Does Homotopy Type Theory Provide a Foundation for           Mathematics?

Ladyman, James; Presnell, Stuart

doi:10.1093/bjps/axw006

Cited by 24 publications

(31 citation statements)

References 18 publications

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“…A mathematician might say that Univalence, like any other axiom, needs no justification beyond its interest or its usefulness. However, if HoTT is to be a foundation for the whole of mathematics (in the sense explained in [Ladyman and Presnell, 2016], in which a 'foundation' goes beyond merely a language or framework in which mathematics can be developed) then each of its definitions, rules, and axioms must be explained and justified in a way that does not require appeal to concepts that must be spelled out using pre-existing mathematics, or to connections with other sophisticated branches of mathematics, or to the intuitions of mathematicians. Moreover, for these purposes the motivation offered for a given axiom must justify that choice of axiom in particular, and not something weaker that follows as a consequence of the axiom.…”

Section: The Justification Of Univalencementioning

confidence: 99%

“…Since the Univalence axiom is formulated in terms of universes, the next section explicates the notion of universes making explicit the conceptual features they are taken to have in [HoTT Book,Section 1.3]. It also explains how to understand universes in terms of our Types-as-Concepts interpretation [Ladyman and Presnell, 2016]. The formal definition of Univalence is given in Section 3, with one detail suppressed.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Following the account of the criteria for a foundation for mathematics outlined in [Ladyman and Presnell, 2016] (involving five interrelated components: a framework, semantics, metaphysics, epistemology, and methodology) Section 5 considers the meaning of Univalence in relation to the explanation of universes given in Section 2. In Section 5.1, consideration of mathematical metaphysics leads to a possible motivation for adopting Univalence which, on closer inspection, turns out to be inadequate for the purposes of this paper.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Section 2.2 briefly considers some of the technical details of implementation. Section 2.3 gives an account of universes that is compatible with the Types-as-Concepts interpretation of HoTT [Ladyman and Presnell, 2016] and explains how universes should be understood in an autonomous account of HoTT.…”

Section: Universesmentioning

confidence: 99%

“…[Awodey, 2014b, p. 9] In this paper we explore the conceptual, foundational and philosophical status of the Univalence axiom in Homotopy Type Theory. In particular, we address the question of whether Univalent HoTT (that is, HoTT with the Univalence axiom) can be given an autonomous presentation as a foundation for mathematics in the sense defined in [Ladyman and Presnell, 2016].…”

Section: Introductionmentioning

confidence: 99%

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Universes and Univalence in Homotopy Type Theory

Ladyman

Presnell

2019

The Review of Symbolic Logic

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The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory (HoTT) research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising.

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Section: The Justification Of Univalencementioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Universesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Universes and Univalence in Homotopy Type Theory

Ladyman

Presnell

2019

The Review of Symbolic Logic

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Univalence and Ontic Structuralism

Chen

2024

Found Phys

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The persistent challenge of formulating ontic structuralism in a rigorous manner, which prioritizes structures over the entities they contain, calls for a transformation of traditional logical frameworks. I argue that Univalent Foundations (UF), which feature the axiom that all isomorphic structures are identical, offer such a foundation and are more attractive than other proposed structuralist frameworks. Furthermore, I delve into the significance in the case of the hole argument and, very briefly, the nature of symmetries.

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Arithmetic, set theory, reduction and explanation

D’Alessandro

2017

Synthese

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Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.[I]n the philosophy of science the notions of explanation and reduction have been extensively discussed, even in formal frameworks, but there exist few successful and exact applications of the notions to actual theories, and, furthermore, any two philosophers of science seem to think differently about the question of how the notions should be reconstructed. On the other hand, philosophers of mathematics and mathematicians have been successful in defining and applying various exact notions of reduction (or interpretation), but they have not seriously studied the questions of explanation and understanding. I hope the rest of the paper will show that this neglect is unwarranted. As philosophers of science have long known in the empirical context, understanding the link between reduction and explanation is a vital part of understanding theory succession, mathematical progress, and the nature and purpose of foundations of mathematics. I aim to illustrate these points by examining a particularly important single case: the reduction of arithmetic to set theory. Although a number of authors have expressed views about the explanatory significance of this reduction, the issues involved haven't yet been weighed as carefully as they deserve to be. This paper sets out to do so, and to draw some considered conclusions.My thesis is that the reduction of arithmetic to set theory is unexplanatory. After clarifying this claim and giving some positive reasons to believe it, I devote the larger part of the paper to showing that the most serious arguments to the contrary-due to Steinhart, Maddy, Kitcher, and Quine-are unsuccessful. Finally, I discuss some consequences of the view for philosophy of mathematics, philosophy of science and the theory of reduction. Intertheoretic reduction and explanation in math...

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Does Homotopy Type Theory Provide a Foundation for Mathematics?

Cited by 24 publications

References 18 publications

Universes and Univalence in Homotopy Type Theory

Universes and Univalence in Homotopy Type Theory

Univalence and Ontic Structuralism

Arithmetic, set theory, reduction and explanation

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