Editing Intuitionism (section)

== History ==

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 [[finitism|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 just as the second volume was going to press, Russell sent Frege a letter outlining his paradox, which demonstrated that one of Frege's rules of self-reference was self-contradictory.  In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell's paradox.{{refn| See {{harvnb|Frege|1960|pages=234–244}}}}

Frege, the story goes, plunged into depression and did not publish the third volume 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 [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] represented the ''intuitionist'' position and [[David Hilbert]] the [[Formalism (mathematics)|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".{{sfn|Turing|1939|page=216}} Later, [[Stephen Cole Kleene]] brought forth a more rational consideration of intuitionism in his ''Introduction to metamathematics'' (1952).{{sfn|Kleene|1991}}

[[Nicolas Gisin]] is adopting intuitionist mathematics to reinterpret [[quantum indeterminacy]], [[information theory]] and the [[time in physics|physics of time]].{{sfn|Wolchover|2020}}