There's a long collection of comments by mathematicians at [1].
A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later.
The main social complaint is that the social foundations of mathematics just changed, in a big way. The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
One comment was "A common piece of wisdom is that mathematicians only have a few tricks in their bag – which they skillfully apply again and again."
Feynman made that point in his autobiography. For various reasons, he had
a different set of mathematical tools than others in his field, and that helped
him to solve problems that stumped others.
A big AI model has access to all known tools. AIs have more breadth than any human.
Some of these breakthrough proofs seem to come from applying techniques
from areas different from what people in the field tend to use. I'm not qualified to comment on math at that level, but the comments by mathematicians seem to hint in that direction.
That may indicate why throwing vast amounts of compute at these problems works - an AI can try everything.
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.
Yup. This is what Terence Tao has been saying about digestion of results for several months now. Mathematics has become a field where results themselves matter less, the social aspect of attracting others to understand your results and explore new problems has become more important.
In a sense it is a little
bit like software engineering, where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
If you think the above comment reflects a change of heart from Terrence Tao, you never actually listened to him, only the strawmans of his opinion online.
Sure, but given the approximate tens of millions of dollars in API costs of solving these major math problems currently, we will also have to acknowledge that getting credit for major mathematical contributions is likely now to be far less about meritocracy and far more about access to capital to kickstart the original proof.
Less capable (but good enough) mathematicians with more capital will have a huge advantage over great mathematicians who are broke.
I'm not a mathematician, but that situation rubs everyone (except those with the most capital) the wrong way.
I never had a chance against Terence Tao. Neither did you. There are a few dozen people on the planet who had a giant advantage in math and could do things no one else could do.
It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
"On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
I will note that this includes the Quasi Riemann Hypothesis. The model that solved NS was significantly dumber and less efficient that OpenAIs more recent math-models and you shouldn't over index on "it's expensive now". AI has advanced so quickly that your <2 month old opinion is out of date by 5+ orders of magnitude!
This is true in experimental sciences, where people who have grants and funding can obtain results using equipment unavailable to others. But their raw data can help much less funded theory groups to formulate new understanding that can lead to new experiments.
As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
> As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
My assumption (based on perceived human nature) is that the raw proofs will not be made available.
If all of the important credit goes to the person who attached understanding to them, I'd assume those with the money to take the first step will keep the AI proofs secret until they believe they have adequately done the follow-up that will give them the lion's share of any credit.
How you get somewhere is important and tools change how you get there.
You can drive 1000 miles and arrive at the destination on the highway quickly.
Or you can take the slow road and walk. You discover unrelated things, make new relationships, see the scenery, listen and smell what is around you and have time to contemplate things which may or not be related.
This particular tool economy robs us of all these things in the rush to the destination.
I'd sacrifice "progress" a little for some of the better bits of humanity.
The thing is that if you don't learn to drive some destinations are unachievable. And destination is more important than the journey unless you travel for pleasure or personal development.
To a first approximation I agree that it’s similar to industrialization. However there’s a key factor which distinguishes this case, and which makes the analogy less than perfect: there isn’t a profit motive in mathematics, so there was no push for this development from inside the field. Instead, the industrialization was entirely exogenous, from companies which do in fact have that profit motive and view math as a means to achieve it.
There might be a “proof motive” in math which one could argue will drive the full automation of theorem proving. But we must recognize that profit is more universally valuable (i.e. it has value outside the industry in which it was created), whereas proofs are only valuable inside the math community (at least the pure math proofs the labs have been publishing thus far).
Two possible walls for advancing math with AI: first, it could be that the kind of problems that AI solves are the ones that don't open new avenues to concepts and ideas, and second, problems complexity is usually not linear, and the new and interesting problems to solve could be at a level of complexity beyond what AI can currently solve. Perhaps mathematicians could work to find ways to help AI cope with those walls.
Yes, this. Unless there is a true singularity where no more questions are ever asked, there will be definition be infinitely many problems to solve, and most of them we have not even glimpsed yet.
Singularity? Probably not.
Tech bro dictatorship? Unfortunately a bit more likely.
None of the above? Then we have a small shot at a great future.
Having a machine that solves hard problems must be a good thing.
Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?
> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.
These all seem totally achievable by LLM agents, or at least, I don’t see an obvious reason why it wouldn’t be. 3blue1brown’s recent discussion on this subject was a bit better, but he spent the plurality of main body of the video assuming this scenario, devoting less time to the other possibilities. To me, it seems dangerous to build a math career on the foundation that AI won’t be able to explain math well in a few years, let alone a few months (which seems entirely plausible given the recent math progress).
