On the Nature of Mathematical Knowledge

Abstract

An important point of contention in the philosophy of mathematics concerns the existence of mathematical objects. Platonists believe they exist independently; nominalists, that they are only linguistic constructs; formalists, that mathematics is not at all a science of objects. I believe the existence of mathematical objects is in fact immaterial for the understanding of the nature of mathematical knowledge. Mathematical truths are formal and only the formal properties of arbitrary domains of objects – whether they exist on their own or are only “intentional correlates” of their theories – matter to mathematics. This perspective has the advantage of making the applicability of mathematics in science less “unreasonable”, connecting it directly to the indifference of formal truth to material context. In this paper I intend to argue for an epistemologically relevant ontologically uncommitted formalist philosophy of mathematics (far from the “rules of the game” variety of formalism) that strips the ontological problem of its philosophical relevance and renders the applicability problem more treatable.