The maths community is now in the antithesis phase, synthesis will take a while ;)
Lee Sedol said in an interview that "losing to AI, in a sense, meant my entire world was collapsing. ... I could no longer enjoy the game. So I retired", and I think there will be folks in the mathematical community who would feel the same when the solutions pages to hard problems are suddenly available.
But on the other hand, people learned a lot from chess engines. After decades of chess computers beating humans, there was still a renewed interest in watching Leela beat Stockfish, with many people trying to understand the strategy Leela used.
If your happiness comes from grinding on a problem and making progress, the prospect of having to dig through a corpus of AI-generated proofs might be hard to swallow. But if you're willing to do that, you will still find beautiful things that only so many people can truly appreciate.
I really hate this overly condescending takes. First of all, what do you know about the internals of math research that allows you to speak with so much confidence. Second, you're not even addressing the issues raised by the letter! This is not about "oh they made a bunch of problems easier". There are huge economical interest behind: who owns and has access to models? are these companies interested in developing research or they just grind PR stunts without worrying about externalities in how research is actually conducted? Etc etc.
If it's worth anything: I have a PhD in (theoretical) mathematics and I entirely stand by stabbles' comment.
There is a real, undeniable possibility of AI becoming better at mathematics in the same way that it became better at chess and Go, and in such a scenario, one may expect the community's response to be comparable.
It makes no sense to compare mathematics with chess. Chess is a sport. No one is interested in watching two machines compete. Chess doesn't have a practical impact. Etc. What you seem to suggest is that AI will be able to completely (or at least in a great part) replace mathematicians. It could be the case in the future, but no one knows right now, and more importantly: tech companies don't even think about it! they don't think on the externalities.
Well, what you are saying that the situation is even better for math than for chess?
Chess is only valuable as an entertainment. So no one really gains anything from computers becoming really good at chess.
But with math, everyone in the world would gain from computers becoming really, really good at it.
Does mathematics still have a practical impact without humans in the loop? I don't think there's one single answer to that question, but I think it's worth considering exactly what that impact may be.
Tech companies are as much the topic of this post as AI, I think that's the immediacy.
> Does mathematics still have a practical impact without humans in the loop?
There's a single answer to that question: yes, math very much has an impact without humans in the loop.
Math has a lot of applications, and those applications don't care whether eg the new faster matrix multiplication algorithm was found and proven correct by a machine or a meatbag.
To a significant extent, the pursuit of mathematics research is a pursuit of human understanding of mathematics, without knowing where it might lead, or whether it might lead anywhere at all. I don't see how the motivation for that goes away on its own, but the institution supporting it is certainly threatened by the potential loss of grant money and graduate student applications.
> No one is interested in watching two machines compete
I’ve watched quite a lot of YouTube videos where two machines compete, so you may not be completely right here
The Square One commentary on the AlphaZero v Stockfish game from 2017 is pretty entertaining: https://www.youtube.com/watch?v=LnVDUQksIDk
In other videos he's called out the influence that this and similar games have had on human players in recent years, particularly around square denial and thorn pawn strategies.
https://tcec-chess.com/
Top chess engine championship is pretty fun to watch.
> What you seem to suggest is that AI will be able to completely (or at least in a great part) replace mathematicians.
There is no bound on the amibitions of AI. AI is set to replace anything done by people, and there won't be any room left for people. There isn't any task done by humans that AI won't be better at.
This is not a tenable outcome.
We should never have built machines with agency, rather than optimization processes that operate as subroutines of humans.
You say it like it's a bad thing.
Replacing people leaves no room for people.
> We should never have built machines with agency...
We haven't quite crossed that bridge, but we do appear to be standing on it.
Stockfish isn't owned by a club of three trillionaires. It does not cost $15 million to achieve a result in Stockfish.
Stockfish does not steal research or scoop researchers.
The concentration of computing resources and capital should be examined by the math community.
