Talk: John Wigglesworth (York)

Title:

Practice-based philosophy of mathematics: A case study in nominalism

Abstract:

Traditional philosophy of mathematics focuses on the metaphysics, semantics, and epistemology of mathematics. Philosophy of mathematical practice tends to focus on different areas of research, many of which emphasize that mathematics is a human activity, and which give a central role to the practitioner. This paper explores connections between these two areas, arguing that methods in mathematical practice can have direct implications for certain approaches to traditional philosophy of mathematics. Looking at the area of mathematical nominalism, we focus on two prominent nominalist approaches: mathematical fictionalism and modal structuralism. While both are nominalist approaches, they are incompatible in important ways, particularly with respect to mathematical truth. This article advances the debate between fictionalism and modal structuralism by emphasizing a specific aspect of mathematical practice, namely the cognitive process of mathematical abstraction. We argue that fictionalism offers a more comprehensive explanation of mathematical abstraction, when compared to modal structuralism. We describe four key components of mathematical abstraction and demonstrate how fictionalism, by appealing to cognitive processes such as pretense, imagination, and make-believe, can provide a mechanism that explains how these components work. In contrast, modal structuralism, as a mainly formal framework, lacks any mechanism to account for this aspect of mathematical practice. For these reasons, by taking mathematical practice into consideration, we conclude that fictionalism has a theoretical edge over modal structuralism.