Abstract

Ante rem structuralism is a version of mathematical structuralism presented by Shapiro (1997) that asserts the existence of mathematical objects. It stands out amongst other structuralist views in that it secures a face value semantics for mathematical statements. In this paper, I propose an alternative ontology where structures are fully explained by relations. I argue that this alternative ontology avoids arguments against ante rem structuralism’s endorsement of indiscernible entities while retaining the convenience of a face value semantics for mathematical expressions. This ontology is based on mereological nihilism and the Huayan Buddhist argument that an object can be fully described by its relations to other objects. With this metaphysical backdrop, I provide rules that configure an infinite set of successor relations into a structure that is functionally equivalent to Shapiro’s natural number structure. To show that the natural number structure of my ontology is functionally equivalent to Shapiro’s, I explain how the same mathematical expressions of Shapiro’s ontology are understood in my ontology and show that the natural number structure of my ontology is compatible with both addition and multiplication. This allows two equally valid semantics for mathematical expressions such that my ultimate semantics justifies the structures rejection of the PII while the conventional semantics of mathematical practice can treat numbers as bona fide objects.