Because if you assume there is going to be an AI-god that can do everything better than us, there's nothing to really do. Like really, what do you want these people to write? "Well, math for humans is over as well as all intellectual activity, better just jerk off watching AI-generated hentai all day until I die"
I theorize that number of problems AI is possible of solving is limited. My theory rests on an assumption that all human knowledge is a result of lived experience. AI as a statistical machine doesn't know lived experience, therefore it can't solve problems that a living person interacting with reality could solve. Therefore human brain > AI, or so long AI is unconscious anyway.
Sure it can solve math problems by just following the implication arrows. But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory, particularly the Riemann hypothesis or its variants. (One result, "no Landau-Siegal zeroes", even had a 10-page elementary proof that everyone missed for a century.) Complexity theory had seemingly more open questions than results. Progress could meaningfully accelerate now that we have a quick test of which of our ideas are good, and which are bad.
The wildcard is that we have no idea what the model after that will be capable of, or the model after that.
It’s amazing for me to see math research going from craft to technology.
CS had this moment decades ago, sans some theoretical parts that are joining the bandwagon now.
I did a bit of math research in grad school collaborating with my math postdoc roommate. I found the aversion to any computation making its way into the craft to be odd decades ago.
Well, now the proof’s in the AI generated pudding.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
Of course if you solve any existing problem, new ones appear. That's not the point. The point is that AI will be able to advance on any problem better than any human.
But in a human way? I don't think so. The 700+ solutions that were put out by openAI are in no way accepted as solved until humans say so. It's kind of like saying SC2 was solved when Alpha beat the best players. The professional community just went on playing much the same. Math should do the same. Just with better tools
Mathematics as a field will not cease to exist, and not everyone will be interested in (or qualified to) managing the fields progress and its applications.
Isn't it also possible that the written proofs themselves are incorrect, say, for example, by mis-specification of constraints? Let's say I publish what I claim is a proof that 2+2=4, but in the Lean specification, I proved 1+1=2.
The main issue would rather that the formalization itself has a bug, eg. those proofs that you're trying to formalize are not "1 + 1 = 2" but theorems which take already multiple pages just to enunciate. And in these enunciations (that are made by humans) there can be errors.
In general, once the proof has been formalized, it's supposed to be the proof of the correct theorem (that's what the Lean proof-checker checks)
Plenty of these AI proofs have been accepted. On the most important result, on the Quasi-Riemann Hypothesis, there is no serious belief it's wrong. Mathematicians have already dissected and reformulated the claims.
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.
the impression i get is that people don’t per se assert that exploitable bugs exist in Lean, but that the volume of (code) proofs published represent a new magnitude of challenge to the integrity of Lean.
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
A single bug can usually allow you to prove anything you want. Saying that this is how the proofs were achieved is a totally different matter though, and probably incorrect.
Hmm, am no expert at Lean, but on one hand, since it's a programming language, guessing it necessarily means like all other programming languages, it has bugs.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
It can possibly have bugs. That said, the "core" of Lean is a somewhat "simple" core that does simple things to check the proof, so it's quite rare to find meaningful bugs
don't disbelieve you, but just looked at the reference language, it seems pretty full featured as a programming language, even with basic classes, polymorphism, scope, namespaces... But maybe most of the mathematicians who actual code with stick with the simpler language features? And also, didn't look to deeply, these could be very simple features that actually aren't very sophisticated.
I keep seeing people point to AI solved games as evidence that AI advancements are less impactful than we might think. Usually it's chess ("today there are more players than ever!"). Most of the time these are bad analogies. Games are played for entertainment, so productivity is meaningless.
I never advanced in math past high school, but my assumption would be that we as a society aren't paying mathematicians to have fun[0]. The discoveries mathematicians make must translate into real improvements in engineering and other life improving fields. So why would we pay mathematicians to work on math much the same way, if a new machine-led way makes these discoveries faster and cheaper?
[0] Being a little facetious. I know I don't know what I don't know.
Many academic disciplines bring no everyday life improvements. They exist because they provide intrinsic value and long-term foundations, hence they aren't merely entertainment. Mathematics/physics happen to be fields that span the entirety of applied/pure spectrum. Those said, your question can be generalized: Should society continue funding human intellectual activities when AI can perform them substantially better?
Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
>Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.
>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.
That depends. Many of these things can only be applied safely once they're understood.
For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.
Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.
Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.
LLM agents are perfectly capable of "advancing" my codebase down bizarre, esoteric rabbitholes. Meanwhile, the value remains undelivered.
Many crystals, likewise, are advancing their fantastic complexities in the deep, dark recesses beneath the crust of the earth, but does it make a sound? Who gives a shit.
Many paperclips, indeed, can be churned out. Even we can be converted into paperclips, remember.
I don’t think we know whether or not that’s true yet. Imagine a problem which requires a paradigm shift or new novel techniques to solve. There is no proof (yet) that an AI could create one.
We chose the problems to value. Literally every step of a proof is equally proved as the starting point. The choice of the starting point is what makes the problem, and it's a normative choice, which is to say it is a social choice.
It's not magical abilities - humans have proved for millenia that they can find novel solutions to problems - we have no idea how that comes to be though (which could be called magic I guess)
At issue with AI is that it's searching an existing corpus for fragments of solutions generated by humans - that's not necessarily novel, and we've not yet worked out how to make the machines actually invent.
But the question is whether the AI will itself be able to decide on which problem to advance, and do it in an ethical and aligned way with humanity. And that if it does, its significance will be understood by humans.
Otherwise we're just creating a black box, easily manipulated by their owners, who would be ruling over human lives. That's not what people want
The thing about much human progress on various problems is that human mathematicians formulate a human comprehensible machinery for the purposes of solving a given problem and then that machinery naturally suggests (to humans) further questions which then give rise to more machinery - And, in a sense, the value is ultimately the machinery, which has tended in the past to be relevant to physics, computer science, logistics, sociology and philosophy.
When AI writes proof no human can understand, it doesn't advance mathematics this way. Maybe AIs can understand their own proofs - meaning not just verify them but extract a machinery/broader-perspective (but I wouldn't count on it). But even machinery an AI isn't necessarily useful for humanity/society.
> The point is that AI will be able to advance on any problem better than any human.
If that was true it would of course be significant.
If you had no morals, 2 trillion dollars to spend, ability to commit intellectual property theft without consequences, and most of the smartest people in the world actively working to advance your cause... ...you really think you couldn't compete with the clankers?
Terry's stance has shifted gradually. This reminds me of Garry Kasporov's response and stance a few decades ago agains deep blue (and deeper blue). I think in a closed system/game (or "field"), this kind of AI superiority is inevitable. However, I do agree with Terry that we need the right form of transition if we want to continue holding the rein or at least playing a part in the subject.
> someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution
among all the discourse, it seems mathematicians are vaguely converging on this higher layer of abstraction being where the human value in math is now. It seems across domains (such as in software too), people speak of “judgement”, “taste” and other fuzzy concepts around defining problems rather than solving them. The lowest level of abstraction where humans win is asking the questions, rather than merely answering them.
As math is so verifiable, the “transition” seems most poignant and quick there. Watching these big name mathematicians discuss this unfolding revolution in real time is fascinating.
What is the current thought on AI creating new emergent forms of mathematics? Even the most stunning results like the proof of the quasi-RH are a clever combination of known mathematics. Humans have created (discovered?) stunning new ideas (e.g. complex numbers) that have opened wide new horizons -- do we think AI will be capable of creting / discovering novel new things like this?
Are humans going to be able to recognize the things AI discovers?
It took us a while to make the planets line up neatly in a heliocentric system. Just last month I read that a whale mother "might" grief while carrying her stillborn child.
The limiting factor is human imagination and comprehension. No offense, but see your comment.
I think AI has already invented new math, but the guys haven't looked for it because they didn't think it possible. And even under their noses, they wouldn't understand what they see.
> It's not what you look at that matters, it's what you see. —Thoreau
LEAN and other tools verify from what do know -- at least stepwise. Anything that deviates from these things should stick out. I suppose some million line LEAN program may have embedded within it something that is novel and not readily apparent, but I imagine truly novel approaches should stick out like a sore thumb.
Could AI create a novel set of axioms (a new math) from which things like RH become a by product? I think that would be hard to miss.
Dozens of agents probably already invented it, but they were given ephemeral containers where noone ever looked or they only wrote it in direct CPU instructions.
That analogy makes no sense to me. Wouldn't OpenAI have the incentive to release results that are incomprehensible, to showcase how beyond humans their models are?
