I never expected this many people (on this thread) arguing semantics and what not. I know that not everyone has morality and ethics, but I didn't realize it was this bad.
I'm afraid of the ripple effect of the agenda pushed by AI companies will have. In future and even now, they say AI has significantly progressed math and scientific research in general. There is truth to this, but the narrative has done more damage (so far) to the students, researchers, and the culture of knowledge transfer in academia. Many graduate students (I know) are having a crisis if any of their research worth it? If AI can (or will) do everything, what's the point of doing experiments and all? This will eventually deter a whole generation of curious minded students from research.
I guess, only time will whether this is for the good or bad. And how good AI models get without new data from research and experiments.
> Many graduate students (I know) are having a crisis if any of their research worth it? If AI can (or will) do everything, what's the point of doing experiments and all? This will eventually deter a whole generation of curious minded students from research.
Those who think it's me or the machine will fail.
Those who realize how much you can accelerate your research with the help of AI will succeed.
I think that the main thing people in this thread are missing, is that it's not about math. AI progression is very likely to affect every single thing humans can do today. Mathematicians are feeling the blow this week, especially as there was wide spread denial in that math community over the capabilities of AI over the last few years, but it's the same problem everywhere.
As a mathematician maybe I am a little more optimistic than this declaration.
I am thinking of Mochizuki's abc conjecture: He worked in relative isolation, and dumped a huge incomprehensible proof on the community (to oversimplify a bit). That's not totally unlike what might happen if AI generates a huge, incomprehensible proof of let's say RH.
Well, what is the result? In the Mochizuki case, it was a lot of skepticism, but it also generated conferences, papers, talks in the hallway, discussions with students, and so on--a flurry of exactly that kind of community process that the declaration says is the main driver of mathematics.
Ultimately we think a fatal flaw was found in Mochizuki's proof, so it didn't lead anywhere in particular. But in our hypothetical "AI lean-verified proof of RH" situation, it would presumably generate substantially more of that community activity we saw in the Mochizuki situation. And if it's correct, that community activity would be productive (expository talks, students given problems to flesh out or generalize, etc).
Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
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.
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.
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.
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.
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.
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.
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...
> 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.
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.
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.
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.
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.
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.
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.
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.
Is this the scenario described in Ted Chiang's short story https://en.wikipedia.org/wiki/The_Evolution_of_Human_Science where scientists are "catching crumbs from the table" trying to decipher the results generated by superhuman intelligence?
It's still an optimistic scenario. Artificial superintelligence may develop hypermathematics of a kind that never will be accesible to human mind, enhanced or not. One can't teach geometry to ants even if you put them on a Moebius strip.
What’s optimistic or non-optimistic specifically about the machine having a system of mathematics beyond our comprehension within it? Why should we care about that in itself?
In the first case, we'd still have a chance to take a glance at the frontier of discovery (even if ordinary human mathematicians had to spent years translating what metahumans achieved).
In the second case all human-level maths would be solved and what lies beyond would be always out of our scope.
>Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
Dr. Tao said the same thing. Somehow, this letter came through. He wants to conduct Math competitions where participants who don’t have formal credentials can contribute to mathematical research through AI.
Title: Terence Tao - SAIR Competitions and the Future of Experimental Mathematics
I have zero formal math training beyond my Grade 12 Pre-Calculus class. Yet with an LLM I have recently devised an architecture with incredible math potential. Math is a language like any other, and without LLM's I never would have developed the techniques that I have.
AI is a tool. It speaks languages I don't (Math, Science, Code). I would love to participate in a Math competition without a hint of any formal advanced math training because my experience so far tells me I will do well.
He sees value in mathematicians using AI to carefully study mathematics, develop an understanding of both old and new things, and help others understand the new things.
He doesn't see value in scrolling through unsolved problems asking an AI to please solve them. In his view, this is a fundamental confusion about what mathematical research is for. Knocking down unsolved problems without developing the community's understanding of them is like prompting Claude to go through a Jira board, write code for all the open tickets, and then close them without merging or deploying the code.
Pretty close, but IMO not quite. A math proof in and of itself is useless unless either:
(A) it furthers human knowledge
(B) it gets used in applied sciences, engineering, etc.
If you merge and deploy code, you have released a tool that can be used. If you ship a gibberish math proof, it's not useful unless someone else can understand and deploy it to some other means. Now, it's possible AI could understand and make use of the math proofs, even if we can't, which refutes some of my hair splitting :)
No. Merged code can perform actions with effects on the world, even if a human being never saw it. Constructing a giant Lean formalization that nobody understands simply doesn't do anything.
Yes. I find it really interesting to consider what the machines do and will think of as intrinsically interesting to them. Will they develop their own theories of beauty, mathematical and otherwise?
But the work the Mochizuki case generated can also be done by AI. AI could generate a landmark proof and then people could use it to solve or simplify intermediate problems and you could use a different AI prompt to try to disprove it if you were really skeptical. From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it.
I don't like nuance here. I think progress is really measured by what humans are able to do and understand, not machines. It is significant if we find problems we struggle to solve. That tells us something. What does it take for humans to solve these problems is related.
The best analogy I can give is if you wanted to climb Mt. Everest you might ask someone for guidance. Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak? This is like the AI versus human gap to me. The helicopter is like using AI to generate a proof. The person who actually climbed Mt. Everest has firsthand knowledge of the experience. Same thing for a difficult proof. The struggle people have is actually valuable here. Likewise, we know people are actually capable of climbing Mt. Everest but if they had only ever rode a helicopter to the top, the knowledge of climbing it would not exist, and surely that is meaningful knowledge given the risks.
So if we rely on AI for proofs I think we lose a sense of what is difficult and why. We lose a sense of what human achievement is. Surely climbing Mt. Everest means more than taking a helicopter up? For students, why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now? This would have the affect of destroying knowledge.
(please do not nitpick the analogy because it's the best but perhaps a clumsy way to describe my thoughts)
> From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it.
Keep in mind those ~88 hours were spread across ~10,000 simultaneous agent instances.
So, roughly 880,000 hours of compute.
Assuming a fifty-year career, and forty-hour workweeks, a human mathematician's career is about 100,000 hours of "compute".
I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already.
The perverse incentives of academia mean this has never occurred.
The perverse incentives of industry mean OpenAI intentionally scooped researchers who were getting close (granted, with AI help).
I'm not trying to dismiss the achievement - if the proof turns out to be solid, it's quite impressive (though much less so if the training data included the recent human breakthrough, which seems pretty plausible).
I'm just pointing out that "88 hours" is a very misleading way of framing this.
> I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already.
> The perverse incentives of academia mean this has never occurred.
This. Mathematicians in their most energetic years are trying to get tenure or land a tenure-track job. They are disincentivized to go all-in on ultra high risk, high-reward problems. The potential downside is just too forbidding. It's much safer to develop a research program in a mainstream field that affords many opportunities for partial progress that can translate to a robust publication record.
I agree entirely with what you're saying, right up until your final question:
> why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now?
I think you answered this yourself earlier:
> I think progress is really measured by what humans are able to do and understand
People want to make this progress. Therefore people will "grind Everest" as a mathematical community, and that is maybe not so hugely different from a lot of previous mathematical work.
There's still ample room for creativity: simplifying, generalizing, asking new questions humans are interested in, ...
> Ultimately we think a fatal flaw was found in Mochizuki's proof, so it didn't lead anywhere in particular. But in our hypothetical "AI lean-verified proof of RH" situation, it would presumably generate substantially more of that community activity we saw in the Mochizuki situation. And if it's correct, that community activity would be productive (expository talks, students given problems to flesh out or generalize, etc).
This also sounds like a vector for trolling the community with complex putative proofs hiding a known flaw.
There was a hash collision bug in the main Lean kernel that was patched, but AFAIK nothing relied on it. You'd have to know what you were doing to accidentally get there...
The difference is the scale. A few incomprehensible long papers per year, sure, we will study it. A flood of AI results closing research directions left and right, that will be a problem.
Why would research be closed in one direction? Even if AI or human says "Tried that, didn't work" or whatever, someone (or something I suppose) might very well retry it in the future, if nothing else to reproduce it didn't work, in theory at least.
AI tends to take nearly finished research directions and push it to the conclusion in one step. If deployed massively, it will pluck all the low hanging fruits causing a drought of near term promising research project. Because people who start promising research directions do not get to see it finish, over the long term fewer people will start new directions, causing the field to slowly whither.
> Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
I think a better comparison is: mathematics just becomes like mining bitcoins.
I think you might have to explain that comparison a bit more to be honest. How are math proofs like bitcoins? A bitcoin has a pre-defined value, a math conjecture / proof is a bit more complicated.
Mochizuki's claimed proof of the abc conjecture was extremely unusual for the reason that nobody was able to extract a single useful idea from the argument. I was starting grad school when it came out, and my immediate visceral response was "if this is what number theory is going to look like in the future, then I will leave mathematics."
The current wave of AI slop mathematics might end up driving the next generation of mathematicians away from the subject for the same reason that Mochizuki would have convinced me to quit if his proof had been accepted by the community. Luckily, my professors had the taste to immediately recognize that it was garbage.
I really like this take, and while I hate math I value it. Your position sounds extremely plausible and it fits with the pattern we see in the community here.
Regardless if it's ai slop or not we still debate the value and attempt to understand. In the process generating new insights and ideas. Life will go on.
It's not just the isolated dumping, it's the fast, isolated, possibly untraceable dumping, without long term support.
It'll basically become slop fatigue if OpenAI starts dumping out proofs faster than the community can keep up, and some turn out to be wrong, never formalize it, don't stay to support it, etc.
I wonder if they will continue to dump proofs, though? Their point has been made, the novelty will wear off, and it maybe won't be a priority use of their resources to spend however many millions on another big proof--they will move on to the next thing to show off I'm sure. At that point, the ones generating proofs will be, I hope, mathematicians (professional and otherwise) that are more interested in the results and community discussion.
(Well that's my hopeful, optimistic take, anyway.)
They aren't going to stop at one, that's for sure. They already claimed they have "made substantial progress" on another millenium problem. Let's say they bag another one (Hodge and/or BSD according to the rumors), if it looks like their internal model could solve P/NP or Riemann Hypothesis, you think they wouldn't take that chance ?
I think AI companies making a point is not the only thing at play. Discovering new maths ultimately leads to new technologies and applications. It may start theoretically but end up being of practical use in the future. Even if humans do not understand it (lose interest, too complex, or just way too many new proofs to go though) AI can use this AI derived math corpus which will help it in other fields.
So it wasted everyone's time, thousands of hours of research trying to disprove something said very loudly. What OpenAI is doing is a DoS of the scientific community: wasting your time trying to check if they're not wrong, and claiming glory in the mean time.
That's true, but the story would have unfolded differently if Mochizuki had a lean-verified proof and was correct. I guess baked into my premise is that AI is producing reliable proofs (in the long term at least).
Even before AI we used to say if you write code that you only barely understand, then it will be to complicated to debug. (and/or maintain)
Mochizuki was still one human and it required legions of other humans to unpack and untangle to confirm that it didn't lead to anywhere in particular.
AI is now capable of constructions so complex that no human or human team can unpack. And its ability to increase that complexity is growing while our human ability is stagnant.
meta-AI analysis cannot help. We (software professionals who use AI regularly) already know that if you run into a situation where a Fable/Astra-generated analysis reaches the limits of our comprehension/complexity due to their subjectivity, throwing more AI at the problem doesn't always converge.
There are many reasons to feel optimistic about AI, and ultimately its general ability to help science and mathematics.
I see no reason to feel optimistic about the future of mathematics and AI based on the current path of frontier labs, unless the misalignment Tao is writing about can be reconciled.
> AI is now capable of constructions so complex that no human or human team can unpack.
How can we possibly know this when we haven't even seriously started on the endeavor of actively reverse engineering these AI-generated proofs? That's a proper job for human mathematicians, because the AIs themselves are demonstrably clueless about what steps in a proof are genuinely interesting and load-bearing from a human POV. This is evidence of a limitation in AIs' capabilities, not of any kind of misaligned behavior. The fact that Tao actually uses that term in his complaint is deeply disappointing.
Not to mention, there are already (pre AI) machine-generated proofs that we've pretty much agreed not to try to explain fully, like the four-color theorem which ends up with brute-force verification of 600+ cases (down from close to 2,000 when first demonstrated)
> there are already (pre AI) machine-generated proofs that we've pretty much agreed not to try to explain fully, like the four-color theorem
Algorithmic verification is a very unsatisfying answer to the problem (e.g., surely it's not just dumb luck that every single case happen to have this exact property), but that's an entirely different issue than saying that no one follows logic of the proof method itself.
For the interested; the saying I believe you are referencing in regards to writing code / debugging is from Brian Kernighan, specifically:
Everyone knows that debugging is twice as hard as writing a program in the first place. So if you're as clever as you can be when you write it, how will you ever debug it?
The first major computer-assisted proof, of the four-color map theorem in 1976, was an example of this. It created a lot of controversy at the time. It used proof by exhaustion, i.e. essentially analyzing every possible relevant case, something that no human could do without the assistance of, at the time, a supercomputer.
I get excited at the idea of a world in which advanced mathematical problems (and their solutions) become much more accessible to a much greater number of people. As a result, making mathematics much more loved at a societal level.
Imagine a world where these most complex mathematical problems are not accessible to a few hundred people, but a few hundred thousands people.
...Those original few hundred gifted mathematicians would have an even more prominent role, and their names and achievements would be known by orders of magnitude more people that they are now.
This is the hope, but I suspect the reality is that we see an ever widening gap between the fortunate and the unfortunate. We're looking at the automation and commodification of all knowledge, and the best models will be kept locked behind closed doors so that they can't be stolen. And, of course, "for our own protection".
Based on current reward models, the frontier AI labs will burn down mathematics as an impressive display of capabilities and in doing so, will make it impossible for people that get paid to do mathematics to stay employed.
If your job is literally to publish papers, and OpenAI and Anthropic decide that making an infinite-paper-printing machine is the best thing to show how effective their tech is, then as a demo, they destroy that industry.
I wouldn't expect them to destroy that industry, imagine global squadrons of academics and mathematicians focusing their attention on LLM's, training algorithms, scaling laws, ... they're gonna try and beat the incumbent frontier AI labs, eye for an eye, tooth for a tooth
I've met a few Ph.D Mathematicians in Academia socially. My unfortunate experience was that they were insufferable,borderline hostile people. I tried to genuinely engage with them too. I've met one Ph.D Mathematician that left the industry whom was very enjoyable to talk to. I have a feeling that my experience was not unique and the Math world is mostly a bunch of too good for everyone on their high horse a-holes that are now being knocked down a peg. They don't like it obviously.
I'm not a fan of knocking down things that work, however I also find it hard to be against death of the gatekeeping old guard of any industry.
I think math is just gonna have to suck it up like every other industry now. Math productivity is longer out of reach of the average grad student. Like every other industry they are no longer untouchable and are gonna have to adjust to the new way of things or market forces will do what they always do which is refuse to fund ineffectiveness.
I've had to accept that tech/IT will never be the same. Just how it is. You can thrash against it all you want.
Tao's critique of AI in the field of mathematics reminds me of what French art critic Charles Baudelaire said in the 19th century about photography [0].
Baudelaire argued that photography became a haven for failed painters, the sorts of hacks that could not finish proper training. Photography, as a mechanical rendering of the world, could only record what already existed; it couldn't transform reality the way a painting could.
He also criticized the public's craze for "rushing" into it, and complained that this technical "progress" was weakening the arts.
I have listened quite a few interviews with Tao and I see him being very careful about criticizing AI. He very often emphasizes the usefulness of it. Where he is critical has a lot of merit. One of the points I clearly remember him saying that having AI be able to solve many of the open problems, regardless of how important there are (there are many open problems that are not that important) greatly reduces the problem space for mathematics students to give new problems to work on.
In a parallel thread omnicognate correctly pointed out that for AI companies it's a direct commercial loss to pour all this money into bruteforcing the solutions to these problems, and that a lot of times the solutions by themselves are not directly commercially valuable. They are doing it for stock price, trying to lure in private capital in preparation for IPOs.
Their models are good, but they are not the moat because Chinese models are good too, so what they are doing, in my opinion, is more harm than good. Mathematics is a science by humans for humans.
>I have listened quite a few interviews with Tao and I see him being very careful about criticizing AI. He very often emphasizes the usefulness of it.
This is tiresome. People should be able to flat-out criticize AI without the implied need to justify themselves all the time or "be careful". Its almost like AI has a trillion-dollar agenda backing it, to the point that you have to add a careful "its really great! But there's this little issue..." for any criticism.
Even those who are very pro AI should have the intellectual honesty of admitting that there are very valid reasons to criticize AI.
Based on the amount of low-effort criticisms out there, clearly nobody is afraid of "big AI". Tao seems careful about his critiques of AI because low-effort off-the-cuff criticism can lead to all sorts of future problems/hypocracy.
I think there's a difference between "photography will change art--we need to be ready" and "photography will change art, therefore stop photography."
There is no doubt that AI has changed the practice of mathematics, just as it has changed the practice of software engineering (and will soon change almost every intellectual job).
Trying to deal with change by saying, "please stop the change" is foolish, IMHO. Mathematicians need to redesign the discipline with AI in mind. But I get that it's easy for me to say that and hard to actually do.
Both of Baudelaire’s criticisms were reasonable, and the same thing happened again when AI image generation showed up. You think the opponents look ridiculous because you’re viewing the history from the winner’s side.
As for “the public,” people had a real demand for photography as a way to record things, which is also why it won. There is no comparable public demand for proving Fermat’s Last Theorem.
I don't think the analogy works because for the past few years Tao has been one of the most vocal advocates of AI in mathematics and has used it extensively in his own research. You can find several of his talks about this on YouTube. It's completely consistent to believe two things at once, that the tools are useful and that the companies are misbehaving.
Genuine Art versus Mechanism, from 1901, (https://www.jstor.org/stable/25505621) is another article that I read a few years ago that other people might find interesting.
Tao doesn't go as far as Baudelaire, but there are some similarities. In particular, Tao has criticized that AI is not being used to create new interesting conjectures, and that the rush to prove old conjectures is not giving human mathematicians enough time to carefully analyze and understand the proofs and the methods used in those proofs.
My answer to both is the same: nothing stops mathematicians from doing both of those things, with or without the help of AI. And we all understand that it will take time to do that. But complaining about the dawn of a new era of advancements seems counterproductive.
> But complaining about the dawn of a new era of advancements seems counterproductive.
you completely misunderstood the critics. your analogy is awful. this is much closer to the industrial revolution in the uk: it brought a lot of progress, but also extreme inequality and concentration of power.
I don't see any meaningful analogy with your cites of Baudelaire. First of all, Baudelaire discusses art, which is very different to science. Several new forms of art have emerged, and then slowly integrated, despite the strong opinions of some. Here, Tao's letter discusses mainly practical aspect of scientific research. It is not about what a "failed mathematician" would do, or if "LLM can only record what already existed"; in fact, it ackowledges that LLM could become a central tool. If you want me to be even more precise, what they're saying (without saying it out loud) is that tech companies have too much power and are being irrespondible with it because they don't even think about the externalities.
> It is not about what a "failed mathematician" would do, or if "LLM can only record what already existed"
This is what I was alluding to:
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce [0]
Of course there's a meaningful analogy. The poster wants to make Tao sound unreasonable and against "progress" while ignoring the very real concerns Tao has.
It's what fanatics do when they want to enforce their view on the world, they have to attack anyone with a reasonable viewpoint because they can't imagine a world where someone tells them they don't like what they're doing.
I do not want to make him sound unreasonable. He is a brilliant man. But I also think that he is a bit shell shocked, much like other luminaries in the past have been when confronted with rapid technological development, so I wanted to highlight some historical similarities, imperfect as they can only be.
As for the rest of your comment, I hope you have a great day.
It's not about whether the conjectures themselves are interesting or not, or whether people simply have enough time. It's that a bare proof made by a machine doesn't actually do much for us. There is not some set of problems that, once finished, will amount to some kind of final, correct system and we can call it a day and, like, utilize it. "Mathematics" is the people doing it (the "mathematical community" Tao references below). This other stuff is kinda just.. expensive exercises to render a result. They are only actually beneficial to us insofar as they exist in a context of research among peers.
You have expressed a few different ideas, so I will address them separately.
> It's not about whether the conjectures themselves are interesting or not, or whether people simply have enough time.
Tao seems to think otherwise, if I am reading him correctly:
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce [0]
Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. [1]
> It's that a bare proof made by a machine doesn't actually do much for us.
I get it, and I think the same can be said about all sorts of human endeavors.
> There is not some set of problems that, once finished, will amount to some kind of final, correct system and we can call it a day and, like, utilize it.
Sure. Although there are certainly practical applications to be found along the way. E.g. proving P=NP would be potentially very significant in the real world. I think we agree.
> "Mathematics" is the people doing it (the "mathematical community" Tao references below)
Sure. And the same can be said again about all sort of human endeavors. But I don't see how that is a reason to stop using AI in those fields, either. It doesn't subtract anything, in the same way that chess engines didn't destroy the love of the game for chess.
And just like in chess, these AIs can be used to gain a deeper understanding. Including, but not limited to, explaining to humans the proof they just came up with.
Are you trying to argue toward some final verdict with regard to LLMs and mathematics? I thought you were just trying to make a comparison to Baudelaire? I think maybe being clearer on this point would help. Even if you argued sufficiently for the latter (which is going to be tough already), it wouldn't really speak to the former. Or at least: that would have to be a separate argument I think.
Also, how, in your words, do you feel like the first quote justifies your point (presumably with regard to the question of "interesting" or not)? And why do you think the second one is more about time itself rather than attribution? Do these things actually contradict the letter above (or the comment on it) in your mind or not?
In general, do you disagree with something here specifically? Or is it kind of a yes/and thing? Does any of this help, in your mind, with the Baudelaire comparison you were at least at one point trying to argue for? Its a bit hard for me to see the argument here, if there is one, just with what you have written. But I am sure I am just not knowledgeable enough to grasp the argument!
My opinion is that the field will adapt. AI will not spell doom for mathematics. On the contrary, it is the beginning of a new era. Math is having its Deep Fritz moment, just as computer science is going through the same.
Some folks are struggling to adapt to this change. Tao actually sounds like he is doing alright compared to most, even if some of his arguments seem a bit weak, as I alluded to in other comments in this thread.
Hopefully it makes some sense. And if it doesn't, at least we had a nice chat.
Haha yes I imagine we all know your opinion is something like that! But isn't it much more fruitful and interesting to form an argument for it? If he is still in the context here: is there something about Baudelaire's critique you were discussing originally that has helped you arrive at this opinion? Do you think he is wrong? Or is he right, but providing the right kind of nuance that we need in order to cope with the revolutionary changes? Genuinely curious!
People aren't ready to discuss AI assisted imagery as art yet. Most discussions lack the nuance that Baudelaire lacks in that critique, which deals with the nature of art and the importance of human intention and input.
Keep in mind that his critique is very recent, and likely applying to a specific use of AI, as opposed to AI as a whole. If you've been following his Mastodon account, he's been happily using LLMs for math purposes for well over a year.
I'm really tired of these arguments (this and "it's just like calculators").
Photography decimated other forms of visual art, so the concern wasn't wrong. But AI threatens the entirety of human intellectual endeavors. I can make do without oil paintings in my home. I'm not sure I want to live in a future where we make do without brains.
Lots of people still make bad music that other people still manage to enjoy (a lot of it has gone multi-platinum!) even though they're not Mozart or Bach.
It did. While you're busy recording an "instagramable" moment, you are not entirely enjoying that moment. Your eyes are on the screen, not on the subject.
In a way, if photography is an ersatz for painting that eventually made imaging available for the masses, then AI could become an ersatz for thinking. But it feels like I'm paraphrasing TFA.
What will happen when AI companies have spent their advertising budget on math problems and whatever else gives the maximum wow effect for the bucks? Probably customers hooked on the vain satisfaction of spending token$ to impress friends.
Wasn’t the criticism of photography kind of correct though? I don’t think people think of photography as an art as much as painting is an art.
There aren’t a lot of photographs that a layman couldn’t in principle also take, but only a few people could replicate a good painting.
We can’t look back with perfect hindsight because both the past and present have deeply ingrained blindspots. They don’t know what it is like to live in a world with perfect edges. We don’t know what it is like to live in a world with no edges. We can read about someone who proclaims that “something will be lost”. We will just think “but I have no need for any of that.” But we don’t even know what it is.
> But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal.
I will quote Richard Feynman - “the prize is in the pleasure of finding the thing out, the kick in the discovery”.
My personal example is Paxos. The algorithm is simple (it fits a single slide). But it is really hard to understand _why_ it works. I spent many many hours thinking about it, until it made sense to me. I was so happy. I wrote about it (https://uvdn7.github.io/understanding-paxos/ ) and I have gotten very good feedbacks.
The goal for me is not just to follow the protocol to build paxos (or raft for a log if you will). The goal is understanding and insight; it is to understand the why and the how. I still remember the moment when it all makes sense to me.
I am not saying that should be the case for everyone in every discipline. But if there is one subject that is mostly pure curiosity driven (instead of worldly impact), it is math. Robbing them the primary motivation is brutal.
If all problems can be solved by a machine, what do we have left to satisfy our curiosity, our desire to explore, and where can we find the pleasure of “figuring the thing out”.
This sounds a lot to me like people in the 90's complaining that computers were destroying chess. Thirty years later, chess is more popular than it ever was, and chess players are better than they ever have been. I wouldn't be surprised if there are now more chess books now than there ever have been. Furthermore, it turns out that a lot of chess books written before computers were just wrong about a lot of things. It turns out having an oracle for the "right" answer in chess, even without an explanation, used properly, allows humans to develop broader, more accurate insights.
The argument here sounds similar. The fear, as I understand this statement to be saying, is that by being given the correct answer, in the form of a 100-page Lean proof, humans will be robbed of the chance to from insights about the structure of mathematics itself. I don't see any reason that humans can't continue to develop insights as they try to digest the 100-page Lean proof into something more manageable; but with more certainty and fewer false starts.
As a chess fan, 100% this. We have known for the last ~15 years who the best human chess player is, and that he will lose against stockfish on his phone. But chess survives because of the human characters involved, the rivalries and dramas, watching two people trying to overcome each other under insane pressure, and sometimes coming up with something astonishing. In short - it's a sport.
I guess part of the problem is that being against being against anything for economic interests doesn't really rally anyone to your cause; everyone has to make a living doing something productive for society, and professions have come and gone all the time due to technological advances. In fact, when one thinks about it, the people that are losing their professions now were major contributors to others losing their form of income. Often people talk about how they can use technological/programming/IT skills to make some secretary or administrative assistant's job obsolete. So most people just don't feel a lot of sympathy when people complain that AI are going to take those people's jobs.
That's correct. As far as I know nodody builds any kind of science or technology on top of chess, but mathematics is at the base of most science and technology. It would be horrible if we prevented AI from solving mathematics problems, just because mathematicians want to solve them by themselves the "hard way".
But computers have destroyed chess as a "sport". Nobody will sit to watch two chess programs compete, or analyze their tactics. Kinda like how now, anybody can construct a "game" over the weekend or a new song or a slop video. The value of each of these decreases to 0 as the slop overwhelms.
>Nobody will sit to watch two chess programs compete, or analyze their tactics.
I know nothing about chess yet I dare say that I'd doubt this. Surely chess enthusiasts would be interested in analyzing how a superior chess program came out victorious, no?
Yes you are right, some people do watch chess engines play. TCEC (Top Chess Engine Championship) [1] streams them. Popular chess YouTubers goes over engine games from time to time too.
That's not proof of interest so much as proof of commerce. It could easily be a money-laundering mechanism.
Mostly nobody cares about professional chess. The number of people who are actually interested in today's game and not the drama are a tiny sliver of that. This is actually great because it means we can train and evaluate both without interference from chess players, possibly even building a stable society.
> I wouldn't be surprised if there are now more chess books now than there ever have been
Well yeah... how would there be fewer??
But the point itself is silly. Few people are putting effort into Maths for the fun of it (and of those that are many derive fun from being the only one who can produce a solution). Chess differs in that it never had any point but the game its self.
The woman who made up the workforce of telephone switch operators would like to have a word.
Meaning - every new technology has both been perceived as a threa and often forced change in society. Agree maybe “it feels different” this time, but don’t you think everybody before us just said the same thing?
Also not clear if this is an actual called action.
> We are witnessing a general threat to intellectual work
This is the crux of it and goes far beyond Mathematics or Computer Science. To get a bunch of humans to do anything, you have to motivate them. Kleos and Timē; renown and stuff. These AI companies threaten to rip this away from everyone but themselves, and this recent millennium prize is the perfect example.
Solving this problem as a human would have led to tremendous Kleos; my name would be written in the annals of mathematics, lecture tours of praise were mine to be had for the rest of my days. This one victory would have earned my recognition throughout history. The greatest a mortal may hope for. Ripped away.
It would also have given me great Timē. The prize money, the professorships, the book deals. Gone.
If all hope of “renown and stuff” in the intellectual realm is now taken by the AI companies, they will remove all human motivation to pursue these endeavours.
Perhaps the glory will come from slaying these fell beasts.
> But there's still play. There's still curiosity. And there's still the drive to understand something for yourself.
Yes, but think about what that implies if those are the only motivations left. Gone are the professions. Gone are the ambitious.
There is plenty of space for people to work on intellectual pleasure pursuits (as there is with art and music), but the death of all intellectual based industries is still something to avoid. Or to mourn.
This letter is complaining that human understanding has been crucial to advancing of mathematics, and AI companies are not bothering with it. But the promise (and horror) of AI mathematics is that, if it succeeds, human understanding becomes irrelevant. That's the goal. So this letter's message will fall on deaf ears.
Keep in mind employees at AI companies are publicly stating that they believe they're risking a >10% chance of human extinction. They're knowingly risking the lives of every man, woman, and child to continue the work. The lives of their own sons and daughters. A person already rationalizing that isn't going to shed a tear for the careers of mathematicians. Just a bug on the windshield.
I understand this stance and where they are coming from, but I can't help but think this sounds very analogous to engineers' arguments against AI-assisted and vibe-coding, especially with regard to cognitive debt. Yet the software industry is plowing ahead, reportedly pushing mountains of unreviewed code to Prod, and the world hasn't ended.
Of course, nobody's really comfortable with it, so this is also a forcing function for the industry to adapt and figure out new techniques to manage complexity and trust. I think the same will happen with Mathematics.
But it is also possible we will end up with three forms of Mathematics: the one we understand, the one we don't, and the one we don't understand but can prove to work. Kind of like magic -- with all the positive and negative connotations of the word.
It is pretty evident that these models will soon exceed our cognitive capabilities. Is it right to hold them back just because we can't keep up? Many of those discoveries will be so beyond us that we can't do anything with them, but that also means they can't hurt us. On the other hand, there could be many discoveries that we can parlay into practically useful applications, even if we don't understand them.
It really has. All of the places facing an unusually high outage rate are places that have seen huge growth in their service usage (Anthropic, GitHub, etc) which is to be expected. The rest of the world has been happily chugging along with coding agents for almost a year now and things seem to still be working just fine.
There's an unpopular branch of mathematics which does not have infinities - finiteism.[1] The constructive version of finitism takes the position that there is no such thing as infinity, just arbitrarily large upper bounds. You can have theorems about arbitrarily large numbers, but you never get
1 + 1/2 + 1/4 + 1/8 ... = 2
The benefit of finitism is that it escapes undecidability.
The big objection to finiteism is that it's a lot more work. Infinity swallows many special cases. Proofs get longer without infinity, and most of the special cases are uninteresting. That's not a problem for AIs.
Someone may start up an AI and make it grind through Hilbert's program for putting mathematics on a fully consistent foundation, starting from a finiteism base. This is a huge, unrewarding job. Great for machine work.
Limits can be defined within finitism as long as the end result is finite. Essentially it’s just a process which lets us get as close to 2 as we want.
A better example would be a limit that equals sqrt(2) which finitists would probably say cannot represent a real object because it is only defined as the end of an infinite process.
It's not necessarily clear that this statement requires infinity, if you're willing to treat "... =" as a shorthand. You might prefer something like "1 + 1/2 + 1/4 + 1/8 ... -> 2" if it's more clear, where "->" means something like "gets as close as you like without ever getting further away than that", but really the "=" sign is already overloaded in all sorts of subtly different ways anyway, so there's not really any trouble using it here. Almost any rigorous definition you can write down of exactly what that statement means would not rely on the use of infinity.
If you allow infinite recursion, you soon get to Godel and undecidable problems.
Finite deterministic systems are decidable, because you can in principle enumerate all the states. The halting problem is decidable for deterministic systems with finite memory. It may be exponentially hard for some programs, but that's quite different from being undecidable.
(This is too long a subject to discuss here, and I haven't worked on constructive mathematics in many years. It's more practical than it was decades ago. You need power tools, which we now have.)
Finitism doesn't escape anything, it just gives you the illusion of safety. Any intellectually honest thinker should accept the possibility that 10 is a nonstandardly large number.
In internet culture there’s this phrase “Hydrogen Bomb vs Coughing Baby”, meant to highlight the absurd power difference between two combatants.
In almost any scenario even tangentially involving mathematics, twenty-five Fields medallists uniting to denounce something would be a veritable Tsar Bomba.
It should give you pause that here they feel like the ailing infant.
My interpretation is they don't care about scooping mathematicians they were just trying to scoop a competitor. They had a short window in which to complicate Anthropic's priority, when Anthropic announced be able to say "ok nice but we did that too". Upon realizing they'd misunderstood, incredibly they said to this academic (who did not resolve NS) okay well just go ahead and claim the prize, so long as you're not Anthropic let's make this a good story.
If you spend $20M working out a Millennium Prize problem, in what universe would you offer that an unrelated effort should take credit? This is a branding game rather over whether software engineers are going to use codex or claude. In that light $20M (or whatever it was) might be worth it to squash even the rumor that claude code is more capable. Engineers look up to mathematics, while at the same time business and probably most engineers think the problem was to solve the problem. GPTs solved one the hardest known problems so they can solve my company's problem.
Some go further looking at these people, very on-the-nosely likened by one commenter here to ants, talking about education and responsibility and "core values" etc and just don't care. There was a major problem at the beginning of the week that is not a problem now and that is uncomplicated progress.
It's not wrong for OpenAI/Anthropic to do math for product development or even just branding but seemingly at no cost now they could work in an arena real mathematicians aren't interested in, versus just mowing the field. Everyone involved on their side should admit the purpose of these demonstrations is not to engage mathematical ideas it's about Claude/Codex. there's no shame in that. Which is better at solving random hard Diophantine systems? That would seem to tell me as much as I need to know insofar as a model's value is represented by raw mathematical power - then, take my money just as well!
Assuming the worst accusations are not true I think there are ways forward going to be acceptable for all. The labs themselves do not represent Terrance Tao as some kind of gate-keeping dinosaur in this. They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes are paid 7-8 figure salaries ultimately for the product, they solve a Millennium Prize problem then pretty quickly seem to move past it.
>they weren't trying to scoop a mathematician they were just trying to scoop a competitor
Well. Obviously, yes. But in doing so they still DID scoop out a mathematician in a very unethical way.
In doing so they showed that they basically have a huge gun they can point at X work you care about and develop, and can cut you across the finish line. And take credit for it. Obviously this already _existed_, but is just much more significant because even a rumor can be converted into a complete takeover of a discovery.
> They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes take home 7-8 figure salaries, they solve a Millennium Prize problem then pretty quickly move past it.
I get your point, they don't really care about solving all the maths problems. But they're still going to solve them for clout and profit motives. Up until it stops wow-ing people... at which point they will have likely decimated the frontier of the field.
And this is kind of the root of the entire concern. They will move into the forest and completely steamroll all the problems, then declare victory and move on, leaving only pavement and asphalt behind.
The root of all these is the culture in mathematics (and science in general) to only reward those who “get there first”. This creates a perverse incentive to compete. When no one can out compete a tireless swarm of AI, no one gets rewarded any more.
But nothing’s stopping anyone to still work out an alternative proof, or a more elegant proof, or just trying to prove for the sake of understanding, just like doing homework without looking at the solution. It’s just that you can’t get paid doing that anymore.
Mathematics is about discovering and understanding the logical implications of assumed axioms under various inference rules.
Alternatively, some claim that mathematics is about understanding these implications.
Under the first definition, AI is already, and forevermore will be faster and better at proving theorems. Just like it is better at checkers, chess, and now go.
The author asserts that AI proofs are incomprehensible to humans, and so under the second definition AI is merely a tool to overcome one hurdle on the way to understanding.
So which is it? The author seems to claim the second definition, but bemoan the end of mathematics under the first.
Dont confuse mathematics with the formal system. If you beleive mathematics = formal system then AI is obviously better at it, and we dont need humans.
But then who decides why a statement is mor important than another? In the eyes of a formal systems all statements are born equal.
I don’t think they make a coherent argument here? This seems to hinge on some argument that because AI doesn’t properly explain its breakthroughs, therefore the breakthroughs are less fertile for human understanding? This makes no sense. Why wouldn’t these under-explained breakthroughs be extremely fertile soil for explanations?
Imagine time traveling back in time and offering Leibniz a packet of proofs from the intervening years, but with the caveat that there would be no explanations. Would he say no?
Im surprised about the sentiment in this discussion.
I totally see the problem Terence is describing. We are loosing a lot in understanding and focus if it continues like that. The solution found for Navier Stokes doesn’t have much „real value“ - but what almost always happened in the past when people worked on the difficult problems, these sparked new ideas / new theorems that broadened our knowledge.
Think back at your grad studies, figuring out a proof as homework was hard, sometimes incredibly hard, but while doing it we gained a lot of understanding how things work. Now asking AI for the solution and „just“ getting it, risks our understanding, our creativity and our ability to connect the dots with other territories. I see it in students nowadays, there is much less understanding, much less creativity in finding solutions. I truly think this „short-path“ solution with the „death of struggle is one of the biggest risks with AI already for human development
This reads like people lamenting a bygone era and making a desperate attempt to bring it back. I'm sorry. Outside of the good ol' boys club, no one cares about some process they've romanticized simply because "that's how its always been done". Absolute nonsense.
We are moving forward and if that means no human wins a fields medal because they didnt spend three decades working on a problem that could be solved in three days, the world will be better for it.
You can still do all that. The only thing going is that in some cases the human isn't going to be able to claim that they made the key insights that first solved the problem.
Are you implying that OpenAI using someones unpublished research without their permission to solve an career defining math problem with their latest model in order to publish first is a problem with the mathematicians?
My read on this document is that people's work isn't being fairly cited more than what does it mean to be a mathematician in this age.
I didn't know this article was about that issue at all. Yeah, if the issue is properly citing work then yes, OpenAI needs to do that. But the article read like it was tackling a completely different issue.
That's one of several issues, obviously a big one, and they do touch on it:
> Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions.
