Univalent foundations as structuralist foundations

Abstract

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.

Abstraction, equality and univalence

“…We could say that constructive abstraction 'forgets' inessential information in a controlled manner, that is, in such a way that the abstraction data can be fully recovered from (or faithfully encoded in) the abstracta. 26 From a philosophical perspective, some authors-notably Awodey [20] and Tsementzis [8,73]-have analysed UF in the light provided by a family of interrelated trends in philosophy of mathematics enveloped by the term mathematical structuralism (and mainly developed, in its different eliminativist and non-eliminativist variants, by Bourbaki, Benacerraf, Putnam, Resnik, Shapiro, Hellman and Parsons among others; see [74] and references therein). The idiosyncratic presentation of UF that we have proposed here is intended to consider UF from the standpoint provided by an alternative (and maybe complementary 27 ) conceptual framework that stresses above all the constructivist elan of UF.…”

A meaning explanation for HoTT

“…For some of the earlier writing that led to the ideas for UF see [35][36][37]. For a discussion of the sense in which UF is a foundation of mathematics and connections to mathematical structuralism, see [4,32,33]. For some philosophical issues associated to HoTT see [11,12,19,28].…”

Estruturalismo na Filosofia da Matemática e a teoria das categorias: Um debate entre Steve Awodey e Geofrey Hellman

“…Uma interessante objeção a essa afirmação foi posta porTsementzis (2017). Segundo essa objeção a linguagem da teoria das categorias na verdade não está em posição melhor do que as teorias de conjuntos tradicionais em fornecer uma linguagem adequada ao estruturalismo, i.e., uma linguagem cujo poder expressivo seja restrito as propriedades estruturais.…”