Quine's double standard: undermining the indispensability argument via the indeterminacy of reference

Abstract

Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indispensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which reference is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I argue that these two arguments are in conflict with each other. Whereas the indispensability argument supports realism about mathematics, the indeterminacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should
be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics.