Abstract

Mathematical pluralism is roughly the view that any coherent foundational theory is equally correct. This is rough because a careful formulation must avoid contradiction (how could a theory and its negation both be correct?), distinguish foundational theories from algebraic ones, differ from fictionalism, and accommodate the indefinite extensibility of coherence. Even after these repairs, however, pluralism faces an insuperable problem of articulation. I consider several attempts to circumvent this problem, and argue that each fails. Their failure is parallel to the self-refutation problem for global relativism. I propose instead to analyze pluralism on the model of Carnap’s Principle of Tolerance (POT), interpreted as a non-factual policy proposal. Drawing on quasi-realism and the deflationism about truth that it presupposes, I show how such a policy can be truth-apt and embedded in complex sentences without inflating it into a factual thesis. The important point is that the considerations that support or refute it are practical, not evidential. Along the way, I isolate an underappreciated shadow of the problem of articulation, the problem of mathematical agreement. Even what follows from what in a fixed logic – indeed, even whether an explicitly given string is a proof in a logic – can depend on the foundation we use to check. So, why do we agree about mathematics to the extent that we do? I close by showing how pluralism leads to a limited non-factualist pluralism about physics. In particular, whether a discrete Ising system behaves deterministically from almost all starting states provably switches between ZF + V=L and ZFC + Large Cardinals (sufficient to imply Projective Determinacy).