Mathematical Conversations

The Univalent Foundations of Voevodsky offer not only a formal language for use in computer verification of proofs, but also a foundation of mathematics alternative to set theory, in which propositions and their proofs are mathematical objects, and which offers a hierarchy of objects more general than sets ("types"), which are worthy of study in their own right. I'll try to give a flavor of that foundation for mathematicians, contrasting it with set theory as a foundation.