Structuralism is a philosophical position that emphasizes the importance of the relations that objects stand in as opposed to what the objects are in themselves. Structuralism has risen to importance both in philosophy of mathematics and philosophy of science. On the mathematical side, mathematical structuralism favors a characterization of the subject matter of mathematics in terms of, say, the structure of natural numbers, not in terms of natural numbers themselves. On the scientific side, structural realists claim that the subject matter of theories in physics such as quantum mechanics and relativity theory is best understood in terms of relations, or even that relations are all there is according to these theories. What has been neglected, however, is the question of how mathematical and physical structures are related, and what the use of mathematics in the application to the physical world is. The goal of this project is to take some steps in the analysis and solution of these questions. In broad terms, we want to contribute to a better philosophical understanding of the prominent role that mathematics plays in empirical science.
We start with the mathematical side of the story by critically analyzing a recent proposal of how to conceive of mathematical structures. Then we focus on how mathematical and physical structure, and mathematics and the empirical domain in general are related. Here we will first have to make some conceptual distinctions between different ways in which mathematics and the world can be said to be related. We will for example explore the idea that the main difference between mathematical and physical structure is causality, i.e. the fact that mathematical structures such as the natural numbers do not cause anything, whereas physical structures are thought to stand in causal relations. We will also scrutinize the plausible idea that the primary function of mathematical structure is to represent, i.e. stand for physical structure in scientific theories. We hope to highlight strengths and weaknesses of this idea. To give an example, it could be that if mathematics is only there to stand for something else, it cannot help in giving explanations, which is not plausible.
As the subject matter of our philosophical investigations is science, we will work with several scientific case studies which will guide us and also help us check whether philosophical reasoning has lead us astray. We will use several smaller case studies, but also big cases, one from the early history of general relativity and one from quantum mechanics.