Fictionalist mathematical structuralism
Mathematics is ubiquitous throughout the sciences and in everyday language and thought. Its applications across all disciplines - science, economics, engineering, computing and more - are central to the future development of a robust, creative, and driven society. 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 in the philosophy of mathematics, viz. ""mathematical structuralism"" and ""mathematical fictionalism"". Both structuralism and fictionalism capture important aspects of how many working mathematicians approach their practice. But in contemporary philosophy of mathematics, structuralism and fictionalism are widely thought of as presenting competing views of the nature of mathematics. In contrast, this project will develop a new theory of mathematical structure that unifies structuralism and fictionalism, thereby shifting the debate and promoting new and fruitful lines of research in the philosophy of mathematics and mathematical practice. Crucially, this research will be informed and supported by interdisciplinary discussions of mathematical practice with mathematicians, mathematical scientists, linguists who work on meaning and reference, 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.