The trailing edge of AI is catching up fast. There's plenty of open source (and even more open weight) AI models and they are getting better and better.
If that's your only objection: in a few years you can prove Rieman's hypothesis on your smartphone, no need for any trillionaires to give you permission. Does that make any change to your argument, or did it not actually matter?
This is the real problem. We're looking at a future where those who control AI have an insurmountable advantage in everything. They can control the amount of intelligence the masses have access to -- for their own safety, of course -- and they will never, ever be able to close the gap.
There's lots of competition in AI. Where do you see the 'insurmountable advantage in everything'?
Say they do. What then? Will people buy from them? What if we decide not to?
At a certain point, you don't have a choice. Before China got into the game, the only way to avoid giving Luxotica money if you wanted a pair of glasses was to essentially not buy glasses. This is the same for many industries -- consolidation behind the scenes.
Huh? I've been able to buy reasonably priced glasses for all my life. (However, I've never lived in the US nor China.)
Who cares about the giant labs? The trailing edge will catch up quickly, and in a year or three you can prove Rieman's hypothesis on your smartphone.
>> "are these companies interested in developing research" judging from the money, resources spent and the value they derive from this the answer is very definitively yes.
What makes you think these companies (and I'm not a fan of all their motives) are not interested in developing research? The motives may be self-serving, but it is undoubtedly and objectively accelerating research.
Chess is kept afloat by a couple of billionaires like Sinquefield, MBS and the guy who sponsors freestyle (Fisher random) chess.
Carlsen is bored by studying engine lines.
The popularity is boosted by YouTubers because chess is very suitable for somewhat higher class content.
I'm not sure we'd want that world for math. Positions will be cut just like archaeologist positions are cut now.
Chess is kept afloat by chess players, not by billionaires. If all the billionaire backers stopped sponsoring tournaments, people like me would still play, still pay for chess club memberships, still pay entry fees for tournaments, and still buy chess books, and so on.
I think the parent comment meant professional, high-level chess. The kind people get played to play, not just do for a hobby. That's absolutely on life support.
I'm not sure what the equivalent would look like in the math field, but it probably involves a lot of mathematicians losing their jobs and the quality of human-produced math decreasing overall.
The quality of the math in general would be fine, since in this scenario cpus will keep producing it. The quality of cpu-cpu chess games is quite high, beyond human understanding in many cases.
Chess is a weird example because it doesn't really have any utility beyond itself. Even pure math sometimes ends up having use in the strangest places. Although if no one understands the frontier math (because no one is getting paid to), I'm not sure it even matters what the quality of the cpu math is?
It's a bit like a tree falling in a forest. If an LLM proves a theorem but no one understands it, did it make a sound?
"Although if no one understands the frontier math (because no one is getting paid to), I'm not sure it even matters what the quality of the cpu math is?"
Presumably AI will be connect the dots to the applications. As the declaration says, this isn't just about math. Human understanding is losing economic value. You can understand stuff on your own time, I guess.
The standard justification for pure math to holders of purse-strings is something like "it might lead to a useful application down the road, like crypto, who knows". That looks pretty inefficient now. We have to entertain the possibility that AI can develop the math needed for any application we put to it. Eg if number theory didn't exist, we could have asked AI for a way to transit messages securely and it would maybe come up with fermats little theorem as part of its solution or maybe come up with an approach we can't conceive of right now seeing as most of us are constrained to available number theory. Like how in the last year when I give an LLM a programming project I see it doesnt even bother with of the many software libraries I and others have written and just codes up the calls it needs on the fly or finds some other ad hoc solution.
When there's a billion people playing something, money will never be an issue for those at the top. Even things like chess.com was able to sponsor a tournament with a million dollar prize pool.
Also I'd argue that chess's utility is ultimately the same as pure math, particularly in esoteric fields. These things are highly unlikely to ever lead to any sort of real world breakthrough or application. The main benefit is an outlet for human logic, creativity, and exploration - which significant self improvement possible along the journey for players.
