Abstract
One of Peirce's most important and influential contributions to pragmatism is his reconceptualization of traditional metaphysical ideas like reality in terms of human inquiry. Whether we consider his definition of reality as the external permanence which gives science its self-corrective, or his definition of reality as the object of the final opinion in the long run, Peirce clearly intends to connect his idea of reality to the practice of inquiry. In this paper, we will consider some of the difficulties of applying the same pragmatic argument when it comes to the reality of mathematics. First, there is the difficulty of discerning what Peirce's position is regarding the reality of mathematics, since there are several disparities in his writings. Second, there is the difficulty of which view regarding the reality of mathematics serves inquiry best. In other words, must a mathematician suppose he studies real entities in order to practice mathematics? In considering this problem, we will examine the Quine-Putnam Indispensability thesis, as well as Angus Kerr-Lawson's bicategorial view. We will conclude that Peirce's philosophy of mathematics only apparently commits him to a bicategorial ontological dualism, and that is more monistic view-which is more agreeable to the Indispensability Thesis-is ultimately preferable.