Explication as Elimination: W. V. Quine and Mathematical Structuralism

Abstract

This chapter examines the development of and motives for Quine’s particular form of mathematical structuralism. It will argue that Quine, unlike many contemporary mathematical structuralists, does not appeal to structuralism as a way of accounting for what the numbers really are in any robust metaphysical sense. Instead, his structuralism is deeply rooted in an earlier structuralist tradition found in scientific philosophers such as Russell and Carnap, which emphasized structuralism as a critique of more metaphysical approaches to philosophy. On this view, a philosophy of mathematics answers, in a sense, only to mathematics itself. An account of mathematical objects requires only that the entities—whatever they are—serving as the mathematical objects satisfy the relevant postulates and theorems. Here we also see how Quine’s early work in the foundations of mathematics leads in a natural way to the more general naturalism of his later philosophy.

The structuralist approach to underdetermination

“…That is, "what Quine claims can, in effect, be expressed so that the semantic structure of any language allows for some nontrivial automorphisms that leave any detectable semantic properties intact"(Peregrin, 2020, p. 86). Also, seeMorris (2020) for an exegesis of Quine's structuralist thoughts in mathematics.…”