Though I think even that's probably too socially utilitarian. I think ultimately the 'real' drive is the same in both fields - it's fun and personally rewarding.
> Though I think even that's probably too socially utilitarian. I think ultimately the 'real' drive is the same in both fields - it's fun and personally rewarding.
That's fine, and no amount of machine excellence will keep you from enjoying recreational chess or recreational math.
I would argue people getting paid to play chess was a short lived phenomenon anyway if you put it in context. The transition there is less related to the introduction of chess engines and more related to the shift in the media landscape.
> It's a bit like a tree falling in a forest. If an LLM proves a theorem but no one understands it, did it make a sound?
But in future most proofs will be for consumption by other AI models in the pursuit of yet other proofs.
It's kind of surprising so many mathematicians act surprised by this given this was clearly where automated proof assistants would lead. I guess they assumed they'd always be the ones guiding them.
> But in future most proofs will be for consumption by other AI models in the pursuit of yet other proofs.
What is the purpose of that?
Its like art being produced for AI to consume. What is gained from that?
It's going to be hard to compete with something that has access to all of math at once and can find connections between elements that appear unrelated to humans.
And at some point AI will start suggesting - or doing - physical experiments.
Presumably some of the proofs will have applications beneficial to humans beyond impressing other mathematicians, and AI will surface them, or use them directly.
Well if it proves useless presumably they'd stop doing it.
But if AI is to recursively self improve understanding and evolving its own foundations, which are clearly mathematical, is essential. There is no need for humans to grasp what is going on in that loop.
Math also has applications, and they don't rely on human mathematicians doing the math.
I mean, was the point of math ever just because some humans enjoyed doing it? Even though a lot of it is theoretical, there's been all sorts of useful things that have come out of it as well due to an improved understanding of the universe through new ways of thinking about it. If it got to the point where no human could understand it and there were no ways to actually use it, I don't think anyone would bother having their computers doing it at all.
> was the point of math ever just because some humans enjoyed doing it?
Yes.
Friends and I often work on Putnam problems and this series:
The (Almost) Impossible Integrals, Sums, and Series by Cornel Ioan Vălean
wouldn't AI solving problems in science, engineering, economics, etc be able to apply the new AI math?
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Mathematics is more than establishing arbitrary facts (although some look like curiosities), it's also defining what interesting research directions are and establishing common language/notation. I think that will stay relevant?
In theory you can automate finding interesting research directions by identifying conjectures with many dependencies. And notation has never been mathematicians' forte, with them trying to cram the entirety of universe into single letters.
Why? How do you define interesting research directions? That used to be defined by testing the limits of human understanding i.e. some people can't figure something out. AI might have very different ideas about what is interesting and I am not sure what humans would get out of putting years into understanding AI proofs for what? What are we doing at that point? Like if you spend years understanding some AI proof of theorem 123456, why is that meaningful? I am actually asking why you think defining interesting research directions will stay relevant. In my opinion, people spend years acquiring knowledge so they can work on problems which is separate.
There's two points about this I am assuming 1) Mathematics actually has a significant subjectivity to it and is community oriented and not just climbing a never ending list of theorems that exists in the universe 2) A lot of mathematical research work is inside of a subfield and isn't directly motivated by applications. Sometimes it is but e.g. people don't work on obscure theorems about elliptic curves because of a dire need for that but more because the community found it interesting.
People become interested in things when they become invested in it personally, because they've contributed to it. So I don't think it will stay relevant...
>but no one understands it, did it make a sound?
Does your "one" only contain humans or does it also contain other AI systems. AI math is not a single monolithic thing, but a distributed one. I see value in sharing proofs even among just AI.
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Also, you learn to be a better chess player by... playing better players. The widespread availability of chess engines has made flawless opponents available to every player.
If your goals are understanding the game, self improvement, building thinking skills-- this is the best chess has ever been. It's only if your goal is to beat every opponent you can find that chess is in a bad place.
