Univalence and Constructive Identity

The non-standard identity concept developed in the Homotopy Type theory allows for an alternative analysis of Frege’s famous Venus example, which explains how empirical evidences justify judgements about identities and accounts for the constructive aspect of such judgements.

💡 Research Summary

This paper, titled “Univalence and Constructive Identity,” explores the philosophical implications of a novel concept of identity emerging from Homotopy Type Theory (HoTT). It argues that this non-standard identity concept provides a powerful alternative framework for analyzing classic philosophical problems, particularly those concerning the cognitive value of identity statements, as famously illustrated by Gottlob Frege.

The paper begins by challenging the traditional, Fregean view of identity as a single, monolithic relation. It introduces HoTT as a branch of intuitionistic type theory where identity is diversified. The technical foundation is laid by explaining the distinction in Martin-Löf’s Dependent Type Theory between definitional identity (a syntactic, substitutable equality) and propositional identity (Id_A(x, y)), which is a proposition requiring proof. Through the “propositions as types” correspondence, a proof of an identity proposition is itself an object (a term) of that identity type. HoTT interprets these proofs as paths in a topological space. Furthermore, the identity between two such proofs is interpreted as a homotopy between paths, leading to a hierarchical structure of higher identity types, modeled by infinite-dimensional groupoids (∞-groupoids).

A cornerstone of this interpretation is Voevodsky’s Univalence Axiom, which posits that the identity type between two types is equivalent to the type of equivalences (isomorphisms) between them. This axiom formally captures the idea that isomorphic structures can be considered identical, enriching the notion of identity with a structural dimension.

The core philosophical contribution of the paper is the application of this HoTT framework to Frege’s classic “Morning Star/Evening Star” (Venus) puzzle. The statements “Morning Star = Morning Star” and “Evening Star = Evening Star” are seen as definitional identities. In contrast, “Morning Star = Evening Star” is a propositional identity, Id_A(MS, ES), which requires empirical proof. The author proposes that establishing this identity corresponds to constructing an invertible map (an equivalence) between the observational data associated with the Morning Star and the Evening Star. Crucially, the HoTT perspective doesn’t stop at finding one such proof (equivalence map). It invites us to consider the space of all such proofs and the higher-order identities (homotopies) between them. Thus, the object “Venus” is not merely a referent pointed to by two names; it is progressively constructed or constituted by the network of evidential transformations (equivalences) and their coherences (higher homotopies). This offers a “constructive” account of identity, where an identity judgment is not a simple recognition of a pre-existing fact but an active process of evidence building and integration.

In the concluding section, the author contrasts this constructive approach with other non-standard theories of identity, such as those motivated by structuralist philosophy (e.g., quasi-set theory), which seek to “weaken” the standard identity relation. The author argues that the HoTT approach is fundamentally different: it does not weaken the core definitional identity but instead constructs a rich, evidence-based superstructure of propositional identity on top of it. This constructive process explains how identity judgments can advance knowledge, thereby addressing Frege’s original problem about cognitive value while providing a novel, dynamic model of how empirical evidence justifies claims of identity. The paper posits that this framework, while presented through a philosophical “dummy” example, holds promise for genuine application in empirical sciences by offering a more nuanced logic of identification and evidence.