Univalent foundations as structuralist foundations

The Univalent Foundations of Mathematics (UF) provide not only an entirely non-Cantorian conception of the basic objects of mathematics (“homotopy types” instead of “sets”) but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal system must satisfy if it is to be regarded as a “structuralist foundation.” I will explain why neither set-theoretic foundations like ZFC nor category-theoretic foundations like ETCS are structuralist in my sense, essentially because of an undue emphasis on ontology at the expense of language. Then I will argue that UF is better-able to live up to the proposed criterion for a structuralist foundation than any currently available foundational proposal. First, by showing that most criteria of identity in the practice of mathematics can be formalized in terms of the preferred criterion of identity between the basic objects of UF (“homotopy equivalence”). Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics. ∗Department of Philosophy, Princeton University. e-mail: [email protected] I would like to thank (in random order) John Burgess, Paul Benacerraf, Hans Halvorson, Steve Awodey, Mike Shulman, Chris Kapulkin, Vladimir Voevodsky, Colin McLarty, David Corfield, Richard Williamson, Urs Schreiber, Harry Crane, as well as audiences in Philadelphia, London and Princeton. Lastly, I would like to single out in thanks an anonymous referee, who provided extremely detailed and illuminating comments that resulted in many significant improvements.