The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory 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. (shrink)

Ladyman and Presnell have recently argued that the Hole argument is naturally resolved when spacetime is represented within homotopy type theory rather than set theory. The core idea behind their proposal is that the argument does not confront us with any indeterminism, since the set-theoretically different representations of spacetime involved in the argument are homotopy type-theoretically identical. In this article, we will offer a new resolution based on ZFC set theory to the argument. It neither relies on a constructive-intuitionistic form (...) of mathematics, as used by Ladyman and Presnell, nor is foundationally problematic, such as the existing set-theoretic suggestions. (shrink)