Univalence and the Concept of Structure in the Philosophy of Mathematics

The paper analyzes the current state of the constructivist and structuralist trends in the philosophy of mathematics. The interrelation of these trends is substantiated by means of the concepts of isomorphism, invariant, and structure within the framework of the type-theoretical approach to the foundations of mathematics. In particular, the paper discusses the axiom of univalence and the modern type-theoretical concept of identity.

Keywords: foundations of mathematics, philosophy of mathematics, type theory, constructivism, structuralism, homotopy type theory, structure, isomorphism, computation.