Identity in Homotopy Type Theory, Part I: The Justification of Path Induction
James Ladyman, Stuart Presnell · Philosophia Mathematica · 2015
Homotopy Type Theory (HoTT) is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as (respectively) spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, motivated from pre-mathematical considerations, without recourse to homotopy theory.