Weaseling away the indispensability argument

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense wi th such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I a rgue that, as a strategy, this response fails: for there is no guarantee th at the nominalist world that go beyond the set of sentences in the nominali st language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the c oncrete world that may not be expressible in nominalistically acceptable la nguage. While I grant that this may make quantification over abstracta indi spensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantif y over abstracta whilst denying their existence and I explain how this appa rently contradictory practice (a practice I call 'weaseling') is in fact co herent, unproblematic and rational. Finally, I examine the view that platon istic theories are simpler or more attractive than their nominalistic refor mulations, and thus that abstract ought to be accepted into our ontology fo r the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderst and the kind of simplicity and attractiveness we seek.