. O gods, from it Laksmi, the mother of the world will spring up. O glorious ones, there is no doubt that because of her you will be delighted. I myself, in the form of a tortoise, shall fully hold the (Mandara) mountain (on my back).
In the second account, narrated bySuta, as a result of the curse the 'mother of the worlds' (Lakshmi) disappears, and the world is ruined by drought and famine, forcing the gods - oppressed by hunger and thirst - to seek refuge withVishnuat the shore of theMilky Ocean(Part 5: 8).Ananta(Vasukiin the first account) is used as a churning rope. OnEkadashiday, the poison Kalakuta emerges, which is swallowed byShiva'meditating upon Vishnu in his heart'. An evil being calledAlaksmi(i.e. a-Laksmi or 'notLaksmi') them emerges and is told to reside in places such as where there is quarrel, gambling, adultery, theft, and so forth (Part 5: 9). The churning continues and auspicious beings and items emerge, including 'the brother of Laksmi, [who] sprang up with nectar. (So also)Tulasi[i.e.Lakshmi], Visnu's wife'. On this occasion, Vishnu assumes the form ofMohinimerely to distribute the nectar amongst the gods, without mention of tricking the asuras (Part 5: 10).
The third account, narrated byShiva, is very similar to the others except with a far greater emphasis onLakshmi, and although the poison Kalakuta emerges and is swallowed by Shiva, there is no mention of Alaksmi or the Mohini avatar (Part 9: 231–232). TheNagaused as a rope for churning is referred to as 'the Lord of the Serpents' (likelyAnanta). Other details include:
As the Ocean of Milk was being churned, the mountain sank deep intoRasatala. At that very instant, the Lord ofRama,Visnu, became a tortoise and lifted it up. That was something really marvellous... The excellent mountain hadadamantinestrength. It rolled on the back, neck, thighs, and space between the knees of the noble-souled tortoise. Due to the friction of these two, submarine [i.e. underwater] fire was generated.
In theSkanda Puranafour accounts of theSamudra Manthanaare given. In the first, the churning of theocean of Milktakes place afterIndrais cursed by the sageBrhaspati, resulting in the disappearance ofLakshmi, misery to all, and ruin of thedevas, defeated in battle by theasuraswho take their precious items such as gems toPatala. On the advice ofBrahma, Indra and the devas make a pact withBali, leader of asuras, to recover the gems from the Ocean of Milk. Unable to move the Mandara mountain to use as a churning rod,Vishnuis asked for help, who arrives onGaruda, takes the mountain to the ocean, and incarnates as Kurma.Vasukiis used as the churning rope. The Kalakuta poison generated envelopes the devas andDaityas- causing ignorance and lust - before enveloping all existence (includingVaikuntha) and reducing thecosmic eggto ash (Part 1: 9).Shivais approached for refuge, and the origin and need to worshipGaneshato 'achieve success in undertaking' is explained before Shiva drinks the poison (Part 1: 10). More information on Ganesha-worship is given before the churning resumes, producing many auspicious items and beings, including Lakshmi (Part 1: 11).Dhanvantariemerges with the nectar of immortality (Amrita), which is taken by the asuras. Vishnu incarnates asMohini, and despite warning Bali that 'Women should never be trusted by a wise man' is still given the nectar which She gives to the devas (Part 1: 12).
In the second account, Indra is again cursed by the sageBrhaspati(Part 7: 8), resulting in the disappearance ofLaksmi, and with her, an absence of 'Penance, Purity, Mercy, Truth... TrueDharma, Prosperity... Strength [and]Sattva(quality of goodness)'. Hunger, poverty, anger, lust, flesh-eating, and perverse-thinking abound, including belief thatadharmaisdharma, and perverse interpretations of theVedasto justify killing animals (Part 7: 9).Vishnuis approached for refuge by the devas and instructs them to churn theOcean of Milk(Part 7: 10). Indra forms a pact with the asuras,Sesais used as a churning rope with the Mandara Mountain, and Vishnu incarnates as Kurma as the base. After a thousand years of churning the poisonHalahalais generated and swallowed by Shiva; the drops that fell are taken by serpents, scorpions, and somemedicinalplants (Part 7: 11). The churning continues for another thousand years, producing auspicious items and beings, including Laksmi (Part 7: 12).Dhanvantariemerges with the pitcher ofAmritawhich is taken by the asuras, and Vishnu assumes 'a marvellously beautiful feminine form that enchanted all the world' (Mohini). Despite warning the asuras not to trust her, Mohini is given the Amrita which is handed to the devas before the asuras are destroyed in battle (Part 7: 13).
In the third brief account, the churning takes place after 'a great loss of gems due to wicked souls' and the loss of righteousness.Vasukiis used as the churning cord as the devas and asuras 'placed the main plant of activity on the back of the (divine) tortoise and churned out the precious gems'. Many auspicious items and beings are generated, including Sura (alcohol; in other accountsVaruni) andDhanvantari. Quarreling ensues between the devas and asuras, and Vishnu incarnates as 'the fascinating form of a woman' (Mohini) to beguile the demons asIndragives them the Sura and via 'sleight of hand' takes theAmrita. Halahala poison is also generated which is consumed byShiva(Part 12: 44).
In the fourth account, the legend is briefly retold byVisvamitra. The details are much the same as the previous accounts, with Vasuki as the cord as the 'Kacchapa (Tortoise incarnation ofVisnu) held up (the mountain)', including the Kalakuta poison drunk byShivaand the incarnation ofMohinito trick the asuras. The notable exception is that the churning first produces a 'hideous' family of three ofRatnas(jewels); rejected by both the devas andDanavas, they are accepted by Ka (i.e.Brahma; Part 18: 210).
Notably, reminiscent the account ofPrajapatiand theTortoisein theTaittiriya Aranyaka(see above), there is also an account, during the time of the universal dissolution, whenBrahma'assumed the form of aKhadyota(Firefly, Glow-worm)' and moved about for a thousand divine years before finding 'the Lord [Vishnu] asleep in the form of a tortoise'. Woken by Brahma, Vishnu 'got up ejecting the three worlds that had been swallowed at the time of the close of the [previous]Kalpa' with all creation - including thedevas,Danavas, moon, sun, and planets - being generated from and by Him. Vishnu also sees the Earth 'was in the great ocean perched on the back of the tortoise' (Part 14: Reva Khanda: 7). Other details include:
When the sons ofKasyapa(i.e.DevasandAsuras) will churn the ocean for (obtaining) nectar, I [Vishnu], assuming the form of a tortoise, will bear on my back Mount Mandara used as the churning rod.
The Samudra Manthana is popular in iconography and even found in South East Asia. Notable depictions include the relief atAngkor Watwith Vishnu and Kurma in the centre and the gods and demons on either side churning the ocean. The earth below the temple represents Kurma in Khymer iconography, the earth goddess being Vishnu's consort. The Vishnu on the top of Mandara symbolizes him as the shining midday Sun.[130]
Kurma is depicted eitherzoomorphicallyas a tortoise.[131]
In the anthropomorphic form, the upper half is that of the four-armed man and the lower half is a fish. The upper half resembles Vishnu and wears the traditional ornaments and thekirita-mukuta(tall conical crown) as worn by Vishnu. He holds in two of his hands theSudarshana chakra(discus) and ashankha(conch), the usual weapons of Vishnu. The other two hands make the gestures ofvaradamudra, which grants boons to the devotee, andabhayamudra, which reassures the devotee of protection. The depiction is similar to Matsya, where the lower half is a fish.[132]
Srikurmamwas initially a Shiva temple, which was converted into a Vaishnava one by the Vaishnava saintRamanuja.[49]The sanctum has an icon of Vishnu, as well as of Kurma with the tail and back to the devotee and face to the west. This is in contradiction to scriptural mandate that the central icon should face the east. According to a legend, the Kurma icon turned to the west back wall in honour of a tribalBhilking who worshipped him from the back of the temple. Nanditha Krishna suggests that a tribal tortoise god could have been assimilated in the Hindu fold by identifying him with Kurma.[133]
There are five temples dedicated to this incarnation of Vishnu in India:
The name of the village mentioned above originates from the historical temple of Kurma calledVaradarajaswamy(Kurma avatara of Vishnu), regarding the deity of this village.[134]
M. Vettam notes that there are tenVayus(Winds) in the body, one of which is called 'Kurma' in regards to opening and closing the eyes.[135]
The 'kurma-Nadi'(orKūrmanāḍī, Sanskrit कूर्मनाडी), meaning 'Tortoise-Nerve' or 'Canal Of The Tortoise', is in relation to steadying the mind (slowing down thoughts) inYogicpractice.[136]'Nadi'itself means 'Vein', 'Artery', 'River', or 'Any Tubular Organ Of The Body' (as well as 'Flute').[137]Although the Kurmanaḍi is generally stated to be located in the upper chest below the throat,[136]S. Lele believes this refers to theMuladhara Chakra, located near theTailbone, based on the root-word 'Nal' (Sanskrit नल्), meaning 'to Bind'.[138][139]
These are all mentioned in theUpanishadsandPuranas.
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Inclassical, medieval, and Renaissance astronomy, thePrimum Mobile(Latin: "first movable") was the outermost movingspherein thegeocentric modelof theuniverse.[1]
The concept was introduced byPtolemyto account for the apparentdaily motionof the heavens around the Earth, producing the east-to-west rising and setting of the sun and stars, and reached Western Europe viaAvicenna.[2]
The Ptolemaic system presented a view of the universe in which apparent motion was taken for real – a viewpoint still maintained in common speech through such everyday terms asmoonriseandsunset.[3]Rotation of the Earth on its polar axis – as seen in aheliocentricsolar system, which (while anticipated byAristarchus) was not to be widely accepted until well afterCopernicus[3]– leads to what earlier astronomers saw as the real movement of all the heavenly bodies around the Earth every 24 hours.[4]
Astronomers believed that the sevennaked-eye planets(including the Moon and the Sun) were carried around thespherical Earthon invisible orbs, while an eighth sphere contained the fixedstars. Motion was provided to the whole system by the Primum Mobile, itself set within theEmpyrean, and the fastest moving of all the spheres.[5]
The total number ofcelestial sphereswas not fixed. In this 16th-century illustration, thefirmament(sphere of fixed stars) is eighth, a "crystalline" sphere (posited to account for the reference to "waters ... above the firmament" inGenesis1:7) is ninth, and the Primum Mobile is tenth. Outside all is theEmpyrean, the "habitation of God and all theelect".
Copernicus accepted existence of the sphere of the fixed stars, and (more ambiguously) that of the Primum Mobile,[6]as too (initially) didGalileo[7]– though he would later challenge its necessity in a heliocentric system.[8]
Francis Baconwas as sceptical of the Primum Mobile as he was of the rotation of the earth.[9]OnceKeplerhad made the sun, not the Primum Mobile, the cause of planetary motion, however,[10]the Primum Mobile gradually declined into the realm of metaphor or literary allusion.
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Theunmoved mover(Ancient Greek:ὃ οὐ κινούμενον κινεῖ,romanized:ho ou kinoúmenon kineî,lit.'that which moves without being moved')[1]orprime mover(Latin:primum movens) is a concept advanced byAristotleas a primarycause(orfirst uncaused cause)[2]or "mover" of all the motion in theuniverse.[3]As is implicit in the name, theunmoved movermoves other things, but is not itself moved by any prior action. In Book 12 (Ancient Greek:Λ) of hisMetaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfectcontemplation: self-contemplation. He also equates this concept with theactive intellect. This Aristotelian concept had its roots incosmologicalspeculations of the earliest Greekpre-Socratic philosophers[4]and became highly influential and widely drawn upon inmedieval philosophyandtheology.St. Thomas Aquinas, for example, elaborated on the unmoved mover in theFive Ways.
Aristotle argues, in Book 8 of thePhysicsand Book 12 of theMetaphysics, "that there must be an immortal, unchanging being, ultimately responsible for all wholeness and orderliness in the sensible world."[5]
In thePhysics(VIII 4–6) Aristotle finds "surprising difficulties" explaining even commonplace change, and in support of his approach of explanation byfour causes, he required "a fair bit of technical machinery".[6]This "machinery" includespotentiality and actuality,hylomorphism,the theory of categories, and "an audacious and intriguing argument, that the bare existence of change requires the postulation of a first cause, an unmoved mover whose necessary existence underpins the ceaseless activity of the world of motion".[7]Aristotle's "first philosophy", orMetaphysics("afterthePhysics"), develops his peculiar theology of the prime mover, asπρῶτον κινοῦν ἀκίνητον: an independent divine eternal unchanging immaterial substance.[8]
Aristotle adopted the geometrical model ofEudoxus of Cnidusto provide a general explanation of the apparent wandering of theclassical planetsarising from uniform circular motions ofcelestial spheres.[9]While the number of spheres in the model itself was subject to change (47 or 55), Aristotle's account ofaether, and ofpotentiality and actuality, required an individual unmoved mover for each sphere.[10]
Simpliciusargues that the first unmoved mover is a cause not only in the sense of being a final cause—which everyone in his day, as in ours, would accept—but also in the sense of being an efficient cause (1360. 24ff.), and his masterAmmoniuswrote a whole book defending the thesis (ibid. 1363. 8–10). Simplicius's arguments include citations ofPlato's views in theTimaeus—evidence not relevant to the debate unless one happens to believe in the essential harmony of Plato and Aristotle—and inferences from approving remarks which Aristotle makes about the role ofNousinAnaxagoras, which require a good deal of reading between the lines. But he does point out rightly that the unmoved mover fits the definition of an efficient cause—"whence the first source of change or rest" (Phys. II. 3, 194b29–30; Simpl. 1361. 12ff.). The examples which Aristotle adduces do not obviously suggest an application to the first unmoved mover, and it is at least possible that Aristotle originated his fourfold distinction without reference to such an entity. But the real question is whether his definition of the efficient cause includes the unmoved mover willy-nilly. One curious fact remains: that Aristotle never acknowledges the alleged fact that the unmoved mover is an efficient cause (a problem of which Simplicius is well aware: 1363. 12–14)...[11]
Despite their apparent function in the celestial model, the unmoved movers were afinal cause,notanefficient causefor the movement of the spheres;[12]they were solely a constant inspiration,[13]and even if taken for an efficient causepreciselydue to being a final cause,[14]the nature of the explanation is purely teleological.[15]
The unmoved mover, if they were anywhere, were said to fill the outer void beyond the sphere of fixed stars:
It is clear then that there is neither place, nor void, nor time, outside the heaven. Hence whatever is there, is of such a nature as not to occupy any place, nor does time age it; nor is there any change in any of the things which lie beyond the outermost motion; they continue through their entire duration unalterable and unmodified, living the best and most self sufficient of lives… From [the fulfilment of the whole heaven] derive the being and life which other things, some more or less articulately but other feebly, enjoy.[16]
The unmoved mover is an immaterial substance (separate and individual beings), having neither parts nor magnitude. As such, it would be physically impossible for them to move material objects of any size by pushing, pulling, or collision. Because matter is, for Aristotle, a substratum in which a potential to change can be actualized, any potentiality must be actualized in an eternal being, but it must not be still because continuous activity is essential for all forms of life. This immaterial form of activity must be intellectual and cannot be contingent upon sensory perception if it is to remain uniform; therefore, eternal substance must think only of thinking itself and exist outside the starry sphere, where even the notion of place is undefined for Aristotle. Their influence on lesser beings is purely the result of an "aspiration or desire,"[17]and each aetheric celestial sphere emulates one of the unmoved movers, as best it can, byuniform circular motion. The first heaven, the outmost sphere of fixed stars, is moved by a desire to emulate the prime mover (first cause),[18][note 1]about whom, the subordinate movers suffer an accidental dependency.
Many of Aristotle's contemporaries complained that oblivious, powerless gods are unsatisfactory.[8]Nonetheless, it was a life which Aristotle enthusiastically endorsed as one most enviable and perfect, the unembellished basis of theology. As the whole of nature depends on the inspiration of the eternal unmoved movers, Aristotle was concerned with establishing the metaphysical necessity of the perpetual motions of the heavens. Through the Sun's seasonal action upon the terrestrial spheres, the cycles of generation and corruption give rise to allnaturalmotion as efficient cause.[15]The intellect,nous, "or whatever else it be that is thought to rule and lead us by nature, and to have cognizance of what is noble and divine" is the highest activity, according to Aristotle (contemplation or speculative thinking,theōríā). It is also the most sustainable, pleasant, self-sufficient activity;[19]something which is aimed at for its own sake. (Unlike politics and warfare, it does not involve doing things we'd rather not do, but rather something we do at our leisure.) This aim is not strictly human: to achieve it means to live following not mortal thoughts but something immortal and divine within humans. According to Aristotle, contemplation is the only type of happy activity that it would not be ridiculous to imagine the gods having. In Aristotle's psychology and biology, the intellect is thesoul(see alsoeudaimonia).
According toGiovanni Reale, the first Unmoved Mover is a living, thinking, andpersonalGod who "possesses the theoretical knowledge alone or in the highest degree...knows not only Himself, but all things in their causes and first principles."[20]
In Book VIII of hisPhysics,[21]Aristotle examines the notions of change or motion, and attempts to show by a challenging argument, that the mere supposition of a 'before' and an 'after', requires afirst principle. He argues that in the beginning, if the cosmos had come to be, its first motion would lack an antecedent state; and, asParmenidessaid, "nothing comes from nothing". Thecosmological argument, later attributed to Aristotle, thereby concludes that God exists. However, if the cosmos had a beginning, Aristotle argued, it would require anefficientfirst cause, a notion that Aristotle took to demonstrate a critical flaw.[22][23][24]
But it is a wrong assumption to suppose universally that we have an adequate first principle in virtue of the fact that something always is so ... ThusDemocritusreduces the causes that explain nature to the fact that things happened in the past in the same way as they happen now: but he does not think fit to seek for a first principle to explain this 'always' ... Let this conclude what we have to say in support of our contention that there never was a time when there was not motion, and never will be a time when there will not be motion.
The purpose of Aristotle'scosmological argumentthat at least one eternal unmoved mover must exist is to support everyday change.[26]
Of things that exist, substances are the first. But if substances can, then all things can perish... and yet, time and change cannot. Now, the only continuous change is that of place, and the only continuous change of place is circular motion. Therefore, there must be an eternal circular motion and this is confirmed by the fixed stars which are moved by the eternal actual substance that's purely actual.[27]
In Aristotle's estimation, an explanation without the temporalactuality and potentialityof an infinite locomotive chain is required for an eternal cosmos with neither beginning nor end: an unmoved eternal substance for whom thePrimum Mobile[note 2]turns diurnally, whereby all terrestrial cycles are driven by day and night, the seasons of the year, the transformation of the elements, and the nature of plants and animals.[10]
Aristotle begins by describing substance, of which he says there are three types: the sensible, subdivided into the perishable, which belongs to physics, and the eternal, which belongs to "another science." He notes that sensible substance is changeable and that there are several types of change, including quality and quantity, generation and destruction, increase and diminution, alteration, and motion. Change occurs when one given state becomes something contrary to it: that is to say, what exists potentially comes to exist actually (seepotentiality and actuality). Therefore, "a thing [can come to be], incidentally, out of that which is not, [and] also all things come to be out of that which is, but ispotentially, and is not actually." That by which something is changed is the mover, that which is changed is the matter, and that into which it is changed is the form.[citation needed]
Substance is necessarily composed of different elements. The proof for this is that there are things that are different from each other and that all things are composed of elements. Since elements combine to form composite substances, and because these substances differ from each other, there must be different elements: in other words, "b or a cannot be the same as ba."[citation needed]
Near the end ofMetaphysics, BookΛ, Aristotle introduces a surprising question, asking "whether we have to suppose one such [mover] or more than one, and if the latter, how many."[28]Aristotle concludes that the number of all the movers equals the number of separate movements, and we can determine these by considering the mathematical science most akin to philosophy, i.e., astronomy. Although the mathematicians differ on the number of movements, Aristotle considers that the number ofcelestial sphereswould be 47 or 55. Nonetheless, he concludes hisMetaphysics, BookΛ, with a quotation from theIliad: "The rule of many is not good; one ruler let there be."[29][30]
John Burnet(1892) noted[31]
The Neoplatonists were quite justified in regarding themselves as the spiritual heirs of Pythagoras; and, in their hands, philosophy ceased to exist as such, and became theology. And this tendency was at work all along; hardly a single Greek philosopher was wholly uninfluenced by it. PerhapsAristotlemight seem to be an exception; but it is probable that, if we still possessed a few such "exoteric" works as theProtreptikosin their entirety, we should find that the enthusiastic words in which he speaks of the "blessed life" in theMetaphysicsand in theEthics(Nicomachean Ethics)were less isolated outbursts of feeling than they appear now. In later days,Apollonios of Tyanashowed in practice what this sort of thing must ultimately lead to. Thetheurgyandthaumaturgyof the late Greek schools were only the fruit of the seed sown by the generation which immediately preceded the Persian War.
Aristotle's principles of being (see section above) influencedAnselm's view of God, whom he called "that than which nothing greater can be conceived." Anselm thought God did not feel emotions such as anger or love but appeared to do so through our imperfect understanding. The incongruity of judging "being" against something that might not exist may have led Anselm to his famous ontological argument for God's existence.
Manymedievalphilosophers used the idea of approaching a knowledge of God through negative attributes. For example, we should not say that God exists in the usual sense of the term; all we can safely say is that God is not nonexistent. We should not say that God is wise, but we can say that God is not ignorant (i.e., in some way, God has some properties of knowledge). We should not say that God is One, but we can state that there is no multiplicity in God's being.
Many later Jewish, Islamic, and Christian philosophers accepted Aristotelian theological concepts. KeyJewish philosophersincludedibn Tibbon,Maimonides, andGersonides, among many others. Their views of God are considered mainstream by many Jews of all denominations, even today. Preeminent among Islamic philosophers who were influenced by Aristotelian theology areAvicennaandAverroes. In Christian theology, the key philosopher influenced by Aristotle was undoubtedlyThomas Aquinas. There had been earlier Aristotelian influences within Christianity (notably Anselm), but Aquinas (who, incidentally, found his Aristotelian influence via Avicenna, Averroes, and Maimonides) incorporated extensive Aristotelian ideas throughout his theology. Through Aquinas and theScholastic Christian theologyof which he was a significant part, Aristotle became "academic theology's great authority in the thirteenth century"[32]and influenced Christian theology that became widespread and deeply embedded. However, notable Christian theologians rejected[a]Aristotelian theological influence, especially the first generation of Christian Reformers,[b]most notablyMartin Luther.[33][34][35]In subsequent Protestant theology, Aristotelian thought quickly reemerged inProtestant scholasticism.
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Inphilosophy, theproblem of the creator of Godis the controversy regarding the hypotheticalcauseresponsible for theexistence of God, assuming God exists. It contests the proposition that the universe cannot exist without acreatorby asserting that the creator of theUniversemust have the same restrictions. This, in turn, may lead to a problem ofinfinite regresswherein each new presumed creator of a creator is presumed to have its own creator. A common challenge totheisticpropositions of a creator deity as a necessaryfirst causeof theuniverseis the question: "Who created God?"[1]Some faith traditions have such an element as part of their doctrines.Jainismposits that the universe is eternal and has always existed.Isma'ilismrejects the idea of God as the first cause due to the doctrine of God's incomparability and source of existence, includingabstract objects.[2]
Oshowrites:
No, don't ask that. That's what all the religions say – don't ask who created God. But this is strange – why not? If the question is valid about existence, why does it become invalid when it is applied to God? And once you ask who created God, you are falling into a regress absurdum.[3]
John Humphreys writes:
... if someone were able to provide the explanation, we would be forced to embark upon what philosophers call an infinite regress. Having established who created God, we would then have to answer the question of who created God's creator.[4]
Alan Lurie writes:
In response to one of my blogs about God's purpose in the creation of the universe, one person wrote, "All you've done is divert the question. If God created the Universe, who created God? That is a dilemma that religious folks desperately try to avoid." The question, "Who created God?", has been pondered by theologians for millennia, and the answer is both surprisingly obvious and philosophically subtle... whatever one thinks about the beginnings of the Universe, there is "something" at the very origin that was not created. This is an inescapable given, a cosmic truth.[5]
Joseph Smithstated in theKing Follett discourse:
God himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens! That is the great secret. If the veil were rent today, and the great God who holds this world in its orbit, and who upholds all worlds and all things by His power, was to make himself visible—I say, if you were to see him today, you would see him like a man in form—like yourselves in all the person, image, and very form as a man... it is necessary we should understand the character and being of God and how He came to be so; for I am going to tell you how God came to be God. We have imagined and supposed that God was God from all eternity. I will refute that idea, and take away the veil... It is the first principle of the gospel to know for a certainty the character of God, and to know that we may converse with Him as one man converses with another, and that He was once a man like us; yea, that God himself, the Father of us all, dwelt on an earth, the same as Jesus Christ Himself did... Is it logic to say that a spirit is immortal and yet has a beginning? Because if a spirit has a beginning, it will have an end.... All the fools and learned and wise men from the beginning of creation who say that man had a beginning prove that he must have an end. If that were so, the doctrine of annihilation would be true. But if I am right, I might with boldness proclaim from the house tops that God never did have power to create the spirit of man at all. God himself could not create himself. Intelligence exists upon a self-existent principle; it is a spirit from age to age, and there is no creation about it. Moreover, all the spirits that God ever sent into the world are susceptible to enlargement.