Indeed, I wonder how a similar letter by Uber drivers would be received -- "navigation is an intrinsically human domain, personal relationships are critical for passengers and drivers to progress in the world, etc etc." Or doctors, for that matter.
We are all going to have to come to terms with entities more capable than we are, and in many cases, letting the real work be done by the AIs will be the right thing to do. For all the huffing and puffing about the "human touch" in medicine, it will eventually become downright irresponsible to consult only with a human doctor. I am not sure if this is the case in mathematics or not, but if it isn't, that suggests math will be relegated to more of a hobby than a cutting edge scientific discipline.
That's how I read it too, Terry Tao, who has been a "pro-AI math guy" is going through the same emotions and confusion that us SWE folks are going through, "oh, wait... this might mean I'm not going to be special anymore!?"
I don't mean to be a dick, but I've talked about it previously. These folks are grieving. I get it, I've lived through this sort of life changing thing before, it sucks... but yeah.
I think this is ridiculously flippant. If software engineering and the hardest math is solved, that means that eventually a majority of professions and knowledge work is solved. This is hugely problematic because of the way our society currently functions. People need jobs to eat, pay for housing, etc.
Dismissing it as "innovations have happened before" is disingenuous. Yes, innovations have happened, but none of those threatened to automate all human work in existence.
Yeah given he had been very pro-AI for years, I expected he made peace with the issue many many years ago (like I did back in 2018), and when this time would come he would explain to other mathematicians how to live with it.
> We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align.
This is Terry Tao talking about AI's impact on Math, but this could just as well be a software engineer talking about AI's impact on software development.
Do mathematicians deserve more job security than software engineers?
It's not about job security. It's about the social and intellectual practice of the discipline.
The threat to mathematics isn't that suddenly the profitability of their profession (lol) is going to go away, it's that people are thinking of AI as a replacement for the human social and intellectual practices that constitute the discipline.
> The threat to mathematics isn't that suddenly the profitability of their profession (lol) is going to go away, it's that people are thinking of AI as a replacement for the human social and intellectual practices that constitute the discipline.
You could say the same about software development.
Software development is a group effort, so it includes social practices, and certainly includes intellectual practices as well.
For the sake of argument, how is this different from the Luddites? The Luddites feared that machines would displace not only human labor, but also the social knowledge, skilled judgment, and craft traditions embedded in their work.
Kinda crazy to think that academia functions as a kind of humane reverse centaurism. Theory X (reverse centaur) before Theory Y (centaur) for managing the development of others.
All “fields medalist” signatories - a rarefied and elitist group indeed.
I wish this letter could be more egalitarian and include the view points of those who AREN’T the beneficiaries of a highly competitive winner-take-all system.
Since the common narrative is that AI frees up labor to do other things (engineering -> trades), maybe we can celebrate that genius mathematicians will now spend time teaching children how to be as smart as them?
In the Economist article Tao links, Hugo Duminil-Copin, draws a comparison: airdropping someone on the summit of Mount Everest is very different from climbing it.
The fundamental issue with AI solving any perceived difficult problem is that we have lost the journey. The sight atop Mount Everest looks much different when you have climbed compared to being dropped from above.
But we don't pour billions of dollars of research funding into mountain climbing because we think it's going to lead to wider breakthroughs in science and technology. And when we need to get people on top of a mountain for an important purpose -- like a military or search and rescue operation, for example -- we absolutely do airdrop them right on the top.
So that raises the question: is mathematics simply a pursuit of passion? Are problems solved "because they're there"? If so, then mathematics can join the ranks of things like mountain climbing, cycling, and weight lifting. But if we are trying to accomplish something important (design better airplanes, find theoretical guarantees about cryptography, factor matrices faster), mathematics needs to become more like a military or search and rescue operation, using the best technology available to secure the outcome we need. Given that the NSF pours billions into scientific research every year, it sure seems like mathematicians want to think of themselves as being in the latter category.
What defines important and why must it be solved in haste? Many issues and other problems arise during the journey in solving all problems; those that are needed and those that are pursuits for their own sake.
If AI gave us the plane to reach Everest without us having gone through the journey of aviation and flight, what would we have lost without that process?
But the most important problems to be solved are not technological challenges but social ones, involving humans and our relationship to one another. An area AI will forever ill-suited to handle.
as someone who loves to go down with a snowboard, I can see value to being taken to the top and then enjoying the ride down. im sure it is not a thing to be ashamed of, as millions do it.
I agree with many of the sentiments here. But an open letter signed exclusively by Fields Medalists that purport to define precisely what the "mathematical community" (who is inside and outside) and what their goals are raises my hackles for some reason.
It is true that the manufacturing of "true/false" statements is not the same as gaining understanding of a problem. However, for many mathematicians, true/false statements are already manufactured by others. Think of a student who is given a conjecture to investigate, with their advisor describing it as "it must be true". Most exercises in a textbook are stated such that the outcome is known before you begin. That's not really a problem -- investigating the conjecture/exercise yields its own dividends, whether or not the outcome is known. It is also the case that defining new directions involve understanding and synthesizing related problems, asking the right questions, and deciding upon the right directions, and it's not clear that AI can do that at all.
The real risk, I think, its the public's (and funding agencies') perception of the importance of "human" mathematics, but that's already a struggle. For example, it's tough to explain to the lay person why it's still important to research group theory -- the main example people cite is RSA encryption, which was invented almost 50 years ago.
Can't all the prestige-maxed mathematicians still study all these famous problems after ai solves them. And even if they convince open-ai to stop dunking on them, some normal user with gpt 7.1 on the normal chat interface will do it in a year.
Even if they stopped anyone from releasing ai proofs for five whole years it would be meaningless seeing as these problems are decades old already. Humans weren't JUST about to solve them until openai stepped on their toes.
Nothing AI companies will change the value of math, as William Thurston said: The product of math is clarity and understanding, not theorems by themselves. What they seek for the IPO is devilish and misleading, and doesn't serve the true purpose of the math.
Back in the day you could think of a cool idea. I don't know maybe a plane that could fly without drag. To even see if this was feasable you had to understand physics, engineering, and then from there you had to have a math person see if it was actually possible.
Now, I can ask ChatGPT about this and get back a proof that shows "a passive airframe cannot sustain zero-drag motion through still, viscous air"
So, I think if anything now, Maths has changed for the better. More ideas can be proven false or true from a get go instead of wasting so much to see if its even feasible to find out it isn't.
Progress if anything is about to leap frog anything we have ever known.
This is a pretty moronic take. You only have to understand the first thing about physics (e.g., a 14-year-old's understanding) to know that a dragless airplane is impossible.
If you are suggesting there is some new model of physics or groundbreaking technology that would allow such a dragless plane, then don’t let me discourage you! But AI won’t help at all, since it will only regurgitate conventional wisdom…
I find this explanation has a lot of applications for programmers within companies. It’s one thing to get your LLM to give an answer, it’s another to bring a group of people into shared understanding of a domain.
Playing a devil's advocate. Why do we need understanding ? To take an example i would say ~99% of the population do not understand how combustion engines or how semiconductors work, what say another 1% ?
Is the fear post-apocalyptic in nature ? We need some human priesthood to carry on tradition ? why ?
Let's assume in the next decade GPT-7 PRO Ultra is cheaply ubiquitous, inspectable, reproducible, transferable, reasonably un-constrained by any institutional interests.
I'm not convinced this is an alignment or technology problem.
If my boss vibe coded an app for the customer and then assigned me to get it working, it would be impossible to maintain. If he gave me enough AI tokens to vibe code the MVP myself and to my design, I wouldn't mind.
I think the same issue is at play here in maths. OpenAI owns the model and they can direct it as they please. They chose to spend lots of money getting a quick result, instead of developing mathematical infrastructure for the next generation of problems. The managers are in charge rather than the experts.
It's good to start a conversation, and the number of Fields Medalists behind this certainly lends a lot of weight to it. But I'm not seeing a strong argument for misaligned incentives beyond the specific plagiarism allegations. The job of an AI company is to build systems that solve problems. The job of a mathematician is to advance the state of human knowledge. If anything, an influx of solved problems should increase the demand for human mathematicians who can convert them into conceptual understanding.
Train an LLM with no advanced math texts: only basic math up to 6th grade, conversational text and literary works.
Interact with it (you cannot refer to anything past 6th grade math since you don't know it yourself) and get it to propose a solution to a real world problem. e.g., come up with RSA to practically secure communication.
This is really well written and exposes a core tension between science and something akin to engineering. The "engineering" of proofs has become "easy" (a compute and $) problem, rather than hard (a time and conception problem).
Without the ability to do things the "hard" way it is difficult to figure out if doing things the "easy" way will help us advance the frontier of math and science.
I may be wrong but historically we had this version of science discovery for a long while (empirical observation and brute force application) rather than first principles leading to applications (tools, the wheel, mills etc). Then somewhere along the way it flipped after Newton and the enlightenment period and started understanding first principles before they become engineering applications.
Perhaps it is not required, and we can just keep doing things the "easy" way like we used to, or we might find ourselves out of the ability to brute force things and then we go back to needing to do this the hard way, at which point this period of AI brute forcing would be seen as a detriment.
Is this really any different than the problem in software engineering - where AI is doing the work of junior programmers and now they aren't getting the development they need?
Seems the same to me. And it'll be the same in all industries soon enough. And then it won't just be the junior people.
Somewhat unrelated: is it wrong to say mathematics is not art, and that there is always a right answer? I know that's not romantic, but maybe it's true.
Before LLMS, programming was something I might've said required creativity and human input to do properly. It's not that creativity or human input isn't valuable anymore, but AI has forced me to realize that coding is much a means to an end, and that all things considered, the end matters much more than the means.
If we can make important mathematics progress faster and better with LLMs, I think it's wise not to fret over an apparent loss of our humanity. Perhaps that's only a loss we want to have.
There's a reason mathematics is generally within liberal arts programs rather than science programs. Mathematics is the art of logic. Yes, sometimes mathematics becomes incredibly useful but most mathematics is never applied.
Compare that with computer science. Most of the work we do in software engineering is in service of an applicable output - software products that facilitate processes or bring in revenue. Turning up the dial on AI gets companies to these outputs faster.
Turning up AI on mathematics helps solve conjectures and can provide new insights. But it has a major misalignment with the purpose of mathematics which is largely intellectualism.
“Mathematics is a part of physics. Physics is an experimental science, a part of natural sciences. Mathematics is the part of physics where experiments are cheap” - Vladimir Arnold
On the matter of computer science having anything to do with computers, please refer to Djikstra.
It’s about computation, not computers - an application of mathematics, predominantly thanks to Turing, Von Neumann, and Claude Shannon’s masters’ thesis; though ofc many others as well but I see them as three individuals who made the minimal structurally necessary contributions - VNA and silicon are one of many possible substrates.
To me, the "meaning" of proof is twofold. First is the understanding which is completely absent from a proof which depends on exhaustive iterations of instances or is asserted by fiat from myriad individually contestable paths. That's what I think makes AI proofs risky: they absence of understanding.
The second is the utility. We get to do navigation because the maths about angles and spheres checks out. The social utility downstream of AI proofs may be huge.
Maths will continue as a field of natural science in understanding the results and uncovering meanings in them. It is normal that the established community is afraid of the change, because it’s their _home_ that’s changing. But it will be a better home to the new generation nonetheless, one that’s not as daunting as the higher maths has always been to many. The concerns raised here will not be a problem at all.
Contrarian take: I think the ability of AI to produce valid mathematical proofs (even inscrutable ones) is absolutely fantastic. Mathematics as a profession does not have a monopoly over math itself any more than professional pianists have a monopoly on who plays piano, when they play, and how.
I have sympathy for any jobs that might be affected (much as my own job has become more tenuous in software engineering). And if the field is disrupted by chaos that makes the research process unproductive, that's bad too and should of course be handled by applying better organization within the institutions that tend to perform mathematical research.
But to a large degree, the notion that "sloppy AI proofs are bad for mathematics research" seems like a total failure of the imagination to me. Attempting to find shorter proofs or more elegant proofs can be turned back in on itself via proof theory. There are proofs in Presburger arithmetic that are doubly exponential in the length of the sentence. Yet a more powerful theory like PA makes quick work of such theorems. The explainability or "subjective beauty" of a proof can be quantified and optimized against. Optimization itself can be optimized against. I really don't understand how this magical ability to know the truth of more theorems much more quickly—even via an "ugly" route—is anything but a net positive.
So what's the actual point here? It's too fast and we can't keep up?
Isn't that just a function of the technology itself, and the same problem being faced by every other field? And going to get exponentially "worse" every year!
Or is the issue that they're bad at explaining things, in a way that produces actual learning? (e.g. AI is amazing for learning but the net effect on students so far appears to be negative.)
I kind of expected a sober stoicism from mathematicians. Feels silly in retrospect. This is just the math version of the "anti-ai" movement by "artists".
I don't know, people. We still really don't know how OpenAI or others are producing these results. It's all very hand wavy and trust-me-bro. How much money/time/compute have they really thrown at these problems? How much human involvement was there? What LLM did they even use? How much regular software was involved? They have given answers to some of those questions but no proof that that's actually what they did. I don't know if it's worth giving them this much credit (which is what we are doing by writing these essays and spending so much time debating). Anthropic wrote a C compiler that turned out to not really be a ready made replacement for GCC. Did they ever do any more work on it? Has anyone else produced a C compiler? It seems like that and these proofs are just demoware that are not (yet? Who knows?) production ready to turn the world upside down. Impressive one-off demos, yes, but companies have been pulling those off for centuries without ever going anywhere afterwards.
A very important point! In Tristan (NYU prof)’s write up he noted evidence of the OpenAI mathematician team doing a lot of correction and guidance along the way. We are never told about this with openness and clarity. At a minimum, complete disclosure and honesty is needed by the companies and about the precise role of their staff members.
I know that this comment section is not astroturfed, but it’s really uncanny how different comments are today compared with thread about solving navier stoke
Is the purpose of mathematical research to understand the ‘truth’ of numbers? or be the person who find that truth? I think people who are interested in finding the truth won’t care where it came from.
I was under the impression that mathematics (and science generally) had the primary goal of helping us understand our universe better than those who came before us.
I don't know how societies set "primary goals". After spending 15 years in a tenure track -- tenured position, I thought setting goals well was important.
I take pleasure in how mathematics and science help me understand the universe better than I understood it before I studied the fields. I believe that my understanding has helped me contribute to society.
Taking a snapshot of the state of AI math right now and concluding that it will be net negative to human understanding and insight in the future is very short sighted. This statement will be used to promote ideas and actions that will ultimately be disastrous for our country.
"We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose"
Suppose we eventually have GPT-7-class models running practically on $100 devices, with their activity transparent, inspectable, and reproducible. At that point, what exactly is left for us to fear from this threat?
I've got to ask, let's say we get a small modular nuclear reactor running practically on a 1 acre lot, its design meltdown proof and waste-free, what then should we fear from this threat ? This is just a thought experiment.
Jokes aside, any productivity-improving technology, even one with no negative externalities, has the potential to cause economic displacement and wealth concentration in proportion to the productivity gains catalyzed. Anthropic did a cool analysis of this for AI here: https://www.anthropic.com/institute/econ-scenarios
The fact that those models are encroaching on things only human minds could do. Personally, as a human, I want there to be things humans are the best at, and intellectual things were the final thing that machines hadn't beaten us at.
This is really only a short-term problem where the AI companies only have the internal models that can solve these. In the “long” term, which could honestly mean months, everyone will have access to Bel/C/D-level models capable of solving these anyway.
This. Like programming, the community will shortly be forced to come to terms with a lot of new self-proclaimed mathematicians “vibe-solving” problems and dumping solutions without understanding them. It’s not really a special case for mathematics.
Given the existence of this technology now and the incentives of the AI companies, both of which are not going away; what's a good future here?
A major part of the complaint is that there's no conceptual understanding and building of new ideas coming out of the AI proofs, thus defeating the purpose of the original pursuit.
If in 2027 the AI models start producing, with every mathematics or science breakthrough they make, well-written documents tailored for human understanding, with intermediate concepts, expositions of failed-but-once-promising paths, etc. Would that be good alignment with the mathematics community?
They can get with the program or be the equivalent of a genius SWE writing assembly on punchcards in 2026.
The only thing I read from this is their ego being bruised by a machine.
If these people cared more about discovery and advancement of human knowledge the only thing they should be doing is celebrating. There's no proof of plagarism but that's an independent issue.
How are they not realizing that in the future children will be able to do impossibly hard math but they will be doing something we can't even think of as of now.
One world class mathematician in the future could be advancing mathematics the equivalent of one Riemann hypothesis A DAY.
How are they not celbrating this as the achievment of the centry? Who cares about plagarism at this scale. It has been solved and it wouldn't have been without AI.
lol what a bad analogy. Why should anyone care how much effort a result takes to achieve? If anything our entire world functions because we reduce that effort as much as possible.
I don't judge you for not growing your own food when you hand me a burger.
It's a turning point for science and beyond. AI has shown itself to be transformative. Even today, it is already changing the how research in math (and other sciences) is conducted. In the near future, whether it is LLMs or some other superior method, its capabilities are only expected to grow. The time to ask the question is now: Will AI be arguably the best tool at scientist's disposal, or will it instead be paraded around as a super brain collective that no human or group of humans can compete with, discouraging entire new generations of future scientists from ever entering the field? The jury is out on this one.
For every benefit that sillycon valley has produced in the recent past, there have been many more harms. I am confident that this will be no different. Of course, benefits and harms depend on one's vantage point.
Could we develop new ways to develop understanding and explore new ideas, such as interacting with the models to explain and understand their proofs, as well as to brainstorm related directions to pursue?
>Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others.
That's just capitalism seeping through a previously unexplored crack into academia, and attempting to do the only thing capitalism knows to do - maximize profits - with no additional concern.
The level of things happening in the last several months is just SciFi level, especially last two weeks.
1. Overlords of AI threatening to destroy someone if their demands aren't met over the Millennia problem. Proving the point that given the chance, owners of the AI companies will immediately use their power to squash or control other people. (See Butlerian Jihad)
2. At least for me, the Exponential progress didn't sound true until this week. If AI indeed solved all those problems and proofs and solutions are correct, we went from "Write me English 101 college essays" to "Help me solve the Navier–Stokes existence and smoothness problem" in a just 3-4 years.
3.Looks like someone's work might have been stolen, by AI company and there are strong evidence. See point 1.
4. War between AI & AI companies.
5. It produces a machine verifiable LEAN proofs for the hardest problems know to us !!! Proving that point that best use of AI so far is to create verifiable systems. I.e the absolutely chaotic AI creates verifiable (orderly) proofs. It is Maxwell Demon: chaos to order.
Touché… In short you still need the humans to understand what’s going on at the end of the day whether it’s a mathematical equation or source code for a computer program relying upon mindless AI isn’t good enough. Humans still have to do the thinking or is it in the brave new world the error checking at the end of the day?
I’m sympathetic to the concern, but I’m still unclear on what the concrete ask is.
If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently? Also why does discovering the answers preclude humans developing ideas from them? I don't get why solving a math problem stops anyone from doing that?
Should they (AI companies) avoid training or evaluating models on open problems? Solve them but not publish the results? Delay publication? Only release proofs after mathematicians have had time to study them? Require some attribution or review process?
The statement makes a strong case that “maximize the number of solved problems” may be the wrong objective, but it seems much less clear about what behavior they actually want from OpenAI, Anthropic, DeepMind, etc.
I’d be interested in the most concrete version of the proposal. Without that, it starts to read a little like: "Please stop getting so good at our thing!"
> If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently?
Just stop doing that. Don't treat unsolved math problems as some cheap benchmark to beat.
Leave the math for mathematicians, and let them use AI in a way that helps the field, not in a way that harms it.
