Philosophy of Mathematics
Our research in philosophy of mathematics addresses general questions regarding the epistemology and metaphysics of mathematics as well as the relationship between mathematics and other sciences. Such general questions are concerned with the following issues:
- Mathematics and the Sciences: Are pure mathematics and empirical science fundamentally similar or different enterprises? Why is mathematics so useful in the sciences? Is there any justification for mathematics aside from its scientific applications? Are there mathematical explanations of physical phenomena? To what extent can we formulate scientific theories without using mathematics? To what extent can mathematics be included in a naturalistic view of science? Can we draw conclusions about mathematical ontology from the role mathematics plays in scientific inquiries?
- Metaphysics and Epistemology: Do mathematical truths and objects (like sets, numbers, graphs or groups) exist? If so, what kinds of things are they? How do we gain epistemic access to them? Is there ultimately only one kind of mathematical object (e.g. sets) from which all the rest are made? Are there metaphysical dependence relations between distinct mathematical objects? How is isomorphism related to identity? Is mathematical structuralism true, and if so what is a structure? What (if anything) can we learn from comparing mathematics to other (a priori) domains such as ethics and metaphysics? Is mathematics a paradigm for human reasoning and knowledge? What is an adequate methodology for drawing mathematical analogies?
- Mathematical Proofs: What makes a mathematical argument a proof? What is the relationship between informal proofs (the kind mathematicians actually write down) and derivations in a formal language? Are there, or could there be, acceptable non-deductive proofs? Why do mathematicians look for many different proofs of the same theorem? Can a picture be a proof? Why do some proofs seem to explain their results, while others merely prove without explaining?
- Mathematical Pluralism and Philosophy of Set Theory: Is mathematical truth a univocal notion, or can we adopt a pluralist view in which there are many, mutually contradictory mathematical truths? If so, how does this affect our understanding of the ontology of mathematics? Can we equate existence in mathematics with mere consistency, as some pluralists suggest? Or should we adopt a more restricted account on which only some mathematical “universes” qualify as members of the “multiverse” of mathematical realities? If so, what properties should these universes have? And do proponents of the so-called “universist” view that there is only one mathematical reality have any convincing responses to the question of pluralism?
- Mathematical Practice: What are mathematicians trying to achieve, what do they value, and why? Which mathematical practices vary or stay the same in different times and places? What role do qualities like beauty, elegance, naturalness, depth, transparency and explanatoriness play in mathematical research? How do definitions, axioms, theories and research programs gain acceptance? How much of working mathematicians’ knowledge comes from testimony, “folk theorems” and other social sources? How do computers contribute to generating and checking proofs? Should the role of computers be restricted to carrying out computations that are infeasibly large for humans, checking existing proofs, and filling in small logical gaps? Or should computers assume all of the roles traditionally taken by human mathematicians, including formulating conjectures and proposing new axioms?
Members of faculty working in philosophy of mathematics:
- Marianna Antonutti Marfori
- Ivano Ciardelli
- William D’Alessandro
- Daniel Di Liscia
- Neil Dewar
- John Dougherty
- Benedict Eastaugh
- Norbert Gratzl
- Silvia Jonas
- Hannes Leitgeb
- Robert Prentner
Doctoral fellows working in philosophy of mathematics: