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Univalent Foundations as Structuralist Foundations

Tsementzis, Dimitris (2016) Univalent Foundations as Structuralist Foundations. [Preprint]

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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.

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Item Type: Preprint
Additional Information: Forthcoming in Synthese
Keywords: Univalent Foundations, Homotopy Type Theory, Structuralism, Foundations of Mathematics
Subjects: Specific Sciences > Mathematics
Depositing User: Dr. Dimitris Tsementzis
Date Deposited: 05 May 2016 02:26
Last Modified: 05 May 2016 02:26
Item ID: 12070
DOI or Unique Handle: 10.1007/s11229-016-1109-x
Subjects: Specific Sciences > Mathematics
Date: 4 May 2016

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