On Mathematics’ ‘Indispensable Explanatory Role’∗

The literature on the indispensability argument for mathematical realism often refers to mathematics’ ‘indispensable explanatory role.’ I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically committing, others ‘thin’ and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments.