References [Enhancements On Off] What's this? Additional Information Daniel R. Grayson Affiliation: S. Vine St.
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Homotopy type theory
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Warren , Homotopy theoretic models of identity types , Math. Although these conventional foundations work and all other mathematics can be developed in terms of them, they do have several issues which are not present in the type-theoretic foundations of mathematics that we discuss. In this course, you will learn about type-theoretic foundations of mathematics. Among other things, you will understand how to unify the roles of logic and set theory into one coherent whole.
It will also become clear why type theory is the basis of programming languages and interactive theorem provers like Coq , in which mathematical theorems have been formalized and verified by computers. HoTT brings homotopy theory to the lowest foundational level, much before topological spaces or real numbers. This allows the formulation of the univalence axiom, which roughly states that isomorphic objects are equal and thereby formalizes the common mathematical practice of identifying isomorphic objects.
So if you're fascinated by mathematical elegance and would like to learn new ways of understanding and doing mathematics, then be welcome! This course is likely to shape your mathematical thinking even if you keep doing mathematics in conventional foundations. In this course, we will roughly follow the material of the recently published and freely available HoTT book.
Finally, there is a collection of videos of lectures! You may also want to look at our lecture notes , which start at a more elementary level than the book and lead up to it.
Any feedback will be welcome! We explain the irrelevance of set-theoretic foundations for mathematical practice. This is related to the philosophy of structuralism, which is one of the key ideas implemented in univalent foundations based on HoTT. We will showcase some of the features of HoTT, for example how to define the circle and other spaces at a very fundamental level without having a notion of set, topology, or real numbers.
Starting with the technical details, we introduce some of the basic concepts of type theory: a collection of inference rules forming a trivial type theory; together with the fules of function types, this forms the simply typed lambda calculus with function types only.
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We discuss the philosophy of constructivism and how it leads to the identification of propositions as types and proofs as algorithms. This is knows as the Curry-Howard correspondence. This correspondence is of central importance for type-theoretic foundations. To respond on your own website, enter the URL of your response which should contain a link to this post's permalink URL.
- Advanced course in foundations of mathematics (/)premildalunsio.cfer Science-University of Verona.
- Univalent Foundations of Mathematics - Program Description | IAS School of Mathematics.
- Colloquium Talk: Homotopy Type Theory and Univalent Foundations.
- Will Computers Redefine the Roots of Math?.
- Open lectures for PhD students in computer science!
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