Intuitionism

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Truth and proof

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill-defined. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he proves is in fact intuitionistically true. This gives rise to intuitionistic logic.