Univalent foundations as structuralist foundations

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



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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.

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