Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is invalid; the refutation of the non-existence does not mean that it is possible to find a constructive proof of existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
Intuitionism takes the validity of a mathematical statement to be equivalent to its having been proved; what other criteria can there be for truth (an intuitionist would argue) if mathematical objects are merely mental constructions? This means that an intuitionist may not believe that a mathematical statement has the same meaning that a classical mathematician would. For example, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one cannot assume that it is always possible to either prove the statement A or its negation. (See also intuitionistic logic.)
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 natural numbers or an arbitrary sequence of rational numbers. This requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively.
Contributors to intuitionism
Branches of intuitionistic mathematics
- Intuitionistic logic
- Intuitionistic arithmetic
- Intuitionistic type theory
- Intuitionistic set theory
- Intuitionistic calculus