Academic research is a marvel because (aside from patents) nobody owns it, in the sense of property. It is given away to be used freely. Researchers want their work used and cited. The primary external reward for publishing is reputation and prestige which translates to remuneration for researchers. And that remuneration can be poor.
Beyond the issue of growing understanding and keeping a bountiful stock of questions to pursue, this scheme seems to be threatened as well.
> to the point that they can solve major outstanding problems in many fields of mathematics
I will die on this hill, humans solve math problems not machines, there's no automatic math prover out there. There are humans attempting to solve this problems either by leveraging these tools or not
At some point someone is going to need to answer for "what happens when all intellect is hoarded by one or two companies?" It's pretty clear that these AI labs are basically stealing everyone's alpha.
Maybe mathematicians should be aligned better, rather than AI?
The current measure of a successful mathematician is the problems they have solved or worked on. At some point in history, the measure of a successful scholar was how well one could copy manuscripts.
Once we have a tool that starts to work well for this task, it's time to define success differently. It's a classic alignment problem! ;)
But seriously, these people should start focusing on finding and proposing more important problems. And the credit of discovery should go to the person who defined a new category of important problems.
It appears to me this is an incredible inflection point in mathematics, a neat forcing function like cryptography was for the development for modern number theory and algebraic geometry.
Fundamental problems with great implications for other fields will be solved by AI because some entity would throw tokens at it. And these would be further built upon.
Having worked on nearly 500 complex lean libraries in math and theoretical physics, with AI assistance, I have some experience from the road to share here. In all my work I have found that AI is terrible at coming up with interesting ideas on its own - even the very best, latest models.
Without human scientific and mathematical intuition you can bet that AI will always take the road most travelled and miss the genuinely new and interesting breakthroughs - in fact it will not even think of them or try them without a human rider whipping it constantly to go down paths it would normally not consider.
And I think that is true in the case of the recent OpenAI blow-up -- it is alleged that even in this case AI did not come up with the winning technique without some human guidance.
I have never - across perhaps many thousands of chats with AI - seen AI push beyond the edge of what is known by itself, without being forced by a human to think outside the box.
I do think it might be possible to encode this process with prompting and agent orchestration methodologies - but even then - without a human - and human intuition - in the loop, I am skeptical.
Therefore I think a more accurate view of AI for math and science today is that it is an extremely powerful tool in the hands of a skilled operator with a strong intuition and roadmap of where to go, and rather boring when left to its devices.
Whether that will change in the future is a question. Nothing says that in principle AI could never do what a human driver does - but I still doubt that AI will replicate the life history and experience that real scientific and mathematical intuition is really made of.
AI trains on a lot of stuff - but it doesn't have hallway conversations, office-hours with teachers, or hard-won experience from all the things that failed that are NOT in the training data...
AI trains mainly on the record of what worked, not what didn't work and never even got published (and only lives in peoples heads) - and what didn't work is arguably as or more important for making breakthroughs and forming real intuition.
So, if AI solves problems in the open, it's accused of plagiarism and contributing nothing of value. But when it solves the problem behind closed doors it's accused of not contributing enough to the field and even depleting it of fertile problems.
For such great minds, they seem to be rather muddled thinkers.
E.g. of the actually three misalignments in this complaint, the jey one seems to be:
"The goals of the AI companies and the goals of the mathematical community are severely misaligned."
And that's wrong. There is no misalignment. Each is independently aligned with its respective interest. And those two interests diverge - just as you'd expect.
This kinda reminds me of the documentary about the top Go master that got beaten by a computer in dramatic fashion and had an existential crisis. Man confronting his own limitations in the realm he previously ruled unchallenged, what a time to be alive.
Nobody is going to care that the math isn't being done in the traditional way. The results speak for themselves, this is now a part of the landscape. No amount of hand-wringing is going to put the cat back in the bag. Adapt or perish.
Other than the fact it's mathematicians signing it, why is mathematics special in this regard: surely this generally applies to a lot of different industries and sectors of research / academia?
Mathematics is being used as a benchmark because there are some high-profile awards in this area I guess, and possibly because 2/3 years ago LLMs were pretty atrocious at it so the level of improvement has been significant.
We should create a platform where people can upload AI slop proofs anonymously without taking credit for it. That way the incentives of planting a thorem flag would go down. And if someone wants to clean an AI slop proof to advance the field, they can do it without cleaning the house for free of the person that planted the flag.
Would OpenAI have uploaded their proof to such a platform? If you know the answer then you know what the problem with what happened is.
The thing is no one is stopping anyone from getting the same knowledge/understanding/insight
If those things are disincentivized because the original problem is “solved” and there is less prestige to motivate people doesn’t that say more about issues with the community of mathematicians than the AI
this sort of reminds me a little of the reaction to Elon/SpaceX in its early days .. when Neil Armstrong and other apollo astronauts went before congress and voiced their concerns about relying on commercial companies for human spaceflight and the dangers etc .. also not trying to sound cynical but hasnt LLM made math more accessable to ppl ..
At this rate AI will be doing all of the mathematics within 5 years, I don’t see why a mathematician would be worried about anything other than that at this point?
If no one understands it, it may as well have not happened. There's not much incentive to understand or internalize the results generated by AI. A human operator gives it a prompt and it produces some lean proof no one wants to (maybe can) read.
Without the community of human mathematicians internalizing the proof, simplifying it, and re-communicating it to others we end up losing the main output of mathematics as an institution.
Why would we assume AI will learn to solve millennium prize problems but will struggle with the easy part of doing the explaining? I think it’s too easy to predict that models 5 years from now will have the same limitations as the they do now, I would be amazed if this was true, GPT3 was the best model available 5 years ago.
I'm not sure it's just a matter of explaining... If it's just explaining, than sure, that's something they already do well.
The questions I have are:
* is the structure of the generated proof even compressible/elegant to humans in a way that lends itself to being understood?
* is it possible to transform the proofs to ones that are elegant without redoing all the work?
* are there incentives to do any of this at scale?
It's possible that a headline grabbing proof of a Millennium Prize problem generates enough incentive for people to simplify and gain understanding from it, but we run into problems when AI becomes the dominant approach for all of math. Although, maybe this is self-limiting? I guess it's possible we just ignore a bunch of AI generated proofs and only keep the ones people find comprehensible in a useful way.
You can always hypothesize that at some point in the future (maybe 5 years? maybe later?) the models will be indistinguishable from humans and there will not be any functional difference at all. It's possible, but we're not there yet. And, as they say, past performance does not guarantee future results. Many technologies plateau at some hard ceiling of performance. Moore's law has had an unusually long run, but it's not a universal rule.
This is a "human alignment" problem in this case. OpenAI acted like complete assholes about this, from the beginning until they announced it. Not ChatGPT, the people that were in charge of the project.
Economist: "Are mathematicians talking their own book out of fear?"
The Economist, who recently used "moral panic" now stoops to Hacker News AI booster level and inverts arguments usually directed against the rich and investors. What is next? The Economist inverting Upton Sinclair's quote to serve its billionaire owners?
Look up the AI investments of the Agnelli family for example.
Another bunch of nerds who now have their panties in a bunch because AI threatens their fragile egos and core identity and who they are..
I’m being somewhat harsh here but come on - human endeavors are messy and it’s surprising how much our egos are getting bruised here over seeing the value of these tools
Dont get me wrong Also, there’s no AI Utopia coming this is it guys, were stuck with oligarch Tech Bro funded AI and big funded Govt AI so forget any egalitarian motives- we have to fight for our rights from other humans as always as well but AI as a technology in itself being able to truly solve unsolved intellectual problems is still a boon for society - who cares who gets credit?
Tao's calls for respect for provenance in mathematics publication are laudable but most likely naive given the closed nature of frontier model training data curation. Anthropic and OpenAI may react with a symbolic and short-lived olive branch, yet provenance is a larger issue that has impacted other fields beyond mathematics. While traditional respect for lineage in mathematics is of value to the academy, industry and science at large will likely be much more Machiavellian about such concerns. Mike McCoy's recent article is also timely (https://mbmccoy.dev/posts/mathematical-conservatory/). The parallels to the music conservatory are quite telling -- academic music describes a musical culture in preservation that has completely lost touch with musical developments beyond the early 20th century. Mathematics may very well evolve separately and with very different values than the academy upholds. The crisis of music at the academy is a cultural disconnect and a serious loss of critical analysis and acknowledgement of widespread and dramatically evolving music practice; however, for mathematics, the impact would have much more severe ramifications for education and human development if the academy forces a schism with AI. As models improve they very well may be inventing mathematics -- science and engineering may grasp for them -- they'll exist with or without attribution. Would be a shame for the academy to land on the wrong side of history and refuse stewardship of upcoming AI-assisted mathematics, including provenance, because of this misalignment. If attribution is important, then the academy will set aside the institutional resources to do it. If you succeed at having model developers participate, I commend it. To let mathematics be born in isolated context windows and only serve narrow, localized engineering purpose without rightful addition to the canon, would be a tragic, yet preventable, loss.
thank you for this commentary, as a violinist / software engineer / "computer scientist" / "mathematician" i can see that studying and doing math will still remain a thing that humans have to do to train their own brains to think and to form better neuronal connections. We may not get paid to do it anymore but it will still be important for our own development for the same reasons orchestras and bands still exist in elementary schools and beyond (at least in many parts of the US).
There's massive cope and then there's this. I've never been a fan of Tao et. al, so I'm glad he's being wiped out. He'll do fine anyway, doing conferences, etc.
Also funny they deliver that on vibecoded site, lol.
i was initially optimistic that ai could be distributed. having this technology in the hands of many would counteract the malign elite who would use ai to oppress the rest of humanity. in the meantime given that intelligence is scaling proportionately with inference time compute, we are concentrating far too much power in the hands of the few who have both their own model and datacenter.
putting this amount of power in the hands of so few would require leaders of immaculate moral integrity. what we have in the US at least is an emergent kleptocracy with obvious dark triad traits: sam altman, dario amodei, elon musk, etc. they are malign and will use this power for their own benefit, at the expense of others. ignore pleas of public benefit. look instead at the evidence: infighting, systemic dishonesty, reckless disregard for safety, political lobbying and manipulation, putting power in the hands of the elite few in the guise of safety, using ai in the military to oppress and inflict violence on others.
really to prevent a bad outcome, we would need to act soon enough to prevent the kleptocracy from corrupting politicians and democracy with them. i would ditch the claude or openai subscription and support open models instead, preferably in countries outside the US to prevent global neocolonialism.
The mathematicians would be wise to re-read The Bitter Lesson, maybe twice a day, until it sinks in. No offense and with all due respect to the Ivory Tower Giants but the whole "oh no you ruined the game because you solved it, I was supposed to play with that in child-like wonder manner and take several years to do so, and by then I would have showed you all the trickery I did to get there and maybe that will be useful to you" over the past 2 weeks is, get this, cope.
Chess. Go. Coding. Now Math. Another one bites the dust. Let's meditate on this lest we forget: Stochastic parrots that generate the next-token cannot reason or produce anything meaningful. Let's protect our jobs at all costs, even if we have to drag all of humanity down. It can't be! Stochastic parrots can not replace the Ivory Tower. No way.
Castle dwellers dismayed at moat-crossing technology.
What's most important about this is that it's a case study of what happens when deeply evolved ecosystems are blown up by disruptive technology. The psychological and social and professional impacts and myriad and traumatic to be on the receiving end.
Mathematics is merely one of the first domains disrupted. It will be unique only for being among the first... absent disruption of the entire civilizational project as a result of the disruption being caused.
Woe for us that we try to navigate this degree of change at a moment when the very worst and ignorant and short sighted hold all the power, economic and political.
Mostly that he really should have seen this coming. These people absolutely do not care what happens to mathematics (or any other field). They compulsively lie and steal and they played Tao and others like a fiddle. He danced to their tune and now that the music has stopped, now do they complain.
He was the poster boy of the mathematician yielding these tools for his own benefit. But he forgot who the owners are.
And yes, at one point it gives you right about them taking advantage of him. He sat down for an interview and found later that they just used the "best clips" out of it for an OpenAI ad.
Let's focus on the subject and try to distinguish the technology and the frontier companies selling it. That's the key point in understanding the whole stance.
I never expected this many people (on this thread) arguing semantics and what not. I know that not everyone has morality and ethics, but I didn't realize it was this bad.
I'm afraid of the ripple effect of the agenda pushed by AI companies will have. In future and even now, they say AI has significantly progressed math and scientific research in general. There is truth to this, but the narrative has done more damage (so far) to the students, researchers, and the culture of knowledge transfer in academia. Many graduate students (I know) are having a crisis if any of their research worth it? If AI can (or will) do everything, what's the point of doing experiments and all? This will eventually deter a whole generation of curious minded students from research.
I guess, only time will whether this is for the good or bad. And how good AI models get without new data from research and experiments.
> Many graduate students (I know) are having a crisis if any of their research worth it? If AI can (or will) do everything, what's the point of doing experiments and all? This will eventually deter a whole generation of curious minded students from research.
Those who think it's me or the machine will fail.
Those who realize how much you can accelerate your research with the help of AI will succeed.
I think that the main thing people in this thread are missing, is that it's not about math. AI progression is very likely to affect every single thing humans can do today. Mathematicians are feeling the blow this week, especially as there was wide spread denial in that math community over the capabilities of AI over the last few years, but it's the same problem everywhere.
Try reading the article before posting.
This story's comments are heavily astroturfed.
Just compare with the comments on https://news.ycombinator.com/item?id=49639408
As a mathematician maybe I am a little more optimistic than this declaration.
I am thinking of Mochizuki's abc conjecture: He worked in relative isolation, and dumped a huge incomprehensible proof on the community (to oversimplify a bit). That's not totally unlike what might happen if AI generates a huge, incomprehensible proof of let's say RH.
Well, what is the result? In the Mochizuki case, it was a lot of skepticism, but it also generated conferences, papers, talks in the hallway, discussions with students, and so on--a flurry of exactly that kind of community process that the declaration says is the main driver of mathematics.
Ultimately we think a fatal flaw was found in Mochizuki's proof, so it didn't lead anywhere in particular. But in our hypothetical "AI lean-verified proof of RH" situation, it would presumably generate substantially more of that community activity we saw in the Mochizuki situation. And if it's correct, that community activity would be productive (expository talks, students given problems to flesh out or generalize, etc).
Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
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.
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.
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
https://tcec-chess.com/
Top chess engine championship is pretty fun to watch.
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.
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.
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.
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.
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...
> 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.
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.
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.
wouldn't AI solving problems in science, engineering, economics, etc be able to apply the new AI math?
>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.
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.
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.
Is this the scenario described in Ted Chiang's short story https://en.wikipedia.org/wiki/The_Evolution_of_Human_Science where scientists are "catching crumbs from the table" trying to decipher the results generated by superhuman intelligence?
It's still an optimistic scenario. Artificial superintelligence may develop hypermathematics of a kind that never will be accesible to human mind, enhanced or not. One can't teach geometry to ants even if you put them on a Moebius strip.
It would be more like Lem's novel where it completely disappears from the human horizon: https://en.wikipedia.org/wiki/Golem_XIV
Which is why if humanity had empathy, it would be working on how to make smarter ants, so that they can learn more advanced geometry.
It's quite possible ants understand a geometry more advanced than our own.
I love that as a goal!
Perhaps indeed a better understanding of what intelligence really is would allow this sort of Uplift (as in Brin's books).
What do you think we're doing?!
Ant Intelligence will be next big thing in machine learning after this bubble bursts
What’s optimistic or non-optimistic specifically about the machine having a system of mathematics beyond our comprehension within it? Why should we care about that in itself?
In the first case, we'd still have a chance to take a glance at the frontier of discovery (even if ordinary human mathematicians had to spent years translating what metahumans achieved).
In the second case all human-level maths would be solved and what lies beyond would be always out of our scope.
Why would this be optimistic? This sounds extremely negative to me…
>Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
Dr. Tao said the same thing. Somehow, this letter came through. He wants to conduct Math competitions where participants who don’t have formal credentials can contribute to mathematical research through AI.
Title: Terence Tao - SAIR Competitions and the Future of Experimental Mathematics
https://www.youtube.com/watch?v=rB9YOi3lb7w
and this:
Daniel Litt - Working with LLMs to do high quality math
https://www.youtube.com/watch?v=0wL8NlhxXcU
So he got exuberant because he is funded by SAIR and the "AI for math" fund.
And embarrassingly they used him for a "coal miners should learn math" moment that just benefits the AI industry.
He has severely reversed course in the past week. Without concrete propositions it remains to be seen how much of the new resistance is for show.
I have zero formal math training beyond my Grade 12 Pre-Calculus class. Yet with an LLM I have recently devised an architecture with incredible math potential. Math is a language like any other, and without LLM's I never would have developed the techniques that I have.
AI is a tool. It speaks languages I don't (Math, Science, Code). I would love to participate in a Math competition without a hint of any formal advanced math training because my experience so far tells me I will do well.
> Dr. Tao said the same thing.
Apparently he has since changed his mind.
Did he say so somewhere? I don't think these ideas are contradictory. It's just an pro AI tooling but anti-slop stance.
Where is the difference?
He sees value in mathematicians using AI to carefully study mathematics, develop an understanding of both old and new things, and help others understand the new things.
He doesn't see value in scrolling through unsolved problems asking an AI to please solve them. In his view, this is a fundamental confusion about what mathematical research is for. Knocking down unsolved problems without developing the community's understanding of them is like prompting Claude to go through a Jira board, write code for all the open tickets, and then close them without merging or deploying the code.
Isn’t it more like it merges the code without a dev reviewing or understanding it?
Pretty close, but IMO not quite. A math proof in and of itself is useless unless either:
If you merge and deploy code, you have released a tool that can be used. If you ship a gibberish math proof, it's not useful unless someone else can understand and deploy it to some other means. Now, it's possible AI could understand and make use of the math proofs, even if we can't, which refutes some of my hair splitting :)No. Merged code can perform actions with effects on the world, even if a human being never saw it. Constructing a giant Lean formalization that nobody understands simply doesn't do anything.
taste
"I believe I did, Bob" lives rent free, every time someone tells another person to fuck themselves using technology.
Thank You!
lmao, thank you, i think
Please consider making more comics. You have it.
Yes. I find it really interesting to consider what the machines do and will think of as intrinsically interesting to them. Will they develop their own theories of beauty, mathematical and otherwise?
>Dr. Tao
Professor Tao.
But the work the Mochizuki case generated can also be done by AI. AI could generate a landmark proof and then people could use it to solve or simplify intermediate problems and you could use a different AI prompt to try to disprove it if you were really skeptical. From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it.
I don't like nuance here. I think progress is really measured by what humans are able to do and understand, not machines. It is significant if we find problems we struggle to solve. That tells us something. What does it take for humans to solve these problems is related.
The best analogy I can give is if you wanted to climb Mt. Everest you might ask someone for guidance. Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak? This is like the AI versus human gap to me. The helicopter is like using AI to generate a proof. The person who actually climbed Mt. Everest has firsthand knowledge of the experience. Same thing for a difficult proof. The struggle people have is actually valuable here. Likewise, we know people are actually capable of climbing Mt. Everest but if they had only ever rode a helicopter to the top, the knowledge of climbing it would not exist, and surely that is meaningful knowledge given the risks.
So if we rely on AI for proofs I think we lose a sense of what is difficult and why. We lose a sense of what human achievement is. Surely climbing Mt. Everest means more than taking a helicopter up? For students, why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now? This would have the affect of destroying knowledge.
