Constructivism (philosophy of mathematics)
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| - | '''Intuition''' is ''[[A priori and a posteriori|a priori]]'' knowledge or experiential belief characterized by its immediacy. Beyond this, the nature of intuition is debated. Roughly speaking, there are two main views. They are: | + | In the [[philosophy of mathematics]], '''constructivism''' asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard mathematics, one can prove the [[existence]] of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a [[contradiction]] from that assumption. This [[proof by contradiction]] is not constructively valid. The constructive viewpoint involves a verificational interpretation of the [[existential quantifier]], which is at odds with its classical interpretation. |
| - | # Intuitions are a priori. This view holds that distinctions are to be made between various sorts of intuition, roughly corresponding to their subject matter (see George Bealer). The only intuitions that are relevant in analytic philosophy are 'rational' intuitions. These are intellectual seemings that something is necessarily the case. They are directed exclusively towards statements that make some kind of necessity claim. For example, a rational intuition is what occurs when it seems to us that a mathematical statement (e.g. 2+2=4) must be true. Intuitions as this view characterizes them are to be distinguished from [[beliefs]], since we can hold beliefs which are not intuitive, or have intuitions for [[propositions]] that we know to be false. | ||
| - | # Intuitions are a species of belief, and based ultimately in experience. This view holds that intuitions are not especially different from beliefs, although they appear subjectively to be more unrevisable than other beliefs. Unlike the previous view, these intuitions are liable to differ between social groups. Evidence for this is shown in various psychological studies (e.g. the one by Stich, Weinburg and Nichols) | ||
| - | In the [[philosophy]] of [[Immanuel Kant]], pure intuition is one of the basic [[cognitive]] faculties, equivalent to what might loosely be called [[perception]]. Kant held that our [[mind]] casts all of our external intuitions in the form of [[space]], and all of our internal intuitions ([[memory]], thought) in the form of time. | + | == Mathematicians who have made major contributions to constructivism == |
| - | + | * [[Leopold Kronecker]] (old constructivism, semi-intuitionism) | |
| - | [[Intuitionism]] is a position advanced by [[Luitzen Egbertus Jan Brouwer]] in [[philosophy of mathematics]] derived from Kant's claim that all [[mathematics|mathematical knowledge]] is knowledge of the pure forms of the intuition - that is, intuition that is not empirical (''Prolegomena, p.7''). [[Intuitionistic logic]] was devised by [[Arend Heyting]] to accommodate this position (and has been adopted by other forms of [[constructivism (mathematics)|constructivism]] in general). It is characterized by rejecting the [[law of excluded middle]]: as a consequence it does not in general accept rules such as [[double negation elimination]] and the use of [[reductio ad absurdum]] to prove the existence of something. | + | * [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] (forefather of intuitionism) |
| - | + | * [[Andrey Markov (Soviet mathematician)|A. A. Markov]] (forefather of Russian school of constructivism) | |
| - | ===Western philosophy=== | + | * [[Arend Heyting]] (formalized intuitionistic logic and theories) |
| - | In the West, intuition does not appear as a separate field of study, and early mention and definition can be traced back to [[Plato]]. In his book ''[[Plato Republic|Republic]]'' he tries to define intuition as a fundamental capacity of human [[reason]] to comprehend the true nature of [[reality]].<ref name="Plato">{{cite web |title=Plato: Education and the Value of Justice |url=http://www.philosophypages.com/hy/2h.htm |accessdate= 22 December 2014 }}</ref> In his works ''[[Meno (Plato)|Meno]]'' and ''[[Phaedo (Plato)|Phaedo]]'', he describes intuition as a pre-existing knowledge residing in the "soul of eternity," and a phenomenon by which one becomes conscious of pre-existing knowledge. He provides an example of mathematical truths, and posits that they are not arrived at by reason. He argues that these truths are accessed using a knowledge already present in a dormant form and accessible to our intuitive capacity. This concept by Plato is also sometimes referred to as [[Anamnesis (philosophy)|anamnesis]]. The study was later continued by his [[Neo-platonism|followers]]. | + | * [[Per Martin-Löf]] (founder of constructive type theories) |
| - | + | * [[Errett Bishop]] (promoted a version of constructivism claimed to be consistent with classical mathematics) | |
