Descriptivism about the Reference of Set-Theoretic Expressions: Revisiting Putnam’s Model-Theoretic Arguments

Abstract

Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. These anchors add enough to the content of mathematical expressions to forestall the radical kind of indeterminacy that model-theoretic arguments are purported to show, while still leaving room for a plausible, moderate kind of indeterminacy.