(please do not nitpick the analogy because it's the best but perhaps a clumsy way to describe my thoughts)
Tangent:
> From my memory I think they said it took 88 hours to solve a Millenium Problem versus the decades of time humans have put into it.
Keep in mind those ~88 hours were spread across ~10,000 simultaneous agent instances.
So, roughly 880,000 hours of compute.
Assuming a fifty-year career, and forty-hour workweeks, a human mathematician's career is about 100,000 hours of "compute".
I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already.
The perverse incentives of academia mean this has never occurred.
The perverse incentives of industry mean OpenAI intentionally scooped researchers who were getting close (granted, with AI help).
I'm not trying to dismiss the achievement - if the proof turns out to be solid, it's quite impressive (though much less so if the training data included the recent human breakthrough, which seems pretty plausible).
I'm just pointing out that "88 hours" is a very misleading way of framing this.
> I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already.
> The perverse incentives of academia mean this has never occurred.
This. Mathematicians in their most energetic years are trying to get tenure or land a tenure-track job. They are disincentivized to go all-in on ultra high risk, high-reward problems. The potential downside is just too forbidding. It's much safer to develop a research program in a mainstream field that affords many opportunities for partial progress that can translate to a robust publication record.
I agree entirely with what you're saying, right up until your final question:
> why bother grinding through all the material of climbing Mt. Everest and then attempting it if the helicopter ride is how things are done now?
I think you answered this yourself earlier:
> I think progress is really measured by what humans are able to do and understand
People want to make this progress. Therefore people will "grind Everest" as a mathematical community, and that is maybe not so hugely different from a lot of previous mathematical work.
There's still ample room for creativity: simplifying, generalizing, asking new questions humans are interested in, ...
That grind is emotional. And it's something AI will never have. The desire to solve a problem.
> Would it be better to ask someone who has climbed Mt. Everest or someone who took a helicopter ride up near the top and then went to the peak?
Depends on if I want to go by helicopter myself.
I think progress is really measured by what humans are able to do and understand, not machines.
Building a machine that solves Millennium problems is pretty cool too. You wouldn't know it from reading these stories, though.
> Ultimately we think a fatal flaw was found in Mochizuki's proof, so it didn't lead anywhere in particular. But in our hypothetical "AI lean-verified proof of RH" situation, it would presumably generate substantially more of that community activity we saw in the Mochizuki situation. And if it's correct, that community activity would be productive (expository talks, students given problems to flesh out or generalize, etc).
This also sounds like a vector for trolling the community with complex putative proofs hiding a known flaw.
Not if it's lean-verified.
Didn't some of the recent proofs exploited a couple of blind spots of lean, and they were invalidated?
Edit: Yup. A bug report to Lean was disguised as a "Collatz" proof in a humorous way. Links below.
- https://x.com/gro_tsen/status/2082483878480977959
- https://infosec.exchange/@0xabad1dea/117002106099986943
There was a hash collision bug in the main Lean kernel that was patched, but AFAIK nothing relied on it. You'd have to know what you were doing to accidentally get there...
The incident in the links I posted exploited several bugs AFAICS, so it's a different story than a single hash collision bug, it seems.
I don't know ... do you have a reference?
Yup, found it:
https://news.ycombinator.com/item?id=49101465
Cool, thanks!
The difference is the scale. A few incomprehensible long papers per year, sure, we will study it. A flood of AI results closing research directions left and right, that will be a problem.
> closing research directions left and right
Why would research be closed in one direction? Even if AI or human says "Tried that, didn't work" or whatever, someone (or something I suppose) might very well retry it in the future, if nothing else to reproduce it didn't work, in theory at least.
AI tends to take nearly finished research directions and push it to the conclusion in one step. If deployed massively, it will pluck all the low hanging fruits causing a drought of near term promising research project. Because people who start promising research directions do not get to see it finish, over the long term fewer people will start new directions, causing the field to slowly whither.
> Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.
I think a better comparison is: mathematics just becomes like mining bitcoins.
I think you might have to explain that comparison a bit more to be honest. How are math proofs like bitcoins? A bitcoin has a pre-defined value, a math conjecture / proof is a bit more complicated.
Mochizuki's claimed proof of the abc conjecture was extremely unusual for the reason that nobody was able to extract a single useful idea from the argument. I was starting grad school when it came out, and my immediate visceral response was "if this is what number theory is going to look like in the future, then I will leave mathematics."
The current wave of AI slop mathematics might end up driving the next generation of mathematicians away from the subject for the same reason that Mochizuki would have convinced me to quit if his proof had been accepted by the community. Luckily, my professors had the taste to immediately recognize that it was garbage.
I really like this take, and while I hate math I value it. Your position sounds extremely plausible and it fits with the pattern we see in the community here. Regardless if it's ai slop or not we still debate the value and attempt to understand. In the process generating new insights and ideas. Life will go on.
It's not just the isolated dumping, it's the fast, isolated, possibly untraceable dumping, without long term support.
It'll basically become slop fatigue if OpenAI starts dumping out proofs faster than the community can keep up, and some turn out to be wrong, never formalize it, don't stay to support it, etc.
I wonder if they will continue to dump proofs, though? Their point has been made, the novelty will wear off, and it maybe won't be a priority use of their resources to spend however many millions on another big proof--they will move on to the next thing to show off I'm sure. At that point, the ones generating proofs will be, I hope, mathematicians (professional and otherwise) that are more interested in the results and community discussion.
(Well that's my hopeful, optimistic take, anyway.)
They aren't going to stop at one, that's for sure. They already claimed they have "made substantial progress" on another millenium problem. Let's say they bag another one (Hodge and/or BSD according to the rumors), if it looks like their internal model could solve P/NP or Riemann Hypothesis, you think they wouldn't take that chance ?
They already told the NYT that they’ve made “substantial progress” on another one of the MP Problems (most likely the Hodge Conjecture).
That said, that’s probably just because of the drama miring their most recent one. After 2 I don’t see why they’d bother anymore.
>That said, that’s probably just because of the drama miring their most recent one. After 2 I don’t see why they’d bother anymore.
P/NP and the Riemann Hypothesis are part of the milllenium problems. They will 100% keep trying to crack those regardless.
I think AI companies making a point is not the only thing at play. Discovering new maths ultimately leads to new technologies and applications. It may start theoretically but end up being of practical use in the future. Even if humans do not understand it (lose interest, too complex, or just way too many new proofs to go though) AI can use this AI derived math corpus which will help it in other fields.
So it wasted everyone's time, thousands of hours of research trying to disprove something said very loudly. What OpenAI is doing is a DoS of the scientific community: wasting your time trying to check if they're not wrong, and claiming glory in the mean time.
That's true, but the story would have unfolded differently if Mochizuki had a lean-verified proof and was correct. I guess baked into my premise is that AI is producing reliable proofs (in the long term at least).
Agreed! Although, if done by a mathematician, it's not a ~complete waste. I think the community learns something along the way.
Is there an established term for the idea of "DoS"? I've taken to calling it slop fatigue.
Denial of service is the established term. Hammering their API (reviewer committees) would be an informal one
OpenAI avoids this by formally verifying the proof.
https://github.com/openai/NavierStokesAndEuler
Even before AI we used to say if you write code that you only barely understand, then it will be to complicated to debug. (and/or maintain)
Mochizuki was still one human and it required legions of other humans to unpack and untangle to confirm that it didn't lead to anywhere in particular.
AI is now capable of constructions so complex that no human or human team can unpack. And its ability to increase that complexity is growing while our human ability is stagnant.
meta-AI analysis cannot help. We (software professionals who use AI regularly) already know that if you run into a situation where a Fable/Astra-generated analysis reaches the limits of our comprehension/complexity due to their subjectivity, throwing more AI at the problem doesn't always converge.
There are many reasons to feel optimistic about AI, and ultimately its general ability to help science and mathematics.
I see no reason to feel optimistic about the future of mathematics and AI based on the current path of frontier labs, unless the misalignment Tao is writing about can be reconciled.
> AI is now capable of constructions so complex that no human or human team can unpack.
How can we possibly know this when we haven't even seriously started on the endeavor of actively reverse engineering these AI-generated proofs? That's a proper job for human mathematicians, because the AIs themselves are demonstrably clueless about what steps in a proof are genuinely interesting and load-bearing from a human POV. This is evidence of a limitation in AIs' capabilities, not of any kind of misaligned behavior. The fact that Tao actually uses that term in his complaint is deeply disappointing.
Not to mention, there are already (pre AI) machine-generated proofs that we've pretty much agreed not to try to explain fully, like the four-color theorem which ends up with brute-force verification of 600+ cases (down from close to 2,000 when first demonstrated)
> there are already (pre AI) machine-generated proofs that we've pretty much agreed not to try to explain fully, like the four-color theorem
Algorithmic verification is a very unsatisfying answer to the problem (e.g., surely it's not just dumb luck that every single case happen to have this exact property), but that's an entirely different issue than saying that no one follows logic of the proof method itself.
For the interested; the saying I believe you are referencing in regards to writing code / debugging is from Brian Kernighan, specifically:
(from, 'The Elements of Programming Style')It's prescient.
It’s a cute statement, but it doesn’t really match reality. Programs written by humans can generally be debugged by humans.
Could AI write programs that humans can’t understand or debug? Probably, but that’s not what Kernighan was describing.
> AI is now capable of constructions so complex that no human or human team can unpack
Can you give an example of this?
The first major computer-assisted proof, of the four-color map theorem in 1976, was an example of this. It created a lot of controversy at the time. It used proof by exhaustion, i.e. essentially analyzing every possible relevant case, something that no human could do without the assistance of, at the time, a supercomputer.
I get excited at the idea of a world in which advanced mathematical problems (and their solutions) become much more accessible to a much greater number of people. As a result, making mathematics much more loved at a societal level.
Imagine a world where these most complex mathematical problems are not accessible to a few hundred people, but a few hundred thousands people. ...Those original few hundred gifted mathematicians would have an even more prominent role, and their names and achievements would be known by orders of magnitude more people that they are now.
This is the hope, but I suspect the reality is that we see an ever widening gap between the fortunate and the unfortunate. We're looking at the automation and commodification of all knowledge, and the best models will be kept locked behind closed doors so that they can't be stolen. And, of course, "for our own protection".
As a laymen, I wish the same. But I also hope it doesn’t disincentivize those that dedicated themselves to the study
Math department administrators fire Terence Tao.
Based on current reward models, the frontier AI labs will burn down mathematics as an impressive display of capabilities and in doing so, will make it impossible for people that get paid to do mathematics to stay employed.
If your job is literally to publish papers, and OpenAI and Anthropic decide that making an infinite-paper-printing machine is the best thing to show how effective their tech is, then as a demo, they destroy that industry.
I wouldn't expect them to destroy that industry, imagine global squadrons of academics and mathematicians focusing their attention on LLM's, training algorithms, scaling laws, ... they're gonna try and beat the incumbent frontier AI labs, eye for an eye, tooth for a tooth
The apocalypse being triggered by frontier labs picking a fight with mathematicians was unexpected.
I've met a few Ph.D Mathematicians in Academia socially. My unfortunate experience was that they were insufferable,borderline hostile people. I tried to genuinely engage with them too. I've met one Ph.D Mathematician that left the industry whom was very enjoyable to talk to. I have a feeling that my experience was not unique and the Math world is mostly a bunch of too good for everyone on their high horse a-holes that are now being knocked down a peg. They don't like it obviously.
I'm not a fan of knocking down things that work, however I also find it hard to be against death of the gatekeeping old guard of any industry.
I think math is just gonna have to suck it up like every other industry now. Math productivity is longer out of reach of the average grad student. Like every other industry they are no longer untouchable and are gonna have to adjust to the new way of things or market forces will do what they always do which is refuse to fund ineffectiveness.
I've had to accept that tech/IT will never be the same. Just how it is. You can thrash against it all you want.
Tao's critique of AI in the field of mathematics reminds me of what French art critic Charles Baudelaire said in the 19th century about photography [0].
Baudelaire argued that photography became a haven for failed painters, the sorts of hacks that could not finish proper training. Photography, as a mechanical rendering of the world, could only record what already existed; it couldn't transform reality the way a painting could.
He also criticized the public's craze for "rushing" into it, and complained that this technical "progress" was weakening the arts.
Do you see some parallels as well?
[0] https://fr.wikisource.org/wiki/Curiosit%C3%A9s_esth%C3%A9tiq...
I have listened quite a few interviews with Tao and I see him being very careful about criticizing AI. He very often emphasizes the usefulness of it. Where he is critical has a lot of merit. One of the points I clearly remember him saying that having AI be able to solve many of the open problems, regardless of how important there are (there are many open problems that are not that important) greatly reduces the problem space for mathematics students to give new problems to work on.
In a parallel thread omnicognate correctly pointed out that for AI companies it's a direct commercial loss to pour all this money into bruteforcing the solutions to these problems, and that a lot of times the solutions by themselves are not directly commercially valuable. They are doing it for stock price, trying to lure in private capital in preparation for IPOs.
Their models are good, but they are not the moat because Chinese models are good too, so what they are doing, in my opinion, is more harm than good. Mathematics is a science by humans for humans.
>I have listened quite a few interviews with Tao and I see him being very careful about criticizing AI. He very often emphasizes the usefulness of it.
This is tiresome. People should be able to flat-out criticize AI without the implied need to justify themselves all the time or "be careful". Its almost like AI has a trillion-dollar agenda backing it, to the point that you have to add a careful "its really great! But there's this little issue..." for any criticism.
Even those who are very pro AI should have the intellectual honesty of admitting that there are very valid reasons to criticize AI.
Based on the amount of low-effort criticisms out there, clearly nobody is afraid of "big AI". Tao seems careful about his critiques of AI because low-effort off-the-cuff criticism can lead to all sorts of future problems/hypocracy.
I think there's a difference between "photography will change art--we need to be ready" and "photography will change art, therefore stop photography."
There is no doubt that AI has changed the practice of mathematics, just as it has changed the practice of software engineering (and will soon change almost every intellectual job).
Trying to deal with change by saying, "please stop the change" is foolish, IMHO. Mathematicians need to redesign the discipline with AI in mind. But I get that it's easy for me to say that and hard to actually do.
Both of Baudelaire’s criticisms were reasonable, and the same thing happened again when AI image generation showed up. You think the opponents look ridiculous because you’re viewing the history from the winner’s side. As for “the public,” people had a real demand for photography as a way to record things, which is also why it won. There is no comparable public demand for proving Fermat’s Last Theorem.
I don't think the analogy works because for the past few years Tao has been one of the most vocal advocates of AI in mathematics and has used it extensively in his own research. You can find several of his talks about this on YouTube. It's completely consistent to believe two things at once, that the tools are useful and that the companies are misbehaving.
Genuine Art versus Mechanism, from 1901, (https://www.jstor.org/stable/25505621) is another article that I read a few years ago that other people might find interesting.
Are you aware that Tao is among the largest proponents of using AI in mathematics? The usage itself is not the point here.
Tao doesn't go as far as Baudelaire, but there are some similarities. In particular, Tao has criticized that AI is not being used to create new interesting conjectures, and that the rush to prove old conjectures is not giving human mathematicians enough time to carefully analyze and understand the proofs and the methods used in those proofs.
My answer to both is the same: nothing stops mathematicians from doing both of those things, with or without the help of AI. And we all understand that it will take time to do that. But complaining about the dawn of a new era of advancements seems counterproductive.
> But complaining about the dawn of a new era of advancements seems counterproductive.
you completely misunderstood the critics. your analogy is awful. this is much closer to the industrial revolution in the uk: it brought a lot of progress, but also extreme inequality and concentration of power.
I am very specifically addressing what I have read Terrence Tao say on AI and mathematics, as that is the subject of this submission.
Are we on the same page?
I don't see any meaningful analogy with your cites of Baudelaire. First of all, Baudelaire discusses art, which is very different to science. Several new forms of art have emerged, and then slowly integrated, despite the strong opinions of some. Here, Tao's letter discusses mainly practical aspect of scientific research. It is not about what a "failed mathematician" would do, or if "LLM can only record what already existed"; in fact, it ackowledges that LLM could become a central tool. If you want me to be even more precise, what they're saying (without saying it out loud) is that tech companies have too much power and are being irrespondible with it because they don't even think about the externalities.
> It is not about what a "failed mathematician" would do, or if "LLM can only record what already existed"
This is what I was alluding to:
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce [0]
[0] https://mathstodon.xyz/@tao/117237320796901560
Of course there's a meaningful analogy. The poster wants to make Tao sound unreasonable and against "progress" while ignoring the very real concerns Tao has.
It's what fanatics do when they want to enforce their view on the world, they have to attack anyone with a reasonable viewpoint because they can't imagine a world where someone tells them they don't like what they're doing.
I do not want to make him sound unreasonable. He is a brilliant man. But I also think that he is a bit shell shocked, much like other luminaries in the past have been when confronted with rapid technological development, so I wanted to highlight some historical similarities, imperfect as they can only be.
As for the rest of your comment, I hope you have a great day.
It's not about whether the conjectures themselves are interesting or not, or whether people simply have enough time. It's that a bare proof made by a machine doesn't actually do much for us. There is not some set of problems that, once finished, will amount to some kind of final, correct system and we can call it a day and, like, utilize it. "Mathematics" is the people doing it (the "mathematical community" Tao references below). This other stuff is kinda just.. expensive exercises to render a result. They are only actually beneficial to us insofar as they exist in a context of research among peers.
https://terrytao.wordpress.com/2026/09/11/a-severe-misalignm...
You have expressed a few different ideas, so I will address them separately.
> It's not about whether the conjectures themselves are interesting or not, or whether people simply have enough time.
Tao seems to think otherwise, if I am reading him correctly:
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce [0]
Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. [1]
> It's that a bare proof made by a machine doesn't actually do much for us.
I get it, and I think the same can be said about all sorts of human endeavors.
> There is not some set of problems that, once finished, will amount to some kind of final, correct system and we can call it a day and, like, utilize it.
Sure. Although there are certainly practical applications to be found along the way. E.g. proving P=NP would be potentially very significant in the real world. I think we agree.
> "Mathematics" is the people doing it (the "mathematical community" Tao references below)
Sure. And the same can be said again about all sort of human endeavors. But I don't see how that is a reason to stop using AI in those fields, either. It doesn't subtract anything, in the same way that chess engines didn't destroy the love of the game for chess.
And just like in chess, these AIs can be used to gain a deeper understanding. Including, but not limited to, explaining to humans the proof they just came up with.
[0] https://mathstodon.xyz/@tao/117237320796901560
[1] https://terrytao.wordpress.com/2026/09/11/a-severe-misalignm...
Are you trying to argue toward some final verdict with regard to LLMs and mathematics? I thought you were just trying to make a comparison to Baudelaire? I think maybe being clearer on this point would help. Even if you argued sufficiently for the latter (which is going to be tough already), it wouldn't really speak to the former. Or at least: that would have to be a separate argument I think.
Also, how, in your words, do you feel like the first quote justifies your point (presumably with regard to the question of "interesting" or not)? And why do you think the second one is more about time itself rather than attribution? Do these things actually contradict the letter above (or the comment on it) in your mind or not?
In general, do you disagree with something here specifically? Or is it kind of a yes/and thing? Does any of this help, in your mind, with the Baudelaire comparison you were at least at one point trying to argue for? Its a bit hard for me to see the argument here, if there is one, just with what you have written. But I am sure I am just not knowledgeable enough to grasp the argument!
My opinion is that the field will adapt. AI will not spell doom for mathematics. On the contrary, it is the beginning of a new era. Math is having its Deep Fritz moment, just as computer science is going through the same.
Some folks are struggling to adapt to this change. Tao actually sounds like he is doing alright compared to most, even if some of his arguments seem a bit weak, as I alluded to in other comments in this thread.
