Univalent foundations as structuralist foundations
- 196 Downloads
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 then explain why both set-theoretic foundations like ZFC and category-theoretic foundations like ETCS satisfy this criterion only to a very limited extent. 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.
KeywordsStructuralism Foundations of mathematics Univalent foundations
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.
- Ahrens, B., Kapulkin, K., & Shulman, M. (2015). Univalent categories and the rezk completion. In: Extended abstracts fall 2013 (pp. 75–76). Heidelberg: Springer.Google Scholar
- Bezem, M., Coquand, T., & Huber, S. (2014). A model of type theory in cubical sets. In: 19th International Conference on Types for Proofs and Programs (TYPES 2013) (Vol. 26, pp. 107–128).Google Scholar
- Burgess, J. (2013). Putting structuralism in its place. Manuscript.Google Scholar
- Burgess, J. (2014). Rigor and structure. Oxford: Oxford University Press.Google Scholar
- Cohen, C., Coquand, T., Huber, S., & Mörtberg, A. (2015). Cubical type theory: A constructive interpretation of the univalence axiom. https://www.math.ias.edu/~amortberg/papers/cubicaltt.pdf.
- Freyd, P. (1976). Properties invariant within equivalence types of categories. Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg) (pp. 55–61).Google Scholar
- Grothendieck, A. (1997). Esquisse d’un programme. London Mathematical Society Lecture Note Series (pp. 5–48). http://www.landsburg.com/grothendieck/EsquisseEng.pdf.
- Hofmann, M., & Streicher, T. (1998). The groupoid interpretation of type theory, 36, 83–111.Google Scholar
- Hogan, D. (2015). Kant and the character of mathematical inference. In C. Posy (Ed.), Kant’s philosophy of mathematics. Berlin: Springer.Google Scholar
- HoTT Book. (2013). Homotopy type theory: Univalent foundations of mathematics. http://homotopytypetheory.org/book.
- Kapranov, M., & Voevodsky, V. (1991). \(\infty \)-groupoids and homotopy types. Cahiers Topologie Géom. Différentielle Catég, 32(1), 29–46. International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990).Google Scholar
- Kapulkin, K., Lumsdaine, P., Voevodsky, V. (2014). The simplicial model of univalent foundations. arXiv:1211.2851v2.
- Lawvere, W. (2005). An elementary theory of the category of sets. Reprints in Theory and Applications of Categories, 12, 1–35.Google Scholar
- Lurie, J. (2009). Higher topos theory (Vol. 170). Princeton, NJ: Princeton University Press.Google Scholar
- Makkai, M. (1995). First order logic with dependent sorts with applications to category theory. http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf.
- Makkai, M. (2013). The theory of abstract sets based on first-order logic with dependent types. http://www.math.mcgill.ca/makkai/Various/MateFest2013.pdf.
- Marquis, J. P. (2008). From a geometrical point of view: A study of the history and philosophy of category theory (Vol. 14). Dordrecht: Springer Science & Business Media.Google Scholar
- Marquis, J.P. (March 2013). Categorical foundations of mathematics: Or how to provide foundations for abstract mathematics. The Review of Symbolic Logic, 6(1), 51–75.Google Scholar
- Martin-Löf, P. (1984). Intuitionistic type theory. Napoli: Bibliopolis.Google Scholar
- Parsons, C. (1992). Kant’s philosophy of arithmetic. In C. J. Posy (Ed.), KantÕs philosophy of mathematics (pp. 293–313). Dordrecht: Springer.Google Scholar
- Putnam, H. (1983). Mathematics without foundations. In H. Putnam & P. Benacerraf (Eds.), Philosophy of mathematics (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
- Quine, W. V. O. (1969). Ontological relativity and other essays. New York: Columbia University Press.Google Scholar
- Schreiber, U. (2013) Differential cohomology in a cohesive \(\infty \)-topos.Google Scholar
- Schreiber, U., & Shulman, M. (2012). Quantum gauge field theory in cohesive homotopy type theory.Google Scholar
- Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford: Oxford University Press.Google Scholar
- Shulman, M. (2014). Homotopy type theory should eat itself (but so far, it’s too big to swallow). http://homotopytypetheory.org/2014/03/03/hott-should-eat-itself.
- Shulman, M. (2015). Brouwer’s fixed-point theorem in real-cohesive homotopy type theory. arXiv preprint arXiv:1509.07584.
- Shulman, M. (2016). Homotopy type theory: A synthetic approach to higher equalities. In E. Landry (Ed.), Categories for the working philosopher. Oxford: Oxford University Press.Google Scholar
- Tsementzis, D. (2016a). Homotopy model theory I: syntax and semantics. arXiv preprint arXiv:1603.03092.
- Tsementzis, D. (2016b). What is a higher level set? (unpublished manuscript)Google Scholar
- Voevodsky, V. (2006). Foundations of mathematics and homotopy theory. https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/VV%20Slides.pdf.
- Voevodsky, V. (2010). Univalent foundations project. http://www.math.ias.edu/vladimir/files/univalent_foundations_project.pdf.
- Voevodsky, V. (2014). An experimental library of formalized mathematics based on univalent foundations. http://arxiv.org/pdf/1401.0053.pdf.
- Voevodsky, V. (2014). Univalent Foundations (lecture at the IAS). https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_IAS.pdf.
- Warren, M. A. (2008). Homotopy theoretic aspects of constructive type theory. Ph.D. thesis, Carnegie Mellon University.Google Scholar