Platonism and intra-mathematical explanation

Abstract

Abstract I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails, it can only do so for one of the three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because some form of noneism—the view according to which non-existent entities possess properties and stand in relations—is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation, and a backing conception of explanation. You can have any two, but not all three.