Rethinking Mathematical Knowledge: A Nativist Approach

Abstract

This paper develops a nativist approach to mathematical knowledge as an alternative account within the philosophy of mathematics. Rather than beginning with the ontological status of mathematical objects, it argues that any satisfactory account of mathematical knowledge should address three closely related questions: how to understand and justify truth-talk in mathematical practice, how mathematical knowledge is acquired, and why some mathematical knowledge has indispensable applications in the sciences. The paper first examines representative nominalist and Platonist responses to these questions. Nominalist accounts, while avoiding the specific epistemic difficulty associated with abstract objects, face challenges in explaining the non-arbitrariness of mathematical practice and the indispensability of mathematics in certain scientific theories. Platonist accounts, while offering a straightforward semantics of mathematical truth, still face the familiar epistemic problem of explaining how we acquire knowledge of abstract mathematical entities and require a fuller account of the applicability of mathematics to the empirical sciences. Against this background, the paper proposes two nativist theses: that some mathematical knowledge is innate, and that mathematics may function more fundamentally as a cognitive pattern than as a cognitive capacity. On this basis, mathematical knowledge is construed as cognitively dependent, while mathematics itself is understood as underlying our representations of the world. The paper concludes that the nativist account provides a framework for reconsidering the legitimacy of truth-talk in mathematical practice, the acquisition of mathematical knowledge, and the applicability and indispensability of some mathematics to sciences.