Defenders of religion have countered that, by definition, God isthe first cause, and thus that the question is improper:
We ask, "If all things have a creator, then who created God?" Actually, only created things have a creator, so it's improper to lump God with his creation. God has revealed himself to us in the Bible as having always existed.[6]
Ray Comfort, author and evangelist, writes:
No person or thing created God. He created "time," and because we dwell in the dimension of time,reasondemands that all things have a beginning and an end. God, however, dwells outside of the dimension of time. He moves through time as we flip through a history book...He dwells in "eternity," having no beginning or end.[7]
Tzvi Freemanwrites on the officialChabad website:
Ibn Sina, the preeminent Arabic philosopher, answered this question a thousand years ago, when he described G-d as non-contingent, absolute existence. If so, to ask "Why is there G-d?" is the equivalent of asking, "Why is there is-ness?"[8]
Atheists counter that there is no reason to assume the universe was created. The question becomes irrelevant if the universe is presumed to have circular time instead of linear time, undergoing an infinite series of big bangs and big crunches on its own.[9]
John Lennox, professor of Mathematics at Oxford writes:[10]
Now Dawkins candidly tells us that he does not like people telling him that they also do not believe in the God in which he does not believe. But we cannot afford to base our arguments on his dislikes. For, whether he likes it or not, he openly invites the charge. After all, it is he who is arguing that God is a delusion. In order to weigh his argument we need first of all to know what he means by God. And his main argument is focused on a created God. Well, several billion of us would share his disbelief in such a god. He needn't have bothered. Most of us have long since been convinced of what he is trying to tell us. Certainly, no Christian would ever dream of suggesting that God was created. Nor, indeed, would Jews or Muslims. His argument, by his own admission, has nothing to say about an eternal God. It is entirely beside the point. Dawkins should shelve it on the shelf marked 'Celestial Teapots' where it belongs. For the God who created and upholds the universe was not created – he is eternal. He was not 'made' and therefore subject to the laws that science discovered; it was he who made the universe with its laws. Indeed, that fact constitutes the fundamental distinction between God and the universe. The universe came to be, God did not.
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https://en.wikipedia.org/wiki/Problem_of_the_creator_of_God
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Pyrrhonismis an Ancient Greek school ofphilosophical skepticismwhich rejects dogma and advocates thesuspension of judgementover the truth of all beliefs. It was founded byAenesidemusin the first century BCE, and said to have been inspired by the teachings ofPyrrhoandTimon of Phliusin the fourth century BCE.[1]
Pyrrhonism is best known today through the surviving works ofSextus Empiricus, writing in the late second century or early third century CE.[2]The publication of Sextus' works in theRenaissanceignited arevival of interest in Skepticismand played a major role inReformationthought and the development ofearly modern philosophy.
Pyrrhonism is named afterPyrrho of Elis, a Greekphilosopherin the 4th century BCE who was credited by the later Pyrrhonists with forming the first comprehensive school ofskeptical thought. However, ancient testimony about the philosophical beliefs of the historical Pyrrho is minimal, and often contradictory:[1]his teachings were recorded by his studentTimon of Phlius, but those works have been lost, and only survive in fragments quoted by later authors, and based on testimonies of later authors such asCicero. Pyrrho's own philosophy as recorded by Timon may have been much more dogmatic than that of the later school who bore his name.[1]While Pyrrhonism would become the dominant form of skepticism in the early Roman period, in theHellenistic period, thePlatonic Academywas the primary advocate of skepticism until the mid-first century BCE,[3]when Pyrrhonism as a philosophical school was founded by Aenesidemus.[1][4]
The goal of Pyrrhonism isataraxia,[5]an untroubled and tranquil condition of soul that results from a suspension of judgement, a mental rest owing to which we neither deny nor affirm anything.
Pyrrhonists dispute that the dogmatists – which includes all of Pyrrhonism's rival philosophies – claim to have foundtruthregarding non-evident matters, and that these opinions about non-evident matters (i.e.,dogma) are what prevent one from attainingeudaimonia. For any of these dogmas, a Pyrrhonist makes arguments for and against such that the matter cannot be concluded, thussuspending judgement, and thereby inducing ataraxia.
Pyrrhonists can be subdivided into those who areephectic(engaged in suspension of judgment),aporetic(engaged in refutation)[6]orzetetic(engaged in seeking).[7]An ephectic merely suspends judgment on a matter, "balancing perceptions and thoughts against one another."[8]It is a less aggressive form of skepticism, in that sometimes "suspension of judgment evidently just happens to the sceptic".[9]An aporetic skeptic, in contrast, works more actively towards their goal, engaging in the refutation of arguments in favor of various possible beliefs in order to reachaporia, an impasse, or state of perplexity,[10]which leads to suspension of judgement.[9]Finally, the zetetic claims to be continually searching for the truth but to have thus far been unable to find it, and thus continues to suspend belief while also searching for reason to cease the suspension of belief.
Although Pyrrhonism's objective is ataraxia, it is best known for itsepistemologicalarguments. The core practice is through setting argument against argument. To aid in this, the Pyrrhonist philosophersAenesidemusandAgrippadeveloped sets of stock arguments known as "modes" or "tropes."
Aenesidemusis considered the creator ofthe ten tropes of Aenesidemus(also known asthe ten modes of Aenesidemus)—although whether he invented thetropesor just systematized them from prior Pyrrhonist works is unknown. The tropes represent reasons for suspension of judgment. These are as follows:[11]
According to Sextus, superordinate to these ten modes stand three other modes: that based on the subject who judges (modes 1, 2, 3 & 4), that based on the object judged (modes 7 & 10), that based on both subject who judges and object judged (modes 5, 6, 8 & 9), and superordinate to these three modes is the mode of relation.[12]
These "tropes" or "modes" are given bySextus Empiricusin hisOutlines of Pyrrhonism. According to Sextus, they are attributed only "to the more recent skeptics" and it is byDiogenes Laërtiusthat we attribute them toAgrippa.[13]Thefive tropes of Agrippaare:
According to the mode deriving from dispute, we find that undecidable dissension about the matter proposed has come about both in ordinary life and among philosophers. Because of this we are not able to choose or to rule out anything, and we end up withsuspension of judgement. In the mode deriving from infinite regress, we say that what is brought forward as a source of conviction for the matter proposed itself needs another such source, which itself needs another, and soad infinitum, so that we have no point from which to begin to establish anything, and suspension of judgement follows. In the mode deriving from relativity, as we said above, the existing object appears to be such-and-such relative to the subject judging and to the things observed together with it, but we suspend judgement on what it is like in its nature. We have the mode from hypothesis when the Dogmatists, being thrown backad infinitum, begin from something which they do not establish but claim to assume simply and without proof in virtue of a concession. The reciprocal mode occurs when what ought to be confirmatory of the object under investigation needs to be made convincing by the object under investigation; then, being unable to take either in order to establish the other, we suspend judgement about both.[14]
With reference to these five tropes, that the first and third are a short summary of the earlier Ten Modes ofAenesidemus.[13]The three additional ones show a progress in the Pyrrhonist system, building upon the objections derived from the fallibility of sense and opinion to more abstract and metaphysical grounds. According toVictor Brochard"the five tropes can be regarded as the most radical and most precise formulation of skepticism that has ever been given. In a sense, they are still irresistible today."[15]
Pyrrhonist decision making is made according to what the Pyrrhonists describe as thecriteria of actionholding to theappearances, without beliefs in accord with the ordinary regimen of life based on:
The Pyrrhonists devised several sayings (Greek ΦΩΝΩΝ[clarification needed]) to help practitioners bring their minds to suspend judgment.[17]Among these are:
Except for the works ofSextus Empiricus, the texts of ancient Pyrrhonism have been lost. There is a summary of thePyrrhonian DiscoursesbyAenesidemus, preserved byPhotius, and a brief summary of Pyrrho's teaching byAristocles, quoting Pyrrho's studentTimonpreserved byEusebius:
'The things themselves are equally indifferent, and unstable, and indeterminate, and therefore neither our senses nor our opinions are either true or false. For this reason then we must not trust them, but be without opinions, and without bias, and without wavering, saying of every single thing that it no more is than is not, or both is and is not, or neither is nor is not.[19]
Pyrrhonism is often contrasted withAcademic skepticism, a similar but distinct form of Hellenistic philosophical skepticism.[9][20][21]While early Academic skepticism was influenced in part by Pyrrho,[22]it grew more and more dogmatic untilAenesidemusbroke with the Academics to revive Pyrrhonism in the first century BCE, denouncing the Academy as "Stoics fighting against Stoics.[23]" Some later Pyrrhonists, such asSextus Empiricus, go so far as to claim that Pyrrhonists are the only real skeptics, dividing all philosophy into the dogmatists, the Academics, and the skeptics.[20]Dogmatists claim to have knowledge, Academic skeptics claim thatknowledge is impossible, while Pyrrhonists assent to neither proposition, suspending judgment on both.[9][20][24]The second century Roman historianAulus Gelliusdescribes the distinction as "...the Academics apprehend (in some sense) the very fact that nothing can be apprehended, and they determine (in some sense) that nothing can be determined, whereas the Pyrrhonists assert that not even that seems to be true, since nothing seems to be true.[25][21]"
Sextus Empiricus also said that the Pyrrhonist school influenced and had substantial overlap with theEmpiric schoolof medicine, but that Pyrrhonism had more in common with theMethodic schoolin that it "follow[s] the appearances and take[s] from these whatever seems expedient."[26]
AlthoughJulian the Philosopher[27]mentions that Pyrrhonism had died out at the time of his writings, other writers mention the existence of later Pyrrhonists. Pseudo-Clement, writing around the same time (c.300-320 CE) mentions Pyrrhonists in hisHomilies[28]andAgathiaseven reports a Pyrrhonist named Uranius as late as the middle of the 6th century CE.[29]
According to Diogenes Laërtius, Pyrrho was said to havetraveled to IndiawithAlexander the Great's army, where Pyrrho was said to have studied with the so-calledmagiand thegymnosophists.[30]This has led some to wonder whether he may have been influenced byBuddhistteachings while in India,[31]most particularly thethree marks of existence.[32]Scholars who argue for such influence mention the fact that even the ancient author Diogenesis Laërtius states as much, when he wrote that Pyrrho “foregathered with the Indian Gymnosophists and with the Magi. This led him to adopt a most noble philosophy."[31]
According toChristopher I. Beckwith's analysis of the 'Aristocles Passage',adiaphora(anatta),astathmēta(dukkha), andanepikrita(anicca) are strikingly similar to the Buddhistthree marks of existence,[32]indicating that Pyrrho's teaching is based on Buddhism. Beckwith contends that the 18 months Pyrrho spent in India were long enough to learn a foreign language, and that the key innovative tenets of Pyrrho's skepticism were only found in Indian philosophy at the time and not in Greece.[33]Other similarities between Pyrrhonism and Buddhism include a version of thetetralemmaamong the Pyrrhonist maxims, and more significantly, the idea ofsuspension of judgementand how that can lead to peace and liberation;ataraxiain Pyrrhonism andnirvāṇain Buddhism.[34][35]
Furthermore, Buddhist philosopherJan Westerhoffsays that "many of Nāgārjuna's arguments concerning causation bear strong similarities to classical skeptical arguments as presented in the third book of Sextus Empiricus'sOutlines of Pyrrhonism,"[36]andThomas McEvilleysuspects that Nagarjuna may have been influenced by Greek Pyrrhonist texts imported into India.[37]McEvilley argues for mutual iteration in theBuddhist logico-epistemological traditionsbetween Pyrrhonism andMadhyamika:
An extraordinary similarity, that has long been noticed, between Pyrrhonism and Mādhyamaka is the formula known in connection with Buddhism as the fourfold negation (Catuṣkoṭi) and which in Pyrrhonic form might be called the fourfold indeterminacy.[38]
McEvilley also notes a correspondence between the Pyrrhonist and Madhyamaka views about truth, comparing Sextus' account[39]of two criteria regarding truth, one which judges between reality and unreality, and another which we use as a guide in everyday life. By the first criteria, nothing is either true or false, but by the second, information from the senses may be considered either true or false for practical purposes. As Edward Conze[40][verification needed]has noted, this is similar to the MadhyamikaTwo Truths doctrine, a distinction between "Absolute truth" (paramārthasatya), "the knowledge of the real as it is without any distortion,"[41]and "Truth so-called" (saṃvṛti satya), "truth as conventionally believed in common parlance.[41][42]
However, other scholars, such asStephen Batchelor[43]and Charles Goodman[44]question Beckwith's conclusions about the degree of Buddhist influence on Pyrrho. Conversely, while critical of Beckwith's ideas, Kuzminsky sees credibility in the hypothesis that Pyrrho was influenced by Buddhism, even if it cannot be safely ascertained with our current information.[31]
While discussing Christopher Beckwith's claims inGreek Buddha: Pyrrho's Encounter with Early Buddhism in Central Asia, Jerker Blomqvist states that:
On the other hand, certain elements that are generally regarded as essential features of Buddhism are entirely absent from ancient Pyrrhonism/scepticism. The concepts of good and bad karma must have been an impossibility in the Pyrrhonist universe, if "things" were ἀδιάφορα, 'without a logical self-identity', and, consequently, could not be differentiated from each other by labels such as 'good' and 'bad' or 'just' and 'unjust'. A doctrine of rebirth, reminiscent of the Buddhist one though favored by Plato and Pythagoras, was totally alien to the Pyrrhonists. The ἀταραξία, 'undisturbedness', that the Pyrrhonists promised their followers, may have a superficial resemblance to the Buddhist nirvana, but ἀταραξία, unlike nirvana, did not involve a liberation from a cycle of reincarnation; rather, it was a mode of life in this world, blessed with μετριοπάθεια, 'moderation of feeling' or 'moderate suffering', not with the absence of any variety of pain. Kuzminski, whom Beckwith hails as a precursor of his, had largely ignored the problem with this disparity between Buddhism and Pyrrhonism.[45]
Ajñana, which upheldradical skepticism, may have been a more powerful influence on Pyrrho than Buddhism. The Buddhists referred to Ajñana's adherents asAmarāvikkhepikasor "eel-wrigglers", due to their refusal to commit to a single doctrine.[46]Scholars includingBarua, Jayatilleke, and Flintoff, contend that Pyrrho was influenced by, or at the very least agreed with, Indian skepticism rather than Buddhism or Jainism, based on the fact that he valuedataraxia, which can be translated as "freedom from worry".[47][48][49]Jayatilleke, in particular, contends that Pyrrho may have been influenced by the first three schools of Ajñana, since they too valued freedom from worry.[50]
The recovery and publication of the works of Sextus Empiricus, particularly a widely influential translation byHenri Estiennepublished in 1562,[51]ignited arevival of interest in Pyrrhonism.[51]Philosophers of the time used his works to source their arguments on how to deal with the religious issues of their day. Major philosophers such asMichel de Montaigne,Marin Mersenne, andPierre Gassendilater drew on the model of Pyrrhonism outlined in Sextus Empiricus' works for their own arguments. This resurgence of Pyrrhonism has sometimes been called the beginning of modern philosophy.[51]Montaigneadopted the image of a balance scale for his motto,[52]which became a modern symbol of Pyrrhonism.[53][54]It has also been suggested that Pyrrhonism provided the skeptical underpinnings thatRené Descartesdrew from in developing his influential method ofCartesian doubtand the associated turn ofearly modern philosophytowardsepistemology.[51]In the 18th century,David Humewas also considerably influenced by Pyrrhonism, using "Pyrrhonism" as a synonym for "skepticism."[55][better source needed].
Friedrich Nietzsche, however, criticized the "ephectics" of the Pyrrhonists as a flaw of early philosophers, whom he characterized as "shy little blunderer[s] and milquetoast[s] with crooked legs" prone to overindulging "his doubting drive, his negating drive, his wait-and-see ('ephectic') drive, his analytical drive, his exploring, searching, venturing drive, his comparing, balancing drive, his will toneutralityandobjectivity, his will to everysine ira et studio: have we already grasped that for the longest time they all went against the first demands ofmoralityandconscience?"[56]
The term "neo-Pyrrhonism" is used to refer to modern Pyrrhonists such asBenson MatesandRobert Fogelin.[57][58]
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"Siphonaptera" is a name used[1]to refer to the following rhyme byAugustus De Morgan(Siphonapterabeing thebiological orderto which fleas belong):
Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and soad infinitum.And the great fleas themselves, in turn, have greater fleas to go on;While these again have greater still, and greater still, and so on.[2]
The rhyme appears in De Morgan'sA Budget of Paradoxes(1872) along with a discussion of the possibilities that all particles may be made of clustered smaller particles, "and so down, for ever", and that planets and stars may be particles of some larger universe, "and so up, for ever".[2]
The lines derive[3]from part ofJonathan Swift's long satirical poem "On Poetry: A Rapsody" of 1733:
The Vermin only teaze and pinchTheir Foes superior by an Inch.So,Nat'ralistsobserve, a FleaHath smaller Fleas that on him prey,And these have smaller yet to bite 'em,And so proceedad infinitum:Thus ev'ry Poet, in his KindIs bit by him that comes behind[.][4]
Lewis Fry Richardsonadapted the poem tometeorologyin 1922:[5]
Big whirls have little whirlsThat feed on theirvelocity,And little whirls have lesser whirlsAnd so on toviscosity...
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https://en.wikipedia.org/wiki/Siphonaptera_(poem)
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Theteleological argument(fromτέλος,telos, 'end, aim, goal') also known asphysico-theological argument,argument from design, orintelligent design argument, is arationalargument for theexistence of Godor, more generally, that complex functionality in the natural world, which looks designed, is evidence of an intelligent creator.[1][2][3][4][5]The earliest recorded versions of this argument are associated withSocratesinancient Greece, although it has been argued that he was taking up an older argument.[6][7]Later,PlatoandAristotledeveloped complex approaches to the proposal that the cosmos has an intelligent cause, but it was theStoicsduring the Roman era who, under their influence, "developed the battery of creationist arguments broadly known under the label 'The Argument from Design'".[8]
Since the Roman era, various versions of the teleological argument have been associated with theAbrahamic religions. In theMiddle Ages, Islamic theologians such asAl-Ghazaliused the argument, although it was rejected as unnecessary byQuranicliteralists, and as unconvincing by manyIslamic philosophers. Later, the teleological argument was accepted bySaint Thomas Aquinas, and included as the fifth of his "Five Ways" of proving the existence of God. In early modern England, clergymen such asWilliam TurnerandJohn Raywere well-known proponents. In the early 18th century,William Derhampublished hisPhysico-Theology, which gave his "demonstration of the being and attributes of God from his works of creation".[9]Later,William Paley, in his 1802Natural Theology or Evidences of the Existence and Attributes of the Deitypublished a prominent presentation of the design argument with his version of thewatchmaker analogyand the first use of the phrase "argument from design".[10]
From its beginning, there have been numerous criticisms of the different versions of the teleological argument. Some have been written as responses to criticisms of non-teleological natural science which are associated with it. Especially important were the general logical arguments presented byDavid Humein hisDialogues Concerning Natural Religion, published in 1779, and the explanation of biological complexity given inCharles Darwin'sOrigin of Species, published in 1859.[11]Since the 1960s, Paley's arguments have been influential in the development of acreation sciencemovement which used phrases such as "design by an intelligent designer", and after 1987 this was rebranded as "intelligent design", promoted by theintelligent design movementwhich refers to anintelligent designer. Both movements have used the teleological argument to argue against the modern scientific understanding ofevolution, and to claim that supernatural explanations should be given equal validity in the public school science curriculum.[12]
Starting already in classical Greece, two approaches to the teleological argument developed, distinguished by their understanding of whether the natural order was literally created or not. The non-creationist approach starts most clearly with Aristotle, although many thinkers, such as theNeoplatonists, believed it was already intended by Plato. This approach is not creationist in a simple sense, because while it agrees that a cosmic intelligence is responsible for the natural order, it rejects the proposal that this requires a "creator" to physically make and maintain this order. The Neoplatonists did not find the teleological argument convincing, and in this they were followed by medieval philosophers such asAl-FarabiandAvicenna. Later,Averroesand Thomas Aquinas considered the argument acceptable, but not necessarily the best argument.
While the concept of an intelligence behind the natural order is ancient, a rational argument that concludes that we can know that the natural world has a designer, or a creating intelligence which has human-like purposes, appears to have begun withclassical philosophy.[6]Religious thinkers inJudaism,Hinduism,Confucianism,IslamandChristianityalso developed versions of the teleological argument. Later, variants on the argument from design were produced inWestern philosophyand byChristian fundamentalism.
Contemporary defenders of the teleological argument are mainly Christians,[13]for exampleRichard SwinburneandJohn Lennox.
