Some philosophers of mathematics have argued that mathematics “cannot be regarded as a priori” (Ferreirós 2016, 310). The argument points to mathematical practices, and in particular to cases in which mathematicians successfully justify the truth of a mathematical proposition. It is argued that these practices often fail to meet the rigorous standards of apriorism.

For example, Lakatos (1976ab) argues that mathematical justifications are tentative, fallible, and sometimes even plainly empirical. Putnam (1979, 64) claims that mathematicians often use “quasi-empirical” justifications that are not so different from those used in the sciences. More recently, such claims have been echoed in historical studies (Ferreirós 2016), in questions about the admissibility of probabilistic and computational proofs (Fallis 1997; McEvoy 2007), and in the distinction between proofs and simil-proofs (De Toffoli 2021).

In this talk I will argue that these attempts to establish aposteriorism are mistaken about the a priori/a posteriori distinction. Mathematical practices cannot give us the kind of evidence needed to settle whether mathematics is a priori or a posteriori. I will argue that doing so requires modal evidence, whereas practices can only give us factual evidence.