The Future of Mathematics?

I watched this video years ago and found it interesting. It's a talk by Kevin Buzzard, a pure mathematician who got really interested in theorem proving software, and he explains his motivation. ▶ this video

Kevin Buzzard (a professional mathematician) has a few good talks about Lean ▶ this video from 2019 at MS Research link (2022 at ICM international math congress)

> pure math (topology and countability) and computer science have almost no overlap in the short term. This is an unexpected opinion. What do you mean by this? I would argue the opposite. Look at Formal Verification[0]. Here's a great video about a popular formal verfication proof assitant: ▶ this video A quote from the mathematician in the video expressing enthusiasm for using computers for mathematics: "All I want to do... I don't want to do mathematics on pen and paper anymore, because I don't trust it, and I don't trust other people that use it; which is everyone." (in case people only read this quote, instead of watching the lecture, the tone is tongue-in-cheek) Formal Verification would seem a literal intersection of pure math (formal proofs[1]) and computer science (formal specification[2]). Of note, one of the popular underlying foundations of an approach to formal verification is Homotopy Type Theory(HoTT)[3][4] which is an asserted effort to show the relationship between type theory and, specifically, topology (homotopy[5]). [0] link : fancy word for 'computer verified mathematical proofs' [1] link : fancy word for 'proof' [2] link : fancy word for 'rules for an algo that checks a proof' [3] link [4] link : a great read, even if you are a novice, or prefer other formal verification foundations [5] link : "It originated as a topic in algebraic topology"

wanted to drop a link to buzzard's very interesting talk ▶ this video starting at 13:00 he talks about that frailness. also loved the bit about "elders" at 40:37

That was a delightful tutorial, I recently watched a talk about the Future of Mathematics[0] with respect to theorem proving and formalization. I enjoy this area a lot and love to see it getting more attention. 0 - ▶ this video

Oh man that was fun! Totally nerd-sniped my day though. Got me to go back and re-watch Kevin Buzzard's talk at MS Research in re: Lean, "The Future of Mathematics?" ▶ this video link

I was watching a video by Kevin Buzzard, he talks about using Lean Theorem Prover, seems pretty interesting ▶ this video

Last week Microsoft Research posted an interesting talk by Kevin Buzzard [1] from Imperial College London about how he wants to formalize all mathematics with theorem provers. He's currently using LEAN [2]. Half of the talk is a discussion of the challenges he's faced convincing other mathematicians that this is valuable, and the other half is about his success using LEAN as a tool to teach undergraduates about formal proofs in general. I think Alloy was mentioned at the end, but not really compared to LEAN. [1]: ▶ this video [2]: link

For a perspective of a mathematician that came to evangelize theorem provers, I recommend Kevin Buzzard's MS 2019-09 presentation [0] about LEAN. He highlights cultural misunderstanding and apathy on both sides of the domain divide. He also references the idea that the people who might make the appropriate tools may not have stayed in academia. So, he's structured his courses around using LEAN with the indirect consequence that power users (undergrads) may choose to become open source committers. [0] ▶ this video One hour of presentation and then 15 min of q/a. My favorite is around 1:04:00 when someone asks a second time why he disprefers coq, and Buzzard complains that it can't represent some advanced quotient type that he'd have to work around. I'm reminded of [1] [1] link Dangling by a trivial feature

At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at link . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought it was interesting and relevant to your question. (Here's also a talk by the professor about his rationale for using Lean: ▶ this video ).

"Are you the high priest and decider of the usefulness of mathematics? To be honest, it almost sounds like some category theorist was super mean to you..." Interesting and totally not ad hominem response... I believe Kevin Buzzard (an actual mathematician) had a few words about this last year: ▶ this video

Lean [1] [2] [3] To try and revitalize my math life a bit... There are some things I quite miss, and learning Lean was my idea to force myself to get back in the swing of things. [1]: link [2]: link [3]: ▶ this video