I mean he's basically admitting that math academia is outdated.
If we don't interview candidates based on results, what is it? Politics. It already was. I had a JEMS paper in grad school and got zero postdoc offers. Many other cases.
The real shift is that math will become a hobbyists game. Make money other ways and then spend your money on humans and tokens to solve your problems.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
This analogy is pants on head stupid. They're talking as though the only thing AI will ever solve is the Langlands program, not as though we have discovered a generic tool.
It's like saying "they drove the car one mile -- and every mile driven leads to more roads that we can trot along, why are you so worried about the future of horses?"
I don't think it was a particularly well-reasoned essay (I thought the classification of finite simple groups is almost evidence for the opposite of what he says), but at the same time I don't think you understood his point.
If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.
The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".
People are allowed to process their anxieties. And processing takes time.
That said publicly doing it on the great modern internet attention/noise factory complicates the story.
Everyone is surprised that a person that some consider to be one of the greatest modern mathematicians can have such obtuse views of technology and its own field.
Sure, or you can think for yourself for five minutes on the implications of a general proof machine, and whether it's likely to continue producing proofs.
If you've forgotten how to think for yourself, you can also ask an LLM.
The host of the blog, Tao, was an early booster. He didn't think through the implications, and seems to have expected it to remain unable to do interesting work. Today he's on a "war footing".
I don't think mathematics are uniquely skilled at predicting the future, and the impacts of AI on their profession.
It is weird. From what I can tell, you and I have quite similar expectations on what AI will be able to do in the future, but for some reason you are much more dismissive of Taos opinion.
I perfectly understand his point and it is not about being selfishly frustrated by a machine being better than him. Is that really your interpretation of the discourse?
Logic is logic, you don't need a specialist's opinion on every aspect of a specialist field. I've had eye opening insights delivered to me on problems I'm working on by people who barely know how to use a computer.
> Under a partnership, the Simons Foundation, Anthropic and OpenAI will commit $15 million in combined funding and resources. The foundation will manage and award funding, including through an open call for proposals, to support workshops and initiatives that help researchers evaluate recent AI-assisted results, learn and use new tools, share perspectives and shape collective standards for AI use.
There's a long collection of comments by mathematicians at [1]. A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later. The main social complaint is that the social foundations of mathematics just changed, in a big way. The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
One comment was "A common piece of wisdom is that mathematicians only have a few tricks in their bag – which they skillfully apply again and again." Feynman made that point in his autobiography. For various reasons, he had a different set of mathematical tools than others in his field, and that helped him to solve problems that stumped others.
A big AI model has access to all known tools. AIs have more breadth than any human. Some of these breakthrough proofs seem to come from applying techniques from areas different from what people in the field tend to use. I'm not qualified to comment on math at that level, but the comments by mathematicians seem to hint in that direction. That may indicate why throwing vast amounts of compute at these problems works - an AI can try everything.
[1] https://proofsandprompts.com/2026/10/08/100-reactions-to-100...
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.
Yup. This is what Terence Tao has been saying about digestion of results for several months now. Mathematics has become a field where results themselves matter less, the social aspect of attracting others to understand your results and explore new problems has become more important.
In a sense it is a little bit like software engineering, where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
I'm so glad that sanity ulitmately won in his mind.
If you think the above comment reflects a change of heart from Terrence Tao, you never actually listened to him, only the strawmans of his opinion online.
He's not out of the woods yet.
Sure, but given the approximate tens of millions of dollars in API costs of solving these major math problems currently, we will also have to acknowledge that getting credit for major mathematical contributions is likely now to be far less about meritocracy and far more about access to capital to kickstart the original proof.
Less capable (but good enough) mathematicians with more capital will have a huge advantage over great mathematicians who are broke.
I'm not a mathematician, but that situation rubs everyone (except those with the most capital) the wrong way.
I never had a chance against Terence Tao. Neither did you. There are a few dozen people on the planet who had a giant advantage in math and could do things no one else could do.
It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
Tens of millions? You're thinking specifically of the proof of Navier-Stokes. According to https://github.com/openai/math/blob/main/README.md:
"On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
I will note that this includes the Quasi Riemann Hypothesis. The model that solved NS was significantly dumber and less efficient that OpenAIs more recent math-models and you shouldn't over index on "it's expensive now". AI has advanced so quickly that your <2 month old opinion is out of date by 5+ orders of magnitude!