Hopefully it makes some sense. And if it doesn't, at least we had a nice chat.
Haha yes I imagine we all know your opinion is something like that! But isn't it much more fruitful and interesting to form an argument for it? If he is still in the context here: is there something about Baudelaire's critique you were discussing originally that has helped you arrive at this opinion? Do you think he is wrong? Or is he right, but providing the right kind of nuance that we need in order to cope with the revolutionary changes? Genuinely curious!
People aren't ready to discuss AI assisted imagery as art yet. Most discussions lack the nuance that Baudelaire lacks in that critique, which deals with the nature of art and the importance of human intention and input.
We can start having a meaningful discussion when people use real reasoning instead of analogy.
See https://news.ycombinator.com/item?id=49664505
Analogy is a meaningful way to discuss. Drawing parallels can illustrate a point and bring up nuances by pointing out where it falls apart.
You don’t have to participate in the discussion, but it will take place regardless of your preferences.
Keep in mind that his critique is very recent, and likely applying to a specific use of AI, as opposed to AI as a whole. If you've been following his Mastodon account, he's been happily using LLMs for math purposes for well over a year.
He posted about using ChatGPT to transcribe PDFs when it first became popular. So he’s been enthusiastic about LLMs for a while.
I'm really tired of these arguments (this and "it's just like calculators").
Photography decimated other forms of visual art, so the concern wasn't wrong. But AI threatens the entirety of human intellectual endeavors. I can make do without oil paintings in my home. I'm not sure I want to live in a future where we make do without brains.
Humans still have brains (and thus human intellect) even in the presence of AI... Even today, people make varying use of their brains.
Lots of people still make bad music that other people still manage to enjoy (a lot of it has gone multi-platinum!) even though they're not Mozart or Bach.
This would be valid if we didn’t live in under capitalism, which reduces the way we spend our time down to dollars and cents.
you can still get oil paintings in your house. you just arent willing to pay for actual art
Had me until the last word. Cameras did take away oil paintings, but not eyes.
It did. While you're busy recording an "instagramable" moment, you are not entirely enjoying that moment. Your eyes are on the screen, not on the subject.
In a way, if photography is an ersatz for painting that eventually made imaging available for the masses, then AI could become an ersatz for thinking. But it feels like I'm paraphrasing TFA.
What will happen when AI companies have spent their advertising budget on math problems and whatever else gives the maximum wow effect for the bucks? Probably customers hooked on the vain satisfaction of spending token$ to impress friends.
Wasn’t the criticism of photography kind of correct though? I don’t think people think of photography as an art as much as painting is an art. There aren’t a lot of photographs that a layman couldn’t in principle also take, but only a few people could replicate a good painting.
It's not exactly correct. Photography is just as much an art, but it's a very different medium and pieces are judged differently.
Realism isn't difficult, so critique shifts towards composition, narrative, context, process, and emotional impact.
Taking a clear photograph is easy with modern equipment, which means the bar for what's considered "good" is high.
i think that's just you
Baader–Meinhof phenomenon:
Baudelaire popped up in this article two days ago.
https://www.noemamag.com/a-new-kind-of-creative-poverty/
The AI people sing from the same sheet.
It’s not a parallel. No one ever claimed photography to be the same as painting or some other medium - it was a new medium that wasn’t respected.
AI is being treated and pushed as a replacement for every medium.
It literally was a replacement for portraits.
Wonder what would've Baudelaire said about generative art as in diffusion-based imagery.
If he didn't understand photography, he wouldn't understand the nuance there either.
Photography vs painting is the first thing that comes to my mind when I listen to AI critics.
We can’t look back with perfect hindsight because both the past and present have deeply ingrained blindspots. They don’t know what it is like to live in a world with perfect edges. We don’t know what it is like to live in a world with no edges. We can read about someone who proclaims that “something will be lost”. We will just think “but I have no need for any of that.” But we don’t even know what it is.
No, there are no parallels. It is just cliche propaganda used by AI boosters.
Wait, what? Who’s the ai booster in this scenario?
Beaudelaire
> But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal.
I will quote Richard Feynman - “the prize is in the pleasure of finding the thing out, the kick in the discovery”.
My personal example is Paxos. The algorithm is simple (it fits a single slide). But it is really hard to understand _why_ it works. I spent many many hours thinking about it, until it made sense to me. I was so happy. I wrote about it (https://uvdn7.github.io/understanding-paxos/ ) and I have gotten very good feedbacks.
The goal for me is not just to follow the protocol to build paxos (or raft for a log if you will). The goal is understanding and insight; it is to understand the why and the how. I still remember the moment when it all makes sense to me.
I am not saying that should be the case for everyone in every discipline. But if there is one subject that is mostly pure curiosity driven (instead of worldly impact), it is math. Robbing them the primary motivation is brutal.
If all problems can be solved by a machine, what do we have left to satisfy our curiosity, our desire to explore, and where can we find the pleasure of “figuring the thing out”.
This sounds a lot to me like people in the 90's complaining that computers were destroying chess. Thirty years later, chess is more popular than it ever was, and chess players are better than they ever have been. I wouldn't be surprised if there are now more chess books now than there ever have been. Furthermore, it turns out that a lot of chess books written before computers were just wrong about a lot of things. It turns out having an oracle for the "right" answer in chess, even without an explanation, used properly, allows humans to develop broader, more accurate insights.
The argument here sounds similar. The fear, as I understand this statement to be saying, is that by being given the correct answer, in the form of a 100-page Lean proof, humans will be robbed of the chance to from insights about the structure of mathematics itself. I don't see any reason that humans can't continue to develop insights as they try to digest the 100-page Lean proof into something more manageable; but with more certainty and fewer false starts.
Well, chess is a sport where humans are supposed to compete. But math, programming, science are mostly not, and AI might affect economy, careers, etc.
As a chess fan, 100% this. We have known for the last ~15 years who the best human chess player is, and that he will lose against stockfish on his phone. But chess survives because of the human characters involved, the rivalries and dramas, watching two people trying to overcome each other under insane pressure, and sometimes coming up with something astonishing. In short - it's a sport.
There is no equivalent in math.
I guess part of the problem is that being against being against anything for economic interests doesn't really rally anyone to your cause; everyone has to make a living doing something productive for society, and professions have come and gone all the time due to technological advances. In fact, when one thinks about it, the people that are losing their professions now were major contributors to others losing their form of income. Often people talk about how they can use technological/programming/IT skills to make some secretary or administrative assistant's job obsolete. So most people just don't feel a lot of sympathy when people complain that AI are going to take those people's jobs.
That's correct. As far as I know nodody builds any kind of science or technology on top of chess, but mathematics is at the base of most science and technology. It would be horrible if we prevented AI from solving mathematics problems, just because mathematicians want to solve them by themselves the "hard way".
=> If chess.com was worth billions and Daniel Rensch was threatening everyone to do what he says.
I like this approach.
A but like whenever the first sprinter hits a new world record other runners follow along.
Knowing that something is possible tends to strengthen our ability to work with it.
We will potentially see the same with math.
But computers have destroyed chess as a "sport". Nobody will sit to watch two chess programs compete, or analyze their tactics. Kinda like how now, anybody can construct a "game" over the weekend or a new song or a slop video. The value of each of these decreases to 0 as the slop overwhelms.
>Nobody will sit to watch two chess programs compete, or analyze their tactics.
I know nothing about chess yet I dare say that I'd doubt this. Surely chess enthusiasts would be interested in analyzing how a superior chess program came out victorious, no?
Yes you are right, some people do watch chess engines play. TCEC (Top Chess Engine Championship) [1] streams them. Popular chess YouTubers goes over engine games from time to time too.
[1] https://tcec-chess.com/
That's not proof of interest so much as proof of commerce. It could easily be a money-laundering mechanism.
Mostly nobody cares about professional chess. The number of people who are actually interested in today's game and not the drama are a tiny sliver of that. This is actually great because it means we can train and evaluate both without interference from chess players, possibly even building a stable society.
People do: https://tcec-chess.com/
erm actually, chess tournaments rose in popularity and more people watch now than ever
> I wouldn't be surprised if there are now more chess books now than there ever have been
Well yeah... how would there be fewer??
But the point itself is silly. Few people are putting effort into Maths for the fun of it (and of those that are many derive fun from being the only one who can produce a solution). Chess differs in that it never had any point but the game its self.
The woman who made up the workforce of telephone switch operators would like to have a word.
Meaning - every new technology has both been perceived as a threa and often forced change in society. Agree maybe “it feels different” this time, but don’t you think everybody before us just said the same thing?
Also not clear if this is an actual called action.
> We are witnessing a general threat to intellectual work
This is the crux of it and goes far beyond Mathematics or Computer Science. To get a bunch of humans to do anything, you have to motivate them. Kleos and Timē; renown and stuff. These AI companies threaten to rip this away from everyone but themselves, and this recent millennium prize is the perfect example.
Solving this problem as a human would have led to tremendous Kleos; my name would be written in the annals of mathematics, lecture tours of praise were mine to be had for the rest of my days. This one victory would have earned my recognition throughout history. The greatest a mortal may hope for. Ripped away.
It would also have given me great Timē. The prize money, the professorships, the book deals. Gone.
If all hope of “renown and stuff” in the intellectual realm is now taken by the AI companies, they will remove all human motivation to pursue these endeavours.
Perhaps the glory will come from slaying these fell beasts.
These are both big motivations but not the only ones.
But there's still play. There's still curiosity. And there's still the drive to understand something for yourself.
> But there's still play. There's still curiosity. And there's still the drive to understand something for yourself.
Yes, but think about what that implies if those are the only motivations left. Gone are the professions. Gone are the ambitious.
There is plenty of space for people to work on intellectual pleasure pursuits (as there is with art and music), but the death of all intellectual based industries is still something to avoid. Or to mourn.
Play and curiosity usually make poor breadwinners.
This letter is complaining that human understanding has been crucial to advancing of mathematics, and AI companies are not bothering with it. But the promise (and horror) of AI mathematics is that, if it succeeds, human understanding becomes irrelevant. That's the goal. So this letter's message will fall on deaf ears.
Keep in mind employees at AI companies are publicly stating that they believe they're risking a >10% chance of human extinction. They're knowingly risking the lives of every man, woman, and child to continue the work. The lives of their own sons and daughters. A person already rationalizing that isn't going to shed a tear for the careers of mathematicians. Just a bug on the windshield.
> This letter is complaining that human understanding has been crucial to advancing of mathematics, and AI companies are not bothering with it.
Ironic or what. Mr Tao may be remembered as Mathematics' Canute.
The point of mathematics is human understanding though.
That's only one of mathematics' many purposes.
Exactly!
What got me interested in memes as a kid is precisely the fact that mathematics is true in a way that is wholy independent of our understanding of it.
> Keep in mind employees at AI companies are publicly stating that they believe they're risking a >10% chance of human extinction.
Are we really going to take what they say in public seriously?
I understand this stance and where they are coming from, but I can't help but think this sounds very analogous to engineers' arguments against AI-assisted and vibe-coding, especially with regard to cognitive debt. Yet the software industry is plowing ahead, reportedly pushing mountains of unreviewed code to Prod, and the world hasn't ended.
Of course, nobody's really comfortable with it, so this is also a forcing function for the industry to adapt and figure out new techniques to manage complexity and trust. I think the same will happen with Mathematics.
But it is also possible we will end up with three forms of Mathematics: the one we understand, the one we don't, and the one we don't understand but can prove to work. Kind of like magic -- with all the positive and negative connotations of the word.
It is pretty evident that these models will soon exceed our cognitive capabilities. Is it right to hold them back just because we can't keep up? Many of those discoveries will be so beyond us that we can't do anything with them, but that also means they can't hurt us. On the other hand, there could be many discoveries that we can parlay into practically useful applications, even if we don't understand them.
Just like LLMs.
> Yet the software industry is plowing ahead, reportedly pushing mountains of unreviewed code to Prod, and the world hasn't ended.
Yes, this has gone so well
It really has. All of the places facing an unusually high outage rate are places that have seen huge growth in their service usage (Anthropic, GitHub, etc) which is to be expected. The rest of the world has been happily chugging along with coding agents for almost a year now and things seem to still be working just fine.
There's an unpopular branch of mathematics which does not have infinities - finiteism.[1] The constructive version of finitism takes the position that there is no such thing as infinity, just arbitrarily large upper bounds. You can have theorems about arbitrarily large numbers, but you never get
The benefit of finitism is that it escapes undecidability.The big objection to finiteism is that it's a lot more work. Infinity swallows many special cases. Proofs get longer without infinity, and most of the special cases are uninteresting. That's not a problem for AIs.
Someone may start up an AI and make it grind through Hilbert's program for putting mathematics on a fully consistent foundation, starting from a finiteism base. This is a huge, unrewarding job. Great for machine work.
[1] https://encyclopediaofmath.org/wiki/Finitism
Limits can be defined within finitism as long as the end result is finite. Essentially it’s just a process which lets us get as close to 2 as we want.
A better example would be a limit that equals sqrt(2) which finitists would probably say cannot represent a real object because it is only defined as the end of an infinite process.
> 1 + 1/2 + 1/4 + 1/8 ... = 2
It's not necessarily clear that this statement requires infinity, if you're willing to treat "... =" as a shorthand. You might prefer something like "1 + 1/2 + 1/4 + 1/8 ... -> 2" if it's more clear, where "->" means something like "gets as close as you like without ever getting further away than that", but really the "=" sign is already overloaded in all sorts of subtly different ways anyway, so there's not really any trouble using it here. Almost any rigorous definition you can write down of exactly what that statement means would not rely on the use of infinity.
See this introduction to limits.[1]
If you allow infinite recursion, you soon get to Godel and undecidable problems. Finite deterministic systems are decidable, because you can in principle enumerate all the states. The halting problem is decidable for deterministic systems with finite memory. It may be exponentially hard for some programs, but that's quite different from being undecidable.
(This is too long a subject to discuss here, and I haven't worked on constructive mathematics in many years. It's more practical than it was decades ago. You need power tools, which we now have.)
[1] https://www.mathsisfun.com/calculus/limits.html
Finitism doesn't escape anything, it just gives you the illusion of safety. Any intellectually honest thinker should accept the possibility that 10 is a nonstandardly large number.
In internet culture there’s this phrase “Hydrogen Bomb vs Coughing Baby”, meant to highlight the absurd power difference between two combatants.
In almost any scenario even tangentially involving mathematics, twenty-five Fields medallists uniting to denounce something would be a veritable Tsar Bomba.
It should give you pause that here they feel like the ailing infant.
Well OpenAI took one step on the back foot at least, withdrawing from sponsoring this math hackathon event https://xcancel.com/danintheory/status/2098125701782372640
My interpretation is they don't care about scooping mathematicians they were just trying to scoop a competitor. They had a short window in which to complicate Anthropic's priority, when Anthropic announced be able to say "ok nice but we did that too". Upon realizing they'd misunderstood, incredibly they said to this academic (who did not resolve NS) okay well just go ahead and claim the prize, so long as you're not Anthropic let's make this a good story.
If you spend $20M working out a Millennium Prize problem, in what universe would you offer that an unrelated effort should take credit? This is a branding game rather over whether software engineers are going to use codex or claude. In that light $20M (or whatever it was) might be worth it to squash even the rumor that claude code is more capable. Engineers look up to mathematics, while at the same time business and probably most engineers think the problem was to solve the problem. GPTs solved one the hardest known problems so they can solve my company's problem.
Some go further looking at these people, very on-the-nosely likened by one commenter here to ants, talking about education and responsibility and "core values" etc and just don't care. There was a major problem at the beginning of the week that is not a problem now and that is uncomplicated progress.
It's not wrong for OpenAI/Anthropic to do math for product development or even just branding but seemingly at no cost now they could work in an arena real mathematicians aren't interested in, versus just mowing the field. Everyone involved on their side should admit the purpose of these demonstrations is not to engage mathematical ideas it's about Claude/Codex. there's no shame in that. Which is better at solving random hard Diophantine systems? That would seem to tell me as much as I need to know insofar as a model's value is represented by raw mathematical power - then, take my money just as well!
Assuming the worst accusations are not true I think there are ways forward going to be acceptable for all. The labs themselves do not represent Terrance Tao as some kind of gate-keeping dinosaur in this. They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes are paid 7-8 figure salaries ultimately for the product, they solve a Millennium Prize problem then pretty quickly seem to move past it.
>they weren't trying to scoop a mathematician they were just trying to scoop a competitor
Well. Obviously, yes. But in doing so they still DID scoop out a mathematician in a very unethical way.
In doing so they showed that they basically have a huge gun they can point at X work you care about and develop, and can cut you across the finish line. And take credit for it. Obviously this already _existed_, but is just much more significant because even a rumor can be converted into a complete takeover of a discovery.
> They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes take home 7-8 figure salaries, they solve a Millennium Prize problem then pretty quickly move past it.
I get your point, they don't really care about solving all the maths problems. But they're still going to solve them for clout and profit motives. Up until it stops wow-ing people... at which point they will have likely decimated the frontier of the field.
And this is kind of the root of the entire concern. They will move into the forest and completely steamroll all the problems, then declare victory and move on, leaving only pavement and asphalt behind.
The root of all these is the culture in mathematics (and science in general) to only reward those who “get there first”. This creates a perverse incentive to compete. When no one can out compete a tireless swarm of AI, no one gets rewarded any more.
But nothing’s stopping anyone to still work out an alternative proof, or a more elegant proof, or just trying to prove for the sake of understanding, just like doing homework without looking at the solution. It’s just that you can’t get paid doing that anymore.
Mathematics is about discovering and understanding the logical implications of assumed axioms under various inference rules.
Alternatively, some claim that mathematics is about understanding these implications.
Under the first definition, AI is already, and forevermore will be faster and better at proving theorems. Just like it is better at checkers, chess, and now go.
The author asserts that AI proofs are incomprehensible to humans, and so under the second definition AI is merely a tool to overcome one hurdle on the way to understanding.
So which is it? The author seems to claim the second definition, but bemoan the end of mathematics under the first.
> Mathematics is about discovering and understanding the logical implications of assumed axioms under various inference rules.
That's like saying that programming is about producing valid programs in various programming languages.
Dont confuse mathematics with the formal system. If you beleive mathematics = formal system then AI is obviously better at it, and we dont need humans.
But then who decides why a statement is mor important than another? In the eyes of a formal systems all statements are born equal.
I don’t think they make a coherent argument here? This seems to hinge on some argument that because AI doesn’t properly explain its breakthroughs, therefore the breakthroughs are less fertile for human understanding? This makes no sense. Why wouldn’t these under-explained breakthroughs be extremely fertile soil for explanations?
Imagine time traveling back in time and offering Leibniz a packet of proofs from the intervening years, but with the caveat that there would be no explanations. Would he say no?
Because the fertile soil comes from the process of reaching the breakthrough. This should be fairly obvious.
I don’t think it’s obvious. In my Leibniz example, mathematical understanding surely would have accelerated right?
> Would he say no?
And that’s the issue the article is trying to explain.
My thought experiment was trying to extract the issue from the zeitgeist around AI, so not really
Im surprised about the sentiment in this discussion.
I totally see the problem Terence is describing. We are loosing a lot in understanding and focus if it continues like that. The solution found for Navier Stokes doesn’t have much „real value“ - but what almost always happened in the past when people worked on the difficult problems, these sparked new ideas / new theorems that broadened our knowledge. Think back at your grad studies, figuring out a proof as homework was hard, sometimes incredibly hard, but while doing it we gained a lot of understanding how things work. Now asking AI for the solution and „just“ getting it, risks our understanding, our creativity and our ability to connect the dots with other territories. I see it in students nowadays, there is much less understanding, much less creativity in finding solutions. I truly think this „short-path“ solution with the „death of struggle is one of the biggest risks with AI already for human development
Some people are just out there doing automated math proofs for reasons. If you want understanding, value, and edification, that is on you, not them.
