Ultimately, mathematical intuitionism gets its name and its epistemological parentage from a conviction of Kant: that intuition reveals basic mathematical principles as true a priori. Intuitionism’s mathematical lineage is that of radical constructivism: constructive in requiring proofs of existential claims to yield provable instances of those claims; radical in seeking a wholesale reconstruction of mathematics. Although partly inspired by Kronecker and Poincaré, twentieth-century intuitionism is dominated by the ‘neo-intuitionism’ of the Dutch mathematician L.E.J. Brouwer. Brouwer’s reworking of analysis, paradigmatic for intuitionism, broke the bounds on traditional constructivism by embracing real numbers given by free choice sequences. Brouwer’s theorem – that every real-valued function on a closed, bounded interval is uniformly continuous – brings intuitionism into seeming conflict with results of conventional mathematics.
Despite Brouwer’s distaste for logic, formal systems for intuitionism were devised and developments in intuitionistic mathematics began to parallel those in metamathematics. A. Heyting was the first to formalize both intuitionistic logic and arithmetic and to interpret the logic over types of abstract proofs. Tarski, Beth and Kripke each constructed a distinctive class of models for intuitionistic logic. Gödel, in his Dialectica interpretation, showed how to view formal intuitionistic arithmetic as a calculus of higher-order functions. S.C. Kleene gave a ‘realizability’ interpretation to the same theory using codes of recursive functions. In the last decades of the twentieth century, applications of intuitionistic higher-order logic and type theory to category theory and computer science have made these systems objects of intense study. At the same time, philosophers and logicians, under the influence of M. Dummett, have sought to enlist intuitionism under the banner of general antirealist semantics for natural languages.
Logical Relativism Through Logical Contexts
