Intuitionism - History of Intuitionism

History of Intuitionism

Intuitionism's history can be traced to two controversies in nineteenth century mathematics.

The first of these was the invention of transfinite arithmetic by Georg Cantor and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker — a confirmed finitist.

The second of these was Gottlob Frege's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell, the discoverer of Russell's paradox. Frege had planned a three volume definitive work, but shortly after the first volume had been published, Russell sent Frege a letter outlining his paradox which demonstrated that one of Frege's rules of self-reference was self-contradictory.

Frege, the story goes, plunged into depression and did not publish the second and third volumes of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and van Heijenoort's commentary.

These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.

In the early twentieth century L. E. J. Brouwer represented the intuitionist position and David Hilbert the formalist position — see van Heijenoort. Kurt Gödel offered opinions referred to as Platonist (see various sources re Gödel). Alan Turing considers: "non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive". (Turing 1939, reprinted in Davis 2004, p. 210) Later, Stephen Cole Kleene brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952).