This is true in experimental sciences, where people who have grants and funding can obtain results using equipment unavailable to others. But their raw data can help much less funded theory groups to formulate new understanding that can lead to new experiments.
As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
> As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
My assumption (based on perceived human nature) is that the raw proofs will not be made available.
If all of the important credit goes to the person who attached understanding to them, I'd assume those with the money to take the first step will keep the AI proofs secret until they believe they have adequately done the follow-up that will give them the lion's share of any credit.
For now. It once took Deep Blue hardware to beat Kasparov. Now most laptops can do it.
This has unfortunately been the case in experimental academia forever.
Isn't it the passage of mathematics from the age of craftsmanship to the age of industrial making ?
My recent take on AI is that people haven't adapted to the new huge potential of the tool, and don't dream big enough (yet).
It's like seeing a car engine for the first time, lamenting about the future of horses, and not realizing it'll also help us fly.
How you get somewhere is important and tools change how you get there.
You can drive 1000 miles and arrive at the destination on the highway quickly.
Or you can take the slow road and walk. You discover unrelated things, make new relationships, see the scenery, listen and smell what is around you and have time to contemplate things which may or not be related.
This particular tool economy robs us of all these things in the rush to the destination.
I'd sacrifice "progress" a little for some of the better bits of humanity.
The thing is that if you don't learn to drive some destinations are unachievable. And destination is more important than the journey unless you travel for pleasure or personal development.
There are many more places you can't drive. I've walked to quite a few of them.
That analogy holds too.
To keep with the analogy, there are far more places you can feasibly drive to than walk to.
(And even more if you can combine them.)
That is plainly rubbish.
This is really well put, I totally agree with you.
There’s an immediate resistance towards change rather than embracing it and finding opportunities to improve and work with it.
To a first approximation I agree that it’s similar to industrialization. However there’s a key factor which distinguishes this case, and which makes the analogy less than perfect: there isn’t a profit motive in mathematics, so there was no push for this development from inside the field. Instead, the industrialization was entirely exogenous, from companies which do in fact have that profit motive and view math as a means to achieve it.
There might be a “proof motive” in math which one could argue will drive the full automation of theorem proving. But we must recognize that profit is more universally valuable (i.e. it has value outside the industry in which it was created), whereas proofs are only valuable inside the math community (at least the pure math proofs the labs have been publishing thus far).
Two possible walls for advancing math with AI: first, it could be that the kind of problems that AI solves are the ones that don't open new avenues to concepts and ideas, and second, problems complexity is usually not linear, and the new and interesting problems to solve could be at a level of complexity beyond what AI can currently solve. Perhaps mathematicians could work to find ways to help AI cope with those walls.
Yes, this. Unless there is a true singularity where no more questions are ever asked, there will be definition be infinitely many problems to solve, and most of them we have not even glimpsed yet.
Singularity? Probably not.
Tech bro dictatorship? Unfortunately a bit more likely.
None of the above? Then we have a small shot at a great future.
Having a machine that solves hard problems must be a good thing.
Solving the hardest problems doesn't always result in the best outcomes unless the answers are shared equitably and implemented compassionately.
Absolutely. This is an aspect of the tech bro dictatorship.
Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?
> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.
These all seem totally achievable by LLM agents, or at least, I don’t see an obvious reason why it wouldn’t be. 3blue1brown’s recent discussion on this subject was a bit better, but he spent the plurality of main body of the video assuming this scenario, devoting less time to the other possibilities. To me, it seems dangerous to build a math career on the foundation that AI won’t be able to explain math well in a few years, let alone a few months (which seems entirely plausible given the recent math progress).
Because if you assume there is going to be an AI-god that can do everything better than us, there's nothing to really do. Like really, what do you want these people to write? "Well, math for humans is over as well as all intellectual activity, better just jerk off watching AI-generated hentai all day until I die"
It’s strange to me seeing multiple talk about how Terry’s viewpoint has evolved or not as a response to this post. He didn’t write this.
I theorize that number of problems AI is possible of solving is limited. My theory rests on an assumption that all human knowledge is a result of lived experience. AI as a statistical machine doesn't know lived experience, therefore it can't solve problems that a living person interacting with reality could solve. Therefore human brain > AI, or so long AI is unconscious anyway.
Sure it can solve math problems by just following the implication arrows. But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
They solved a millennium prize problem… they disproved the exponential time hypothesis. Are those low hanging fruits?
Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory, particularly the Riemann hypothesis or its variants. (One result, "no Landau-Siegal zeroes", even had a 10-page elementary proof that everyone missed for a century.) Complexity theory had seemingly more open questions than results. Progress could meaningfully accelerate now that we have a quick test of which of our ideas are good, and which are bad.
The wildcard is that we have no idea what the model after that will be capable of, or the model after that.
It’s amazing for me to see math research going from craft to technology.
CS had this moment decades ago, sans some theoretical parts that are joining the bandwagon now.
I did a bit of math research in grad school collaborating with my math postdoc roommate. I found the aversion to any computation making its way into the craft to be odd decades ago.
Well, now the proof’s in the AI generated pudding.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
Of course if you solve any existing problem, new ones appear. That's not the point. The point is that AI will be able to advance on any problem better than any human.
But in a human way? I don't think so. The 700+ solutions that were put out by openAI are in no way accepted as solved until humans say so. It's kind of like saying SC2 was solved when Alpha beat the best players. The professional community just went on playing much the same. Math should do the same. Just with better tools
Math is too boring as a spectator sport. It's funded because of its practical use.
There's going to be some hobbyists doing math in their basements, but there's unlikely to be a future for professional mathematicians.
> It's funded because of its practical use.
That's exactly what the Simons foundation was set up to reject, and it's one of most important funding streams for math
Haha, the breaks from reality there are two, one professionals do not spend most of their time inventing proofs, and two, practical use? :-)
Mathematics as a field will not cease to exist, and not everyone will be interested in (or qualified to) managing the fields progress and its applications.
Mathematics as field won't. It is the future of mathematicians that is uncertain.
The proofs may not yet be understood, but to claim they should not yet be accepted as solved is to claim there is a bug in Lean, which is a big claim.
Isn't it also possible that the written proofs themselves are incorrect, say, for example, by mis-specification of constraints? Let's say I publish what I claim is a proof that 2+2=4, but in the Lean specification, I proved 1+1=2.
To be pedantic, the proofs would be correct, they would just be proving the wrong thing.
To address what you actually meant: it's possible but specifications are way easier to verify.
The main issue would rather that the formalization itself has a bug, eg. those proofs that you're trying to formalize are not "1 + 1 = 2" but theorems which take already multiple pages just to enunciate. And in these enunciations (that are made by humans) there can be errors.
In general, once the proof has been formalized, it's supposed to be the proof of the correct theorem (that's what the Lean proof-checker checks)
Plenty of these AI proofs have been accepted. On the most important result, on the Quasi-Riemann Hypothesis, there is no serious belief it's wrong. Mathematicians have already dissected and reformulated the claims.
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.
the impression i get is that people don’t per se assert that exploitable bugs exist in Lean, but that the volume of (code) proofs published represent a new magnitude of challenge to the integrity of Lean.
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
No. Lean being bug-free is a far bigger claim.
A bug which has allowed 700 proofs from diverse fields of math to appear as proven, and noone has figured it out yet?
A single bug can usually allow you to prove anything you want. Saying that this is how the proofs were achieved is a totally different matter though, and probably incorrect.
Hmm, am no expert at Lean, but on one hand, since it's a programming language, guessing it necessarily means like all other programming languages, it has bugs.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
It can possibly have bugs. That said, the "core" of Lean is a somewhat "simple" core that does simple things to check the proof, so it's quite rare to find meaningful bugs
don't disbelieve you, but just looked at the reference language, it seems pretty full featured as a programming language, even with basic classes, polymorphism, scope, namespaces... But maybe most of the mathematicians who actual code with stick with the simpler language features? And also, didn't look to deeply, these could be very simple features that actually aren't very sophisticated.
There's been many bugs found in Lean already, and there are probably still a few. So that's not the main issue.
I keep seeing people point to AI solved games as evidence that AI advancements are less impactful than we might think. Usually it's chess ("today there are more players than ever!"). Most of the time these are bad analogies. Games are played for entertainment, so productivity is meaningless.
I never advanced in math past high school, but my assumption would be that we as a society aren't paying mathematicians to have fun[0]. The discoveries mathematicians make must translate into real improvements in engineering and other life improving fields. So why would we pay mathematicians to work on math much the same way, if a new machine-led way makes these discoveries faster and cheaper?
[0] Being a little facetious. I know I don't know what I don't know.