The argument from intelligent design appears to have begun withSocrates, although the concept of a cosmic intelligence is older andDavid Sedleyhas argued that Socrates was developing an older idea, citingAnaxagoras of Clazomenae, born about 500 BC, as a possible earlier proponent.[14][15][16]The proposal that the order of nature showed evidence of having its own human-like "intelligence" goes back to the origins of Greek natural philosophy and science, and its attention to the orderliness of nature, often with special reference to the revolving of the heavens. Anaxagoras is the first person who is definitely known to have explained such a concept using the word "nous" (which is the original Greek term that leads to modern English "intelligence" via its Latin and French translations). Aristotle reports an earlier philosopher fromClazomenaenamedHermotimuswho had taken a similar position.[17]AmongstPre-Socratic philosophersbefore Anaxagoras, other philosophers had proposed a similar intelligent ordering principle causing life and the rotation of the heavens. For exampleEmpedocles, likeHesiodmuch earlier, described cosmic order and living things as caused by a cosmic version oflove,[18]andPythagorasandHeraclitusattributed the cosmos with "reason" (logos).[19]In hisPhilebus28cPlatohas Socrates speak of this as a tradition, saying that "all philosophers agree—whereby they really exalt themselves—that mind (nous) is king of heaven and earth. Perhaps they are right." and later states that the ensuing discussion "confirms the utterances of those who declared of old that mind (nous) always rules the universe".[20]
Xenophon's report in hisMemorabiliamight be the earliest clear account of an argument that there is evidence in nature of intelligent design.[15]The word traditionally translated and discussed as "design" isgnōmēand Socrates is reported by Xenophon to have pressed doubting young men to look at things in the market, and consider whether they could tell which things showed evidence ofgnōmē, and which seemed more to be by blind chance, and then to compare this to nature and consider whether it could be by blind chance.[14][16]In Plato'sPhaedo, Socrates is made to say just before dying that his discovery of Anaxagoras' concept of a cosmicnousas the cause of the order of things, was an important turning point for him. But he also expressed disagreement with Anaxagoras' understanding of the implications of his own doctrine, because of Anaxagoras'materialistunderstanding ofcausation. Socrates complained that Anaxagoras restricted the work of the cosmicnousto the beginning, as if it were uninterested and all events since then just happened because of causes like air and water.[21]Socrates, on the other hand, apparently insisted that the demiurge must be "loving", particularly concerning humanity. (In this desire to go beyond Anaxagoras and make the cosmicnousa more active manager, Socrates was apparently preceded byDiogenes of Apollonia.);[15]McPherran (1996:290); and[22]
Plato'sTimaeusis presented as a description of someone who is explaining a "likely story" in the form of a myth, and so throughout history commentators have disagreed about which elements of the myth can be seen as the position of Plato.[16]: 132Sedley (2007) nevertheless calls it "the creationist manifesto" and points out that although some of Plato's followers denied that he intended it, in classical times writers such as Aristotle,Epicurus, theStoics, andGalenall understood Plato as proposing the world originated in an "intelligent creative act".[16]: 133Plato has a character explain the concept of a "demiurge" with supreme wisdom and intelligence as the creator of the cosmos in his work.[citation needed]
Plato's teleological perspective is also built upon the analysis ofa prioriorder and structure in the world that he had already presented inThe Republic. The story does not propose creationex nihilo; rather, the demiurge made order from the chaos of the cosmos, imitating the eternal Forms.[23]
Plato's world of eternal and unchangingForms, imperfectly represented in matter by a divine Artisan, contrasts sharply with the various mechanisticWeltanschauungen, of whichatomismwas, by the 4th century at least, the most prominent…This debate was to persist throughout the ancient world. Atomistic mechanism got a shot in the arm fromEpicurus…while theStoicsadopted a divine teleology…The choice seems simple: either show how a structured, regular world could arise out of undirected processes, or inject intelligence into the system.[24]
Plato's student and friend Aristotle (c. 384 – 322 BC), continued the Socratic tradition of criticising natural scientists such asDemocrituswho sought (as in modern science) to explain everything in terms of matter and chance motion. He was very influential in the future development of classical creationism, but was not a straightforward "creationist" because he required no creation interventions in nature, meaning he "insulated god from any requirement to intervene in nature, either as creator or as administrator".[16]: 204Instead of direct intervention by a creator it is "scarcely an exaggeration to say that for Aristotle the entire functioning of the natural world, as also the heavens, is ultimately to be understood as a shared striving towards godlikeactuality".[16]: 171And whereas the myth in theTimaeussuggests that all living things are based on one single paradigm, not one for each species, and even tells a story of "devolution" whereby other living things devolved from humans, it was Aristotle who presented the influential idea that each type of normal living thing must be based on a fixed paradigm or form for that species.[16]
Aristotle felt that biology was a particularly important example of a field where materialist natural science ignored information which was needed in order to understand living things well. For example birds use wings for the purpose of flight.[25]Therefore the most complete explanation in regard to the natural, as well as the artificial, is for the most part teleological.[26]In fact, proposals that species had changed by chance survival of the fittest, similar to what is now called "natural selection", were already known to Aristotle, and he rejected these with the same logic.[26][27][28][29][30]He conceded that monstrosities (new forms of life) could come about by chance,[31][32]but he disagreed with those who ascribed all nature purely to chance[33]because he believed science can only provide a general account of that which is normal, "always, or for the most part".[34]The distinction between what is normal, or by nature, and what is "accidental", or not by nature, is important in Aristotle's understanding of nature. As pointed out by Sedley, "Aristotle is happy to say (PhysicsII 8, 199a33-b4) without the slightest fear of blasphemy, crafts make occasional mistakes; therefore, by analogy, so can nature."[16]: 186According to Aristotle the changes which happen by nature are caused by their "formal causes", and for example in the case of a bird's wings there is also afinal causewhich is the purpose of flying. He explicitly compared this to human technology:
If then what comes from art is for the sake of something, it is clear that what come from nature is too…This is clear most of all in the other animals, which do nothing by art, inquiry, or deliberation; for which reason some people are completely at a loss whether it is by intelligence or in some other way that spiders, ants, and such things work.…It is absurd to think that a thing does not happen for the sake of something if we do not see what sets it in motion deliberating.…This is most clear when someone practices medicine himself on himself; for nature is like that.
The question of how to understand Aristotle's conception of nature having a purpose and direction something like human activity is controversial in the details.Martha Nussbaumfor example has argued that in his biology this approach was practical and meant to show nature only being analogous to human art, explanations of an organ being greatly informed by knowledge of its essential function.[26]Nevertheless, Nussbaum's position is not universally accepted. In any case, Aristotle was not understood this way by his followers in the Middle Ages, who saw him as consistent with monotheistic religion and a teleological understanding of all nature. Consistent with the medieval interpretation, in hisMetaphysicsand other works Aristotle clearly argued a case for there being one highest god or "prime mover" which was the ultimate cause, though specifically not the material cause, of the eternal forms or natures which cause the natural order, including all living things.[citation needed]He clearly refers to this entity having anintellectthat humans somehow share in, which helps humans see the true natures or forms of things without relying purely on sense perception of physical things, including living species. This understanding of nature, and Aristotle's arguments against materialist understandings of nature, were very influential in the Middle Ages in Europe. The idea of fixed species remained dominant in biology until Darwin, and a focus upon biology is still common today in teleological criticisms of modern science.[citation needed]
It was theStoicswho "developed the battery of creationist arguments broadly known under the label 'The Argument from Design'".[16]: xviiiCicero (c. 106 – c. 43 BC) reported the teleological argument of the Stoics inDe Natura Deorum(On the Nature of the Gods) Book II, which includes an early version of the watchmaker analogy, which was later developed by William Paley. He has one of the characters in the dialogue say:
When you see a sundial or a water-clock, you see that it tells the time by design and not by chance. How then can you imagine that the universe as a whole is devoid of purpose and intelligence, when it embraces everything, including these artifacts themselves and their artificers?
Another very important classical supporter of the teleological argument wasGalen, whose compendious works were one of the major sources of medical knowledge until modern times, both in Europe and the medieval Islamic world. He was not a Stoic, but like them he looked back to the Socratics and was constantly engaged in arguing against atomists such as the Epicureans. Unlike Aristotle (who was however a major influence upon him), and unlike the Neoplatonists, he believed there was really evidence for something literally like the "demiurge" found in Plato'sTimaeus, which worked physical upon nature. In works such as hisOn the Usefulness of Partshe explained evidence for it in the complexity of animal construction. His work shows "early signs of contact and contrast between the pagan and the Judaeo-Christian tradition of creation", criticizing the account found in the Bible. "Moses, he suggests, would have contented himself with saying that God ordered the eyelashes not to grow and that they obeyed. In contrast to this, the Platonic tradition's Demiurge is above all else a technician." Surprisingly, neither Aristotle nor Plato, but Xenophon are considered by Galen, as the best writer on this subject. Galen shared with Xenophon a scepticism of the value of books about most speculative philosophy, except for inquiries such as whether there is "something in the world superior in power and wisdom to man". This he saw as having an everyday importance, a usefulness for living well. He also asserted that Xenophon was the author who reported the real position of Socrates, including his aloofness from many types of speculative science and philosophy.[36]
Galen's connection of the teleological argument to discussions about the complexity of living things, and his insistence that this is possible for a practical scientist, foreshadows some aspects of modern uses of the teleological argument.[citation needed]
As an appeal togeneral revelation,Paul the Apostle(AD 5–67), argues inRomans1:18–20,[37]that because it has been made plain to all from what has been created in the world, it is obvious that there is a God.[38]
Marcus Minucius Felix(c. late 2nd to 3rd century), an Early Christian writer, argued for the existence of God based on the analogy of an ordered house in hisThe Orders of Minucius Felix: "Supposing you went into a house and found everything neat, orderly and well-kept, surely you would assume it had a master, and one much better than the good things, his belongings; so in this house of the universe, when throughout heaven and earth you see the marks of foresight, order and law, may you not assume that the lord and author of the universe is fairer than the stars themselves or than any portions of the entire world ?"[39]
Augustine of Hippo(AD 354–430) inThe City of Godmentioned the idea that the world's "well-ordered changes and movements", and "the fair appearance of all visible things" was evidence for the world being created, and "that it could not have been created save by God".[40]
Early Islamic philosophy played an important role in developing the philosophical understandings of God among Jewish and Christian thinkers in the Middle Ages, but concerning the teleological argument one of the lasting effects of this tradition came from its discussions of the difficulties which this type of proof has. Various forms of the argument from design have been used by Islamic theologians and philosophers from the time of the earlyMutakallimuntheologians in the 9th century, although it is rejected by fundamentalist or literalist schools, for whom the mention of God in theQu'ranshould be sufficient evidence. The argument from design was also seen as an unconvincing sophism by the early Islamic philosopherAl-Farabi, who instead took the "emanationist" approach of theNeoplatonistssuch as Plotinus, whereby nature is rationally ordered, but God is not like a craftsman who literally manages the world. Later,Avicennawas also convinced of this, and proposed instead a cosmological argument for the existence of God.[41]
The argument was however later accepted by both the Aristotelian philosopherAverroes(Ibn Rushd) and his great anti-philosophy opponentAl-Ghazali. Averroes' term for the argument wasDalīl al-ˁināya, which can be translated as "argument from providence". Both of them however accepted the argumentbecausethey believed it is explicitly mentioned in the Quran.[42]Despite this, like Aristotle, the Neoplatonists, and Al-Farabi, Averroes proposed that order and continual motion in the world is caused by God's intellect. Whether Averroes was an "emanationist" like his predecessors has been a subject of disagreement and uncertainty. But it is generally agreed that what he adapted from those traditions, agreed with them about the fact that God does not create in the same way as a craftsman.[43][44]
In fact then, Averroes treated the teleological argument as one of two "religious" arguments for the existence of God. The principal demonstrative proof is, according to Averroes, Aristotle's proof from motion in the universe that there must be a first mover which causes everything else to move.[45]Averroes' position that the most logically valid proof should be physical rather than metaphysical (because then metaphysics would be proving itself) was in conscious opposition to the position of Avicenna. Later Jewish and Christian philosophers such asThomas Aquinaswere aware of this debate, and generally took a position closer to Avicenna.
An example of the teleological argument inJewish philosophyappears when the medieval Aristotelian philosopherMaimonidescites the passage inIsaiah40:26, where the "Holy One" says: "Lift up your eyes on high, and behold who hath created these things, that bringeth out their host by number:"[46]However, Barry Holtz calls this "a crude form of the argument from design", and that this "is only one possible way of reading the text". He asserts that "Generally, in the biblical texts the existence of God is taken for granted."[47]
Maimonides also recalled thatAbraham(in themidrash, or explanatory text, ofGenesis Rabbah39:1) recognized the existence of "one transcendent deity from the fact that the world around him exhibits an order and design".[48]The midrash makes an analogy between the obviousness that a building has an owner, and that the world is looked after by God. Abraham says "Is it conceivable that the world is without a guide?"[49]Because of these examples, the 19th century philosopherNachman Krochmalcalled the argument from design "a cardinal principle of the Jewish faith".[48]
The American orthodox rabbi,Aryeh Kaplan, retells a legend about the 2nd century ADRabbi Meir. When told by a philosopher that he did not believe that the world was created by God, the rabbi produced a beautiful poem that he claimed had come into being when a cat accidentally knocked over a pot of ink, "spilling ink all over the document. This poem was the result." The philosopher exclaims that would be impossible: "There must be an author. There must be a scribe." The rabbi concludes, "How could the universe ... come into being by itself? There must be an Author. There must be a Creator."[50]
Thomas Aquinas (1225–1274), whose writings became widely accepted within Catholic western Europe, was heavily influenced by Aristotle, Averroes, and other Islamic and Jewish philosophers. He presented a teleological argument in hisSumma Theologica. In the work, Aquinas presented five ways in which he attempted to prove the existence of God: thequinque viae. These arguments feature onlya posterioriarguments, rather than literal reading of holy texts.[51]He sums up his teleological argument as follows:
The fifth way is taken from the governance of the world. We see that things which lack knowledge, such as natural bodies, act for an end, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result. Hence it is plain that they achieve their end, not fortuitously, but designedly. Now whatever lacks knowledge cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is directed by the archer. Therefore, some intelligent being exists by whom all natural things are directed to their end; and this being we call God.
Aquinas notes that the existence offinal causes, by which a cause is directed toward an effect, can only be explained by an appeal to intelligence. However, as natural bodies aside from humans do not possess intelligence, there must, he reasons, exist a being that directs final causes at every moment. That being is what we call God.[52]
Isaac Newtonaffirmed his belief in the truth of the argument when, in 1713, he wrote these words in an appendix to the second edition of hisPrincipia:
This most elegant system of the sun, planets, and comets could not have arisen without the design and dominion of an intelligent and powerful being.[53]
This view, that "God is known from his works", was supported and popularized by Newton's friendsRichard Bentley,Samuel ClarkeandWilliam Whistonin theBoyle lectures, which Newton supervised.[54]Newton wrote to Bentley, just before Bentley delivered the first lecture, that:
when I wrote my treatise about our Systeme I had an eye upon such Principles as might work with considering men for the beliefe [sic] of a Deity, and nothing can rejoice me more than to find it useful for that purpose.[55]
The German philosopherGottfried Leibnizdisagreed with Newton's view of design in the teleological argument. In theLeibniz–Clarke correspondence, Samuel Clarke argued Newton's case that God constantly intervenes in the world to keep His design adjusted, while Leibniz thought that the universe was created in such a way that God would not need to intervene at all. As quoted by Ayval Leshem, Leibniz wrote:
According to [Newton's] doctrine, God Almighty wants [i.e. needs] to wind up his watch from time to time; otherwise it would cease to move. He had not it seems, sufficient foresight to make it a perpetual motion[56]
Leibniz considered the argument from design to have "only moral certainty" unless it was supported by his own idea ofpre-established harmonyexpounded in hisMonadology.[57]Bertrand Russellwrote that "The proof from the pre-established harmony is a particular form of the so-called physico-theological proof, otherwise known as the argument from design." According to Leibniz, the universe is completely made from individual substances known asmonads, programmed to act in a predetermined way.[58]Russell wrote:
In Leibniz's form, the argument states that the harmony of all the monads can only have arisen from a common cause. That they should all exactly synchronize, can only be explained by a Creator who pre-determined their synchronism.[59]
The 17th-centuryDutchwritersLessiusandGrotiusargued that the intricate structure of the world, like that of a house, was unlikely to have arisen by chance.[60]The empiricistJohn Locke, writing in the late 17th century, developed the Aristotelian idea that, excluding geometry, all science must attain its knowledgea posteriori—through sensual experience.[61]In response to Locke, Anglican Irish BishopGeorge Berkeleyadvanced a form ofidealismin which things only continue to exist when they are perceived.[62]When humans do not perceive objects, they continue to exist because God is perceiving them. Therefore, in order for objects to remain in existence, God must exist omnipresently.[63]
David Hume, in the mid-18th century, referred to the teleological argument in hisA Treatise of Human Nature. Here, he appears to give his support to the argument from design. John Wright notes that "Indeed, he claims that the whole thrust of his analysis of causality in the Treatise supports the Design argument", and that, according to Hume, "we are obliged 'to infer an infinitely perfect Architect.'"[64]
However, later he was more critical of the argument in hisAn Enquiry Concerning Human Understanding. This was presented as a dialogue between Hume and "a friend who loves sceptical paradoxes", where the friend gives a version of the argument by saying of its proponents, they "paint in the most magnificent colours the order, beauty, and wise arrangement of the universe; and then ask if such a glorious display of intelligence could come from a random coming together of atoms, or if chance could produce something that the greatest genius can never sufficiently admire".[65]
Hume also presented arguments both for and against the teleological argument in hisDialogues Concerning Natural Religion. The character Cleanthes, summarizing the teleological argument, likens the universe to a man-made machine, and concludes by the principle of similar effects and similar causes that it must have a designing intelligence:
Look round the world: contemplate the whole and every part of it: You will find it to be nothing but one great-machine, subdivided into an infinite number of lesser machines, which again admit of subdivisions to a degree beyond what human senses and faculties can trace and explain. All these various machines, and even their most minute parts, are adjusted to each other with an accuracy, which ravishes into admiration all men who have ever contemplated them. The curious adapting of means to ends, throughout all nature, resembles exactly, though it much exceeds, the productions of human contrivance; of human design, thought, wisdom, and intelligence. Since therefore the effects resemble each other, we are led to infer, by all the rules of analogy, that the causes also resemble; and that the Author of Nature is somewhat similar to the mind of man; though possessed of much larger faculties, proportioned to the grandeur of the work which he has executed. By this argumenta posteriori,and by this argument alone, do we prove at once the existence of a Deity, and his similarity to human mind and intelligence.[66]
On the other hand, Hume's sceptic, Philo, is not satisfied with the argument from design. He attempts a number of refutations, including one that arguably foreshadows Darwin's theory, and makes the point that if God resembles a human designer, then assuming divine characteristics such as omnipotence and omniscience is not justified. He goes on to joke that far from being the perfect creation of a perfect designer, this universe may be "only the first rude essay of some infant deity... the object of derision to his superiors".[66]
Starting in 1696 with hisArtificial Clockmaker,William Derhampublished a stream of teleological books. The best known of these arePhysico-Theology(1713);Astro-Theology(1714); andChristo-Theology(1730).Physico-Theology, for example, was explicitly subtitled "A demonstration of the being and attributes of God from his works of creation". Anatural theologian, Derham listed scientific observations of the many variations in nature, and proposed that these proved "the unreasonableness of infidelity". At the end of the section on Gravity for instance, he writes: "What else can be concluded, but that all was made with manifest Design, and that all the whole Structure is the Work of some intelligent Being; some Artist, of Power and Skill equivalent to such a Work?"[67]Also, of the "sense of sound" he writes:[68]
For who but an intelligent Being, what less than an omnipotent and infinitely wise God could contrive, and make such a fine Body, such a Medium, so susceptible of every Impression, that the Sense of Hearing hath occasion for, to empower all Animals to express their Sense and Meaning to others.
Derham concludes: "For it is a Sign a Man is a wilful, perverse Atheist, that will impute so glorious a Work, as the Creation is, to any Thing, yea, a mere Nothing (as Chance is) rather than to God.[69]Weber (2000) writes that Derham'sPhysico-Theology"directly influenced" William Paley's later work.[70]
The power, and yet the limitations, of this kind of reasoning is illustrated in microcosm by the history ofLa Fontaine'sfable ofThe Acorn and the Pumpkin, which first appeared in France in 1679. The light-hearted anecdote of how a doubting peasant is finally convinced of the wisdom behind creation arguably undermines this approach.[71]However, beginning withAnne Finch's conversion of the story into a polemic against atheism, it has been taken up by a succession of moral writers as presenting a valid argument for the proposition that "The wisdom of God is displayed in creation."[72]
Thewatchmaker analogy, framing the teleological argument with reference to a timepiece, dates at least back to the Stoics, who were reported by Cicero in hisDe Natura Deorum(II.88), using such an argument againstEpicureans, whom, they taunt, would "think more highly of the achievement ofArchimedesin making a model of the revolutions of the firmament than of that of nature in creating them, although the perfection of the original shows a craftsmanship many times as great as does the counterfeit".[73]It was also used byRobert Hooke[74]andVoltaire, the latter of whom remarked:[75][76]
L'univers m'embarrasse, et je ne puis songerQue cette horloge existe, et n'ait point d'horloger
The Universe troubles me, and much less can I thinkThat this clock exists and should have no clockmaker.
William Paleypresented his version of the watchmaker analogy at the start of hisNatural Theology(1802).[77]
[S]uppose I found a watch upon the ground, and it should be inquired how the watch happened to be in that place, I should hardly think...that, for anything I knew, the watch might have always been there. Yet why should not this answer serve for the watch as well as for [a] stone [that happened to be lying on the ground]?... For this reason, and for no other; namely, that, if the different parts had been differently shaped from what they are, if a different size from what they are, or placed after any other manner, or in any order than that in which they are placed, either no motion at all would have been carried on in the machine, or none which would have answered the use that is now served by it.
According toAlister McGrath, Paley argued that "The same complexity and utility evident in the design and functioning of a watch can also be discerned in the natural world. Each feature of a biological organism, like that of a watch, showed evidence of being designed in such a way as to adapt the organism to survival within its environment. Complexity and utility are observed; the conclusion that they were designed and constructed by God, Paley holds, is as natural as it is correct."[78]
Natural theology strongly influenced British science, with the expectation as expressed byAdam Sedgwickin 1831 that truths revealed by science could not conflict with the moral truths of religion.[79]These natural philosophers saw God as the first cause, and sought secondary causes to explain design in nature: the leading figure SirJohn Herschelwrote in 1836 that by analogy with otherintermediate causes"the origination of fresh species, could it ever come under our cognizance, would be found to be a natural in contradistinction to a miraculous process".[80][81]
As a theology student,Charles Darwinfound Paley's arguments compelling. However, he later developed his theory ofevolutionin his 1859 bookOn the Origin of Species, which offers an alternate explanation of biological order. In his autobiography, Darwin wrote that "The old argument of design in nature, as given by Paley, which formerly seemed to me so conclusive, fails, now that the law of natural selection has been discovered".[82]Darwin struggled with theproblem of eviland of suffering in nature, but remained inclined to believe that nature depended upon "designed laws" and commendedAsa Gray's statement about "Darwin's great service to Natural Science in bringing back to it Teleology: so that, instead of Morphology versus Teleology, we shall have Morphology wedded to Teleology."[83]
Darwin owned he was "bewildered" on the subject, but was "inclined to look at everything as resulting from designed laws, with the details, whether good or bad, left to the working out of what we may call chance:"[84]
But I own that I cannot see, as plainly as others do, & as I shd wish to do, evidence of design & beneficence on all sides of us. There seems to me too much misery in the world. I cannot persuade myself that a beneficent & omnipotent God would have designedly created the Ichneumonidae with the express intention of their feeding within the living bodies of caterpillars, or that a cat should play with mice. Not believing this, I see no necessity in the belief that the eye was expressly designed.
In 1928 and 1930,F. R. Tennantpublished hisPhilosophical Theology, which was a "bold endeavour to combine scientific and theological thinking".[85]He proposed a version of the teleological argument based on the accumulation of the probabilities of each individualbiological adaptation. "Tennant concedes that naturalistic accounts such as evolutionary theory may explain each of the individual adaptations he cites, but he insists that in this case the whole exceeds the sum of its parts: naturalism can explain each adaptation but not their totality."[86]TheRoutledge Encyclopedia of Philosophynotes that "Critics have insisted on focusing on the cogency of each piece of theistic evidence – reminding us that, in the end, ten leaky buckets hold no more water than one." Also, "Some critics, such asJohn Hickand D.H. Mellor, have objected to Tennant's particular use of probability theory and have challenged the relevance of any kind of probabilistic reasoning to theistic belief."[86]
Richard Swinburne's "contributions to philosophical theology have sought to apply more sophisticated versions of probability theory to the question of God's existence, a methodological improvement on Tennant's work but squarely in the same spirit".[86]He usesBayesian probability"taking account not only of the order and functioning of nature but also of the 'fit' between human intelligence and the universe, whereby one can understand its workings, as well as human aesthetic, moral, and religious experience".[87]Swinburne writes:[88]
[T]he existence of order in the world confirms the existence of God if and only if the existence of this order in the world is more probable if there is a God than if there is not. ... the probability of order of the right kind is very much greater if there is a God, and so that the existence of such order adds greatly to the probability that there is a God.
Swinburne acknowledges that his argument by itself may not give a reason to believe in the existence of God, but in combination with other arguments such ascosmological argumentsand evidence frommystical experience, he thinks it can.
While discussing Hume's arguments,Alvin Plantingaoffered a probability version of the teleological argument in his bookGod and Other Minds:[89]
Every contingent object such that we know whether or not it was the product of intelligent design, was the product of intelligent design.The universe is a contingent object.So probably the universe is designed.
Following Plantinga, Georges Dicker produced a slightly different version in his book aboutGeorge Berkeley:[90]
A. The world ... shows amazing teleological order.B. All Objects exhibiting such order ... are products of intelligent design.C. Probably the world is a result of intelligent design.D. Probably, God exists and created the world.
TheEncyclopædia Britannicahas the following criticism of such arguments:[87]
It can of course be said that any form in which the universe might be is statistically enormously improbable as it is only one of a virtual infinity of possible forms. But its actual form is no more improbable, in this sense, than innumerable others. It is only the fact that humans are part of it that makes it seem so special, requiring a transcendent explanation.
A modern variation of the teleological argument is built upon the concept of thefine-tuned universe: According to the websiteBiologos:[91]
Fine-tuning refers to the surprising precision of nature's physical constants, and the beginning state of the Universe. To explain the present state of the universe, even the best scientific theories require that the physical constants of nature and the beginning state of the Universe have extremely precise values.
