By Adelaide Gill

Faculty Mentor: Dr. Reno

Abstract

This paper takes up Jerrold J. Katz’s rational realism and presses on its central tension from the outset. If mathematical truths hold with necessity and do not depend on the world, then they resist any account that ties them to experience or convention, yet they still demand an account of how we come to know them. I argue that attempts to ground mathematics in empirical practice or linguistic convention cannot sustain the force of necessity that even the simplest mathematical judgments carry, and that Katz is right to place their veracity in reason itself, though his account leaves the route from thought to object underdeveloped.

To sharpen this route and respond to contemporary criticism, the paper returns to the rationalist lineage that precedes Katz to recover the claim that thought does not merely register mathematical truth but gives the conditions under which it can appear at all. On this view, necessity does not hover over mathematics as a feature to be explained after the fact. It enters with the act of understanding itself and sets the terms within which anything can count as a mathematical object.

From here I recast Katz’s position so that knowledge of abstract objects no longer turns on a separate faculty or an obscure form of access. Instead, it unfolds from the constraints that govern intelligible thought. These constraints do not create mathematical objects, but they do fix the space in which such objects can be grasped. Mathematics thus emerges neither as invention nor as discovery in the empirical sense, but as the articulation of what reason already commits us to.