Many academic disciplines bring no everyday life improvements. They exist because they provide intrinsic value and long-term foundations, hence they aren't merely entertainment. Mathematics/physics happen to be fields that span the entirety of applied/pure spectrum. Those said, your question can be generalized: Should society continue funding human intellectual activities when AI can perform them substantially better?
Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
>Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.
>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.
That depends. Many of these things can only be applied safely once they're understood.
For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.
Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.
Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.
> The point is that AI will be able to advance on any problem better than any human.
Define "better".
Define "advance"!
LLM agents are perfectly capable of "advancing" my codebase down bizarre, esoteric rabbitholes. Meanwhile, the value remains undelivered.
Many crystals, likewise, are advancing their fantastic complexities in the deep, dark recesses beneath the crust of the earth, but does it make a sound? Who gives a shit.
Many paperclips, indeed, can be churned out. Even we can be converted into paperclips, remember.
All but a negligible amount of novel mathematical results going forward will have had substantial LLM involvement
Did any ai made any new interesting question or conjecture?
No.
I don’t think we know whether or not that’s true yet. Imagine a problem which requires a paradigm shift or new novel techniques to solve. There is no proof (yet) that an AI could create one.
Why do you believe human brains are magic? We may need some breakthroughs, but trillions of investment tends to produce those.
And it's becoming less clear over time that a breakthrough will be needed.
> Why do you believe human brains are magic?
We chose the problems to value. Literally every step of a proof is equally proved as the starting point. The choice of the starting point is what makes the problem, and it's a normative choice, which is to say it is a social choice.
Why do you believe that LLMs are good at the same things humans are?
I believe that we're investing trillions into making it so.
It's not magical abilities - humans have proved for millenia that they can find novel solutions to problems - we have no idea how that comes to be though (which could be called magic I guess)
At issue with AI is that it's searching an existing corpus for fragments of solutions generated by humans - that's not necessarily novel, and we've not yet worked out how to make the machines actually invent.
We have to come to terms with the reality, and we must find a way of existing with this new super tool
But how do you know which problems are worth making progress on? Traditionally, you'd find those by working on solving previous problems.
How do you build a society where everyone wants to live in it and has a means to?
How do you reach the conclusion that society is something that admits rational design?
What?
But the question is whether the AI will itself be able to decide on which problem to advance, and do it in an ethical and aligned way with humanity. And that if it does, its significance will be understood by humans.
Otherwise we're just creating a black box, easily manipulated by their owners, who would be ruling over human lives. That's not what people want
The thing about much human progress on various problems is that human mathematicians formulate a human comprehensible machinery for the purposes of solving a given problem and then that machinery naturally suggests (to humans) further questions which then give rise to more machinery - And, in a sense, the value is ultimately the machinery, which has tended in the past to be relevant to physics, computer science, logistics, sociology and philosophy.
When AI writes proof no human can understand, it doesn't advance mathematics this way. Maybe AIs can understand their own proofs - meaning not just verify them but extract a machinery/broader-perspective (but I wouldn't count on it). But even machinery an AI isn't necessarily useful for humanity/society.
ANY problem?
> The point is that AI will be able to advance on any problem better than any human.
If that was true it would of course be significant.
If you had no morals, 2 trillion dollars to spend, ability to commit intellectual property theft without consequences, and most of the smartest people in the world actively working to advance your cause... ...you really think you couldn't compete with the clankers?
Terry's stance has shifted gradually. This reminds me of Garry Kasporov's response and stance a few decades ago agains deep blue (and deeper blue). I think in a closed system/game (or "field"), this kind of AI superiority is inevitable. However, I do agree with Terry that we need the right form of transition if we want to continue holding the rein or at least playing a part in the subject.
> someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution
among all the discourse, it seems mathematicians are vaguely converging on this higher layer of abstraction being where the human value in math is now. It seems across domains (such as in software too), people speak of “judgement”, “taste” and other fuzzy concepts around defining problems rather than solving them. The lowest level of abstraction where humans win is asking the questions, rather than merely answering them.
As math is so verifiable, the “transition” seems most poignant and quick there. Watching these big name mathematicians discuss this unfolding revolution in real time is fascinating.
> Much of the reaction seems to reflect a feeling that the subject that they love is under attack, and is being “hollowed out”.
Sums up the reaction from a lot of developers too.
What is the current thought on AI creating new emergent forms of mathematics? Even the most stunning results like the proof of the quasi-RH are a clever combination of known mathematics. Humans have created (discovered?) stunning new ideas (e.g. complex numbers) that have opened wide new horizons -- do we think AI will be capable of creting / discovering novel new things like this?