This reads like people lamenting a bygone era and making a desperate attempt to bring it back. I'm sorry. Outside of the good ol' boys club, no one cares about some process they've romanticized simply because "that's how its always been done". Absolute nonsense.
We are moving forward and if that means no human wins a fields medal because they didnt spend three decades working on a problem that could be solved in three days, the world will be better for it.
And who is "we" in your story?
Yeah who gives a shit about knowledge or understanding or insight.
You can still do all that. The only thing going is that in some cases the human isn't going to be able to claim that they made the key insights that first solved the problem.
"how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place."
That sounds like an "us problem", not an AI or OpenAI/Anthropic problem.
Are you implying that OpenAI using someones unpublished research without their permission to solve an career defining math problem with their latest model in order to publish first is a problem with the mathematicians?
My read on this document is that people's work isn't being fairly cited more than what does it mean to be a mathematician in this age.
>Are you implying that OpenAI using someones unpublished research without their permission to solve an career defining math problem
Good thing they never did that then
I didn't know this article was about that issue at all. Yeah, if the issue is properly citing work then yes, OpenAI needs to do that. But the article read like it was tackling a completely different issue.
That's one of several issues, obviously a big one, and they do touch on it:
> Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions.
Yes. I forgot about that section. That is probably the one thing that the AI companies can and should do better on.
This is the real issue, though this article does not really hit on it explicitly just alludes to it.
Yes I think they should be more direct. Not sure why they weren't
Indeed, I wonder how a similar letter by Uber drivers would be received -- "navigation is an intrinsically human domain, personal relationships are critical for passengers and drivers to progress in the world, etc etc." Or doctors, for that matter.
We are all going to have to come to terms with entities more capable than we are, and in many cases, letting the real work be done by the AIs will be the right thing to do. For all the huffing and puffing about the "human touch" in medicine, it will eventually become downright irresponsible to consult only with a human doctor. I am not sure if this is the case in mathematics or not, but if it isn't, that suggests math will be relegated to more of a hobby than a cutting edge scientific discipline.
That's how I read it too, Terry Tao, who has been a "pro-AI math guy" is going through the same emotions and confusion that us SWE folks are going through, "oh, wait... this might mean I'm not going to be special anymore!?"
I don't mean to be a dick, but I've talked about it previously. These folks are grieving. I get it, I've lived through this sort of life changing thing before, it sucks... but yeah.
I think this is ridiculously flippant. If software engineering and the hardest math is solved, that means that eventually a majority of professions and knowledge work is solved. This is hugely problematic because of the way our society currently functions. People need jobs to eat, pay for housing, etc.
Dismissing it as "innovations have happened before" is disingenuous. Yes, innovations have happened, but none of those threatened to automate all human work in existence.
Our current society will certainly not survive. Read Mark Fisher’s capitalist realism. You’re right that all most works are “cooked.”
Yeah given he had been very pro-AI for years, I expected he made peace with the issue many many years ago (like I did back in 2018), and when this time would come he would explain to other mathematicians how to live with it.
I am a bit disappointed by him.
It sucks. But yeah. Trillion-dollar industry wants your livelihood. This is but a force of nature.
> We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align.
This is Terry Tao talking about AI's impact on Math, but this could just as well be a software engineer talking about AI's impact on software development.
Do mathematicians deserve more job security than software engineers?
It's not about job security. It's about the social and intellectual practice of the discipline.
The threat to mathematics isn't that suddenly the profitability of their profession (lol) is going to go away, it's that people are thinking of AI as a replacement for the human social and intellectual practices that constitute the discipline.
> The threat to mathematics isn't that suddenly the profitability of their profession (lol) is going to go away, it's that people are thinking of AI as a replacement for the human social and intellectual practices that constitute the discipline.
You could say the same about software development.
Software development is a group effort, so it includes social practices, and certainly includes intellectual practices as well.
For the sake of argument, how is this different from the Luddites? The Luddites feared that machines would displace not only human labor, but also the social knowledge, skilled judgment, and craft traditions embedded in their work.
Kinda crazy to think that academia functions as a kind of humane reverse centaurism. Theory X (reverse centaur) before Theory Y (centaur) for managing the development of others.
All “fields medalist” signatories - a rarefied and elitist group indeed.
I wish this letter could be more egalitarian and include the view points of those who AREN’T the beneficiaries of a highly competitive winner-take-all system.
Since the common narrative is that AI frees up labor to do other things (engineering -> trades), maybe we can celebrate that genius mathematicians will now spend time teaching children how to be as smart as them?
Agree. That is also the part of the letter that does not make a lot of sense to me.
In the Economist article Tao links, Hugo Duminil-Copin, draws a comparison: airdropping someone on the summit of Mount Everest is very different from climbing it.
The fundamental issue with AI solving any perceived difficult problem is that we have lost the journey. The sight atop Mount Everest looks much different when you have climbed compared to being dropped from above.
But we don't pour billions of dollars of research funding into mountain climbing because we think it's going to lead to wider breakthroughs in science and technology. And when we need to get people on top of a mountain for an important purpose -- like a military or search and rescue operation, for example -- we absolutely do airdrop them right on the top.
So that raises the question: is mathematics simply a pursuit of passion? Are problems solved "because they're there"? If so, then mathematics can join the ranks of things like mountain climbing, cycling, and weight lifting. But if we are trying to accomplish something important (design better airplanes, find theoretical guarantees about cryptography, factor matrices faster), mathematics needs to become more like a military or search and rescue operation, using the best technology available to secure the outcome we need. Given that the NSF pours billions into scientific research every year, it sure seems like mathematicians want to think of themselves as being in the latter category.
What defines important and why must it be solved in haste? Many issues and other problems arise during the journey in solving all problems; those that are needed and those that are pursuits for their own sake.
If AI gave us the plane to reach Everest without us having gone through the journey of aviation and flight, what would we have lost without that process?
But the most important problems to be solved are not technological challenges but social ones, involving humans and our relationship to one another. An area AI will forever ill-suited to handle.
as someone who loves to go down with a snowboard, I can see value to being taken to the top and then enjoying the ride down. im sure it is not a thing to be ashamed of, as millions do it.
I agree with many of the sentiments here. But an open letter signed exclusively by Fields Medalists that purport to define precisely what the "mathematical community" (who is inside and outside) and what their goals are raises my hackles for some reason.
It is true that the manufacturing of "true/false" statements is not the same as gaining understanding of a problem. However, for many mathematicians, true/false statements are already manufactured by others. Think of a student who is given a conjecture to investigate, with their advisor describing it as "it must be true". Most exercises in a textbook are stated such that the outcome is known before you begin. That's not really a problem -- investigating the conjecture/exercise yields its own dividends, whether or not the outcome is known. It is also the case that defining new directions involve understanding and synthesizing related problems, asking the right questions, and deciding upon the right directions, and it's not clear that AI can do that at all.
The real risk, I think, its the public's (and funding agencies') perception of the importance of "human" mathematics, but that's already a struggle. For example, it's tough to explain to the lay person why it's still important to research group theory -- the main example people cite is RSA encryption, which was invented almost 50 years ago.
Can't all the prestige-maxed mathematicians still study all these famous problems after ai solves them. And even if they convince open-ai to stop dunking on them, some normal user with gpt 7.1 on the normal chat interface will do it in a year.
Even if they stopped anyone from releasing ai proofs for five whole years it would be meaningless seeing as these problems are decades old already. Humans weren't JUST about to solve them until openai stepped on their toes.
Nothing AI companies will change the value of math, as William Thurston said: The product of math is clarity and understanding, not theorems by themselves. What they seek for the IPO is devilish and misleading, and doesn't serve the true purpose of the math.
Back in the day you could think of a cool idea. I don't know maybe a plane that could fly without drag. To even see if this was feasable you had to understand physics, engineering, and then from there you had to have a math person see if it was actually possible.
Now, I can ask ChatGPT about this and get back a proof that shows "a passive airframe cannot sustain zero-drag motion through still, viscous air"
So, I think if anything now, Maths has changed for the better. More ideas can be proven false or true from a get go instead of wasting so much to see if its even feasible to find out it isn't.
Progress if anything is about to leap frog anything we have ever known.
This is a pretty moronic take. You only have to understand the first thing about physics (e.g., a 14-year-old's understanding) to know that a dragless airplane is impossible.
If you are suggesting there is some new model of physics or groundbreaking technology that would allow such a dragless plane, then don’t let me discourage you! But AI won’t help at all, since it will only regurgitate conventional wisdom…
Well, there is physics for an alcubierre drive, so these moronic takes have answers too.
I find this explanation has a lot of applications for programmers within companies. It’s one thing to get your LLM to give an answer, it’s another to bring a group of people into shared understanding of a domain.
Playing a devil's advocate. Why do we need understanding ? To take an example i would say ~99% of the population do not understand how combustion engines or how semiconductors work, what say another 1% ? Is the fear post-apocalyptic in nature ? We need some human priesthood to carry on tradition ? why ?
Let's assume in the next decade GPT-7 PRO Ultra is cheaply ubiquitous, inspectable, reproducible, transferable, reasonably un-constrained by any institutional interests.
What say the 1% ?
Lifted up my comment for addition visibility
Lots of Fields medalists signing this. Interesting to see one not there: Timothy Gowers.
Interesting indeed. He talked about this imminent crisis a few months ago.
https://xcancel.com/wtgowers/status/2052830948685676605
nor Andrew Wiles
I'm not convinced this is an alignment or technology problem.
If my boss vibe coded an app for the customer and then assigned me to get it working, it would be impossible to maintain. If he gave me enough AI tokens to vibe code the MVP myself and to my design, I wouldn't mind.
I think the same issue is at play here in maths. OpenAI owns the model and they can direct it as they please. They chose to spend lots of money getting a quick result, instead of developing mathematical infrastructure for the next generation of problems. The managers are in charge rather than the experts.
It's good to start a conversation, and the number of Fields Medalists behind this certainly lends a lot of weight to it. But I'm not seeing a strong argument for misaligned incentives beyond the specific plagiarism allegations. The job of an AI company is to build systems that solve problems. The job of a mathematician is to advance the state of human knowledge. If anything, an influx of solved problems should increase the demand for human mathematicians who can convert them into conceptual understanding.
I note here an isomorphism with my essay from earlier this year, The deliverable is you! Programming as theory building:
A program is not the only output of programming. The other, arguably far more important output, is the programmer.
When you write the program — with your own hands — the program is proof that you have a solid mental model of the program.
When you let the computer write the program for you, the program is proof of … nothing.
https://nekolucifer.substack.com/p/the-deliverable-is-you-pr...
An experiment I would like to see:
Train an LLM with no advanced math texts: only basic math up to 6th grade, conversational text and literary works.
Interact with it (you cannot refer to anything past 6th grade math since you don't know it yourself) and get it to propose a solution to a real world problem. e.g., come up with RSA to practically secure communication.
This is really well written and exposes a core tension between science and something akin to engineering. The "engineering" of proofs has become "easy" (a compute and $) problem, rather than hard (a time and conception problem).
Without the ability to do things the "hard" way it is difficult to figure out if doing things the "easy" way will help us advance the frontier of math and science.
I may be wrong but historically we had this version of science discovery for a long while (empirical observation and brute force application) rather than first principles leading to applications (tools, the wheel, mills etc). Then somewhere along the way it flipped after Newton and the enlightenment period and started understanding first principles before they become engineering applications.
Perhaps it is not required, and we can just keep doing things the "easy" way like we used to, or we might find ourselves out of the ability to brute force things and then we go back to needing to do this the hard way, at which point this period of AI brute forcing would be seen as a detriment.
Is this really any different than the problem in software engineering - where AI is doing the work of junior programmers and now they aren't getting the development they need?
Seems the same to me. And it'll be the same in all industries soon enough. And then it won't just be the junior people.
All the same problem: what do people do now?
It’s better to compare this to computer science, especially, theoretical one. LLMs stop people from exploring new languages, architectures and so on.
Well... computer science really is math already. :)
Somewhat unrelated: is it wrong to say mathematics is not art, and that there is always a right answer? I know that's not romantic, but maybe it's true.
Before LLMS, programming was something I might've said required creativity and human input to do properly. It's not that creativity or human input isn't valuable anymore, but AI has forced me to realize that coding is much a means to an end, and that all things considered, the end matters much more than the means.
If we can make important mathematics progress faster and better with LLMs, I think it's wise not to fret over an apparent loss of our humanity. Perhaps that's only a loss we want to have.
There's a reason mathematics is generally within liberal arts programs rather than science programs. Mathematics is the art of logic. Yes, sometimes mathematics becomes incredibly useful but most mathematics is never applied.
Compare that with computer science. Most of the work we do in software engineering is in service of an applicable output - software products that facilitate processes or bring in revenue. Turning up the dial on AI gets companies to these outputs faster.
Turning up AI on mathematics helps solve conjectures and can provide new insights. But it has a major misalignment with the purpose of mathematics which is largely intellectualism.
You are referring to Boubakhism, not mathematics.
“Mathematics is a part of physics. Physics is an experimental science, a part of natural sciences. Mathematics is the part of physics where experiments are cheap” - Vladimir Arnold
On the matter of computer science having anything to do with computers, please refer to Djikstra.
It’s about computation, not computers - an application of mathematics, predominantly thanks to Turing, Von Neumann, and Claude Shannon’s masters’ thesis; though ofc many others as well but I see them as three individuals who made the minimal structurally necessary contributions - VNA and silicon are one of many possible substrates.
Also in the service of others around us.
I think it could be useful to know something like P=NP, even if we don't understand why.
I have never, and I mean never, seen such a declaration have any effect whatsoever.
To me, the "meaning" of proof is twofold. First is the understanding which is completely absent from a proof which depends on exhaustive iterations of instances or is asserted by fiat from myriad individually contestable paths. That's what I think makes AI proofs risky: they absence of understanding.
The second is the utility. We get to do navigation because the maths about angles and spheres checks out. The social utility downstream of AI proofs may be huge.
I don't like this disjunction.
Maths will continue as a field of natural science in understanding the results and uncovering meanings in them. It is normal that the established community is afraid of the change, because it’s their _home_ that’s changing. But it will be a better home to the new generation nonetheless, one that’s not as daunting as the higher maths has always been to many. The concerns raised here will not be a problem at all.
A Severe Misalignment of AI in human-centered Mathematics
Jacob Tsimerman fields medalist notably missing from the list now works at OpenAI but recently admitted to be "grieving" for mathematics
1. https://youtu.be/6uIJdXmB4vE?si=5QMN5Dos7EE7WlrB&t=154
Contrarian take: I think the ability of AI to produce valid mathematical proofs (even inscrutable ones) is absolutely fantastic. Mathematics as a profession does not have a monopoly over math itself any more than professional pianists have a monopoly on who plays piano, when they play, and how.
I have sympathy for any jobs that might be affected (much as my own job has become more tenuous in software engineering). And if the field is disrupted by chaos that makes the research process unproductive, that's bad too and should of course be handled by applying better organization within the institutions that tend to perform mathematical research.
But to a large degree, the notion that "sloppy AI proofs are bad for mathematics research" seems like a total failure of the imagination to me. Attempting to find shorter proofs or more elegant proofs can be turned back in on itself via proof theory. There are proofs in Presburger arithmetic that are doubly exponential in the length of the sentence. Yet a more powerful theory like PA makes quick work of such theorems. The explainability or "subjective beauty" of a proof can be quantified and optimized against. Optimization itself can be optimized against. I really don't understand how this magical ability to know the truth of more theorems much more quickly—even via an "ugly" route—is anything but a net positive.
Yes. Mathematicians don't really get logic, and Tao is no exception.
So what's the actual point here? It's too fast and we can't keep up?
Isn't that just a function of the technology itself, and the same problem being faced by every other field? And going to get exponentially "worse" every year!
Or is the issue that they're bad at explaining things, in a way that produces actual learning? (e.g. AI is amazing for learning but the net effect on students so far appears to be negative.)
I am curious what does the first and probably the last human who solved a millennium problem thinks about this.
I kind of expected a sober stoicism from mathematicians. Feels silly in retrospect. This is just the math version of the "anti-ai" movement by "artists".
I don't know, people. We still really don't know how OpenAI or others are producing these results. It's all very hand wavy and trust-me-bro. How much money/time/compute have they really thrown at these problems? How much human involvement was there? What LLM did they even use? How much regular software was involved? They have given answers to some of those questions but no proof that that's actually what they did. I don't know if it's worth giving them this much credit (which is what we are doing by writing these essays and spending so much time debating). Anthropic wrote a C compiler that turned out to not really be a ready made replacement for GCC. Did they ever do any more work on it? Has anyone else produced a C compiler? It seems like that and these proofs are just demoware that are not (yet? Who knows?) production ready to turn the world upside down. Impressive one-off demos, yes, but companies have been pulling those off for centuries without ever going anywhere afterwards.
A very important point! In Tristan (NYU prof)’s write up he noted evidence of the OpenAI mathematician team doing a lot of correction and guidance along the way. We are never told about this with openness and clarity. At a minimum, complete disclosure and honesty is needed by the companies and about the precise role of their staff members.
I know that this comment section is not astroturfed, but it’s really uncanny how different comments are today compared with thread about solving navier stoke
Is the purpose of mathematical research to understand the ‘truth’ of numbers? or be the person who find that truth? I think people who are interested in finding the truth won’t care where it came from.
I was under the impression that mathematics (and science generally) had the primary goal of helping us understand our universe better than those who came before us.
I don't know how societies set "primary goals". After spending 15 years in a tenure track -- tenured position, I thought setting goals well was important.
I take pleasure in how mathematics and science help me understand the universe better than I understood it before I studied the fields. I believe that my understanding has helped me contribute to society.
Taking a snapshot of the state of AI math right now and concluding that it will be net negative to human understanding and insight in the future is very short sighted. This statement will be used to promote ideas and actions that will ultimately be disastrous for our country.
At some point we will lose track of all the ai discoveries that are worth remembering.
Academia with the publication system had a way of retrieving old discoveries and build upon them.
If my LLM session found something groundbreaking in between the billion tokens it produced, how would you ever know?
I think this is why Terence Tao created https://palomar-registry.org (I have no affiliation with them besides also having sent in a result there)
LLMs are excellent at doing searches of the math literature; presumably they'd also be good at doing searches of AI-generated results.
"We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose"
Suppose we eventually have GPT-7-class models running practically on $100 devices, with their activity transparent, inspectable, and reproducible. At that point, what exactly is left for us to fear from this threat?
I've got to ask, let's say we get a small modular nuclear reactor running practically on a 1 acre lot, its design meltdown proof and waste-free, what then should we fear from this threat ? This is just a thought experiment.
Jokes aside, any productivity-improving technology, even one with no negative externalities, has the potential to cause economic displacement and wealth concentration in proportion to the productivity gains catalyzed. Anthropic did a cool analysis of this for AI here: https://www.anthropic.com/institute/econ-scenarios
What's the threat? I would use it to write a dependent type theory that is JIT compiled and use it to rewrite emacs.
The fact that those models are encroaching on things only human minds could do. Personally, as a human, I want there to be things humans are the best at, and intellectual things were the final thing that machines hadn't beaten us at.
100% on board with this take.
This is really only a short-term problem where the AI companies only have the internal models that can solve these. In the “long” term, which could honestly mean months, everyone will have access to Bel/C/D-level models capable of solving these anyway.
