Arithmetical pluralism and the objectivity of syntax

Abstract

Arithmetical pluralism is the view that there is not one true arithmetic but rather many apparently conflicting arithmetical theories, each true in its own language. While pluralism has recently attracted considerable interest, it has also faced significant criticism. One powerful objection, which can be extracted from Parsons (2008), appeals to a categoricity result to argue against the possibility of seemingly conflicting true arithmetics. Another salient objection raised by Putnam (1994) and Koellner (2009) draws upon the arithmetization of syntax to argue that arithmetical pluralism is inconsistent with the objectivity of syntax. First, we review these arguments and explain why they ultimately fail. We then offer a novel, more sophisticated argument that avoids the pitfalls of both. Our argument combines strategies from both objections to show that pluralism about arithmetic entails pluralism about syntax. Finally, we explore the viability of pluralism in light of our argument and conclude that a stable pluralist position is coherent. This position allows for the possibility of rival packages of arithmetic and syntax theories, provided that they systematically co‐vary with one another.