Are humans going to be able to recognize the things AI discovers?
It took us a while to make the planets line up neatly in a heliocentric system. Just last month I read that a whale mother "might" grief while carrying her stillborn child.
The limiting factor is human imagination and comprehension. No offense, but see your comment.
I think AI has already invented new math, but the guys haven't looked for it because they didn't think it possible. And even under their noses, they wouldn't understand what they see.
> It's not what you look at that matters, it's what you see. —Thoreau
LEAN and other tools verify from what do know -- at least stepwise. Anything that deviates from these things should stick out. I suppose some million line LEAN program may have embedded within it something that is novel and not readily apparent, but I imagine truly novel approaches should stick out like a sore thumb.
Could AI create a novel set of axioms (a new math) from which things like RH become a by product? I think that would be hard to miss.
Dozens of agents probably already invented it, but they were given ephemeral containers where noone ever looked or they only wrote it in direct CPU instructions.
> Are humans going to be able to recognize the things AI discovers?
They have been so far.
That's like saying that all serial killers make a mistake that gets them caught, which is true only of those that were caught (and studied.)
That analogy makes no sense to me. Wouldn't OpenAI have the incentive to release results that are incomprehensible, to showcase how beyond humans their models are?
I mean he's basically admitting that math academia is outdated.
If we don't interview candidates based on results, what is it? Politics. It already was. I had a JEMS paper in grad school and got zero postdoc offers. Many other cases.
The real shift is that math will become a hobbyists game. Make money other ways and then spend your money on humans and tokens to solve your problems.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
This analogy is pants on head stupid. They're talking as though the only thing AI will ever solve is the Langlands program, not as though we have discovered a generic tool.
It's like saying "they drove the car one mile -- and every mile driven leads to more roads that we can trot along, why are you so worried about the future of horses?"
I don't think it was a particularly well-reasoned essay (I thought the classification of finite simple groups is almost evidence for the opposite of what he says), but at the same time I don't think you understood his point.
If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.
The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".
People are allowed to process their anxieties. And processing takes time. That said publicly doing it on the great modern internet attention/noise factory complicates the story.
Everyone is surprised that a person that some consider to be one of the greatest modern mathematicians can have such obtuse views of technology and its own field.
Note, this is a guest post.
I think I’ll take the professional mathematician perception of how AI affects his field over random HN commenter saying it’s “pants on head stupid”
Sure, or you can think for yourself for five minutes on the implications of a general proof machine, and whether it's likely to continue producing proofs.
If you've forgotten how to think for yourself, you can also ask an LLM.
The host of the blog, Tao, was an early booster. He didn't think through the implications, and seems to have expected it to remain unable to do interesting work. Today he's on a "war footing".
I don't think mathematics are uniquely skilled at predicting the future, and the impacts of AI on their profession.
Neither are mindless AI boosters
https://fortune.com/article/ai-godfather-radiologists-obsole...
Ironic that you resort to “think for yourself” as your silly name calling go to btw
It is weird. From what I can tell, you and I have quite similar expectations on what AI will be able to do in the future, but for some reason you are much more dismissive of Taos opinion.
I perfectly understand his point and it is not about being selfishly frustrated by a machine being better than him. Is that really your interpretation of the discourse?
Logic is logic, you don't need a specialist's opinion on every aspect of a specialist field. I've had eye opening insights delivered to me on problems I'm working on by people who barely know how to use a computer.
You are welcome to opine on anything but there’s a reason we’re reading his blog and you’re down here in the comments
On the flip side, the professional mathematician has all the incentive to pearl clutch and rent seek.
Wake me up when AI solves Collatz.
[dead]
[flagged]
> Under a partnership, the Simons Foundation, Anthropic and OpenAI will commit $15 million in combined funding and resources. The foundation will manage and award funding, including through an open call for proposals, to support workshops and initiatives that help researchers evaluate recent AI-assisted results, learn and use new tools, share perspectives and shape collective standards for AI use.
"Normal technology"?
[flagged]
Can we stop with this ridiculous narrative? That didn't happen.
Can you elaborate or provide a good source to explain why this is not the case?
Have the pro-AI people finally found a new mathematician? Otherwise it is always the same 10 people on Tao's blog citing each other.
This guest author runs a seminar to lure people into AI:
https://math.uchicago.edu/~fcale/AI.html
Lure people? It isn't a cult.