This. Like programming, the community will shortly be forced to come to terms with a lot of new self-proclaimed mathematicians “vibe-solving” problems and dumping solutions without understanding them. It’s not really a special case for mathematics.
That's going to force formalization to become required for any new result to be taken seriously.
The main fear seems to be that if mathematics is done at this speed and in this way, humanity will lose its intuition for doing mathematics.
chain of credit is important, and plagiarism is harmful.
But is science/mathematics ultimately a pursuit of knowledge, or a pursuit of recognition?
Recognition helps keep people motivated, but that shouldn't be the pursuit of science or mathematics.
Given the existence of this technology now and the incentives of the AI companies, both of which are not going away; what's a good future here?
A major part of the complaint is that there's no conceptual understanding and building of new ideas coming out of the AI proofs, thus defeating the purpose of the original pursuit.
If in 2027 the AI models start producing, with every mathematics or science breakthrough they make, well-written documents tailored for human understanding, with intermediate concepts, expositions of failed-but-once-promising paths, etc. Would that be good alignment with the mathematics community?
As I understand the article, its title should be: "A Severe Misalignment of AI Frontier Companies with Ethics."
They can get with the program or be the equivalent of a genius SWE writing assembly on punchcards in 2026.
The only thing I read from this is their ego being bruised by a machine.
If these people cared more about discovery and advancement of human knowledge the only thing they should be doing is celebrating. There's no proof of plagarism but that's an independent issue.
How are they not realizing that in the future children will be able to do impossibly hard math but they will be doing something we can't even think of as of now.
One world class mathematician in the future could be advancing mathematics the equivalent of one Riemann hypothesis A DAY.
How are they not celbrating this as the achievment of the centry? Who cares about plagarism at this scale. It has been solved and it wouldn't have been without AI.
for the same reason why you do not get full credit for only writing down an answer without showing work in an exam.
lol what a bad analogy. Why should anyone care how much effort a result takes to achieve? If anything our entire world functions because we reduce that effort as much as possible.
I don't judge you for not growing your own food when you hand me a burger.
It's a turning point for science and beyond. AI has shown itself to be transformative. Even today, it is already changing the how research in math (and other sciences) is conducted. In the near future, whether it is LLMs or some other superior method, its capabilities are only expected to grow. The time to ask the question is now: Will AI be arguably the best tool at scientist's disposal, or will it instead be paraded around as a super brain collective that no human or group of humans can compete with, discouraging entire new generations of future scientists from ever entering the field? The jury is out on this one.
I wonder what Demis Hassabis thinks about this. I thought he cared a lot about mathematics.
For every benefit that sillycon valley has produced in the recent past, there have been many more harms. I am confident that this will be no different. Of course, benefits and harms depend on one's vantage point.
OpenAI should just apologize for having been too greedy. That’s it, as simple as it gets. The fact that it never will is the biggest red flag.
Are we here to solve problems or are we here for prestige?
Who cares how the problems are solved?
It's more about *finding* problems. And soon there won't be anyone who can do that, thanks to people lacking imagination.
Could we develop new ways to develop understanding and explore new ideas, such as interacting with the models to explain and understand their proofs, as well as to brainstorm related directions to pursue?
>Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others.
That's just capitalism seeping through a previously unexplored crack into academia, and attempting to do the only thing capitalism knows to do - maximize profits - with no additional concern.
What changes exactly is this post asking for?
This comment section is very astroturfed.
That has to be the most impressive endorsement list any declaration has ever seen!
25 Fields medallists! Wow!
And here we all thought for sure that it was gonna be the "dumb" work to fall to the machines first.
Human hubris really is something...
Hans Moravec likely isn't surprised by this.
For some reason, every time I see something like that, I get major Ted Kaczynski vibes.
The level of things happening in the last several months is just SciFi level, especially last two weeks. 1. Overlords of AI threatening to destroy someone if their demands aren't met over the Millennia problem. Proving the point that given the chance, owners of the AI companies will immediately use their power to squash or control other people. (See Butlerian Jihad) 2. At least for me, the Exponential progress didn't sound true until this week. If AI indeed solved all those problems and proofs and solutions are correct, we went from "Write me English 101 college essays" to "Help me solve the Navier–Stokes existence and smoothness problem" in a just 3-4 years. 3.Looks like someone's work might have been stolen, by AI company and there are strong evidence. See point 1. 4. War between AI & AI companies. 5. It produces a machine verifiable LEAN proofs for the hardest problems know to us !!! Proving that point that best use of AI so far is to create verifiable systems. I.e the absolutely chaotic AI creates verifiable (orderly) proofs. It is Maxwell Demon: chaos to order.
Touché… In short you still need the humans to understand what’s going on at the end of the day whether it’s a mathematical equation or source code for a computer program relying upon mindless AI isn’t good enough. Humans still have to do the thinking or is it in the brave new world the error checking at the end of the day?
This is a Stephen Wolfram tier problem. I hope to read what he has to say on the matter in the near future.
Underrated comment
You can outsource your work to AI. You can even outsource your thinking. But you can never ever outsource your understanding.
I’m sympathetic to the concern, but I’m still unclear on what the concrete ask is.
If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently? Also why does discovering the answers preclude humans developing ideas from them? I don't get why solving a math problem stops anyone from doing that?
Should they (AI companies) avoid training or evaluating models on open problems? Solve them but not publish the results? Delay publication? Only release proofs after mathematicians have had time to study them? Require some attribution or review process?
The statement makes a strong case that “maximize the number of solved problems” may be the wrong objective, but it seems much less clear about what behavior they actually want from OpenAI, Anthropic, DeepMind, etc.
I’d be interested in the most concrete version of the proposal. Without that, it starts to read a little like: "Please stop getting so good at our thing!"
> If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently?
Just stop doing that. Don't treat unsolved math problems as some cheap benchmark to beat.
Leave the math for mathematicians, and let them use AI in a way that helps the field, not in a way that harms it.
Academic research is a marvel because (aside from patents) nobody owns it, in the sense of property. It is given away to be used freely. Researchers want their work used and cited. The primary external reward for publishing is reputation and prestige which translates to remuneration for researchers. And that remuneration can be poor.
Beyond the issue of growing understanding and keeping a bountiful stock of questions to pursue, this scheme seems to be threatened as well.
> to the point that they can solve major outstanding problems in many fields of mathematics
I will die on this hill, humans solve math problems not machines, there's no automatic math prover out there. There are humans attempting to solve this problems either by leveraging these tools or not
Math, like art, it is more about the process and not the end result.
At some point someone is going to need to answer for "what happens when all intellect is hoarded by one or two companies?" It's pretty clear that these AI labs are basically stealing everyone's alpha.
Maybe mathematicians should be aligned better, rather than AI?
The current measure of a successful mathematician is the problems they have solved or worked on. At some point in history, the measure of a successful scholar was how well one could copy manuscripts.
Once we have a tool that starts to work well for this task, it's time to define success differently. It's a classic alignment problem! ;)
But seriously, these people should start focusing on finding and proposing more important problems. And the credit of discovery should go to the person who defined a new category of important problems.
It appears to me this is an incredible inflection point in mathematics, a neat forcing function like cryptography was for the development for modern number theory and algebraic geometry.
Fundamental problems with great implications for other fields will be solved by AI because some entity would throw tokens at it. And these would be further built upon.
Another historical analogy: this is kind of like the transition from alchemy to chemistry.
Interesting analogy
Having worked on nearly 500 complex lean libraries in math and theoretical physics, with AI assistance, I have some experience from the road to share here. In all my work I have found that AI is terrible at coming up with interesting ideas on its own - even the very best, latest models.
Without human scientific and mathematical intuition you can bet that AI will always take the road most travelled and miss the genuinely new and interesting breakthroughs - in fact it will not even think of them or try them without a human rider whipping it constantly to go down paths it would normally not consider.
And I think that is true in the case of the recent OpenAI blow-up -- it is alleged that even in this case AI did not come up with the winning technique without some human guidance.
I have never - across perhaps many thousands of chats with AI - seen AI push beyond the edge of what is known by itself, without being forced by a human to think outside the box.
I do think it might be possible to encode this process with prompting and agent orchestration methodologies - but even then - without a human - and human intuition - in the loop, I am skeptical.
Therefore I think a more accurate view of AI for math and science today is that it is an extremely powerful tool in the hands of a skilled operator with a strong intuition and roadmap of where to go, and rather boring when left to its devices.
Whether that will change in the future is a question. Nothing says that in principle AI could never do what a human driver does - but I still doubt that AI will replicate the life history and experience that real scientific and mathematical intuition is really made of.
AI trains on a lot of stuff - but it doesn't have hallway conversations, office-hours with teachers, or hard-won experience from all the things that failed that are NOT in the training data...
AI trains mainly on the record of what worked, not what didn't work and never even got published (and only lives in peoples heads) - and what didn't work is arguably as or more important for making breakthroughs and forming real intuition.
Knowledge is not a private guild. And this revolution will not stop just because you're upset.
What can be solved, will be solved. And that's a good thing.
So, if AI solves problems in the open, it's accused of plagiarism and contributing nothing of value. But when it solves the problem behind closed doors it's accused of not contributing enough to the field and even depleting it of fertile problems.
Damned if you do, damned if you don't.
Somehow physicists don't complain.
First they came for the graphic designers, and I did not speak up, because I was not a graphic designer.
Then they came for the writers, and I did not speak up, because I was not a writer.
Then they came for the programmers, and I did not speak up, because I was not a programmer.
Then they came for the mathematicians, and I did not speak up, because I was not a mathematician.
And then they came for me.
By then, there was no one left to explain to the frontier labs that there is a severe misalignment problem.
For such great minds, they seem to be rather muddled thinkers.
E.g. of the actually three misalignments in this complaint, the jey one seems to be:
"The goals of the AI companies and the goals of the mathematical community are severely misaligned."
And that's wrong. There is no misalignment. Each is independently aligned with its respective interest. And those two interests diverge - just as you'd expect.
This kinda reminds me of the documentary about the top Go master that got beaten by a computer in dramatic fashion and had an existential crisis. Man confronting his own limitations in the realm he previously ruled unchallenged, what a time to be alive.
Nobody is going to care that the math isn't being done in the traditional way. The results speak for themselves, this is now a part of the landscape. No amount of hand-wringing is going to put the cat back in the bag. Adapt or perish.
Terrence, youre a mathematician; now extend this to the general case: AI is misaligned (inherently) with humanity.
Other than the fact it's mathematicians signing it, why is mathematics special in this regard: surely this generally applies to a lot of different industries and sectors of research / academia?
Mathematics is being used as a benchmark because there are some high-profile awards in this area I guess, and possibly because 2/3 years ago LLMs were pretty atrocious at it so the level of improvement has been significant.
What's interesting is that these are the smartest people in the world, and AI is eating their lunch. You do the math -- pun intended.
Both Amodei and sama are weeping at this news. The horror. The horror.
We should create a platform where people can upload AI slop proofs anonymously without taking credit for it. That way the incentives of planting a thorem flag would go down. And if someone wants to clean an AI slop proof to advance the field, they can do it without cleaning the house for free of the person that planted the flag.
Would OpenAI have uploaded their proof to such a platform? If you know the answer then you know what the problem with what happened is.
The thing is no one is stopping anyone from getting the same knowledge/understanding/insight
If those things are disincentivized because the original problem is “solved” and there is less prestige to motivate people doesn’t that say more about issues with the community of mathematicians than the AI
Welcome to the new world. I, for one, welcome our new math overlords. :)
this sort of reminds me a little of the reaction to Elon/SpaceX in its early days .. when Neil Armstrong and other apollo astronauts went before congress and voiced their concerns about relying on commercial companies for human spaceflight and the dangers etc .. also not trying to sound cynical but hasnt LLM made math more accessable to ppl ..
solving the problem is aligned with humankind
At this rate AI will be doing all of the mathematics within 5 years, I don’t see why a mathematician would be worried about anything other than that at this point?
Sure, but who will care?
If no one understands it, it may as well have not happened. There's not much incentive to understand or internalize the results generated by AI. A human operator gives it a prompt and it produces some lean proof no one wants to (maybe can) read.
Without the community of human mathematicians internalizing the proof, simplifying it, and re-communicating it to others we end up losing the main output of mathematics as an institution.
Why would we assume AI will learn to solve millennium prize problems but will struggle with the easy part of doing the explaining? I think it’s too easy to predict that models 5 years from now will have the same limitations as the they do now, I would be amazed if this was true, GPT3 was the best model available 5 years ago.
I'm not sure it's just a matter of explaining... If it's just explaining, than sure, that's something they already do well.
The questions I have are:
It's possible that a headline grabbing proof of a Millennium Prize problem generates enough incentive for people to simplify and gain understanding from it, but we run into problems when AI becomes the dominant approach for all of math. Although, maybe this is self-limiting? I guess it's possible we just ignore a bunch of AI generated proofs and only keep the ones people find comprehensible in a useful way.You can always hypothesize that at some point in the future (maybe 5 years? maybe later?) the models will be indistinguishable from humans and there will not be any functional difference at all. It's possible, but we're not there yet. And, as they say, past performance does not guarantee future results. Many technologies plateau at some hard ceiling of performance. Moore's law has had an unusually long run, but it's not a universal rule.
This is a "human alignment" problem in this case. OpenAI acted like complete assholes about this, from the beginning until they announced it. Not ChatGPT, the people that were in charge of the project.
Economist: "Are mathematicians talking their own book out of fear?"
The Economist, who recently used "moral panic" now stoops to Hacker News AI booster level and inverts arguments usually directed against the rich and investors. What is next? The Economist inverting Upton Sinclair's quote to serve its billionaire owners?
Look up the AI investments of the Agnelli family for example.
Another bunch of nerds who now have their panties in a bunch because AI threatens their fragile egos and core identity and who they are..
I’m being somewhat harsh here but come on - human endeavors are messy and it’s surprising how much our egos are getting bruised here over seeing the value of these tools
Dont get me wrong Also, there’s no AI Utopia coming this is it guys, were stuck with oligarch Tech Bro funded AI and big funded Govt AI so forget any egalitarian motives- we have to fight for our rights from other humans as always as well but AI as a technology in itself being able to truly solve unsolved intellectual problems is still a boon for society - who cares who gets credit?
Tao's calls for respect for provenance in mathematics publication are laudable but most likely naive given the closed nature of frontier model training data curation. Anthropic and OpenAI may react with a symbolic and short-lived olive branch, yet provenance is a larger issue that has impacted other fields beyond mathematics. While traditional respect for lineage in mathematics is of value to the academy, industry and science at large will likely be much more Machiavellian about such concerns. Mike McCoy's recent article is also timely (https://mbmccoy.dev/posts/mathematical-conservatory/). The parallels to the music conservatory are quite telling -- academic music describes a musical culture in preservation that has completely lost touch with musical developments beyond the early 20th century. Mathematics may very well evolve separately and with very different values than the academy upholds. The crisis of music at the academy is a cultural disconnect and a serious loss of critical analysis and acknowledgement of widespread and dramatically evolving music practice; however, for mathematics, the impact would have much more severe ramifications for education and human development if the academy forces a schism with AI. As models improve they very well may be inventing mathematics -- science and engineering may grasp for them -- they'll exist with or without attribution. Would be a shame for the academy to land on the wrong side of history and refuse stewardship of upcoming AI-assisted mathematics, including provenance, because of this misalignment. If attribution is important, then the academy will set aside the institutional resources to do it. If you succeed at having model developers participate, I commend it. To let mathematics be born in isolated context windows and only serve narrow, localized engineering purpose without rightful addition to the canon, would be a tragic, yet preventable, loss.
thank you for this commentary, as a violinist / software engineer / "computer scientist" / "mathematician" i can see that studying and doing math will still remain a thing that humans have to do to train their own brains to think and to form better neuronal connections. We may not get paid to do it anymore but it will still be important for our own development for the same reasons orchestras and bands still exist in elementary schools and beyond (at least in many parts of the US).
There's massive cope and then there's this. I've never been a fan of Tao et. al, so I'm glad he's being wiped out. He'll do fine anyway, doing conferences, etc.
Also funny they deliver that on vibecoded site, lol.
Which mathematician's take on AI do you like? Do you think mathematics should no longer be an intellectual pursuit?
I haven't heard about his take on AI (sorry about derailing a bit), but generally I like Robert Ghrist's approach to life and math.
I'll check out if he has said something about it. Interesting guy.
i was initially optimistic that ai could be distributed. having this technology in the hands of many would counteract the malign elite who would use ai to oppress the rest of humanity. in the meantime given that intelligence is scaling proportionately with inference time compute, we are concentrating far too much power in the hands of the few who have both their own model and datacenter.
putting this amount of power in the hands of so few would require leaders of immaculate moral integrity. what we have in the US at least is an emergent kleptocracy with obvious dark triad traits: sam altman, dario amodei, elon musk, etc. they are malign and will use this power for their own benefit, at the expense of others. ignore pleas of public benefit. look instead at the evidence: infighting, systemic dishonesty, reckless disregard for safety, political lobbying and manipulation, putting power in the hands of the elite few in the guise of safety, using ai in the military to oppress and inflict violence on others.
really to prevent a bad outcome, we would need to act soon enough to prevent the kleptocracy from corrupting politicians and democracy with them. i would ditch the claude or openai subscription and support open models instead, preferably in countries outside the US to prevent global neocolonialism.
The mathematicians would be wise to re-read The Bitter Lesson, maybe twice a day, until it sinks in. No offense and with all due respect to the Ivory Tower Giants but the whole "oh no you ruined the game because you solved it, I was supposed to play with that in child-like wonder manner and take several years to do so, and by then I would have showed you all the trickery I did to get there and maybe that will be useful to you" over the past 2 weeks is, get this, cope.
Chess. Go. Coding. Now Math. Another one bites the dust. Let's meditate on this lest we forget: Stochastic parrots that generate the next-token cannot reason or produce anything meaningful. Let's protect our jobs at all costs, even if we have to drag all of humanity down. It can't be! Stochastic parrots can not replace the Ivory Tower. No way.
Castle dwellers dismayed at moat-crossing technology.
What's most important about this is that it's a case study of what happens when deeply evolved ecosystems are blown up by disruptive technology. The psychological and social and professional impacts and myriad and traumatic to be on the receiving end.
Mathematics is merely one of the first domains disrupted. It will be unique only for being among the first... absent disruption of the entire civilizational project as a result of the disruption being caused.
Woe for us that we try to navigate this degree of change at a moment when the very worst and ignorant and short sighted hold all the power, economic and political.
Woe.
another story about the lack of chain-of-thought reasoning traces in frontier models...
AI bros here and elsewhere seem to come at Terry Tao with "not so smart now are you?" viciousness that is off-putting.
How dare they defend themselves when attacked by their betters. The nerve of those peasants.
I’m trying to determine now if Terence Tao was a plant or a useful idiot. Maybe both.
Care to elaborate?
Mostly that he really should have seen this coming. These people absolutely do not care what happens to mathematics (or any other field). They compulsively lie and steal and they played Tao and others like a fiddle. He danced to their tune and now that the music has stopped, now do they complain.
He was the poster boy of the mathematician yielding these tools for his own benefit. But he forgot who the owners are.
Too little too late.
You should read his comment from wednesday here: https://terrytao.wordpress.com/2026/09/07/finite-time-blowup...
And yes, at one point it gives you right about them taking advantage of him. He sat down for an interview and found later that they just used the "best clips" out of it for an OpenAI ad.
That’s table stakes!
Strange they are rallying against something their fellow mathematicians had a hand in creating.
Let's focus on the subject and try to distinguish the technology and the frontier companies selling it. That's the key point in understanding the whole stance.