30 March 2021: Lavinia Picollo (joint work with Dan Waxman)

Mathematical Pluralism Revisited 

Abstract: Benacerraf (1973) posed the following challenge for mathematical realism: how can we have knowledge of mathematical objects, given that they lie outside the causal realm? Balaguer (1995) famously argued that full-blooded Platonism, the view that every consistent mathematical theory truly describes a part of the mathematical realm, can offer a successful naturalistic response to the challenge. Pluralist views such as Balaguer’s have attracted much interest but have also been the subject of significant criticism, most saliently from Putnam (1979) and Koellner (2009). These critics argue that,  due to the possibility of arithmetizing the syntax of mathematical languages, one cannot coherently say that mathematics is a matter of ‘taste’ whilst consistency is a matter of fact. In response, Warren (2015) argued that Putnam’s and Koellner’s argument relies on a misunderstanding, and that it is in fact coherent to maintain a pluralist conception of mathematical truth while supposing that consistency is a matter of fact. We argue that it is not. We put forward a modified version of Putnam’s and Koellner’s argument that isn’t subject to Warren’s criticisms