| - | In his book ''[[Meditations on First Philosophy]]'', [[René Descartes|Descartes]] refers to an intuition as a pre-existing knowledge gained through rational reasoning or discovering truth through contemplation. This definition is commonly referred to as rational intuition. Later philosophers, such as [[David Hume|Hume]], have more ambiguous interpretations of intuition. Hume claims intuition is a recognition of relationships (relation of time, place, and causation) while he states that "the resemblance" (recognition of relations) "will strike the eye" (which would not require further examination) but goes on to state, "or rather in mind" – attributing intuition to power of mind, contradicting the theory of [[empiricism]]. | + | * [[Paul Lorenzen]] (developed constructive analysis) |
| - | + | ||
| - | [[Immanuel Kant]] finds intuition is thought of as basic sensory information provided by the [[cognitive]] faculty of sensibility (equivalent to what might loosely be called [[perception]]). Kant held that our [[mind]] casts all of our external intuitions in the form of [[space]], and all of our internal intuitions ([[memory]], thought) in the form of time. [[Intuitionism]] is a position advanced by [[Luitzen Egbertus Jan Brouwer]] in [[philosophy of mathematics]] derived from Kant's claim that all [[mathematics|mathematical knowledge]] is knowledge of the pure forms of the intuition – that is, intuition that is not empirical. [[Intuitionistic logic]] was devised by [[Arend Heyting]] to accommodate this position (and has been adopted by other forms of [[constructivism (mathematics)|constructivism]] in general). It is characterized by rejecting the [[law of excluded middle]]: as a consequence it does not in general accept rules such as [[double negation elimination]] and the use of [[reductio ad absurdum]] to prove the existence of something.{{citation needed|date=August 2015}} | + | |
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| - | Intuitions are customarily appealed to independently of any particular theory of how intuitions provide evidence for claims, and there are divergent accounts of what sort of mental state intuitions are, ranging from mere spontaneous judgment to a special presentation of a necessary truth. However, in recent years a number of philosophers, especially [[George Bealer]] have tried to defend appeals to intuition against [[Willard Van Orman Quine|Quinean]] doubts about [[conceptual analysis]]. A different challenge to appeals to intuition has recently come from [[experimental philosophy|experimental philosophers]], who argue that appeals to intuition must be informed by the methods of social science.{{citation needed|date=August 2015}} | + | |
| - | + | ||
| - | The [[metaphilosophy|metaphilosophical]] assumption that philosophy depends on intuitions has recently been challenged by some philosophers. [[Timothy Williamson]] has argued that intuition plays no special role in philosophy practice, and that skepticism about intuition cannot be meaningfully separated from a general [[skepticism]] about judgment. On this view, there are no qualitative differences between the methods of philosophy and common sense, the sciences or mathematics. | + | |
| + | == Branches == | ||
| + | * [[Constructive logic]] | ||
| + | * [[Constructive type theory]] | ||
| + | * [[Constructive analysis]] | ||
| + | * [[Constructive non-standard analysis]] | ||
| == See also == | == See also == | ||
| - | *[[Intuition (Bergson)]] | + | * [[Computability theory]] |
| - | *[[Intuition (knowledge)]] | + | * [[Constructive proof]] |
| - | *[[Intuitionist logic]] | + | * [[Finitism]] |
| + | * [[Game semantics]] | ||
| + | * [[Intuitionism]] | ||
| + | * [[Intuitionistic type theory]] | ||
| {{GFDL}} | {{GFDL}} | ||
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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. This proof by contradiction is not constructively valid. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
[edit]
Mathematicians who have made major contributions to constructivism
- Leopold Kronecker (old constructivism, semi-intuitionism)
- L. E. J. Brouwer (forefather of intuitionism)
- A. A. Markov (forefather of Russian school of constructivism)
- Arend Heyting (formalized intuitionistic logic and theories)
- Per Martin-Löf (founder of constructive type theories)
- Errett Bishop (promoted a version of constructivism claimed to be consistent with classical mathematics)
- Paul Lorenzen (developed constructive analysis)
[edit]
Branches
- Constructive logic
- Constructive type theory
- Constructive analysis
- Constructive non-standard analysis
[edit]
See also
- Computability theory
- Constructive proof
- Finitism
- Game semantics
- Intuitionism
- Intuitionistic type theory
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