Intuitionism

Intuitionism is a philosophy of mathematics that treats mathematical objects as mental constructions and accepts a mathematical statement only when it can be constructed or proved.

Mathematical constructivism: A philosophy of mathematics requiring mathematical objects and results to be constructed or explicitly justified. Intuitionism is a distinctive philosophical form of the broader constructive approach.
Heyting algebra: A lattice structure that provides algebraic semantics for intuitionistic propositional logic. Its operations model intuitionistic connectives without requiring classical Boolean complementation.
L. E. J. Brouwer: A Dutch mathematician who founded intuitionism and challenged unrestricted use of classical logic in mathematics. His philosophy made mathematical construction, rather than formal deduction alone, foundational.
Classical logic: A system of logic that includes the law of excluded middle and double-negation elimination. Its unrestricted principles are precisely what intuitionistic logic revises.
Intuitionistic analysis: The study of analysis using intuitionistic logic and constructive standards of mathematical proof. It shows how rejecting classical principles affects familiar results about real numbers and continuity.
Constructive proof: A proof that establishes a claim by providing an explicit construction or effective method. Intuitionistic proofs must provide constructions rather than merely rule out counterexamples.
Kripke semantics: A relational semantics in which truth can persist across ordered states of information or possibility. It models intuitionistic truth as knowledge grows, explaining why excluded middle can fail.
Arend Heyting: A Dutch mathematician who formalized intuitionistic logic and developed intuitionistic mathematics. His systems translated Brouwer's philosophical position into explicit logical rules.
Classical mathematics: Mathematics developed using classical logic, including principles not accepted universally by intuitionists. It permits existence and case arguments that may not supply intuitionistically acceptable constructions.
Constructive mathematics: Mathematics in which proofs provide explicit constructions or computationally meaningful evidence. Intuitionist foundations helped establish construction as a standard for mathematical existence proofs.

Linked from 19 pages