Steve Awodey: Structuralism, Invariance, and Univalence (Workshop CSPM, April 26, 2013 in Toulouse)
Recent advances in foundations have led to some developments that are significant for the philosophy of mathematics, particularly structuralism. Specifically, the discovery of an interpretation of Martin-Löf's constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics, with both intrinsic geometric content and a computational implementation. Leading homotopy theorist Vladimir Voevodsky has initiated an ambitious new program of foundations on this basis, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice, according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to the framework of homotopical type theory.