Some Links to Notes on Intuitionism

Obviously Brouwer is responsible for the philosophy of intuitionism. In 1923, he criticized excluded middle and showed intuitionistic counterexamples for basic theorems of analysis (Heine-Borel and Bolzano-Weiestrass). Later he reconstructed portions of set theory, topology, and analysis in terms of intuitionism.

Even before Brouwer, Kronecker and Poincare expressed constructivist attitudes. Goldblatt explains the origin of intuitionism in Chapter 8 of his Topoi book.

In 1925, Kolmogorov provided the first formalization of intuitionistic logic and showed how to interpret classical logic in intuitionistic logic by a double negation transformation. Heyting's formalization of intuitionistic logic would later become standard.

Hilbert railed against Brouwer's intuitionism. For example, here is a quote from his 1927 "The foundations of mathematics".

Weyl supported Brouwer's intuitionism in 1920, but did not strictly maintain this support later. He did respond to Hilbert's rejection of Poincare's complaint about the metamathematical use of induction to prove consistency. Weyl also suggested that a defeat of intuitionism may be considered a defeat of phenomonology.

In 1927, Brouwer (sort of) responded to Hilbert by giving four "insights" which indicate where intuitionism and formalism could have a dialogue.