Abstract

Kant’s philosophy of mathematics presents two fundamental problems of interpretation: (1) Kant claims that mathematical truths or “judgments” are synthetic a priori; and (2) Kant maintains that intuition is required for generating and/or understanding mathematical statements. Both of these problems arise for us because of developments in mathematics since Kant. In particular, the axiomatization of geometry--Kant’s paradigm of mathematical thinking--has made it seem to some commentators as, for example, Russell, that both (1) and (2) are false (Russell 1919, p. 145).2 If virtually all of mathematics, including geometry, is axiomatizable, it would seem that mathematics results in analytic judgments that are totally independent of sensibility, the source, according to Kant, of intuition. In this paper I will address both of these difficulties. I shall argue that Kant’s understanding of both “synthetic” and “intuition” make his position immune to these criticisms.