Project

quantum mechanics; general relativity; mathematical structuralism; applicability of mathematics; structural realism; metaphysics of science; causation

Lead
Lay summary
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.

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Last update: 21.02.2013

Self-organised

Number	Title	Start	Funding scheme

This project is situated in the context of structural realism in philosophy of science. In the debates surrounding ontic structural realism (OSR), it has become increasingly clear that there is a need to clarify the distinction between mathematical and physical structures. Our main goal is to take some steps in the analysis and solution of this problem. We will divide the problem into the following parts. First, we will discuss how we should conceive of mathematical structures. Second, we will scrutinize the idea that the distinction between mathematical and physical structures can be spelled out in causal terms, that is, we characterize physical structures.It has proven useful to broaden the context in which we tackle these questions. For the question of how mathematical structures are best characterized, we critically discuss a solution from philosophy of mathematics, namely mathematical structuralism, using case studies from graph theory and complex number. Then, the recent debate on the applicability of mathematics by authors such as Mark Steiner, Mark Colyvan and Christopher Pincock is a fruitful setting for the general question of how mathematical and physical domain are related. The most important philosophical lessons from this debate are the following points: a) Several senses in which mathematics can be said to be related to the world should be carefully distinguished. Especially the distinction between the metaphysical relation between acausal-mathematical and causal-physical structures on the one hand and the descriptive adequacy of mathematics in application, e.g. in explanations, can help to disentangle the purported `mystery' of how mathematics is helpful in application; b) it has sometimes been claimed that the role of mathematics is a merely representational one, that is, the relation between mathematics and the world is like the relation between a map and the city it depicts. We are sceptical that this is universally true, and advocate a more pluralistic picture of the role of mathematics, e.g. as explanatory.We will confront our philosophical findings with scientific practice in several case studies. First, we will use `toy examples' from the debate on the applicability of mathematics such as the `Königsberg bridges' case and from the debate on the indispensability of mathematics, e.g. periodic cicadas. Second, we will present two `real life' case studies from fundamental physics, one from the early history of general relativity, one from quantum mechanics. These case studies will connect our results to discussions in philosophy of physics.Our research will result in the PhD thesis of Tim Räz, in publications by Michael Esfeld and Tim Räz including a joint publication.