Also, the fine-tuning of the Universe is the apparent delicate balance of conditions necessary for human life. In this view, speculation about a vast range of possible conditions in which life cannot exist is used to explore the probability of conditions in which life can and does exist. For example, it can be argued that if the force of theBig Bangexplosion had been different by 1/10 to the sixtieth power or thestrong interaction forcewas only 5% different, life would be impossible.[92]Noted physicistStephen Hawkingestimates that "if the rate of the universe's expansion one second after the Big Bang had been smaller by even one part in a hundred thousand million million, the universe would have re-collapsed into a hot fireball due to gravitational attraction".[93]In terms of a teleological argument, the intuition in relation to a fine-tuned universe would be that God must have been responsible, if achieving such perfect conditions is so improbable.[91][92]However, in regard to fine-tuning,Kenneth Einar Himmawrites: "The mere fact that it is enormously improbable that an event occurred... by itself, gives us no reason to think that it occurred by design ... As intuitively tempting as it may be..."[92]Himma attributes the "Argument from Suspicious Improbabilities", a formalization of "the fine-tuning intuition" toGeorge N. Schlesinger:
To understand Schlesinger's argument, consider your reaction to two different events. If John wins a 1-in-1,000,000,000 lottery game, you would not immediately be tempted to think that John (or someone acting on his behalf) cheated. If, however, John won three consecutive 1-in-1,000 lotteries, you would immediately be tempted to think that John (or someone acting on his behalf) cheated. Schlesinger believes that the intuitive reaction to these two scenarios is epistemically justified. The structure of the latter event is such that it... justifies a belief that intelligent design is the cause... Despite the fact that the probability of winning three consecutive 1-in-1,000 games is exactly the same as the probability of winning one 1-in-1,000,000,000 game, the former event... warrants an inference of intelligent design.
Himma considers Schlesinger's argument to be subject to the same vulnerabilities he noted in other versions of the design argument:[92]
While Schlesinger is undoubtedly correct in thinking that we are justified in suspecting design in the case [of winning] three consecutive lotteries, it is because—and only because—we know two related empirical facts about such events. First, we already know that there exist intelligent agents who have the right motivations and causal abilities to deliberately bring about such events. Second, we know from past experience with such events that they are usually explained by the deliberate agency of one or more of these agents. Without at least one of these two pieces of information, we are not obviously justified in seeing design in such cases…[T]he problem for the fine-tuning argument is that we lack both of the pieces that are needed to justify an inference of design. First, the very point of the argument is to establish the fact that there exists an intelligent agency that has the right causal abilities and motivations to bring the existence of a universe capable of sustaining life. Second, and more obviously, we do not have any past experience with the genesis of worlds and are hence not in a position to know whether the existence of fine-tuned universes are usually explained by the deliberate agency of some intelligent agency. Because we lack this essential background information, we are not justified in inferring that there exists an intelligent Deity who deliberately created a universe capable of sustaining life.
Antony Flew, who spent most of his life as an atheist, converted todeismlate in life, and postulated "an intelligent being as involved in some way in the design of conditions that would allow life to arise and evolve".[94]He concluded that the fine-tuning of the universe was too precise to be the result of chance, so accepted the existence of God. He said that his commitment to "go where the evidence leads" meant that he ended up accepting the existence of God.[95]Flew proposed the view, held earlier byFred Hoyle, that the universe is too young for life to have developed purely by chance and that, therefore, an intelligent being must exist which was involved in designing the conditions required for life to evolve.[94]
Would you not say to yourself, "Some super-calculating intellect must have designed the properties of the carbon atom, otherwise the chance of my finding such an atom through the blind forces of nature would be utterly minuscule." Of course you would ... A common sense interpretation of the facts suggests that a superintellect has monkeyed with physics, as well as with chemistry and biology, and that there are no blind forces worth speaking about in nature. The numbers one calculates from the facts seem to me so overwhelming as to put this conclusion almost beyond question.[96]
Robin Collinsargues that the universe is fine-tuned for scientific discoverability, and that this fine-tuning cannot be explained by the multiverse hypothesis.[97]According to Collins, the universe's laws, fundamental parameters, and initial conditions must be just right for the universe to be as discoverable as ours. According to Collins, examples of fine-tuning for discoverability include:
A version of the argument from design is central to bothcreation scienceandintelligent design,[12]but unlike Paley's openness todeisticdesign through God-given laws, proponents seek scientific confirmation of repeated miraculous interventions in the history of life, and argue that theirtheistic scienceshould be taught in science classrooms.[98]
The teaching ofevolutionwas effectively barred from United States public school curricula by the outcome of the 1925Scopes Trial, but in the 1960s theNational Defense Education Actled to theBiological Sciences Curriculum Studyreintroducing the teaching of evolution. In response, there was a resurgence ofcreationism, now presented as "creation science", based on biblical literalism but with Bible quotes optional. ("Explicit references to the Bible were optional: Morris's 1974 bookScientific Creationismcame in two versions, one with Bible quotes, and one without.")[12]
A 1989 survey found that virtually all literature promoting creation science presented the design argument, withJohn D. Morrissaying "any living thing gives such strong evidence for design by an intelligent designer that only a willful ignorance of the data (II Peter 3:5) could lead one to assign such intricacy to chance". Such publications introduced concepts central to intelligent design, includingirreducible complexity(a variant of the watchmaker analogy) andspecified complexity(closely resembling a fine-tuning argument). TheUnited States Supreme Courtruling onEdwards v. Aguillardbarred the teaching of "Creation Science" in public schools because it breached theseparation of church and state, and a group of creationists rebranded Creation Science as "intelligent design" which was presented as a scientific theory rather than as a religious argument.[12]
Scientists disagreed with the assertion that intelligent design is scientific, and its introduction into the science curriculum of aPennsylvaniaschool district led to the 2005Kitzmiller v. Dover Area School Districttrial, which ruled that the "intelligent design" arguments are essentially religious in nature and not science.[99]The court took evidence from theologianJohn F. Haught, and ruled that "ID is not a new scientificargument, but is rather an old religious argument for the existence of God. He traced this argument back to at least Thomas Aquinas in the 13th century, who framed the argument as a syllogism: Wherever complex design exists, there must have been a designer; nature is complex; therefore nature must have had an intelligent designer." "This argument for the existence of God was advanced early in the 19th century by Reverend Paley": "The only apparent difference between the argument made by Paley and the argument for ID, as expressed by defense expert witnesses Behe and Minnich, is that ID's 'official position' does not acknowledge that the designer is God."[100]
Proponents of theintelligent design movementsuch as Cornelius G. Hunter, have asserted that the methodologicalnaturalismupon which science is based is religious in nature.[101]They commonly refer to it as 'scientific materialism' or as 'methodological materialism' and conflate it with 'metaphysical naturalism'.[102]They use this assertion to support their claim that modern science is atheistic, and contrast it with their preferred approach of a revivednatural philosophywhich welcomes supernatural explanations for natural phenomena and supportstheistic science. This ignores the distinction between science and religion, established in Ancient Greece, in which science can not use supernatural explanations.[99]
Intelligent design advocate andbiochemistMichael Beheproposed a development of Paley's watch analogy in which he argued in favour of intelligent design. Unlike Paley, Behe only attempts to prove the existence of an intelligent designer, rather than the God ofclassical theism. Behe uses the analogy of a mousetrap to proposeirreducible complexity: he argues that if a mousetrap loses just one of its parts, it can no longer function as a mousetrap. He argues that irreducible complexity in an object guarantees the presence of intelligent design. Behe claims that there are instances of irreducible complexity in the natural world and that parts of the world must have been designed.[103]This negative argument against step by step evolution ignores longstanding evidence that evolution proceeds throughchanges of functionfrom preceding systems. The specific examples Behe proposes have been shown to have simplerhomologueswhich could act as precursors with different functions. His arguments have been rebutted, both in general and in specific cases by numerous scientific papers.[citation needed][examples needed]In response, Behe and others, "ironically, given the absence of any detail in their own explanation, complain that the proffered explanations lack sufficient detail to be empirically tested".[12]
William Lane Craighas proposed a nominalist argument influenced by thephilosophy of mathematics. This argument revolves around the fact that, by using mathematical concepts, we can discover much about the natural world. For example, Craig writes,Peter Higgs, and any similar scientist, "can sit down at his desk and, by pouring [sic] over mathematical equations, predict the existence of a fundamental particle which, thirty years later, after investing millions of dollars and thousands of man-hours, experimentalists are finally able to detect." He names mathematics the 'language of nature', and refutes two possible explanations for this. Firstly, he suggests, the idea that they are abstract entities brings about the question of their application. Secondly, he responds to the problem of whether they are merely useful fictions by suggesting that this asks why these fictions are so useful. CitingEugene Wigneras an influence on his thought, he summarizes his argument as follows:[104][105][106]
1. If God did not exist, the applicability of mathematics would be just a happy coincidence.2. The applicability of mathematics is not just a happy coincidence.3. Therefore, God exists.
University of ChicagogeneticistJames A. Shapiro, writing in theBoston Review, states that advancements in genetics and molecular biology, and "the growing realization that cells have molecular computing networks which process information about internal operations and about the external environment to make decisions controlling growth, movement, and differentiation", have implications for the teleological argument. Shapiro states that these "natural genetic engineering" systems, can produce radical reorganizations of the "genetic apparatus within a single cell generation".[107]Shapiro suggests what he calls a 'Third Way'; a non-creationist, non-Darwinian type of evolution:
What significance does an emerging interface between biology and information science hold for thinking about evolution? It opens up the possibility of addressing scientifically rather than ideologically the central issue so hotly contested by fundamentalists on both sides of the Creationist-Darwinist debate: Is there any guiding intelligence at work in the origin of species displaying exquisite adaptations…[107]
In his book,Evolution: A View from the 21st Century, Shapiro refers to this concept of "natural genetic engineering", which he says, has proved troublesome, because many scientists feel that it supports the intelligent design argument. He suggests that "function-oriented capacities [can] be attributed to cells", even though this is "the kind of teleological thinking that scientists have been taught to avoid at all costs".[108]
The metaphysical theologianNorris Clarkeshared an argument to his fellow professors atFordham Universitythat was popularised byPeter Kreeftin his "Twenty Arguments for the Existence of God" (1994). The argument states that as components are ordered universally in relation to one another, and are defined by these connections (for example, every two hydrogen atoms are ordered to form a compound with one oxygen atom.) Therefore, none of the parts are self-sufficient, and cannot be explained individually. However, the whole cannot be explained either because it is composed of separate beings and is not a whole. From here, three conclusions can be found: firstly, as the system cannot in any way explain itself, it requires an efficient cause. Secondly, it must be an intelligent mind because the unity transcends every part, and thus must have been conceived as an idea, because, by definition, only an idea can hold together elements without destroying or fusing their distinctness. An idea cannot exist without a creator, so there must be an intelligent mind. Thirdly, the creative mind must be transcendent, because if it were not, it would rely upon the system of space and time, despite having created it. Such an idea is absurd. As a conclusion, therefore, the universe relies upon a transcendent creative mind.[109]
The original development of the argument from design was in reaction to atomistic, explicitly non-teleological understandings of nature. Socrates, as reported by Plato and Xenophon, was reacting to such natural philosophers. While less has survived from the debates of the Hellenistic and Roman eras, it is clear from sources such asCiceroandLucretius, that debate continued for generations, and several of the striking metaphors used still today, such as the unseen watchmaker, and theinfinite monkey theorem, have their roots in this period. While the Stoics became the most well-known proponents of the argument from design, the atomistic counter arguments were refined most famously by theEpicureans. On the one hand, they criticized the supposed evidence for intelligent design, and the logic of the Stoics. On the defensive side, they were faced with the challenge of explaining how un-directed chance can cause something which appears to be a rational order. Much of this defence revolved around arguments such as the infinite monkey metaphor. Democritus had already apparently used such arguments at the time of Socrates, saying that there will be infinite planets, and only some having an order like the planet we know. But the Epicureans refined this argument, by proposing that the actual number of types of atoms in nature is small, not infinite, making it less coincidental that after a long period of time, certain orderly outcomes will result.[16]
These were not the only positions held in classical times. A more complex position also continued to be held by some schools, such as the Neoplatonists, who, like Plato and Aristotle, insisted that Nature did indeed have a rational order, but were concerned about how to describe the way in which this rational order is caused. According to Plotinus for example, Plato's metaphor of a craftsman should be seen only as a metaphor, and Plato should be understood as agreeing with Aristotle that the rational order in nature works through a form of causation unlike everyday causation. In fact, according to this proposal each thing already has its own nature, fitting into a rational order, whereby the thing itself is "in need of, and directed towards, what is higher or better".[110]
Louis Loeb writes thatDavid Hume, in hisEnquiry, "insists that inductive inference cannot justify belief in extended objects". Loeb also quotes Hume as writing:
It is only when two species of objects are found to be constantly conjoined, that we can infer the one from the other.…If experience and observation and analogy be, indeed, the only guides which we can reasonably follow in inference of this nature; both the effect and cause must bear a similarity and resemblance to other effects and causes...which we have found, in many instances, to be conjoined with another.…[The proponents of the argument] always suppose the universe, an effect quite singular and unparalleled, to be the proof of a Deity, a cause no less singular and unparalleled.
Loeb notes that "we observe neither God nor other universes, and hence no conjunction involving them. There is no observed conjunction to ground an inference either to extended objects or to God, as unobserved causes."[111]
Hume also presented a criticism of the argument in hisDialogues Concerning Natural Religion. The characterPhilo, a religious sceptic, voices Hume's criticisms of the argument. He argues that the design argument is built upon a faulty analogy as, unlike with man-made objects, we have not witnessed the design of a universe, so do not know whether the universe was the result of design. Moreover, the size of the universe makes the analogy problematic: although our experience of the universe is of order, there may be chaos in other parts of the universe.[112]Philo argues:
A very small part of this great system, during a very short time, is very imperfectly discovered to us; and do we thence pronounce decisively concerning the origin of the whole?
Philo also proposes that the order in nature may be due to nature alone. If nature contains a principle of order within it, the need for a designer is removed. Philo argues that even if the universe is indeed designed, it is unreasonable to justify the conclusion that the designer must be an omnipotent, omniscient, benevolent God – the God of classical theism.[112]It is impossible, he argues, to infer the perfect nature of a creator from the nature of its creation. Philo argues that the designer may have been defective or otherwise imperfect, suggesting that the universe may have been a poor first attempt at design.[113]Hume also pointed out that the argument does not necessarily lead to the existence of one God: “why may not several deities combine in contriving and framing the world?” (p. 108).[66]
Wesley C. Salmondeveloped Hume's insights, arguing that all things in the universe which exhibit order are, to our knowledge, created by material, imperfect, finite beings or forces. He also argued that there are no known instances of an immaterial, perfect, infinite being creating anything. Using the probability calculus ofBayes Theorem, Salmon concludes that it is very improbable that the universe was created by the type of intelligent being theists argue for.[114]
Nancy Cartwrightaccuses Salmon ofbegging the question. One piece of evidence he uses in his probabilistic argument – that atoms and molecules are not caused by design – is equivalent to the conclusion he draws, that the universe is probably not caused by design. The atoms and molecules are what the universe is made up of and whose origins are at issue. Therefore, they cannot be used as evidence against the theistic conclusion.[115]
Referring to it as thephysico-theologicalproof,Immanuel Kantdiscussed the teleological argument in hisCritique of Pure Reason. Even though he referred to it as "the oldest, clearest and most appropriate to human reason", he nevertheless rejected it, heading section VI with the words, "On the impossibility of a physico-theological proof."[116][117]In accepting some of Hume's criticisms, Kant wrote that the argument "proves at most intelligence only in the arrangement of the 'matter' of the universe, and hence the existence not of a 'Supreme Being', but of an 'Architect'". Using the argument to try to prove the existence of God required "a concealed appeal to theOntological argument".[118]
In hisTraité de métaphysiqueVoltaireargued that, even if the argument from design could prove the existence of a powerful intelligent designer, it would not prove that this designer is God.[119]
…from this sole argument I cannot conclude anything further than that it is probable that an intelligent and superior being has skillfully prepared and fashioned the matter. I cannot conclude from that alone that this being has made matter out of nothing and that he is infinite in every sense.
Richard Dawkinsis harshly critical of intelligent design in his bookThe God Delusion.In this book, he contends that an appeal to intelligent design can provide no explanation for biology because it not onlybegs the questionof the designer's own origin but raises additional questions: an intelligent designer must itself be far more complex and difficult to explain than anything it is capable of designing.[120]He believes the chances of life arising on a planet like the Earth are many orders of magnitude less probable than most people would think, but theanthropic principleeffectively counters skepticism with regard to improbability. For example AstronomerFred Hoylesuggested that potential for life on Earth was no more probable than aBoeing 747being assembled by a hurricane from the scrapyard. Dawkins argues that a one-time event is indeed subject to improbability but once under way, natural selection itself is nothing like random chance. Furthermore, he refers to his counter argument to the argument from improbability by that same name:[120]
The argument from improbability is the big one. In the traditional guise of the argument from design, it is easily today's most popular argument offered in favour of the existence of God and it is seen, by an amazingly large number of theists, as completely and utterly convincing. It is indeed a very strong and, I suspect, unanswerable argument—but in precisely the opposite direction from the theist's intention. The argument from improbability, properly deployed, comes close to proving that God doesnotexist. My name for the statistical demonstration that God almost certainly does not exist is the Ultimate Boeing 747 gambit.
The creationist misappropriation of the argument from improbability always takes the same general form, and it doesn't make any difference…[if called] 'intelligent design' (ID). Some observed phenomenon—often a living creature or one of its more complex organs, but it could be anything from a molecule up to the universe itself—is correctly extolled as statistically improbable. Sometimes the language of information theory is used: the Darwinian is challenged to explain the source all the information in living matter, in the technical sense of information content as a measure of improbability or 'surprise value'... However statistically improbable the entity you seek to explain by invoking a designer, the designer himself has got to be at least as improbable. God is the Ultimate Boeing 747.
…The whole argument turns on the familiar question 'Who made God?'…A designer God cannot be used to explain organized complexity because any God capable of designing anything would have to be complex enough to demand the same kind of explanation in his own right. God presents an infinite regress from which he cannot help us to escape. This argument... demonstrates that God, though not technically disprovable, is very very improbable indeed.[120]
Dawkins considered the argument from improbability to be "much more powerful" than the teleological argument, or argument from design, although he sometimes implies the terms are used interchangeably. He paraphrases St. Thomas' teleological argument as follows: "Things in the world, especially living things, look as though they have been designed. Nothing that we know looks designed unless it is designed. Therefore there must have been a designer, and we call him God."[120]
PhilosopherEdward Fesercontends that Dawkins fundamentally misunderstands the teleological argument, particularly Aquinas' version, and refutes astraw man.[121][122]
Thephilosopher of biologyMichael Rusehas argued that Darwin treated the structure of organisms as if they had a purpose: "the organism-as-if-it-were-designed-by God picture was absolutely central to Darwin's thinking in 1862, as it always had been".[123]He refers to this as "the metaphor of design ... Organisms give the appearance of being designed, and thanks to Charles Darwin's discovery of natural selection we know why this is true." In his review of Ruse's book, R.J. Richards writes, "Biologists quite routinely refer to the design of organisms and their traits, but properly speaking it'sapparentdesign to which they refer – an 'as if' design."[124]Robert Foleyrefers to this as "the illusion of purpose, design, and progress". He adds, "there is no purpose in a fundamentally causative manner in evolution but that the processes of selection and adaptation give the illusion of purpose through the utter functionality and designed nature of the biological world".[125]
Richard Dawkins suggests that while biology can at first seem to be purposeful and ordered, upon closer inspection its true function becomes questionable. Dawkins rejects the claim that biology serves any designed function, claiming rather that biology only mimics such purpose. In his bookThe Blind Watchmaker, Dawkins states that animals are the most complex things in the known universe: "Biology is the study of complicated things that give the appearance of having been designed for a purpose." He argues that natural selection should suffice as an explanation of biological complexity without recourse todivine providence.[126]
However, theologianAlister McGrathhas pointed out that the fine-tuning of carbon is even responsible for nature's ability to tune itself to any degree.
[The entire biological] evolutionary process depends upon the unusual chemistry of carbon, which allows it to bond to itself, as well as other elements, creating highly complex molecules that are stable over prevailing terrestrial temperatures, and are capable of conveying genetic information (especially DNA).…Whereas it might be argued that nature creates its own fine-tuning, this can only be done if the primordial constituents of the universe are such that an evolutionary process can be initiated. The unique chemistry of carbon is the ultimate foundation of the capacity of nature to tune itself.[91][127]
Proponents ofintelligent designcreationism, such asWilliam A. Dembskiquestion the philosophical assumptions made by critics with regard to what a designer would or would not do. Dembski claims that such arguments are not merely beyond the purview of science: often they are tacitly or overtly theological while failing to provide a serious analysis of the hypothetical objective's relative merit. Some critics, such asStephen Jay Gouldsuggest that any purported 'cosmic' designer would only produce optimal designs, while there are numerous biological criticisms to demonstrate that such an ideal is manifestly untenable. Against these ideas, Dembski characterizes both Dawkins' and Gould's argument as a rhetoricalstraw man.[128]He suggests a principle ofconstrained optimizationmore realistically describes the best any designer could hope to achieve:
Not knowing the objectives of the designer, Gould was in no position to say whether the designer proposed a faulty compromise among those objectives… In criticizing design, biologists tend to place a premium on functionalities of individual organisms and see design as optimal to the degree that those individual functionalities are maximized. But higher-order designs of entire ecosystems might require lower-order designs of individual organisms to fall short of maximal function.[128]
Some theologiansoppose the usage of human reason and science in attaining knowledge of Godaltogether, asserting the primacy of faith in this endeavour.[citation needed]
The design claim can be challenged as anargument from analogy. Supporters of design suggest that natural objects and man-made objects have many similar properties, and man-made objects have a designer. Therefore, it is probable that natural objects must be designed as well. However, proponents must demonstrate that all the available evidence has been taken into account.[129]Eric Rust argues that, when speaking of familiar objects such as watches, "we have a basis to make an inference from such an object to its designer". However, the "universe is a unique and isolated case" and we have nothing to compare it with, so "we have no basis for making an inference such as we can with individual objects. ... We have no basis for applying to the whole universe what may hold of constituent elements in the universe."[130]
George H. Smith, in his bookAtheism: The Case Against God, points out what he considers to be a flaw in the argument from design:[131]
Now consider the idea that nature itself is the product of design. How could this be demonstrated? Nature…provides the basis of comparison by which we distinguish between designed objects and natural objects. We are able to infer the presence of design only to the extent that the characteristics of an object differ from natural characteristics. Therefore, to claim that nature as a whole was designed is to destroy the basis by which we differentiate between artifacts and natural objects.
The teleological argument assumes that one can infer the existence of intelligent design merely by examination, and because life is reminiscent of something a human might design, it too must have been designed. However, considering "snowflakes and crystals of certain salts", "[i]n no case do we find intelligence". "There are other ways that order and design can come about" such as by "purely physical forces."[132]
Most professional biologistssupportthemodern evolutionary synthesis, not merely as an alternative explanation for the complexity of life but a better explanation with more supporting evidence.[133]Living organisms obey the same physical laws as inanimate objects. Oververy long periods of timeself-replicating structures arose and later formedDNA.[134]
In response to such objections, Andrew Loke of Hong Kong Baptist University argues that these can be avoided by formulating the argument deductively as an argument by exclusion concerning the possible explanations for the highly ordered laws of nature rather than as an argument based on analogy, and that the objection from evolution is invalid because evolution requires the highly ordered laws of nature to be already in place.[135]
Nyaya, the Hindu school of logic, had a version of the argument from design. P.G. Patil writes that, in this view, it is not the complexity of the world from which one can infer the existence of a creator, but the fact that "the world is made up of parts". In this context, it is the Supreme Soul,Ishvara, who created all the world.
The argument is in five parts:[136]
However, other Hindu schools, such asSamkhya, deny that the existence of God can ever be proved, because such a creator can never be perceived.Krishna Mohan Banerjee, in hisDialogues on the Hindu Philosophy, has the Samkhya speaker saying, "the existence of God cannot be established because there is no proof. ... nor can it be proved by Inference, because you cannot exhibit an analogous instance."[137]
Buddhismdenies the existence of a creator god, and rejects the Nyayasyllogismfor the teleological argument as being "logically flawed". Buddhists argue that "the 'creation' of the world cannot be shown to be analogous to the creation of a human artifact, such as a pot".[138]
The 18th century German philosopherChristian Wolffonce thought thatConfuciuswas a godless man, and that "the ancient Chinese had no natural religion, since they did not know the creator of the world". However, later, Wolff changed his mind to some extent. "On Wolff's reading, Confucius's religious perspective is thus more or less the weak deistic one ofHume'sCleanthes."[139]
The Taoist writings of the 6th-century-BC philosopherLaozi(also known as Lao Tzu) have similarities with modern naturalist science. B. Schwartz notes that, inTaoism, "The processes of nature are not guided by a teleological consciousness ... the tao [dao] is not consciously providential.[140]
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Transfinite inductionis an extension ofmathematical inductiontowell-ordered sets, for example to sets ofordinal numbersorcardinal numbers. Its correctness is a theorem ofZFC.[1]
LetP(α){\displaystyle P(\alpha )}be apropertydefined for all ordinalsα{\displaystyle \alpha }. Suppose that wheneverP(β){\displaystyle P(\beta )}is true for allβ<α{\displaystyle \beta <\alpha }, thenP(α){\displaystyle P(\alpha )}is also true.[2]Then transfinite induction tells us thatP{\displaystyle P}is true for all ordinals.
