The Future of Mathematics?

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

> 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"

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) [this video] [this video]