ABSTRACT
The alleged existence of mathematical explanations of physical phenomena presents a prima facie difficulty for mathematical nominalism. If the best explanation of a physical phenomenon appeals to abstract mathematical objects, then nominalists owe an account of why they are able to accept such an explanation given that they do not accept the existence of abstracta. In some cases, where an explanation is couched in mathematical terms, it is open for a nominalist to argue that all the explanatory work resides in the explanation’s nominalistic content, and that it is this content that the nominalist believes when she accepts the mathematically expressed explanation. However, Christopher Pincock suggests that there are some cases of mathematical explanations where the explanatory work does not reside in the nominalistic content of the explanations in question, but rather in more general structural features that the mathematics allows us to recognize. I agree with Pincock that in such cases mathematics does play a genuine explanatory role. However, I argue, we can understand how such structural explanations work from a nominalistic perspective, as the success of these explanations requires only that (a) the theorems of pure mathematics tell us what would have to be true in any system of objects satisfying the axioms of a given mathematical theory, and (b) the axioms of that theory are approximately true of the physical system to which the theory is applied.
