FICTIONALIST MATHEMATICAL STRUCTURALISM
A Marie Skłodowska-Curie project at the University of York
The main goal of this research project is to establish a comprehensive approach to the philosophy of mathematics that draws on mathematical practice. The abstract character of mathematics raises difficult philosophical questions about the nature of mathematical entities (such as numbers, functions, and sets), how we have knowledge of such entities, and how they can be applied in the real world. This research project tackles such questions by integrating two perspectives on the philosophy of mathematics, viz. "mathematical structuralism" and "mathematical fictionalism". Crucially, this research will be informed and supported by interdisciplinary discussions of mathematical practice with mathematicians, mathematical scientists, and psychologists whose research focuses on mathematical cognition. Drawing on aspects of their research, this project will build a new and distinctive positive theory of the nature of mathematics. With this comprehensive theory established, the project will show how it can be deployed to make progress on long-standing problems in the philosophy of mathematics and the philosophy of science.
Research on the project is based on three interrelated subprojects:
1. Structuralism and fictionalism in mathematical practice: The aim here is to explore the extent to which these approaches in the philosophy of mathematics align with mathematical practice.
2. A fictionalist theory of mathematical structure: Building on the previous investigations, this subproject develops versions of structuralism, based on structural abstraction principles, that are nominalistically acceptable.
3. Applications of fictionalist mathematical structuralism: With a comprehensive fictionalist theory of mathematical structure in hand, this subproject focuses on applications of this theory to problems in the philosophy of mathematics and the philosophy of science.