Usually the proof is broken down into three cases:
All three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered alimit ordinaland then may sometimes be treated in proofs in the same case as limit ordinals.
Transfinite recursionis similar to transfinite induction; however, instead of proving that something holds for all ordinal numbers, we construct a sequence of objects, one for each ordinal.
As an example, abasisfor a (possibly infinite-dimensional)vector spacecan be created by starting with the empty set and for each ordinalα > 0choosing a vector that is not in thespanof the vectors{vβ∣β<α}{\displaystyle \{v_{\beta }\mid \beta <\alpha \}}. This process stops when no vector can be chosen.
More formally, we can state the Transfinite Recursion Theorem as follows:
Transfinite Recursion Theorem (version 1). Given a class function[3]G:V→V(whereVis theclassof all sets), there exists a uniquetransfinite sequenceF: Ord →V(where Ord is the class of all ordinals) such that
As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is the following:
Transfinite Recursion Theorem (version 2). Given a setg1, and class functionsG2,G3, there exists a unique functionF: Ord →Vsuch that
Note that we require the domains ofG2,G3to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
More generally, one can define objects by transfinite recursion on anywell-founded relationR. (Rneed not even be a set; it can be aproper class, provided it is aset-like relation; i.e. for anyx, the collection of allysuch thatyRxis a set.)
Proofs or constructions using induction and recursion often use theaxiom of choiceto produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.[4]For example, many results aboutBorel setsare proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.
The following construction of theVitali setshows one way that the axiom of choice can be used in a proof by transfinite induction:
The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. After that step, the axiom of choice is not used again.
Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify auniquevalue forAα+1, given the sequence up toα, but will specify only aconditionthatAα+1must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions ofcountablelength, the weakeraxiom of dependent choiceis sufficient. Because there are models ofZermelo–Fraenkel set theoryof interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.
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Turtle Islandis a name forEarth[1]orNorth America, used by someAmerican Indigenous peoples, as well as by someIndigenous rights activists. The name is based on acreation mythcommon to severalindigenous peoples of the Northeastern Woodlandsof North America.[2]
A number of contemporary works continue to use and/or tell the Turtle Island creation story.[2][3]
TheLenapestory of the "Great Turtle" was first recorded by Europeans between 1678 and 1680 byJasper Danckaerts.The story is sharedby otherNortheastern Woodlands tribes, notably theIroquois peoples.[2][4]
The Lenape believe that before creation there was nothing, an empty dark space. However, in this emptiness, there existed a spirit of their creator, Kishelamàkânk. Eventually in that emptiness, he fell asleep. While he slept, he dreamt of the world as we know it today, the Earth with mountains, forests, and animals. He also dreamt up man, and he saw the ceremonies man would perform. Then he woke up from his dream to the same nothingness he was living in before. Kishelamàkânk then started to create the Earth as he had dreamt it.
First, he created helper spirits, the Grandfathers of the North, East, and West, and the Grandmother of the South. Together, they created the Earth just as Kishelamàkânk had dreamt it. One of their final acts was creating a special tree. From the roots of this tree came the first man, and when the tree bent down and kissed the ground, woman sprang from it.
All the animals and humans did their jobs on the Earth, until a problem eventually arose. There was a tooth of a giant bear that could give the owner magical powers, and the humans started to fight over it. Eventually, the wars got so bad that people moved away, and made new tribes and new languages. Kishelamàkânk saw this fighting and decided to send down a spirit, Nanapush, to bring everyone back together. He went on top of a mountain and started the first Sacred Fire, which gave off a smoke that caused all the people of the world to come investigate what it was. When they all came, Nanapush created a pipe with a sumac branch and a soapstone bowl, and the creator gave him Tobacco to smoke with. Nanapush then told the people that whenever they fought with each other, to sit down and smoke tobacco in the pipe, and they would make decisions that were good for everyone.
The same bear tooth later caused a fight between two evil spirits, a giant toad and an evil snake. The toad was in charge of all the waters, and amidst the fighting he ate the tooth and the snake. The snake then proceeded to bite his side, releasing a great flood upon the Earth. Nanapush saw this destruction and began climbing a mountain to avoid the flood, all the while grabbing animals that he saw and sticking them in his sash. At the top of the mountain there was a cedar tree that he started to climb, and as he climbed he broke off limbs of the tree. When he got to the top of the tree, he pulled out his bow, played it and sang a song that made the waters stop. Nanapush then asked which animal he could put the rest of the animals on top of in the water. The turtle volunteered saying he'd float and they could all stay on him, and that's why they call the land Turtle Island.
Nanapush then decided the turtle needed to be bigger for everyone to live on, so he asked the animals if one of them would dive down into the water to get some of the old Earth. The beaver tried first, but came up dead and Nanapush had to revive him. The loon tried second, but its attempt ended with the same fate. Lastly, the muskrat tried. He stayed down the longest, and came up dead as well, but he had some Earth on his nose that Nanapush put on the Turtles back. Because of his accomplishment, Nanapush told the muskrat he was blessed and his kind would always thrive in the land.
Nanapush then took out his bow and again sang, and the turtle started to grow. It kept growing, and Nanapush sent out animals to try to get to the edge to see how long it had grown. First, he sent the bear, and the bear returned in two days saying he had reached the end. Next, he sent out the deer, who came back in two weeks saying he had reached the end. Finally, he sent the wolf, and the wolf never returned because the land had gotten so big. Lenape tradition said wolves howl to call their ancestor back home.[5]
According to theoral traditionof theHaudenosaunee (or "Iroquois"), "the earth was the thought of [a ruler of] a great island which floats in space [and] is a place of eternal peace."[6][2]Sky Womanfell down to the earth when it was covered with water, or more specifically, when there was a "great cloud sea".[1]Various animals tried to swim to the bottom of the ocean to bring back dirt to create land.Muskratsucceeded in gathering dirt,[1]which was placed on the back of aturtle. This dirt began to multiply and also caused the turtle to grow bigger. The turtle continued to grow bigger and bigger and thedirtcontinued to multiply until it became a huge expanse of land.[1][7][8]Thus, when Iroquois cultures refer to the earth, they often call itTurtle Island.[8]
According to Converse and Parker, the Iroquois faith shared with other religions the "belief that the Earth is supported by a gigantic turtle."[1]In theSeneca language, the mythical turtle is calledHah-nu-nah,[1]while the name for an everyday turtle isha-no-wa.[9]
In Susan M. Hill's version of the story, the muskrat or other animals die in their search for land for the Sky Woman (named Mature Flower in Hills's telling). This is a representation of the Haudenosaunee beliefs of death and chaos as forces of creation, as we all give our bodies to the land to become soil, which in turn continues to support life. This concept plays out again when the Mature Flower's daughter dies during childbirth, becoming the first person to be buried on the turtle's back and whose burial post helped grow various plants such ascornandstrawberries.[10]This, according to Hill, also shows how soil, and the land itself, has the ability to act and shape creation. Some tellings do not include this expanded edition as part of the Creation Story, however, these differences are important to note when considering Haudenosaunee traditions and relationships.
The name Turtle Island has been used by many Indigenous cultures in North America, and both native and non-native activists, especially since the 1970s when the term came into wider usage.[7]American author and ecologistGary Snyderuses the term to refer to North America, writing that it synthesizes both indigenous and colonizer cultures, by translating the indigenous name into the colonizer's languages (the Spanish "Isla Tortuga" being proposed as a name as well). Snyder argues that understanding North America under the name of Turtle Island will help shift conceptions of the continent.[11]Turtle Island has been used by writers and musicians, including Snyder for hisPulitzer Prize-winningbook of poetry,Turtle Island; theTurtle Island Quartetjazz string quartet; Tofurky manufacturerTurtle Island Foods; and the Turtle Island Research Cooperative inBoise, Idaho.[12][13]
TheCanadian Association of University Teachershas put into practice the acknowledgment of indigenous territory and claims, particularly at institutions located withinunceded landor covered by perpetual decrees such as theHaldimand Tract. At Canadian universities, many courses, student and academic meetings, as well as convocation and other celebrations begin with a spoken acknowledgement of the traditional Indigenous territories, sometimes including reference to Turtle Island, in which they are taking place.[3][14]
There are a number of contemporary works which continue to use and/or tell the story of the Turtle Island creation story.
Thomas King's book tells us that "the truth about stories is they're all we are."[15]King's book explores the power of story both in native lives and in the lives of every person on this planet. Every chapter opens with a telling of the story of the world on the back of a turtle in space, and in each chapter, it is slightly altered to show how stories change through tellers and audiences. Their fluidity is itself a characteristic of the story as they traverse through time.[15]
King provides us with his own telling of the story using a woman named Charm as his Sky Woman. Charm is from a different planet and is described as being curious to a fault, often asking the animals of her planet questions they deem to be too nosy. When she becomes pregnant, she develops a craving for Red Fern Root, which can only be found underneath the oldest tree. While digging for the Red Fern Root she digs so deep she makes a hole in the planet, and in her curiosity falls through all the way to earth. King tells us that this is a young Earth from before land was created, and in order to save Charm from falling hard and fast into the water and upsetting the stillness of the water, all the water birds fly up to catch her. With no land to set her on they offer her the back of the turtle. When Charm is almost ready to give birth the animals fear that the turtle will be too crowded, so she asks the animals to dive down to find mud so that she can use its magic to build dry land. Many animals try but most fail, until the otter dives down for days before finally surfacing, passed out from exhaustion, clutching mud in its paws. Charm creates land from the mud, magic, and the turtle's back and gives birth to twins which keep the earth in balance. One twin flattened out the land, created light, and created woman, while the other made valleys and mountains, shadows, and man.
King emphasizes that the Turtle Island creation story creates "a world in which creation is a shared activity...a world that begins in chaos and moves toward harmony."[15]He explains that understanding and continuing to tell this story creates a world that values these ideas and relationships with nature. Without that understanding, we fail to uphold the relationships forged by Charm, the twins, and the animals that created the earth.
Robin Wall Kimmerer's book,Braiding Sweetgrass, addresses the need for us to understand our reciprocal relationships with nature in order for us to understand and use ecology as a means to save the earth. The version of the story from Kimmerer starts off with the Sky Woman falling from a hole in the sky, cradling something tightly in her hands. Geese rise up to soften her landing and place her on the back of a turtle so that she does not drown. All the animals congregate to help find dirt for the sky woman so that she can build her habitat, some giving their lives in the search. Finally, the muskrat surfaces, dead but clutching a handful of soil for the Sky Woman, who takes the offering gratefully and uses seeds from The Tree of Life to begin her garden using her gratitude and the gifts from the animals, thus creating Turtle Island as we know it. Through the Sky Woman story, Kimmerer tells us that we cannot "begin to move toward ecological and cultural sustainability if we cannot even imagine what the path feels like."[16]
Christopher B. Teuton book provides a comprehensive look into Cherokee oral traditions and art to bring them into the contemporary moment. He put together his collection with three friends, also master storytellers, who get together to swap stories from around the 14 Cherokee states.[17]The first chapter of the bookBeginningsstarts with a telling of the Sky Woman story. Notably, this telling of Turtle Island has the water beetle dive for the earth necessary for the sky woman, where often you will see a muskrat or otter. Turtle Island is a running theme throughout the book, as it is the beginning of life and story.
We Are Water Protectorsis a children's storybook written by Carole Lindstrom in 2020 in response to the building of theDakota Access Pipeline, represented as a large black snake in the book. The book says that water is the source of all life, and it is all of ours duty to protect our water sources so that we can preserve not only ourselves but those of animals and the environment. The story draws important meanings from the Turtle Island creation story such as water as the origin of life and closes with a drawing of the main character returning the turtle to the water saying "We are stewards of the earth. Our spirits are not to be broken."[18]
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https://en.wikipedia.org/wiki/Turtle_Island_(Native_American_folklore)
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TheWorld Turtle, also called theCosmic Turtleor theWorld-Bearing Turtle, is amythemeof a giantturtle(ortortoise) supporting or containingthe world. It occurs inHinduism,Chinese mythology, and themythologies of some of the indigenous peoples of the Americas. Thecomparative mythologyof theWorld-Tortoisediscussed byEdward Burnett Tylor(1878: 341) includes the counterpartWorld Elephant.
The World Turtle in Hinduism is known asAkūpāra(Sanskrit: अकूपार), or sometimesChukwa. An example of a reference to the World Turtle in Hindu literature is found inJñānarāja(the author ofSiddhantasundara, writing c. 1500): "A vulture, whichever has only little strength, rests in the sky holding a snake in its beak for a prahara [three hours]. Why can [the deity] in the form of a tortoise, who possesses an inconceivable potency, not hold the Earth in the sky for akalpa[billions of years]?"[1][dead link]The British philosopherJohn Lockemade reference to this in his 1689 tract,An Essay Concerning Human Understanding, which compares one who would say that properties inhere in "substance" to the Indian, who said the world was on an elephant, which was on a tortoise, "but being again pressed to know what gave support to the broad-backed tortoise, replied—something, he knew not what".[2]
Brewer's Dictionary of Phrase and Fablelists, without citation,Maha-pudma and Chukwaas names from a "popular rendition of a Hindu myth in which the tortoise Chukwa supports the elephantMaha-pudma, which in turn supports the world".[3]
In the Chinese mythology, the creator goddessNüwacut the legs off the giant sea turtleAo(simplified Chinese:鳌;traditional Chinese:鰲;pinyin:áo) and used them to prop up the sky afterGong GongdamagedMount Buzhou, which had previously supported the heavens.[4]
TheLenapecreation story of the "Great Turtle" was first recorded between 1678 and 1680 byJasper Danckaerts. The belief is shared by otherindigenous peoples of the Northeastern Woodlands, most notably those of theHaudenosanee confederacy,[5]and theAnishinaabeg.[6]
The Jesuit Relationscontain aHuronstory concerning the World Turtle:
"When the Father was explaining to them [some Huron seminarists] some circumstance of the passion of our Lord, and speaking to them of the eclipse of the Sun, and of the trembling of the earth which was felt at that time, they replied that there was talk in their own country of a greatearthquakewhich had happened in former times; but they did not know either the time or the cause of that disturbance. 'There is still talk,' (said they) 'of a very remarkable darkening of the Sun, which was supposed to have happened because the great turtle which upholds the earth, in changing its position or place, brought its shell before the Sun, and thus deprived the world of sight.'"[7]
The usilosimapundu ofZulufolklore also bears some similarities to the world turtle. It is a creature so large that it contains many countries and that one side of it experiences a different season than the other side.[8]
TheDiscworldbook series, created byTerry Pratchett, takes place on a fictional world that is a flat disc sitting on top of four elephants astride the shell of a giant turtle namedGreat A'Tuin.
In the bookMonday Begins on SaturdaybyArkady and Boris Strugatsky, a disc upon elephants on a turtle is said to have been discovered by a pupil who entered an ideal world of imagination.
In the bookItbyStephen King, Pennywise's archenemy is a giant turtle named Maturin. Maturin also appears in King'sWizard And Glass, the fourth book inThe Dark Towerseries.
In the start of the first chapter of the bookA Brief History of TimebyStephen Hawking, an old woman says, "What you have told us is rubbish. The world is really aflat plate supported on the back of a giant tortoise."[9]
The filmStrange Worldis revealed to take place on and inside a World Turtle, with the characters trying to stop an infection from killing it.[10]
In thePokémon Scarlet and Violetvideogame expansion, The Indigo Disk, the legendary PokémonTerapagoscan undergo terastallization bearing the Stellar Type. In this form, Terapagos resembles the world as the ancients saw it.
InUrusei Yatsura 2: Beautiful Dreamera giant turtle is carrying the world which is in some sort of time knot.
In the bookA Wild Sheep ChasebyHaruki Murakami, the narrator references this idea: "The "world"--the world always makes me think of a tortoise and elephants tirelessly supporting a gigantic disc."
The television seriesWhat We Do in the Shadows (TV series)references character Nandor the Relentless's belief in the World Turtle in the episode "The Casino". A B-plot of the episode involves character Colin Robinson teaching Nandor about theBig Bang Theory.
The young adult novelTurtles All the Way Downand subsequentfilm adaptationderives its name from the World Turtle and discusses it.
The television seriesIt’s Always Sunny in Philadelphiareferences this idea in the episode “Charlie Rules the World”, as Frank Reynolds, arguing with Dennis Reynolds about what is real, claims that they could be in “a turtle’s dream in outer space.”
Sturgill Simpson’s “Turtles All the Way Down” is a modern country psychedelic ballad from his 2014 album, Metamodern Sounds in Country Music. Sturgill comes to a conclusion, choosing to encourage listeners to live their life the way they please, and don’t waste their time trying to find the answers, because “it’s turtles all the way down the line.”
In "Metal Gear Solid 2: Sons of Liberty" an illustration of a World Turtle appears if the game is paused during the Arsenal Gear section.
Theregress argumentinepistemologyand theinfinite regressinphilosophyoften use the expression "turtles all the way down" to indicate an explanatory failure based on an explanation that needs a potentially infinite series of additional explanations to support it.[citation needed]
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https://en.wikipedia.org/wiki/World_Turtle
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Yertle the Turtle and Other Storiesis apicture bookcollection byTheodor Seuss Geisel, published under his more commonly knownpseudonymofDr. Seuss. It was first released byRandom HouseBooks on April 12, 1958, and is written in Seuss's trademark style, using a type ofmetercalledanapestic tetrameter. Though it contains three short stories, it is mostly known for its first story, "Yertle the Turtle", in which the eponymous Yertle, king of the pond, stands on his subjects in an attempt to reach higher than the Moon—until the bottom turtle burps and he falls into the mud, ending his rule.
Though the book included "burp", a word then considered to be relatively rude, it was a success upon publication, and has since sold more than a million copies. In 2001, it was listed at 125 on thePublishers Weeklylist of the best-selling children's books of all time.
The eponymous story revolves around Yertle theTurtle, the king of the pond (located on the faraway island of Sala-ma-sond), where all the turtles swim happily. Dissatisfied with the stone that serves as his throne (it's too small for him to rule the landscape beyond the pond), Yertle commands the other turtles to stack themselves beneath him so that he can see farther and expand his kingdom, each time marveling at what he believes he now rules. However, the stacked turtles are in pain. A turtle named Mack, who has acheckerboard-style shell and is at the bottom of the pile, is bearing the brunt of the suffering. Mack asks Yertle for a respite, but Yertle just tells him to be quiet. Yertle decides to further expand his kingdom and commands more and more turtles to add to his throne and rises above everything he sees. Mack makes a second request for a respite because the increased weight is now causing extreme pain and hunger to the turtles at the bottom of the pile. Again, Yertle yells at Mack to be quiet. Yertle then notices the moon rising above him as the night approaches. Furious, he decides to call for even more turtles in an attempt to rise above it. Before he can give the command, Mack decides he has had enough. He burps, which shakes up Yertle's throne and tosses the turtle king off the turtle stack and into the water, leaving him "King of the Mud" and allowing the others to once again swim free, "as turtles, and maybe all creatures, should be."[1]
The second story recounts the tale of the "girl-bird" Gertrude McFuzz, who only has one small, plain tail feather and envies Lolla Lee Lou, who has two feathers. She goes to her uncle, Doctor Dake, for something that will make her tail grow. He tries to tell her that her tail is just right for her species, but she throws a tantrum. He gives in, and he tells her where she can find berries that will make her tail grow. The first berry makes her tail exactly like Lolla Lee Lou's, but greed overtakes her. Now wanting to surpass Lolla Lee Lou, she eats the entire vine, causing her tail to grow to an enormous size. But the added weight of too many feathers does not allow her to fly, run, or even walk. Panicked, she yelps repeatedly, while being stuck on the hill. Her uncle, having heard her painful cries for help, sends for many other birds to carry her home and pluck out her tail feathers, which takes a few weeks, causing her to be sore. Though she has only one feather left—as before—she now has "enough, because now she is smarter".[1]
The third and final story tells of arabbitand abear, who both boast that they are the "best of the beasts", because of the range of their hearing and smelling abilities, respectively. However, they are humbled by a worm who claims he can see all around the world—right back to his own hill, where he sees the rabbit and bear, whom he calls "the two biggest fools that have ever been seen". Then the worm "dived in his hole and went back to his work".[1]
A stack of turtles drawn similarly to those featured in "Yertle the Turtle" first appeared on March 20, 1942, in a cartoon for the New York City newspaperPM, where Seuss worked as aneditorial cartoonist. The illustration shows two stacks of turtles forming the letter "V" on top of a large turtle labelled "Dawdling Producers", with a caption reading "You Can't Build A Substantial V Out of Turtles!"[2]
Seuss has stated that the titular character Yertle representedAdolf Hitler, with Yertle's despotic rule of the pond and takeover of the surrounding area parallel to Hitler'sregime in Germanyand invasion of various parts of Europe.[3][4]Though Seuss made a point of not beginning the writing of his stories with a moral in mind, stating that "kids can see a moral coming a mile off", he was not against writing about issues; he said "there's an inherent moral in any story" and remarked that he was "subversive as hell".[5][6]"Yertle the Turtle" has variously been described as "autocraticrule overturned",[7]"a reaction against thefascismof World War II",[8]and "subversive ofauthoritarianrule".[9]
All three stories inYertlewere originally published inRedbookmagazine in the early 1950s, as part of a series of stories that Dr. Seuss wrote for the magazine. These stories proved to be popular, and Geisel decided to put some of them in a book. On September 14, 1956, Geisel signed a contract with Random House for such a book, which would include the story "How Officer Pat Saved the Whole Town" and have the titleHow Officer Pat Saved the Town and Other Stories.Officer Patwas planned to be published in the autumn of 1957, but it never did get published. On December 18, 1957, theOfficer Patcontract was dissolved, and Geisel signed another contract for the publication ofYertlein 1958. The "Officer Pat" story was eventually included inHorton and the Kwuggerbug and More Lost Stories, which was published posthumously in 2014.[10]
The last lines of "Yertle the Turtle" read: "And the turtles, of course ...all the turtles are free / As turtles, and maybe, all creatures should be".[1]When questioned about why he wrote "maybe" rather than "surely", Seuss replied that he did not want to sound "didactic or like a preacher on a platform", and that he wanted the reader "to say 'surely' in their minds instead of my having to say it".[6]
The use of the word "burp"—"plain little Mack did a plain little thing.He burped!"—was also an issue before publication. According to Seuss, the publishers atRandom House, including the president, had to meet to decide whether or not they could use "burp" because "nobody had ever burped before on the pages of a children's book".[3][11]However, despite the publishers' initial worries, it eventually proved to be a hit—in 2001,Publishers Weeklyreported that it was 125th on the list of best-selling hardcover children's books in the United States, at just over one million copies.[12]
"This Book is forThe Bartletts of Norwich, Vt.and forThe Sagmasters of Cincinnati, Ohio"
The book is dedicated to the Sagmaster family as a tribute to Joseph Sagmaster, who had introduced Seuss to his first wife,Helen Palmer, when they were both attendingOxford University. Sagmaster is quoted as saying that bringing the two together was "the happiest inspiration I've ever had".[13]
AlthoughYertle the Turtle and Other Storieshas not been directly adapted, several characters from the book have appeared in other media. Yertle is a main antagonist in the first season of the 1996–1998 puppetry television seriesThe Wubbulous World of Dr. Seuss(performed byAnthony Asbury), and in Lynn Ahrens and Stephen Flaherty's Broadway musicalSeussical, Yertle serves as a judge and Gertrude McFuzz acts as Horton's love interest. The story was also turned into a dance number in the 1994 filmIn Search of Dr. Seuss.
Yertle the Turtle and Other Storiesis a 1992 animation directed by Ray Messecar and narrated byJohn Lithgow.[14]
TheRed Hot Chili Peppersadapted the story in the song "Yertle the Turtle" on their second album,Freaky Styley, released in 1985.
