Univalent foundations as structuralist foundations

Synthese 194 (9):3583-3617 (2017)
  Copy   BIBTEX

Abstract

The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics 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. Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 89,654

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Categories without Structures.Andrei Rodin - 2011 - Philosophia Mathematica 19 (1):20-46.
Categorical Foundations and Mathematical Practice.C. McLarty - 2012 - Philosophia Mathematica 20 (1):111-113.
Set-Theoretic Foundations.Stewart Shapiro - 2000 - The Proceedings of the Twentieth World Congress of Philosophy 6:183-196.
Foundations as truths which organize mathematics.Colin Mclarty - 2013 - Review of Symbolic Logic 6 (1):76-86.
Structuralism in the Foundations of Mathematics.Jane Terry Nutter - 1980 - Dissertation, State University of New York at Buffalo
Reflections on mathematics.Edward N. Zalta - 2007 - In V. F. Hendricks & Hannes Leitgeb (eds.), Philosophy of Mathematics: Five Questions. Automatic Press/VIP.
The foundations of mathematics.Ian Stewart & David Tall - 1977 - New York: Oxford University Press. Edited by David Orme Tall.

Analytics

Added to PP
2016-05-08

Downloads
124 (#132,773)

6 months
17 (#105,573)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Maddy On The Multiverse.Claudio Ternullo - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78.
What is a Higher Level Set?Dimitris Tsementzis - 2016 - Philosophia Mathematica:nkw032.
A meaning explanation for HoTT.Dimitris Tsementzis - 2020 - Synthese 197 (2):651-680.

Add more citations

References found in this work

Ontological relativity and other essays.Willard Van Orman Quine (ed.) - 1969 - New York: Columbia University Press.
What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
Rigor and Structure.John P. Burgess - 2015 - Oxford, England: Oxford University Press UK.

View all 29 references / Add more references