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

The main reason code isn't considered a formal specification is that it's not intended to be either formal or a specification. That doesn't mean you cannot write a formal specification in code. If anything code allows for a higher level of formality than all but the most formal mathematics. Which is why some mathematicians are advocating that mathematical proofs should be specified as code [1]. You shouldn't confuse the existence of these languages to mean that any code not capable of expressing mathematical proofs is not formal. It might be limited, but not informal. [1]: [VIDEO]

I asked a similar question in the comments on a youtube video, and Mario Carneiro, one of the people who work on Lean, answered me. He said he does work on this area, and it kind of works (at least for the theorems you would want to translate?), but in addition to the proofs being very not nice to read, the way the statements of the theorems are translated are, by default, also rather not nice (being statements about specific formulations of the objects (such as the set of integers) as they are expressed/defined in the language being translated from, instead of referring to the corresponding objects that have been defined in the language being translated to. However, he says that it doesn’t take all that much work to manually massage it until the statement of the theorem is expressed in the way that you want it to be. I’m on my phone right now, and the phone youtube app doesn’t give me a permalink, but I’ll try and edit this comment soon on laptop to include link to the comment section. Edit : here are links to what Mario Carneiro said in the thread click the read all replies to read the full thread) [VIDEO] [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]: [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: [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?" [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"

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: [VIDEO] ).

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)

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. [VIDEO]