In 1961, RCA Camden Records released "Yertle the Turtle and Other Stories" with the three stories on the A side and "Bartholomew and the Oobleck" on the B side. The liner notes state "set to dramatic action personally by "Dr. Seuss" with music featuring Marvin Miller".[15]
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https://en.wikipedia.org/wiki/Yertle_the_Turtle_and_Other_Stories
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Instatistics, acontingency table(also known as across tabulationorcrosstab) is a type oftablein amatrixformat that displays the multivariatefrequency distributionof the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The termcontingency tablewas first used byKarl Pearsonin "On the Theory of Contingency and Its Relation to Association and Normal Correlation",[1]part of theDrapers' CompanyResearch Memoirs Biometric Series Ipublished in 1904.
A crucial problem ofmultivariate statisticsis finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of theconditional independencesare revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can useinformation theoryconcepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies.
Apivot tableis a way to create contingency tables using spreadsheet software.
Suppose there are two variables, sex (male or female) andhandedness(right- or left-handed). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male right-handed and left-handed, female right-handed and left-handed. Such a contingency table is shown below.
The numbers of the males, females, and right- and left-handed individuals are calledmarginal totals. The grand total (the total number of individuals represented in the contingency table) is the number in the bottom right corner.
The table allows users to see at a glance that the proportion of men who are right-handed is about the same as the proportion of women who are right-handed although the proportions are not identical. The strength of the association can be measured by theodds ratio, and the population odds ratio estimated by thesample odds ratio. Thesignificanceof the difference between the two proportions can be assessed with a variety of statistical tests includingPearson's chi-squared test, theG-test,Fisher's exact test,Boschloo's test, andBarnard's test, provided the entries in the table represent individuals randomly sampled from the population about which conclusions are to be drawn. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), it is said that there is acontingencybetween the two variables. In other words, the two variables arenotindependent. If there is no contingency, it is said that the two variables areindependent.
The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 × 2 contingency table. In principle, any number of rows and columns may be used. There may also be more than two variables, but higher order contingency tables are difficult to represent visually. The relation betweenordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare. For more on the use of a contingency table for the relation between two ordinal variables, seeGoodman and Kruskal's gamma.
The degree of association between the two variables can be assessed by a number of coefficients. The following subsections describe a few of them. For a more complete discussion of their uses, see the main articles linked under each subsection heading.
The simplest measure of association for a 2 × 2 contingency table is theodds ratio. Given two events, A and B, the odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the odds ratio is 1; if the odds ratio is greater than 1, the events are positively associated; if the odds ratio is less than 1, the events are negatively associated.
The odds ratio has a simple expression in terms of probabilities; given the joint probability distribution:
the odds ratio is:
A simple measure, applicable only to the case of 2 × 2 contingency tables, is thephi coefficient(φ) defined by
whereχ2is computed as inPearson's chi-squared test, andNis the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or −1 (complete association or complete inverse association), provided it is based on frequency data represented in 2 × 2 tables. Then its sign equals the sign of the product of themain diagonalelements of the table minus the product of the off–diagonal elements. φ takes on the minimum value −1.0 or the maximum value of +1.0if and only ifevery marginal proportion is equal to 0.5 (and two diagonal cells are empty).[2]
Two alternatives are thecontingency coefficientC, andCramér's V.
The formulae for theCandVcoefficients are:
kbeing the number of rows or the number of columns, whichever is less.
Csuffers from the disadvantage that it does not reach a maximum of 1.0, notably the highest it can reach in a 2 × 2 table is 0.707 . It can reach values closer to 1.0 in contingency tables with more categories; for example, it can reach a maximum of 0.870 in a 4 × 4 table. It should, therefore, not be used to compare associations in different tables if they have different numbers of categories.[3]
Ccan be adjusted so it reaches a maximum of 1.0 when there is complete association in a table of any number of rows and columns by dividingCbyk−1k{\displaystyle {\sqrt {\frac {k-1}{k}}}}wherekis the number of rows or columns, when the table is square[citation needed], or byr−1r×c−1c4{\displaystyle {\sqrt[{\scriptstyle 4}]{{r-1 \over r}\times {c-1 \over c}}}}whereris the number of rows andcis the number of columns.[4]
Another choice is thetetrachoric correlation coefficientbut it is only applicable to 2 × 2 tables.Polychoric correlationis an extension of the tetrachoric correlation to tables involving variables with more than two levels.
Tetrachoric correlation assumes that the variable underlying eachdichotomousmeasure is normally distributed.[5]The coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."[6]
The tetrachoric correlation coefficient should not be confused with thePearson correlation coefficientcomputed by assigning, say, values 0.0 and 1.0 to represent the two levels of each variable (which is mathematically equivalent to the φ coefficient).
Thelambda coefficientis a measure of the strength of association of the cross tabulations when the variables are measured at thenominal level. Values range from 0.0 (no association) to 1.0 (the maximum possible association).
Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.
Theuncertainty coefficient, or Theil's U, is another measure for variables at the nominal level. Its values range from −1.0 (100% negative association, or perfect inversion) to +1.0 (100% positive association, or perfect agreement). A value of 0.0 indicates the absence of association.
Also, the uncertainty coefficient is conditional and an asymmetrical measure of association, which can be expressed as
This asymmetrical property can lead to insights not as evident in symmetrical measures of association.[7]
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https://en.wikipedia.org/wiki/Cross-tabulation
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Instatistics, acontingency table(also known as across tabulationorcrosstab) is a type oftablein amatrixformat that displays the multivariatefrequency distributionof the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The termcontingency tablewas first used byKarl Pearsonin "On the Theory of Contingency and Its Relation to Association and Normal Correlation",[1]part of theDrapers' CompanyResearch Memoirs Biometric Series Ipublished in 1904.
A crucial problem ofmultivariate statisticsis finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of theconditional independencesare revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can useinformation theoryconcepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies.
Apivot tableis a way to create contingency tables using spreadsheet software.
Suppose there are two variables, sex (male or female) andhandedness(right- or left-handed). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male right-handed and left-handed, female right-handed and left-handed. Such a contingency table is shown below.
The numbers of the males, females, and right- and left-handed individuals are calledmarginal totals. The grand total (the total number of individuals represented in the contingency table) is the number in the bottom right corner.
The table allows users to see at a glance that the proportion of men who are right-handed is about the same as the proportion of women who are right-handed although the proportions are not identical. The strength of the association can be measured by theodds ratio, and the population odds ratio estimated by thesample odds ratio. Thesignificanceof the difference between the two proportions can be assessed with a variety of statistical tests includingPearson's chi-squared test, theG-test,Fisher's exact test,Boschloo's test, andBarnard's test, provided the entries in the table represent individuals randomly sampled from the population about which conclusions are to be drawn. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), it is said that there is acontingencybetween the two variables. In other words, the two variables arenotindependent. If there is no contingency, it is said that the two variables areindependent.
The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 × 2 contingency table. In principle, any number of rows and columns may be used. There may also be more than two variables, but higher order contingency tables are difficult to represent visually. The relation betweenordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare. For more on the use of a contingency table for the relation between two ordinal variables, seeGoodman and Kruskal's gamma.
The degree of association between the two variables can be assessed by a number of coefficients. The following subsections describe a few of them. For a more complete discussion of their uses, see the main articles linked under each subsection heading.
The simplest measure of association for a 2 × 2 contingency table is theodds ratio. Given two events, A and B, the odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the odds ratio is 1; if the odds ratio is greater than 1, the events are positively associated; if the odds ratio is less than 1, the events are negatively associated.
The odds ratio has a simple expression in terms of probabilities; given the joint probability distribution:
the odds ratio is:
A simple measure, applicable only to the case of 2 × 2 contingency tables, is thephi coefficient(φ) defined by
whereχ2is computed as inPearson's chi-squared test, andNis the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or −1 (complete association or complete inverse association), provided it is based on frequency data represented in 2 × 2 tables. Then its sign equals the sign of the product of themain diagonalelements of the table minus the product of the off–diagonal elements. φ takes on the minimum value −1.0 or the maximum value of +1.0if and only ifevery marginal proportion is equal to 0.5 (and two diagonal cells are empty).[2]
Two alternatives are thecontingency coefficientC, andCramér's V.
The formulae for theCandVcoefficients are:
kbeing the number of rows or the number of columns, whichever is less.
Csuffers from the disadvantage that it does not reach a maximum of 1.0, notably the highest it can reach in a 2 × 2 table is 0.707 . It can reach values closer to 1.0 in contingency tables with more categories; for example, it can reach a maximum of 0.870 in a 4 × 4 table. It should, therefore, not be used to compare associations in different tables if they have different numbers of categories.[3]
Ccan be adjusted so it reaches a maximum of 1.0 when there is complete association in a table of any number of rows and columns by dividingCbyk−1k{\displaystyle {\sqrt {\frac {k-1}{k}}}}wherekis the number of rows or columns, when the table is square[citation needed], or byr−1r×c−1c4{\displaystyle {\sqrt[{\scriptstyle 4}]{{r-1 \over r}\times {c-1 \over c}}}}whereris the number of rows andcis the number of columns.[4]
Another choice is thetetrachoric correlation coefficientbut it is only applicable to 2 × 2 tables.Polychoric correlationis an extension of the tetrachoric correlation to tables involving variables with more than two levels.
Tetrachoric correlation assumes that the variable underlying eachdichotomousmeasure is normally distributed.[5]The coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."[6]
The tetrachoric correlation coefficient should not be confused with thePearson correlation coefficientcomputed by assigning, say, values 0.0 and 1.0 to represent the two levels of each variable (which is mathematically equivalent to the φ coefficient).
Thelambda coefficientis a measure of the strength of association of the cross tabulations when the variables are measured at thenominal level. Values range from 0.0 (no association) to 1.0 (the maximum possible association).
Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.
Theuncertainty coefficient, or Theil's U, is another measure for variables at the nominal level. Its values range from −1.0 (100% negative association, or perfect inversion) to +1.0 (100% positive association, or perfect agreement). A value of 0.0 indicates the absence of association.
Also, the uncertainty coefficient is conditional and an asymmetrical measure of association, which can be expressed as
This asymmetrical property can lead to insights not as evident in symmetrical measures of association.[7]
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Data drilling(alsodrilldown) refers to any of various operations and transformations on tabular, relational, and multidimensional data. The term has widespread use in various contexts, but is primarily associated with specializedsoftwaredesigned specifically fordata analysis.
There are certain operations that are common to applications that allow data drilling. Among them are:
Queryoperations:
Tabular query operations consist of standard operations on data tables.
Among these operations are:
Consider the following example:
Fred and Wilma table (Fig 001):
The preceding is an example of a simple flat file table formatted as comma-separated values. The table includes first name, last name, gender and home country for various people named fred or wilma. Although the example is formatted this way, it is important to emphasize that tabular query operations (as well as all data drilling operations) can be applied to any conceivabledata type, regardless of the underlying formatting. The only requirement is that the data be readable by the software application in use.
A pivot query allows multiple representations of data according to different dimensions. This query type is similar to tabular query, except it also allows data to be represented in summary format, according to a flexible user-selectedhierarchy. This class of data drilling operation is formally, (and loosely) known by different names, includingcrosstab query,pivot table,data pilot,selective hierarchy,intertwingularityand others.
To illustrate the basics of pivot query operations, consider theFred and Wilma table (Fig 001). A quick scan of the data reveals that the table has redundant information. This redundancy could be consolidated using an outline or atree structureor in some other way. Moreover, once consolidated, the data could have many different alternate layouts.
Using a simple text outline as output, the following alternate layouts are all possible with a pivot query:
Summarize by gender (Fig 001):
Summarize by home, lname (Fig 001):
Pivot query operations are useful for summarizing a corpus of data in multiple ways, thereby illustrating different representations of the same basic information. Although this type of operation appears prominently inspreadsheetsand desktopdatabasesoftware, its flexibility is arguably under-utilized. There are many applications that allow only a 'fixed' hierarchy for representing data, and this represents a substantial limitation.
Drillupis the opposite of drilldown. For example, if you drilldown to see the revenue of one product, then you might want to drillup to see the revenue of all products.[1]
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https://en.wikipedia.org/wiki/Data_drilling
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Extract, transform, load(ETL) is a three-phasecomputingprocess where data isextractedfrom an input source,transformed(includingcleaning), andloadedinto an output data container. The data can be collected from one or more sources and it can also be output to one or more destinations. ETL processing is typically executed usingsoftware applicationsbut it can also be done manually by system operators. ETL software typically automates the entire process and can be run manually or on recurring schedules either as single jobs or aggregated into a batch of jobs.
A properly designed ETL system extracts data from source systems and enforces data type and data validity standards and ensures it conforms structurally to the requirements of the output. Some ETL systems can also deliver data in a presentation-ready format so that application developers can build applications and end users can make decisions.[1]
The ETL process is often used indata warehousing.[2]ETL systems commonly integrate data from multiple applications (systems), typically developed and supported by differentvendorsor hosted on separate computer hardware. The separate systems containing the original data are frequently managed and operated by differentstakeholders. For example, a cost accounting system may combine data from payroll, sales, and purchasing.
Data extraction involves extracting data from homogeneous or heterogeneous sources; data transformation processes data by data cleaning and transforming it into a proper storage format/structure for the purposes of querying and analysis; finally, data loading describes the insertion of data into the final target database such as anoperational data store, adata mart,data lakeor a data warehouse.[3][4]
ETL processing involves extracting the data from the source system(s). In many cases, this represents the most important aspect of ETL, since extracting data correctly sets the stage for the success of subsequent processes. Most data-warehousing projects combine data from different source systems. Each separate system may also use a different data organization and/orformat. Common data-source formats includerelational databases,flat-file databases,XML, andJSON, but may also include non-relational database structures such asIBM Information Management Systemor other data structures such asVirtual Storage Access Method (VSAM)orIndexed Sequential Access Method (ISAM), or even formats fetched from outside sources by means such as aweb crawlerordata scraping. The streaming of the extracted data source and loading on-the-fly to the destination database is another way of performing ETL when no intermediate data storage is required.
An intrinsic part of the extraction involves data validation to confirm whether the data pulled from the sources has the correct/expected values in a given domain (such as a pattern/default or list of values). If the data fails the validation rules, it is rejected entirely or in part. The rejected data is ideally reported back to the source system for further analysis to identify and to rectify incorrect records or performdata wrangling.
In thedata transformationstage, a series of rules or functions are applied to the extracted data in order to prepare it for loading into the end target.
An important function of transformation isdata cleansing, which aims to pass only "proper" data to the target. The challenge when different systems interact is in the relevant systems' interfacing and communicating. Character sets that may be available in one system may not be in others.
In other cases, one or more of the following transformation types may be required to meet the business and technical needs of the server or data warehouse:
The load phase loads the data into the end target, which can be any data store including a simple delimited flat file or adata warehouse. Depending on the requirements of the organization, this process varies widely. Some data warehouses may overwrite existing information with cumulative information; updating extracted data is frequently done on a daily, weekly, or monthly basis. Other data warehouses (or even other parts of the same data warehouse) may add new data in a historical form at regular intervals – for example, hourly. To understand this, consider a data warehouse that is required to maintain sales records of the last year. This data warehouse overwrites any data older than a year with newer data. However, the entry of data for any one year window is made in a historical manner. The timing and scope to replace or append are strategic design choices dependent on the time available and thebusinessneeds. More complex systems can maintain a history andaudit trailof all changes to the data loaded in the data warehouse.
As the load phase interacts with a database, the constraints defined in the database schema – as well as in triggers activated upon data load – apply (for example, uniqueness,referential integrity, mandatory fields), which also contribute to the overall data quality performance of the ETL process.
A real-life ETL cycle may consist of additional execution steps, for example:
ETL processes can involve considerable complexity, and significant operational problems can occur with improperly designed ETL systems.
The range of data values or data quality in an operational system may exceed the expectations of designers at the time validation and transformation rules are specified.Data profilingof a source during data analysis can identify the data conditions that must be managed by transform rules specifications, leading to an amendment of validation rules explicitly and implicitly implemented in the ETL process.
Data warehouses are typically assembled from a variety of data sources with different formats and purposes. As such, ETL is a key process to bring all the data together in a standard, homogeneous environment.
Design analysis[5]should establish thescalabilityof an ETL system across the lifetime of its usage – including understanding the volumes of data that must be processed withinservice level agreements. The time available to extract from source systems may change, which may mean the same amount of data may have to be processed in less time. Some ETL systems have to scale to process terabytes of data to update data warehouses with tens of terabytes of data. Increasing volumes of data may require designs that can scale from dailybatchto multiple-day micro batch to integration withmessage queuesor real-time change-data-capture for continuous transformation and update.
Unique keysplay an important part in all relational databases, as they tie everything together. A unique key is a column that identifies a given entity, whereas aforeign keyis a column in another table that refers to a primary key. Keys can comprise several columns, in which case they are composite keys. In many cases, the primary key is an auto-generated integer that has no meaning for thebusiness entitybeing represented, but solely exists for the purpose of the relational database – commonly referred to as asurrogate key.
As there is usually more than one data source getting loaded into the warehouse, the keys are an important concern to be addressed. For example: customers might be represented in several data sources, with theirSocial Security numberas the primary key in one source, their phone number in another, and a surrogate in the third. Yet a data warehouse may require the consolidation of all the customer information into onedimension.
A recommended way to deal with the concern involves adding a warehouse surrogate key, which is used as a foreign key from the fact table.[6]
Usually, updates occur to a dimension's source data, which obviously must be reflected in the data warehouse.
If the primary key of the source data is required for reporting, the dimension already contains that piece of information for each row. If the source data uses a surrogate key, the warehouse must keep track of it even though it is never used in queries or reports; it is done by creating alookup tablethat contains the warehouse surrogate key and the originating key.[7]This way, the dimension is not polluted with surrogates from various source systems, while the ability to update is preserved.
The lookup table is used in different ways depending on the nature of the source data.
There are 5 types to consider;[7]three are included here:
ETL vendors benchmark their record-systems at multiple TB (terabytes) per hour (or ~1 GB per second) using powerful servers with multiple CPUs, multiple hard drives, multiple gigabit-network connections, and much memory.
In real life, the slowest part of an ETL process usually occurs in the database load phase. Databases may perform slowly because they have to take care of concurrency, integrity maintenance, and indices. Thus, for better performance, it may make sense to employ:
Still, even using bulk operations, database access is usually the bottleneck in the ETL process. Some common methods used to increase performance are:
Whether to do certain operations in the database or outside may involve a trade-off. For example, removing duplicates usingdistinctmay be slow in the database; thus, it makes sense to do it outside. On the other side, if usingdistinctsignificantly (x100) decreases the number of rows to be extracted, then it makes sense to remove duplications as early as possible in the database before unloading data.
A common source of problems in ETL is a big number of dependencies among ETL jobs. For example, job "B" cannot start while job "A" is not finished. One can usually achieve better performance by visualizing all processes on a graph, and trying to reduce the graph making maximum use ofparallelism, and making "chains" of consecutive processing as short as possible. Again, partitioning of big tables and their indices can really help.
Another common issue occurs when the data are spread among several databases, and processing is done in those databases sequentially. Sometimes database replication may be involved as a method of copying data between databases – it can significantly slow down the whole process. The common solution is to reduce the processing graph to only three layers:
This approach allows processing to take maximum advantage of parallelism. For example, if you need to load data into two databases, you can run the loads in parallel (instead of loading into the first – and then replicating into the second).
Sometimes processing must take place sequentially. For example, dimensional (reference) data are needed before one can get and validate the rows for main"fact" tables.
Some ETL software implementations includeparallel processing. This enables a number of methods to improve overall performance of ETL when dealing with large volumes of data.
ETL applications implement three main types of parallelism:
All three types of parallelism usually operate combined in a single job or task.
An additional difficulty comes with making sure that the data being uploaded is relatively consistent. Because multiple source databases may have different update cycles (some may be updated every few minutes, while others may take days or weeks), an ETL system may be required to hold back certain data until all sources are synchronized. Likewise, where a warehouse may have to be reconciled to the contents in a source system or with the general ledger, establishing synchronization and reconciliation points becomes necessary.
Data warehousing procedures usually subdivide a big ETL process into smaller pieces running sequentially or in parallel. To keep track of data flows, it makes sense to tag each data row with "row_id", and tag each piece of the process with "run_id". In case of a failure, having these IDs help to roll back and rerun the failed piece.
Best practice also calls forcheckpoints, which are states when certain phases of the process are completed. Once at a checkpoint, it is a good idea to write everything to disk, clean out some temporary files, log the state, etc.
An established ETL framework may improve connectivity andscalability.[citation needed]A good ETL tool must be able to communicate with the many differentrelational databasesand read the various file formats used throughout an organization. ETL tools have started to migrate intoenterprise application integration, or evenenterprise service bus, systems that now cover much more than just the extraction, transformation, and loading of data. Many ETL vendors now havedata profiling,data quality, andmetadatacapabilities. A common use case for ETL tools include converting CSV files to formats readable by relational databases. A typical translation of millions of records is facilitated by ETL tools that enable users to input csv-like data feeds/files and import them into a database with as little code as possible.
ETL tools are typically used by a broad range of professionals – from students in computer science looking to quickly import large data sets to database architects in charge of company account management, ETL tools have become a convenient tool that can be relied on to get maximum performance. ETL tools in most cases contain a GUI that helps users conveniently transform data, using a visual data mapper, as opposed to writing large programs to parse files and modify data types.
While ETL tools have traditionally been for developers and IT staff, research firm Gartner wrote that the new trend is to provide these capabilities to business users so they can themselves create connections and data integrations when needed, rather than going to the IT staff.[8]Gartner refers to these non-technical users as Citizen Integrators.[9]
Inonline transaction processing(OLTP) applications, changes from individual OLTP instances are detected and logged into a snapshot, or batch, of updates. An ETL instance can be used to periodically collect all of these batches, transform them into a common format, and load them into a data lake or warehouse.[1]
Data virtualizationcan be used to advance ETL processing. The application of data virtualization to ETL allowed solving the most common ETL tasks ofdata migrationand application integration for multiple dispersed data sources. Virtual ETL operates with the abstracted representation of the objects or entities gathered from the variety of relational, semi-structured, andunstructured datasources. ETL tools can leverage object-oriented modeling and work with entities' representations persistently stored in a centrally locatedhub-and-spokearchitecture. Such a collection that contains representations of the entities or objects gathered from the data sources for ETL processing is called a metadata repository and it can reside in memory or be made persistent. By using a persistent metadata repository, ETL tools can transition from one-time projects to persistent middleware, performing data harmonization anddata profilingconsistently and in near-real time.
Extract, load, transform(ELT) is a variant of ETL where the extracted data is loaded into the target system first.[10]The architecture for the analytics pipeline shall also consider where to cleanse and enrich data[10]as well as how to conform dimensions.[1]Some of the benefits of an ELT process include speed and the ability to more easily handle both unstructured and structured data.[11]
Ralph KimballandJoe Caserta's book The Data Warehouse ETL Toolkit, (Wiley, 2004), which is used as a textbook for courses teaching ETL processes in data warehousing, addressed this issue.[12]
Cloud-based data warehouses likeAmazon Redshift, GoogleBigQuery,Microsoft Azure Synapse AnalyticsandSnowflake Inc.have been able to provide highly scalable computing power. This lets businesses forgo preload transformations and replicate raw data into their data warehouses, where it can transform them as needed usingSQL.
After having used ELT, data may be processed further and stored in a data mart.[13]
Most data integration tools skew towards ETL, while ELT is popular in database and data warehouse appliances. Similarly, it is possible to perform TEL (Transform, Extract, Load) where data is first transformed on a blockchain (as a way of recording changes to data, e.g., token burning) before extracting and loading into another data store.[14]
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AGROUP BYstatement inSQLspecifies that a SQLSELECTstatement partitions result rows into groups, based on their values in one or several columns. Typically, grouping is used to apply some sort ofaggregate functionfor each group.[1][2]
The result of a query using aGROUP BYstatement contains one row for each group. This implies constraints on the columns that can appear in the associatedSELECTclause. As a general rule, theSELECTclause may only contain columns with a unique value per group. This includes columns that appear in theGROUP BYclause as well as aggregates resulting in one value per group.[3]
Returns a list of Department IDs along with the sum of their sales for the date of January 1, 2000.
In the following example one can ask "How manyunitswere sold in eachregionfor everyship date?":
The following code returns the data of the abovepivot tablewhich answers the question "How many units were sold in each region for every ship date?":
SinceSQL:1999,GROUP BYcan be extendedWITH ROLLUPto add a result line with a super-agregator result. In the above example, it corresponds to theGrand totalline.
Common grouping (aggregation) functions include:
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AnOLAP cubeis amulti-dimensional arrayof data.[1]Online analytical processing(OLAP)[2]is a computer-based technique of analyzing data to look for insights. The termcubehere refers to a multi-dimensional dataset, which is also sometimes called ahypercubeif the number of dimensions is greater than three.
