Ante rem structuralism and the semantics of instantial terms

Ante rem structures were posited as the subject matter of mathematics in order to resolve a problem of referential indeterminacy within mathematical discourse. Nevertheless, ante rem structuralists are inevitably committed to the existence of indiscernible entities, and this commitment produces an exactly analogous problem. If it cannot be sorted out, then the postulation of ante rem structures is futile. In a recent paper, Stewart Shapiro argued that the problem may be solved by analysing some of the singular terms of mathematics not as genuinely referring expressions, but as instantial terms. In this paper, I discuss several competing accounts of the semantics of terms of this kind, and argue that they are all untenable for the ante rem structuralist. Shapiro, then, still owes us an account of the semantics of instantial terms that suits the ante rem structuralist project. Without it, the ante rem structuralist is still unable to determine the reference of the singular terms of mathematics.