Mathematical Intuitionism

Mathematical intuitionism is the philosophy of mathematics associated with L.E.J. Brouwer, holding that mathematics is a mental construction and that mathematical objects exist only insofar as they are constructible by the human mind. On this view, a mathematical statement is true only if there exists a constructive proof of it — a proof that exhibits the object or procedure in question, rather than merely ruling out its non-existence by contradiction.

Intuitionism rejects the law of excluded middle as a general principle: to assert that "either P or not-P" holds for arbitrary P is, for the intuitionist, to claim that every mathematical question is in principle decidable — a claim that has not been and cannot be established. Brouwer's insight was that classical logic, developed for reasoning about finite domains, had been illegitimately extended to the infinite.

The pragmatist challenge intuitionism has never fully answered: if mathematics is a mental construction, how does it achieve the intersubjective stability that makes mathematical communication possible? Two minds constructing the same number — do they construct the same object? Brouwer's answer, involving temporal intuition and the "creating subject," remains one of the most contested foundations in all of philosophy of mathematics.