Kantian roots of modern mathematics


Many pivotal aspects of modern mathematics have their roots in concepts and methodologies which were articulated decisively by Kant (even if they didn’t originate with him).


Kantian aspects of Hilbert’s program

Hilbert’s philosophical background was broadly Kantian, as was Bernays’s, who was closely affiliated with the neo-Kantian school of philosophy around Leonard Nelson in Göttingen. Hilbert’s characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. Indeed, in Kant’s epistemology, immediacy is a defining characteristic of intuitive knowledge. The question is, what kind of intuition is at play? Mancosu (1998b) identifies a shift in this regard. He argues that whereas the intuition involved in Hilbert’s early papers was a kind of perceptual intuition, in later writings (e.g., Bernays 1928a) it is identified as a form of pure intuition in the Kantian sense. However, at roughly the same time Hilbert (1928, 469) still identifies the kind of intuition at play as perceptual. In (1931b, 266–267), Hilbert sees the finite mode of thought as a separate source of a priori knowledge in addition to pure intuition (e.g., of space) and reason, claiming that he has “recognized and characterized the third source of knowledge that accompanies experience and logic.” Both Bernays and Hilbert justify finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible. SEP

Poincare was deeply engaged with kantian and post-kantian thinking