It's not that easy. Playing stockfish is like playing tennis against the wall (for untitled players at least).
Even if you don't blunder anything, you'll still find yourself in a worse position without any clue as of what went wrong and why.
Whereas when playing humans, they can usually explain their approach and when they noticed errors in your play.
I mean, I'm not a great player but I've learned a lot from working through games with stockfish using a git repo and a small script that lets me rewind to different moves and try different approaches. It may not explain its moves but if you're thinking through what's happened you can usually debug your game anyway.
> Playing stockfish is like playing tennis against the wall (for untitled players at least).
It's the same for Magnus Carlsen. Even with Queen odds, Stockfish is literally unbeatable for the best players in the world. It's just too strong at evaluating all kinds of random tangent moves (and ensuing positional advantage) which no human player can possibly pay attention due to the time required.
Stockfish vs any human is like Carlsen vs other players by about 3-5 orders of magnitude[0]. It's that stark.
[0] A wild pun appears.
EDIT: To avoid having to respond to each responder, fair comments about Queen odds. Maybe I was thinking Rook odds? Also, I kinda lumped Stockfish in with all the other engines, but I realize there are other engines with different properties ofc.
Well, that's actually not true at all. Stockfish is not a very good odds player and at queen odds is easily beatable even by bad players like me. It will just trade down into more trivial and easier to win positions that it perceives as "less bad", since everything is super-losing anyway when you start down a queen.
Leela odds networks, on the other hand, are an entirely different beast. I cannot beat Leela queen odds, much less rook or minor piece odds, and even GMs struggle against Leela knight odds.
Without odds though, yeah, Stockfish is just incomprehensibly strong by human standards. All top chess engines are, but Stockfish moreso.
No worries, I know stockfish is unbeatable by humans.
But sometimes, these GMs can flag it, which counts as a win (especially when it's proxied by a cheater). Sometimes they can also explain the idea that cost them the game, so they've learned something maybe.
Whereas us scrubs literally cannot do anything at all for reasons completely beyond our understanding.
> Whereas us scrubs literally cannot do anything at all for reasons completely beyond our understanding
Computer moves are typically much more concrete than human moves: a human will play based on pattern matching ("intuition") and can only make explicit calculation of a small fraction of possibilities, after which decisions are guided by guesswork. The computers are unbeatable in practice because they can calculate concretely in seconds what might take an expert human long intensive study to notice, and they don't make the same kinds of oversights humans can make.
But if you stop and explore a particular position for an extended time, and if you have an intermediate level of chess skill, you too can probably often (usually?) figure out why it's doing something. Sometimes understanding the computer's reasons takes searching multiple branches of a tree several unlikely looking moves deep, but the collection of threats the computer was preemptively thwarting, traps it was setting, etc. are comprehensible to humans with enough effort, especially in games between the computer and a human.
The frustrating thing about playing against the computer is that it notices and thwarts every plan you might come up with, before you make up the plan yourself, and it doesn't make (human-apparent) mistakes, so the game ends up feeling hopeless. Nothing you try works on it, and if your idea is even slightly inaccurate it will be exploited.
You put it better than I could. The lesson of "in this exact position you can kick the pieces for 7 moves to get a fork, so instead you should play a4" is not something that I can implement into my games
I mean, this is the problem with analogies and trying to use them to prove things, right? People working through problems from an analysis book with their friend (or an LLM) is not the same as research mathematics. People playing in a chess club is not the same as what makes for a good chess tournament. Lumping everything together is just making this branch of the conversation less relevant.
Chess is a fun game. That's why it's been around for 1000+ years.
There was a renaissance during Covid and due to 'The Queen's Gambit' where it gained much more mainstream popularity, but... Chess AI was already far far (like 1000+ Elo) ahead of human players at that point.
The thing is... chess is humans playing (communicating) with humans and that's what keeps it interesting. Check out the view counts of chess AI tourneys vs. human tourneys.
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