Reflections on Mathematics

Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and epistemological issues concerning mathematics. My answers will be framed from within a single conceptual framework. By ‘conceptual framework’, I mean an explicit and formal listing of primitive notions and first principles, set within a well-understood background logic. In what follows, I shall assume the primitive notions and first principles of the (formalized and) axiomatized theory of abstract objects, which I shall sometimes refer to as ‘object theory’. 1 These notions and principles are mathematics-free, consisting only of metaphysical and logical primitives. The first principles assert the existence, and comprehend a domain, of abstract objects, and in this domain we can identify (either by definition or by other means) logical objects, natural mathematical objects, and theoretical mathematical objects. These formal principles and identifications will help us to articulate answers not only to the five questions explicitly before us, but also to some of the other fundamental questions in the philosophy of mathematics raised below