#math #logic #Symmetry The Univalent Foundations program is a research initiative in mathematics and computer science that aims to develop a new foundation for mathematics based on the principles of homotopy type theory. It was initiated by [[Vladimir Voevodsky]], who proposed the idea of univalent foundations in 2006 as a way to provide a more intuitive and computationally friendly framework for formalizing mathematics. The program seeks to unify various branches of mathematics under one cohesive framework by treating mathematical objects as types and proofs as elements of these types. This approach draws inspiration from ideas in category theory, dependent type theory, and algebraic topology. Homotopy type theory provides a way to reason about spaces and paths between them, allowing for a rich and precise formalization of mathematical concepts. One of the key features of univalent foundations is the concept of "univalence," which states that isomorphic structures can be considered equal. This allows for more flexible reasoning about mathematical objects and enables new connections between different areas of mathematics. The program also emphasizes the use of formal proof verification techniques to ensure the [[Correctness]] of mathematical arguments. The Univalent Foundations program has led to significant developments in both mathematics and computer science. It has contributed to the development of proof assistants such as [[Coq]] and Lean, which are powerful tools for formalizing and verifying mathematical proofs. The program has also influenced research in [[Homotopy Theory|homotopy theory]], [[Category theory|category theory]], and [[Logic]]. Overall, the Univalent Foundations program aims to provide a solid foundation for mathematics that is both rigorous and computationally tractable. By using homotopy type theory as its basis, it seeks to bridge gaps between different areas of mathematics and facilitate new ways of thinking about mathematical concepts. ## References ```dataview Table title as Title, authors as Authors where contains(subject, "Univalent Foundation") or contains(subject, "univalent foundation") or contains(title, "Univalent Foundation") or contains(title, "univalent foundation") ```