A cube can be considered a multi-dimensional generalization of a two- or three-dimensionalspreadsheet. For example, a company might wish to summarize financial data by product, by time-period, and by city to compare actual and budget expenses. Product, time, city and scenario (actual and budget) are the data's dimensions.[3]
Cubeis a shorthand formultidimensional dataset, given that data can have an arbitrary number ofdimensions. The termhypercubeis sometimes used, especially for data with more than three dimensions. A cube is not a "cube" in the strict mathematical sense, as the sides are not all necessarily equal. But this term is used widely.
ASliceis a term for a subset of the data, generated by picking a value for one dimension and only showing the data for that value (for instance only the data at one point in time). Spreadsheets are only 2-dimensional, so by (continued) slicing or other techniques, it becomes possible to visualise multidimensional data in them.
Each cell of the cube holds a number that represents somemeasureof the business, such as sales, profits, expenses, budget and forecast.
OLAP data is typically stored in astar schemaorsnowflake schemain arelationaldata warehouseor in a special-purpose data management system. Measures are derived from the records in thefact tableand dimensions are derived from thedimension tables.
The elements of a dimension can be organized as ahierarchy,[4]a set of parent-child relationships, typically where a parent member summarizes its children. Parent elements can further be aggregated as the children of another parent.[5]
For example, May 2005's parent is Second Quarter 2005 which is in turn the child of Year 2005. Similarly cities are the children of regions; products roll into product groups and individual expense items into types of expenditure.
Conceiving data as a cube with hierarchical dimensions leads to conceptually straightforward operations to facilitate analysis. Aligning the data content with a familiar visualization enhances analyst learning and productivity.[5]The user-initiated process of navigating by calling for page displays interactively, through the specification of slices via rotations and drill down/up is sometimes called "slice and dice". Common operations include slice and dice, drill down, roll up, and pivot.
Sliceis the act of picking a rectangular subset of a cube by choosing a single value for one of its dimensions, creating a new cube with one fewer dimension.[5]The picture shows a slicing operation: The sales figures of all sales regions and all product categories of the company in the year 2005 and 2006 are "sliced" out of the data cube.
Dice: The dice operation produces a subcube by allowing the analyst to pick specific values of multiple dimensions.[6]The picture shows a dicing operation: The new cube shows the sales figures of a limited number of product categories, the time and region dimensions cover the same range as before.
Drill Down/Upallows the user to navigate among levels of data ranging from the most summarized (up) to the most detailed (down).[5]The picture shows a drill-down operation: The analyst moves from the summary category "Outdoor protective equipment" to see the sales figures for the individual products.
Roll-up: A roll-up involves summarizing the data along a dimension. The summarization rule might be anaggregate function, such as computing totals along a hierarchy or applying a set of formulas such as "profit = sales - expenses".[5]General aggregation functions may be costly to compute when rolling up: if they cannot be determined from the cells of the cube, they must be computed from the base data, either computing them online (slow) or precomputing them for possible rollouts (large space). Aggregation functions that can be determined from the cells are known asdecomposable aggregation functions, and allow efficient computation.[7]For example, it is easy to supportCOUNT, MAX, MIN,andSUMin OLAP, since these can be computed for each cell of the OLAP cube and then rolled up, since on overall sum (or count etc.) is the sum of sub-sums, but it is difficult to supportMEDIAN, as that must be computed for every view separately: the median of a set is not the median of medians of subsets.
Pivotallows an analyst to rotate the cube in space to see its various faces. For example, cities could be arranged vertically and products horizontally while viewing data for a particular quarter. Pivoting could replace products with time periods to see data across time for a single product.[5][8]
The picture shows a pivoting operation: The whole cube is rotated, giving another perspective on the data.
Indatabase theory, an OLAP cube is[9]an abstract representation of aprojectionof anRDBMSrelation. Given arelationof orderN, consider a projection that subtendsX,Y, andZas the key andWas theresidualattribute. Characterizing this as afunction,
the attributesX,Y, andZcorrespond to the axes of the cube, while theWvalue corresponds to the data element that populates each cell of the cube.
Insofar as two-dimensional output devices cannot readily characterize three dimensions, it is more practical to project "slices" of the data cube (we sayprojectin the classic vector analytic sense of dimensional reduction, not in theSQLsense, although the two are conceptually similar),
which may suppress a primary key, but still have some semantic significance, perhaps a slice of the triadic functional representation for a givenZvalue of interest.
The motivation[9]behindOLAPdisplays harks back to thecross-tabbed reportparadigm of 1980sDBMS, and to earliercontingency tablesfrom 1904. The result is a spreadsheet-style display, where values ofXpopulate row $1; values ofYpopulate column $A; and values ofg: (X,Y) →Wpopulate the individual cells at intersections ofX-labeled columns andY-labeled rows, "southeast", so to speak, of $B$2, with $B$2 itself included.
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Apivot tableis atableof values which are aggregations of groups of individual values from a more extensive table (such as from adatabase,spreadsheet, orbusiness intelligence program) within one or more discrete categories. The aggregations or summaries of the groups of the individual terms might include sums, averages, counts, or other statistics. A pivot table is the outcome of the statistical processing of tabularized raw data and can be used for decision-making.
Althoughpivot tableis a generic term,Microsoftheld a trademark on the term in the United States from 1994 to 2020.[1]
In their bookPivot Table Data Crunching,[2]Bill Jelen and Mike Alexander refer toPito Salasas the "father of pivot tables". While working on a concept for a new program that would eventually becomeLotus Improv, Salas noted that spreadsheets have patterns of data. A tool that could help the user recognize these patterns would help to build advanced data models quickly. With Improv, users could define and store sets of categories, then change views by dragging category names with the mouse. This core functionality would provide the model for pivot tables.
Lotus Developmentreleased Improv in 1991 on theNeXTplatform. A few months after the release of Improv,Brio Technologypublished a standaloneMacintoshimplementation, called DataPivot (with technology eventually patented in 1999).[3]Borlandpurchased the DataPivot technology in 1992 and implemented it in their own spreadsheet application,Quattro Pro.
In 1993 the Microsoft Windows version of Improv appeared. Early in 1994Microsoft Excel5[4]brought a new functionality called a "PivotTable" to market. Microsoft further improved this feature in later versions of Excel:
In 2007 Oracle Corporation madePIVOTandUNPIVOToperators available inOracle Database11g.[6]
For typical data entry and storage, data usually appear inflattables, meaning that they consist of only columns and rows, as in the following portion of a sample spreadsheet showing data on shirt types:
While tables such as these can contain many data items, it can be difficult to get summarized information from them. A pivot table can help quickly summarize the data and highlight the desired information. The usage of a pivot table is extremely broad and depends on the situation. The first question to ask is, "What am I seeking?" In the example here, let us ask, "How manyUnitsdid we sell in eachRegionfor everyShip Date?":
A pivot table usually consists ofrow,columnanddata(orfact) fields. In this case, the column isship date, the row isregionand the data we would like to see is (sum of)units. These fields allow several kinds ofaggregations, including: sum, average,standard deviation, count, etc. In this case, the total number of units shipped is displayed here using asumaggregation.
Using the example above, the software will find all distinct values forRegion. In this case, they are:North,South,East,West. Furthermore, it will find all distinct values forShip date. Based on the aggregation type,sum, it will summarize the fact, the quantities ofUnit, and display them in a multidimensional chart. In the example above, the first datum is 66. This number was obtained by finding all records where bothRegionwasEastandShip Datewas2005-01-31, and adding theUnitsof that collection of records (i.e., cells E2 to E7) together to get a final result.
Pivot tables are not created automatically. For example, in Microsoft Excel one must first select the entire data in the original table and then go to the Insert tab and select "Pivot Table" (or "Pivot Chart"). The user then has the option of either inserting the pivot table into an existing sheet or creating a new sheet to house the pivot table. A pivot table field list is provided to the user which lists all the column headers present in the data. For instance, if a table represents sales data of a company, it might include Date of sale, Sales person, Item sold, Color of item, Units sold, Per unit price, and Total price. This makes the data more readily accessible.
The fields that would be created will be visible on the right hand side of the worksheet. By default, the pivot table layout design will appear below this list.
Pivot Table fields are the building blocks of pivot tables. Each of the fields from the list can be dragged on to this layout, which has four options:
Some uses of pivot tables are related to the analysis of questionnaires with optional responses but some implementations of pivot tables do not allow these use cases. For example the implementation inLibreOffice Calcsince 2012 is not able to process empty cells.[7][8]
Report filter is used to apply a filter to an entire table. For example, if the "Color of Item" field is dragged to this area, then the table constructed will have a report filter inserted above the table. This report filter will have drop-down options (Black, Red, and White in the example above). When an option is chosen from thisdrop-down list("Black" in this example), then the table that would be visible will contain only the data from those rows that have the "Color of Item= Black".
Column labels are used to apply a filter to one or more columns that have to be shown in the pivot table. For instance if the "Salesperson" field is dragged to this area, then the table constructed will have values from the column "Sales Person",i.e., one will have a number of columns equal to the number of "Salesperson". There will also be one added column of Total. In the example above, this instruction will create five columns in the table — one for each salesperson, and Grand Total. There will be a filter above the data — column labels — from which one can select or deselect a particular salesperson for the pivot table.
This table will not have any numerical values as no numerical field is selected but when it is selected, the values will automatically get updated in the column of "Grand total".
Row labels are used to apply a filter to one or more rows that have to be shown in the pivot table. For instance, if the "Salesperson" field is dragged on this area then the other output table constructed will have values from the column "Salesperson",i.e., one will have a number of rows equal to the number of "Sales Person". There will also be one added row of "Grand Total". In the example above, this instruction will create five rows in the table — one for each salesperson, and Grand Total. There will be a filter above the data — row labels — from which one can select or deselect a particular salesperson for the Pivot table.
This table will not have any numerical values, as no numerical field is selected, but when it is selected, the values will automatically get updated in the Row of "Grand Total".
This usually takes a field that has numerical values that can be used for different types of calculations. However, using text values would also not be wrong; instead of Sum, it will give a count. So, in the example above, if the "Units sold" field is dragged to this area along with the row label of "Salesperson", then the instruction will add a new column, "Sum of units sold", which will have values against each salesperson.
Pivot tables or pivot functionality are an integral part of manyspreadsheet applicationsand somedatabase software, as well as being found in other data visualization tools andbusiness intelligencepackages.
Programming languages and libraries suited to work with tabular data contain functions that allow the creation and manipulation of pivot tables.
Excel pivot tables include the feature to directly query anonline analytical processing(OLAP) server for retrieving data instead of getting the data from an Excel spreadsheet. On this configuration, a pivot table is a simple client of an OLAP server. Excel's PivotTable not only allows for connecting to Microsoft's Analysis Service, but to anyXML for Analysis(XMLA) OLAP standard-compliant server.
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Some branches ofeconomicsandgame theorydeal withindivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an item cannot be divided among two or more agents.
It is usually assumed that every agent assigns subjectiveutilityto every subset of the items. This can be represented in one of two ways:
A cardinal utility function implies a preference relation:u(A)>u(B){\displaystyle u(A)>u(B)}impliesA≻B{\displaystyle A\succ B}andu(A)≥u(B){\displaystyle u(A)\geq u(B)}impliesA⪰B{\displaystyle A\succeq B}. Utility functions can have several properties.[1]
Monotonicitymeans that an agent always (weakly) prefers to have extra items. Formally:
Monotonicity is equivalent to thefree disposalassumption: if an agent may always discard unwanted items, then extra items can never decrease the utility.
Additivity (also calledlinearityormodularity) means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. This property is relevant only for cardinal utility functions. It says that for every setA{\displaystyle A}of items,
assuming thatu(∅)=0{\displaystyle u(\emptyset )=0}. In other words,u{\displaystyle u}is anadditive function. An equivalent definition is: for any sets of itemsA{\displaystyle A}andB{\displaystyle B},
An additive utility function is characteristic ofindependent goods. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right.
Submodularitymeans that "the whole is not more than the sum of its parts (and may be less)." Formally, for all setsA{\displaystyle A}andB{\displaystyle B},
In other words,u{\displaystyle u}is asubmodular set function.
An equivalent property isdiminishing marginal utility, which means that for any setsA{\displaystyle A}andB{\displaystyle B}withA⊆B{\displaystyle A\subseteq B}, and everyx∉B{\displaystyle x\notin B}:[2]
A submodular utility function is characteristic ofsubstitute goods. For example, an apple and a bread loaf can be considered substitutes: the utility a person receives from eating an apple is smaller if he has already ate bread (and vice versa), since he is less hungry in that case. A typical utility function for this case is given at the right.
Supermodularityis the opposite of submodularity: it means that "the whole is not less than the sum of its parts (and may be more)". Formally, for all setsA{\displaystyle A}andB{\displaystyle B},
In other words,u{\displaystyle u}is asupermodular set function.
An equivalent property isincreasing marginal utility, which means that for all setsA{\displaystyle A}andB{\displaystyle B}withA⊆B{\displaystyle A\subseteq B}, and everyx∉B{\displaystyle x\notin B}:
A supermoduler utility function is characteristic ofcomplementary goods. For example, an apple and a knife can be considered complementary: the utility a person receives from an apple is larger if he already has a knife (and vice versa), since it is easier to eat an apple after cutting it with a knife. A possible utility function for this case is given at the right.
A utility function isadditiveif and only if it is both submodular and supermodular.
Subadditivitymeans that for every pair of disjoint setsA,B{\displaystyle A,B}
In other words,u{\displaystyle u}is asubadditive set function.
Assumingu(∅){\displaystyle u(\emptyset )}is non-negative, every submodular function is subadditive.
However, there are non-negative subadditive functions that are not submodular.
For example, assume that there are 3 identical items,X,Y{\displaystyle X,Y}, and Z, and the utility depends only on their quantity. The table on the right describes a utility function that is subadditive but not submodular, since
Superadditivitymeans that for every pair of disjoint setsA,B{\displaystyle A,B}
In other words,u{\displaystyle u}is asuperadditive set function.
Assumingu(∅){\displaystyle u(\emptyset )}is non-positive, every supermodular function is superadditive.
However, there are non-negative superadditive functions that are not supermodular.
For example, assume that there are 3 identical items,X,Y{\displaystyle X,Y}, and Z, and the utility depends only on their quantity. The table on the right describes a utility function that is non-negative and superadditive but not supermodular, since
A utility function withu(∅)=0{\displaystyle u(\emptyset )=0}is said to beadditiveif and only if it is both superadditive and subadditive.
With the typical assumption thatu(∅)=0{\displaystyle u(\emptyset )=0}, every submodular function is subadditive and every supermodular function is superadditive.
Without any assumption on the utility from the empty set, these relations do not hold.
In particular, if a submodular function is not subadditive, thenu(∅){\displaystyle u(\emptyset )}must be negative.
For example, suppose there are two items,X,Y{\displaystyle X,Y}, withu(∅)=−1{\displaystyle u(\emptyset )=-1},u({X})=u({Y})=1{\displaystyle u(\{X\})=u(\{Y\})=1}andu({X,Y})=3{\displaystyle u(\{X,Y\})=3}.
This utility function is submodular and supermodular and non-negative except on the empty set, but is not subadditive, since
Also, if a supermodular function is not superadditive, thenu(∅){\displaystyle u(\emptyset )}must be positive.
Suppose instead thatu(∅)=u({X})=u({Y})=u({X,Y})=1{\displaystyle u(\emptyset )=u(\{X\})=u(\{Y\})=u(\{X,Y\})=1}.
This utility function is non-negative, supermodular, and submodular, but is not superadditive, since
Unit demand (UD) means that the agent only wants a single good. If the agent gets two or more goods, he uses the one of them that gives him the highest utility, and discards the rest. Formally:
A unit-demand function is an extreme case of a submodular function. It is characteristic of goods that are pure substitutes. For example, if there are an apple and a pear, and an agent wants to eat a single fruit, then his utility function is unit-demand, as exemplified in the table at the right.
Gross substitutes (GS) means that the agents regards the items assubstitute goodsorindependent goodsbut notcomplementary goods. There are many formal definitions to this property, all of which are equivalent.
SeeGross substitutes (indivisible items)for more details.
Hence the following relations hold between the classes:
See diagram on the right.
A utility function describes the happiness of an individual. Often, we need a function that describes the happiness of an entire society. Such a function is called asocial welfare function, and it is usually anaggregate functionof two or more utility functions. If the individual utility functions areadditive, then the following is true for the aggregate functions:
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https://en.wikipedia.org/wiki/Utility_functions_on_indivisible_goods#Aggregates_of_utility_functions
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XML for Analysis(XMLA) is an industry standard for data access in analytical systems, such asonline analytical processing(OLAP) anddata mining. XMLA is based on other industry standards such asXML,SOAPandHTTP. XMLA is maintained byXMLA CouncilwithMicrosoft,HyperionandSAS Institutebeing the XMLA Council founder members.
The XMLA specification was first proposed byMicrosoftas a successor forOLE DB for OLAPin April 2000. By January 2001 it was joined byHyperionendorsing XMLA. The 1.0 version of the standard was released in April 2001, and in September 2001 the XMLA Council was formed. In April 2002SASjoined Microsoft and Hyperion as founding member of XMLA Council.[1]With time, more than 25 companies joined with their support for the standard.
XMLA consists of only twoSOAPmethods: execute and discover.[2]It was designed in such a way to preserve simplicity.
Execute method has two parameters:
The result of Execute command could beMultidimensional DatasetorTabular Rowset.
Discover method was designed to model all the discovery methods possible inOLEDBincluding various schema rowset, properties, keywords, etc. Discover method allows users to specify both what needs to be discovered and the possible restrictions or properties.
The result of Discover method is a rowset.
XMLA specifiesMDXMLas the query language. In the XMLA 1.1 version, the only construct in MDXML is anMDXstatement enclosed in the <Statement> tag.[3]
Below is an example of XMLA Execute request with MDX query in command.
XMLA has a notion ofsession state. It is maintained through predefinedSOAPheaders
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https://en.wikipedia.org/wiki/XML_for_Analysis
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AggregateIQ(AIQ) previously known as SCL Canada is a Canadianpolitical consultancyandtechnology company, based inVictoria, British Columbia.[1]
AIQ was founded in 2013 by Zack Massingham, a former university administrator and Jeff Silvester.[2]As of February 2017, AIQ employed 20 people and was based in downtown Victoria, British Columbia.[3]
AIQ has attracted controversy over its involvement in theVote LeaveandBeLeavecampaigns in 2016 and theCambridge Analyticascandal that broke out in 2018.
Two years after theBrexitvote in 2016, it was revealed that AggregateIQ had been paid £3.5 million by four pro-Brexitcampaigning groups -Vote Leave,BeLeave,Veterans for Britain, and Northern Ireland'sDemocratic Unionist Party- to design software aimed at aggregating personal data and influencing voters through messaging on social media.[4]Under UK law, co-ordination between groups during an election is prohibited.[1]In May 2018, aFacebookexecutive testified before theHouse of CommonsSelect Committee forDigital, Culture, Media and Sportthat Vote Leave and BeLeave were targeting exactly the same audiences on Facebook via AIQ.[5]
Prior to the Brexit campaign, AIQ had worked withJohn Boltonbefore he becameDonald Trump's national security adviser, and with US senatorsThom TillisandTed Cruzon their senatorial campaigns.[4]As part of Cambridge Analytica's work for the Cruz campaign, AIQ createdRipon, a customized campaign software platform that became the prototype used by pro-Brexit campaign groups, including VoteLeave and BeLeave.[1]
On 6 April 2018,Facebooksuspended AggregateIQ from its platform due to concerns over its possible affiliation withSCL Group, the parent company ofCambridge Analytica.[6][4][7][8]Facebook stated, "In light of recent reports that AggregateIQ may be affiliated with SCL and may, as a result, have improperly received FB user data, we have added them to the list of entities we have suspended from our platform while we investigate."[4]
On 20 September 2018, AggregateIQ became the first company to be served a formal notice by the UK'sInformation Commissioner's Officefor breaching the European Union'sGeneral Data Protection Regulation. The company has launched an appeal against the notice.[9]
AIQ has also been reprimanded by thePrivacy Commissioner of Canadaand the Privacy Commissioner ofBritish Columbia, who stated in a report issued in November 2019 that the company had violated privacy laws in its handling of British voters' data during the Vote Leave campaign. The report noted, “When the company used and disclosed the personal information of Vote Leave supporters to Facebook... it went beyond the purposes for which Vote Leave had consent to use that information.”[10]
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https://en.wikipedia.org/wiki/AggregateIQ
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Inmathematics, aunary operationis anoperationwith only oneoperand, i.e. a single input.[1]This is in contrast tobinary operations, which use two operands.[2]An example is anyfunctionf:A→A{\displaystyle f:A\rightarrow A}, whereAis aset; the functionf{\displaystyle f}is a unary operation onA.
Common notations areprefix notation(e.g.¬,−),postfix notation(e.g.factorialn!), functional notation (e.g.sinxorsin(x)), andsuperscripts(e.g.transposeAT). Other notations exist as well, for example, in the case of thesquare root, ahorizontal barextending the square root sign over the argument can indicate the extent of the argument.
Obtaining theabsolute valueof a number is a unary operation. This function is defined as|n|={n,ifn≥0−n,ifn<0{\displaystyle |n|={\begin{cases}n,&{\mbox{if }}n\geq 0\\-n,&{\mbox{if }}n<0\end{cases}}}where|n|{\displaystyle |n|}is the absolute value ofn{\displaystyle n}.
Negationis used to find the negative value of a single number. Here are some examples:
For any positive integern, the product of the integers less than or equal tonis a unary operation calledfactorial. In the context ofcomplex numbers, thegamma functionis a unary operation extension of factorial.
Intrigonometry, thetrigonometric functions, such assin{\displaystyle \sin },cos{\displaystyle \cos }, andtan{\displaystyle \tan }, can be seen as unary operations. This is because it is possible to provide only one term as input for these functions and retrieve a result. By contrast, binary operations, such asaddition, require two different terms to compute a result.
Below is a table summarizing common unary operators along with their symbols, description, and examples:[3]
InJavaScript, these operators are unary:[4]
In theCfamily of languages, the following operators are unary:[5][6]
In theUnix shell(Bash/Bourne Shell), e.g., the following operators are unary:[7][8]
In thePowerShell, the following operators are unary:[9]
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https://en.wikipedia.org/wiki/Unary_operation
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Inmathematics, aunary functionis afunctionthat takes oneargument. Aunary operatorbelongs to a subset of unary functions, in that itscodomaincoincides with itsdomain. In contrast, a unary function's domain need not coincide with its range.
Thesuccessor function, denotedsucc{\displaystyle \operatorname {succ} }, is a unary operator. Its domain and codomain are thenatural numbers; its definition is as follows:
In someprogramming languagessuch asC, executing this operation is denoted bypostfixing++to the operand, i.e. the use ofn++is equivalent to executing the assignmentn:=succ(n){\displaystyle n:=\operatorname {succ} (n)}.
Many of theelementary functionsare unary functions, including thetrigonometric functions,logarithmwith a specified base,exponentiationto a particular power or base, andhyperbolic functions.
Thismathematics-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Unary_function
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Inmathematics, abinary operationordyadic operationis a rule for combining twoelements(calledoperands) to produce another element. More formally, a binary operation is anoperationofaritytwo.
More specifically, abinary operationon asetis abinary functionthat maps everypairof elements of the set to an element of the set. Examples include the familiararithmetic operationslikeaddition,subtraction,multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such asvector addition,matrix multiplication, andconjugation in groups.
A binary function that involves several sets is sometimes also called abinary operation. For example,scalar multiplicationofvector spacestakes a scalar and a vector to produce a vector, andscalar producttakes two vectors to produce a scalar.
Binary operations are the keystone of moststructuresthat are studied inalgebra, in particular insemigroups,monoids,groups,rings,fields, andvector spaces.
More precisely, a binary operation on asetS{\displaystyle S}is amappingof the elements of theCartesian productS×S{\displaystyle S\times S}toS{\displaystyle S}:[1][2][3]
Iff{\displaystyle f}is not afunctionbut apartial function, thenf{\displaystyle f}is called apartial binary operation. For instance, division is a partial binary operation on the set of allreal numbers, because one cannotdivide by zero:a0{\displaystyle {\frac {a}{0}}}is undefined for every real numbera{\displaystyle a}. In bothmodel theoryand classicaluniversal algebra, binary operations are required to be defined on all elements ofS×S{\displaystyle S\times S}. However,partial algebras[4]generalizeuniversal algebrasto allow partial operations.
