The Future of Mathematics?

I was watching a video by Kevin Buzzard, he talks about using Lean Theorem Prover, seems pretty interesting [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]: [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] [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

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?" [VIDEO] [LINK]

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 - [VIDEO]

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

> 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: [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"

Kevin Buzzard (a professional mathematician) has a few good talks about Lean [VIDEO] from 2019 at MS Research [LINK] (2022 at ICM international math congress)