An introduction to the TCC course “Formalising Mathematics”
Mathematics in type theory.
An explanation of how to set up mathematics using universes, types, and terms
Embracing AI and formalization: Experimenting with tomorrow’s mathematical tools
Embracing AI and formalization: Experimenting with tomorrow’s mathematical tools. By Jarod Alper
Conceptual mathematics: a first introduction to categories
Conceptual mathematics by F. W. Lawvere, 2009, Cambridge University Press edition, in English - 2nd ed.
The Technological Turn in Mathematics
Quickly evolving technologies, such as Interactive Theorem Provers (ITPs), Automated Theorem Provers (ATPs), and Large Language Models (LLMs), all falling under the general heading 'AI for mathematics,' are transforming mathematical practice in profound ways. This chapter explores the implications of these innovations, focusing on their impact on how mathematical knowledge is created and shared. It also discusses how they are reshaping the social dimension of mathematics, altering collaboration dynamics, trust relationships, and the collective production of knowledge. For instance, tools like ITPs facilitate large-scale collaborations and make new types of teamwork possible, where trust is not a necessary ingredient. ITPs also help us mitigate our human fallibility, yet they raise questions about the nature of formalization and the relationship between traditional and formal mathematics. Technologies such as LLMs are reshaping the division of epistemic labour between humans and machines and urge philosophers of mathematics to ask questions about the value of their work.
The End of Mathematics — Daniel Litt
I'm currently returning to Toronto from a summit on the future of mathematics, at OpenAI. Sebastian Bubeck asked me to talk a bit about the future we'd all like to avoid, where humans are mathematically disempowered. Jacob Tsimerman advised us to try to prioritize detail over correctness, and I have no doubt that I succeeded in deprioritizing correctness.
Regnestykker
Sheets with calculus exercises suitable for kids that are learning the first basic arithmetic operations.
Homotopy Type Theory: Univalent Foundations of Mathematics
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
Mathematical Beauty, Truth and Proof in the Age of AI | Quanta Magazine
Mathematicians have started to prepare for a profound shift in what it means to do mathematics.
We’re gonna need a lot more mathematicians
[This is a guest post by Amit Sahai. This blog post was initially written in a different file format and converted using AI. — T.] When I was an undergraduate student, I remember talking with…
Formalizing 100 Theorems
There is a "top 100" of mathematical theorems on the web, which is a rather arbitrary list (and most of the theorems seem rather elementary), but still is nice to look at. On the current page I keep track of which theorems from this list have been formalized. Currently the fraction that already has been formalized seems to be
Iconic Math
As an introduction to a forthcoming book (late 2025), this page links to three narrated videos about unary boundary logic:
Brilliant | Your Personal Tutor for Math and Coding
Visual, interactive tutoring in math and coding, from 5th grade through college. Master concepts in algebra, calculus, Python, computer science, and more.
My Journey Back to Math: A Short Review of MathAcademy
Personal journey of rediscovering math through MathAcademy and learning in public
GitHub - janaagaard75/calculus-exercises: Sheets with calculus exercises suitable for kids that are learning the first basic arithmetic operations.
Sheets with calculus exercises suitable for kids that are learning the first basic arithmetic operations. - janaagaard75/calculus-exercises