Sometimes, especially incomputer science, the term binary operation is used for anybinary function.
Typical examples of binary operations are theaddition(+{\displaystyle +}) andmultiplication(×{\displaystyle \times }) ofnumbersandmatricesas well ascomposition of functionson a single set.
For instance,
Many binary operations of interest in both algebra and formal logic arecommutative, satisfyingf(a,b)=f(b,a){\displaystyle f(a,b)=f(b,a)}for all elementsa{\displaystyle a}andb{\displaystyle b}inS{\displaystyle S}, orassociative, satisfyingf(f(a,b),c)=f(a,f(b,c)){\displaystyle f(f(a,b),c)=f(a,f(b,c))}for alla{\displaystyle a},b{\displaystyle b}, andc{\displaystyle c}inS{\displaystyle S}. Many also haveidentity elementsandinverse elements.
The first three examples above are commutative and all of the above examples are associative.
On the set of real numbersR{\displaystyle \mathbb {R} },subtraction, that is,f(a,b)=a−b{\displaystyle f(a,b)=a-b}, is a binary operation which is not commutative since, in general,a−b≠b−a{\displaystyle a-b\neq b-a}. It is also not associative, since, in general,a−(b−c)≠(a−b)−c{\displaystyle a-(b-c)\neq (a-b)-c}; for instance,1−(2−3)=2{\displaystyle 1-(2-3)=2}but(1−2)−3=−4{\displaystyle (1-2)-3=-4}.
On the set of natural numbersN{\displaystyle \mathbb {N} }, the binary operationexponentiation,f(a,b)=ab{\displaystyle f(a,b)=a^{b}}, is not commutative since,ab≠ba{\displaystyle a^{b}\neq b^{a}}(cf.Equation xy= yx), and is also not associative sincef(f(a,b),c)≠f(a,f(b,c)){\displaystyle f(f(a,b),c)\neq f(a,f(b,c))}. For instance, witha=2{\displaystyle a=2},b=3{\displaystyle b=3}, andc=2{\displaystyle c=2},f(23,2)=f(8,2)=82=64{\displaystyle f(2^{3},2)=f(8,2)=8^{2}=64}, butf(2,32)=f(2,9)=29=512{\displaystyle f(2,3^{2})=f(2,9)=2^{9}=512}. By changing the setN{\displaystyle \mathbb {N} }to the set of integersZ{\displaystyle \mathbb {Z} }, this binary operation becomes a partial binary operation since it is now undefined whena=0{\displaystyle a=0}andb{\displaystyle b}is any negative integer. For either set, this operation has aright identity(which is1{\displaystyle 1}) sincef(a,1)=a{\displaystyle f(a,1)=a}for alla{\displaystyle a}in the set, which is not anidentity(two sided identity) sincef(1,b)≠b{\displaystyle f(1,b)\neq b}in general.
Division(÷{\displaystyle \div }), a partial binary operation on the set of real or rational numbers, is not commutative or associative.Tetration(↑↑{\displaystyle \uparrow \uparrow }), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.
Binary operations are often written usinginfix notationsuch asa∗b{\displaystyle a\ast b},a+b{\displaystyle a+b},a⋅b{\displaystyle a\cdot b}or (byjuxtapositionwith no symbol)ab{\displaystyle ab}rather than by functional notation of the formf(a,b){\displaystyle f(a,b)}. Powers are usually also written without operator, but with the second argument assuperscript.
Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses. They are also called, respectively,Polish notation∗ab{\displaystyle \ast ab}andreverse Polish notationab∗{\displaystyle ab\ast }.
A binary operationf{\displaystyle f}on a setS{\displaystyle S}may be viewed as aternary relationonS{\displaystyle S}, that is, the set of triples(a,b,f(a,b)){\displaystyle (a,b,f(a,b))}inS×S×S{\displaystyle S\times S\times S}for alla{\displaystyle a}andb{\displaystyle b}inS{\displaystyle S}.
For example,scalar multiplicationinlinear algebra. HereK{\displaystyle K}is afieldandS{\displaystyle S}is avector spaceover that field.
Also thedot productof two vectors mapsS×S{\displaystyle S\times S}toK{\displaystyle K}, whereK{\displaystyle K}is a field andS{\displaystyle S}is a vector space overK{\displaystyle K}. It depends on authors whether it is considered as a binary operation.
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https://en.wikipedia.org/wiki/Binary_operation
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Inmathematics, abinary function(also calledbivariate function, orfunction of two variables) is afunctionthat takes two inputs.
Precisely stated, a functionf{\displaystyle f}is binary if there existssetsX,Y,Z{\displaystyle X,Y,Z}such that
whereX×Y{\displaystyle X\times Y}is theCartesian productofX{\displaystyle X}andY.{\displaystyle Y.}
Set-theoretically, a binary function can be represented as asubsetof theCartesian productX×Y×Z{\displaystyle X\times Y\times Z}, where(x,y,z){\displaystyle (x,y,z)}belongs to the subsetif and only iff(x,y)=z{\displaystyle f(x,y)=z}.
Conversely, a subsetR{\displaystyle R}defines a binary function if and only iffor anyx∈X{\displaystyle x\in X}andy∈Y{\displaystyle y\in Y},there existsauniquez∈Z{\displaystyle z\in Z}such that(x,y,z){\displaystyle (x,y,z)}belongs toR{\displaystyle R}.f(x,y){\displaystyle f(x,y)}is then defined to be thisz{\displaystyle z}.
Alternatively, a binary function may be interpreted as simply afunctionfromX×Y{\displaystyle X\times Y}toZ{\displaystyle Z}.
Even when thought of this way, however, one generally writesf(x,y){\displaystyle f(x,y)}instead off((x,y)){\displaystyle f((x,y))}.
(That is, the same pair of parentheses is used to indicate bothfunction applicationand the formation of anordered pair.)
Division ofwhole numberscan be thought of as a function. IfZ{\displaystyle \mathbb {Z} }is the set ofintegers,N+{\displaystyle \mathbb {N} ^{+}}is the set ofnatural numbers(except for zero), andQ{\displaystyle \mathbb {Q} }is the set ofrational numbers, thendivisionis a binary functionf:Z×N+→Q{\displaystyle f:\mathbb {Z} \times \mathbb {N} ^{+}\to \mathbb {Q} }.
In a vector spaceVover a fieldF,scalar multiplicationis a binary function. A scalara∈Fis combined with a vectorv∈Vto produce a new vectorav∈V.
Another example is that of inner products, or more generally functions of the form(x,y)↦xTMy{\displaystyle (x,y)\mapsto x^{\mathrm {T} }My}, wherex,yare real-valued vectors of appropriate size andMis a matrix. IfMis apositive definite matrix, this yields aninner product.[1]
Functions whose domain is a subset ofR2{\displaystyle \mathbb {R} ^{2}}are often also called functions of two variables even if their domain does not form a rectangle and thus the cartesian product of two sets.[2]
In turn, one can also derive ordinary functions of one variable from a binary function.
Given any elementx∈X{\displaystyle x\in X}, there is a functionfx{\displaystyle f^{x}}, orf(x,⋅){\displaystyle f(x,\cdot )}, fromY{\displaystyle Y}toZ{\displaystyle Z}, given byfx(y)=f(x,y){\displaystyle f^{x}(y)=f(x,y)}.
Similarly, given any elementy∈Y{\displaystyle y\in Y}, there is a functionfy{\displaystyle f_{y}}, orf(⋅,y){\displaystyle f(\cdot ,y)}, fromX{\displaystyle X}toZ{\displaystyle Z}, given byfy(x)=f(x,y){\displaystyle f_{y}(x)=f(x,y)}. In computer science, this identification between a function fromX×Y{\displaystyle X\times Y}toZ{\displaystyle Z}and a function fromX{\displaystyle X}toZY{\displaystyle Z^{Y}}, whereZY{\displaystyle Z^{Y}}is the set of all functions fromY{\displaystyle Y}toZ{\displaystyle Z}, is calledcurrying.
The various concepts relating to functions can also be generalised to binary functions.
For example, the division example above issurjective(oronto) because every rational number may be expressed as a quotient of an integer and a natural number.
This example isinjectivein each input separately, because the functionsfxandfyare always injective.
However, it's not injective in both variables simultaneously, because (for example)f(2,4) =f(1,2).
One can also considerpartialbinary functions, which may be defined only for certain values of the inputs.
For example, the division example above may also be interpreted as a partial binary function fromZandNtoQ, whereNis the set of all natural numbers, including zero.
But this function is undefined when the second input is zero.
Abinary operationis a binary function where the setsX,Y, andZare all equal; binary operations are often used to definealgebraic structures.
Inlinear algebra, abilinear transformationis a binary function where the setsX,Y, andZare allvector spacesand the derived functionsfxandfyare alllinear transformations.
A bilinear transformation, like any binary function, can be interpreted as a function fromX×YtoZ, but this function in general won't be linear.
However, the bilinear transformation can also be interpreted as a single linear transformation from thetensor productX⊗Y{\displaystyle X\otimes Y}toZ.
The concept of binary function generalises toternary(or3-ary)function,quaternary(or4-ary)function, or more generally ton-ary functionfor anynatural numbern.
A0-ary functiontoZis simply given by an element ofZ.
One can also define anA-ary functionwhereAis anyset; there is one input for each element ofA.
Incategory theory,n-ary functions generalise ton-ary morphisms in amulticategory.
The interpretation of ann-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the originaln-ary morphism will work in amonoidal category.
The construction of the derived morphisms of one variable will work in aclosed monoidal category.
The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.
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https://en.wikipedia.org/wiki/Binary_function
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Inmathematics, aternary operationis ann-aryoperationwithn= 3. A ternary operation on asetAtakes any given three elements ofAand combines them to form a single element ofA.
Incomputer science, aternary operatoris anoperatorthat takes threeargumentsas input and returns one output.[1]
ThefunctionT(a,b,c)=ab+c{\displaystyle T(a,b,c)=ab+c}is an example of a ternary operation on theintegers(or on any structure where+{\displaystyle +}and×{\displaystyle \times }are both defined). Properties of this ternary operation have been used to defineplanar ternary ringsin the foundations ofprojective geometry.
In theEuclidean planewith pointsa,b,creferred to an origin, the ternary operation[a,b,c]=a−b+c{\displaystyle [a,b,c]=a-b+c}has been used to definefree vectors.[2]Since (abc) =dimpliesb–a=c–d, thedirected line segmentsb–aandc–dareequipollentand are associated with the same free vector. Any three points in the planea, b, cthus determine aparallelogramwithdat the fourth vertex.
Inprojective geometry, the process of finding aprojective harmonic conjugateis a ternary operation on three points. In the diagram, pointsA,BandPdetermine pointV, the harmonic conjugate ofPwith respect toAandB. PointRand the line throughPcan be selected arbitrarily, determiningCandD. DrawingACandBDproduces the intersectionQ, andRQthen yieldsV.
SupposeAandBare given sets andB(A,B){\displaystyle {\mathcal {B}}(A,B)}is the collection ofbinary relationsbetweenAandB.Composition of relationsis always defined whenA=B, but otherwise a ternary composition can be defined by[p,q,r]=pqTr{\displaystyle [p,q,r]=pq^{T}r}whereqT{\displaystyle q^{T}}is theconverse relationofq. Properties of this ternary relation have been used to set the axioms for aheap.[3]
InBoolean algebra,T(A,B,C)=AC+(1−A)B{\displaystyle T(A,B,C)=AC+(1-A)B}defines the formula(A∨B)∧(¬A∨C){\displaystyle (A\lor B)\land (\lnot A\lor C)}.
In computer science, a ternary operator is anoperatorthat takes three arguments (or operands).[1]The arguments and result can be of different types. Manyprogramming languagesthat useC-like syntax[4]feature a ternary operator,?:, which defines aconditional expression. In some languages, this operator is referred to as theconditional operator.
InPython, the ternary conditional operator readsx if C else y. Python also supports ternary operations calledarray slicing, e.g.a[b:c]return an array where the first element isa[b]and last element isa[c-1].[5]OCamlexpressions provide ternary operations against records, arrays, and strings:a.[b]<-cwould mean the stringawhere indexbhas valuec.[6]
Themultiply–accumulate operationis another ternary operator.
Another example of a ternary operator isbetween, as used inSQL.
TheIcon programming languagehas a "to-by" ternary operator: the expression1 to 10 by 2generates theoddintegers from 1 through 9.
In Excel formulae, the form is =if(C, x, y).
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https://en.wikipedia.org/wiki/Ternary_operation
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Inmathematics, specifically incategory theory,F-algebrasgeneralize the notion ofalgebraic structure. Rewriting the algebraic laws in terms ofmorphismseliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a singlefunctorF, thesignature.
F-algebras can also be used to representdata structuresused inprogramming, such aslistsandtrees.
The main related concepts areinitialF-algebras which may serve to encapsulate the induction principle, and thedualconstructionF-coalgebras.
IfC{\displaystyle C}is acategory, andF:C→C{\displaystyle F:C\rightarrow C}is anendofunctorofC{\displaystyle C}, then anF{\displaystyle F}-algebrais a tuple(A,α){\displaystyle (A,\alpha )}, whereA{\displaystyle A}is anobjectofC{\displaystyle C}andα{\displaystyle \alpha }is aC{\displaystyle C}-morphismF(A)→A{\displaystyle F(A)\rightarrow A}. The objectA{\displaystyle A}is called thecarrierof the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple.
Ahomomorphismfrom anF{\displaystyle F}-algebra(A,α){\displaystyle (A,\alpha )}to anF{\displaystyle F}-algebra(B,β){\displaystyle (B,\beta )}is aC{\displaystyle C}-morphismf:A→B{\displaystyle f:A\rightarrow B}such thatf∘α=β∘F(f){\displaystyle f\circ \alpha =\beta \circ F(f)}, according to the followingcommutative diagram:
Equipped with these morphisms,F{\displaystyle F}-algebras constitute a category.
The dual construction areF{\displaystyle F}-coalgebras, which are objectsA∗{\displaystyle A^{*}}together with a morphismα∗:A∗→F(A∗){\displaystyle \alpha ^{*}:A^{*}\rightarrow F(A^{*})}.
Classically, agroupis a setG{\displaystyle G}with agroup lawm:G×G→G{\displaystyle m:G\times G\rightarrow G}, withm(x,y)=x⋅y{\displaystyle m(x,y)=x\cdot y}, satisfying three axioms: the existence of an identity element, the existence of an inverse for each element of the group, and associativity.
To put this in a categorical framework, first define the identity and inverse as functions (morphisms of the setG{\displaystyle G}) bye:1→G{\displaystyle e:1\rightarrow G}withe(∗)=1{\displaystyle e(*)=1}, andi:G→G{\displaystyle i:G\rightarrow G}withi(x)=x−1{\displaystyle i(x)=x^{-1}}. Here1{\displaystyle 1}denotes the set with one element1={∗}{\displaystyle 1=\left\{*\right\}}, which allows one to identify elementsx∈G{\displaystyle x\in G}with morphisms1→G{\displaystyle 1\rightarrow G}.
It is then possible to write the axioms of a group in terms of functions (note how the existential quantifier is absent):
Then this can be expressed with commutative diagrams:[1][2]
Now use thecoproduct(thedisjoint unionof sets) to glue the three morphisms in one:α=e+i+m{\displaystyle \alpha =e+i+m}according to
Thus a group is anF{\displaystyle F}-algebra whereF{\displaystyle F}is the functorF(G)=1+G+G×G{\displaystyle F(G)=1+G+G\times G}. However the reverse is not necessarily true. SomeF{\displaystyle F}-algebra whereF{\displaystyle F}is the functorF(G)=1+G+G×G{\displaystyle F(G)=1+G+G\times G}are not groups.
The above construction is used to definegroup objectsover an arbitrary category withfinite productsand aterminal object1{\displaystyle 1}. When the category admits finitecoproducts, the group objects areF{\displaystyle F}-algebras. For example,finite groupsareF{\displaystyle F}-algebras in the category offinite setsandLie groupsareF{\displaystyle F}-algebras in the category ofsmooth manifoldswithsmooth maps.
Going one step ahead ofuniversal algebra, most algebraic structures areF-algebras. For example,abelian groupsareF-algebras for the same functorF(G) = 1 +G+G×Gas for groups, with an additional axiom for commutativity:m∘t=m, wheret(x,y) = (y,x) is the transpose onGxG.
MonoidsareF-algebras of signatureF(M) = 1 +M×M. In the same vein,semigroupsareF-algebras of signatureF(S) =S×S
Rings,domainsandfieldsare alsoF-algebras with a signature involving two laws +,•:R×R→ R, an additive identity 0: 1 →R, a multiplicative identity 1: 1 →R, and an additive inverse for each element -:R→R. As all these functions share the samecodomainRthey can be glued into a single signature function 1 + 1 +R+R×R+R×R→R, with axioms to express associativity,distributivity, and so on. This makes ringsF-algebras on thecategory of setswith signature 1 + 1 +R+R×R+R×R.
Alternatively, we can look at the functorF(R) = 1 +R×Rin thecategory of abelian groups. In that context, the multiplication is a homomorphism, meaningm(x+y,z) =m(x,z) +m(y,z) andm(x,y+z) =m(x,y) +m(x,z), which are precisely the distributivity conditions. Therefore, a ring is anF-algebra of signature 1 +R×Rover the category of abelian groups which satisfies two axioms (associativity and identity for the multiplication).
When we come tovector spacesandmodules, the signature functor includes ascalar multiplicationk×E→E, and the signatureF(E) = 1 +E+k×Eis parametrized bykover the category of fields, or rings.
Algebras over a fieldcan be viewed asF-algebras of signature 1 + 1 +A+A×A+A×A+k×Aover the category of sets, of signature 1 +A×Aover thecategory of modules(a module with an internal multiplication), and of signaturek×Aover thecategory of rings(a ring with a scalar multiplication), when they are associative and unitary.
Not all mathematical structures areF-algebras. For example, aposetPmay be defined in categorical terms with a morphisms:P×P→ Ω, on asubobject classifier(Ω = {0,1} in the category of sets ands(x,y)=1 precisely whenx≤y). The axioms restricting the morphismsto define a poset can be rewritten in terms of morphisms. However, as the codomain ofsis Ω and notP, it is not anF-algebra.
However,lattices, which are partial orders in which every two elements have a supremum and an infimum, and in particulartotal orders, areF-algebras. This is because they can equivalently be defined in terms of the algebraic operations:x∨y= inf(x,y) andx∧y= sup(x,y), subject to certain axioms (commutativity, associativity, absorption and idempotency). Thus they areF-algebras of signaturePxP+PxP. It is often said that lattice theory draws on both order theory and universal algebra.
Consider the functorF:Set→Set{\displaystyle F:\mathrm {\bf {Set}} \to \mathrm {\bf {Set}} }that sends a setX{\displaystyle X}to1+X{\displaystyle 1+X}. Here,Set{\displaystyle \mathrm {\bf {Set}} }denotes the category of sets,+{\displaystyle +}denotes the usualcoproductgiven by thedisjoint union, and1{\displaystyle 1}is a terminal object (i.e. anysingletonset). Then, the setN{\displaystyle \mathbb {N} }ofnatural numberstogether with the function[zero,succ]:1+N→N{\displaystyle [\mathrm {zero} ,\mathrm {succ} ]:1+\mathbb {N} \to \mathbb {N} }—which is the coproduct of the functionszero:1↦0{\displaystyle \mathrm {zero} :1\mapsto 0}andsucc:n↦n+1{\displaystyle \mathrm {succ} :n\mapsto n+1}—is anF-algebra.
If the category ofF-algebras for a given endofunctorFhas aninitial object, it is called aninitial algebra. The algebra(N,[zero,succ]){\displaystyle (\mathbb {N} ,[\mathrm {zero} ,\mathrm {succ} ])}in the above example is an initial algebra. Variousfinitedata structuresused inprogramming, such aslistsandtrees, can be obtained as initial algebras of specific endofunctors.
Types defined by usingleast fixed pointconstruct with functorFcan be regarded as an initialF-algebra, provided thatparametricityholds for the type.[3]
See alsoUniversal algebra.
In adualway, a similar relationship exists between notions ofgreatest fixed pointand terminalF-coalgebra. These can be used for allowingpotentially infiniteobjects while maintainingstrong normalization property.[3]In the strongly normalizingCharityprogramming language (i.e. each program terminates in it),coinductivedata types can be used to achieve surprising results, enabling the definition oflookupconstructs to implement such“strong”functions like theAckermann function.[4]
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Incomputer science, and in particularfunctional programming, ahylomorphismis arecursivefunction, corresponding to thecompositionof ananamorphism(which first builds a set of results; also known as 'unfolding') followed by acatamorphism(which thenfoldsthese results into a finalreturn value). Fusion of these two recursive computations into a single recursive pattern then avoids building the intermediatedata structure. This is an example ofdeforestation, aprogram optimizationstrategy. A related type of function is ametamorphism, which is a catamorphism followed by an anamorphism.
A hylomorphismh:A→C{\displaystyle h:A\rightarrow C}can be defined in terms of its separate anamorphic and catamorphic parts.
The anamorphic part can be defined in terms of aunaryfunctiong:A→B×A{\displaystyle g:A\rightarrow B\times A}defining the list of elements inB{\displaystyle B}by repeated application ("unfolding"), and apredicatep:A→Boolean{\displaystyle p:A\rightarrow {\text{Boolean}}}providing the terminating condition.
The catamorphic part can be defined as a combination of an initial valuec∈C{\displaystyle c\in C}for the fold and a binaryoperator⊕:B×C→C{\displaystyle \oplus :B\times C\rightarrow C}used to perform the fold.
Thus a hylomorphism
may be defined (assuming appropriate definitions ofp{\displaystyle p}&g{\displaystyle g}).
An abbreviated notation for the above hylomorphism ish=[[(c,⊕),(g,p)]]{\displaystyle h=[\![(c,\oplus ),(g,p)]\!]}.
Listsare common data structures as they naturally reflect linear computational processes. These processes arise in repeated (iterative) function calls. Therefore, it is sometimes necessary to generate a temporary list of intermediate results before reducing this list to a single result.
One example of a commonly encountered hylomorphism is the canonicalfactorialfunction.
In the previous example (written inHaskell, apurely functional programming language) it can be seen that this function, applied to any given valid input, will generate a linear call treeisomorphicto a list. For example, givenn= 5 it will produce the following:
In this example, the anamorphic part of the process is the generation of the call tree which is isomorphic to the list[1, 1, 2, 3, 4, 5]. The catamorphism, then, is the calculation of theproductof theelementsof this list. Thus, in the notation given above, the factorial function may be written asfactorial=[[(1,×),(g,p)]]{\displaystyle {\text{factorial}}=[\![(1,\times ),(g,p)]\!]}wheregn=(n,n−1){\displaystyle g\ n=(n,n-1)}andpn=(n=0){\displaystyle p\ n=(n=0)}.
However, the term 'hylomorphism' does not apply solely to functions acting upon isomorphisms of lists. For example, a hylomorphism may also be defined by generating a non-linear call tree which is then collapsed. An example of such a function is the function to generate thenthtermof theFibonacci sequence.
This function, again applied to any valid input, will generate a call tree which is non-linear. In the example on the right, the call tree generated by applying thefibonaccifunction to the input4.
This time, the anamorphism is the generation of the call tree isomorphic to the tree withleaf nodes0, 1, 1, 0, 1and the catamorphism thesummationof these leaf nodes.
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https://en.wikipedia.org/wiki/Hylomorphism_(computer_science)
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Informal methodsofcomputer science, aparamorphism(fromGreekπαρά, meaning "close together")
is an extension of the concept ofcatamorphismfirst introduced byLambert Meertens[1]to deal with a form which “eats its argument and keeps it too”,[2][3]as exemplified by thefactorialfunction. Itscategorical dualis theapomorphism.
It is a more convenient version of catamorphism in that it gives the combining step function immediate access not only to the result value recursively computed from each recursive subobject, but the original subobject itself as well.
Example Haskell implementation, for lists:
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https://en.wikipedia.org/wiki/Paramorphism
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Informal methodsofcomputer science, anapomorphism(fromἀπό—Greekfor "apart") is thecategorical dualof aparamorphismand an extension of the concept ofanamorphism(coinduction). Whereas a paramorphism modelsprimitive recursionover aninductive data type, an apomorphism models primitivecorecursionover a coinductive data type.
The term "apomorphism" was introduced inFunctional Programming with Apomorphisms (Corecursion).[1]
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