How Should We Understand the Modal Potentialist’s Modality?

Abstract

Modal potentialism argues that mathematics has a generative nature, and aims to formalise mathematics accordingly using quantified modal logic. This paper shows that Øystein Linnebo’s approach to modal potentialism in his book Thin Objects is incoherent. In particular, he is committed to the legitimacy of introducing a primitive modal predicate of formulae. However, as with the semantic paradoxes, natural principles for such a predicate are inconsistent; no such predicate can underpin an account of modal potentialism. Hence, Linnebo’s intended interpretation of the primitive modality and his formal framework do not match up.