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Intuitionism, also known as constructivism, is an approach to mathematics in which mathematical theorems are proven by the construction of the mathematical object whose existence is asserted. |
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Intuitionism, or neointuitionism (opposed to preintuitionism?), is an approach to mathematics studying constructive mental activity of humans. Any mathematical object is considered to be a product of a construction of a mind, and therefore, an existence of an object is equivalent to the possibility of its construction. It contrasts with classical approach stating that the existence can be proved by refuting its non-existence and applying the law of the excluded middle. |
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Constructive mathematics is opposed to classical mathematics, in which it is perfectly legal to prove an object exists by assuming it does not and deriving a contradiction. Such proofs are called 'non-constructive'. Formally, inuititionistic mathematics is equal to classical mathematics, but without the logical law of the excluded middle. |
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Intuitionism also rejects the [abstraction of actual infinity]?, i.e. it does not consider as given objects infinite entities such as the set of all naturals or an arbitrary sequence of rationals. This requires the reconstruction of the most part of set theory and calculus, leading to theories highly differing from their originals. |
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The Dutch mathematician [L.E.J. Brouwer]? is considered the father of intuitionism. |
Mathematicians having contributed to intuitionism* [L. E. J. Brouwer]? * [A. Heyting]? * Stephen Kleene |
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Although few mathematicians nowadays can be regarded intuitionists, a constructivist proof is considered superior to a non-constructivist proof of the same theorem. |
Branches of intuitionistic mathematics[Intuitionistic logic]? -- [Intuitionistic arithmetic]? -- [Intuitionistic set theory]? -- [Intuitionistic calculus]? |
See also[Mathematical constructivism]?, ultraintuitionism? |
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