> 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 feel like the hyper-competitiveness of academia generally is the elephant in the room.
The disaster for a problem-solution-oriented mathematician isn't that once problem-solution is easy they'll just switch to theory-explanation and machinery-building. It is that they will have to compete (and won't be able to compete) with students and post-docs who've been doing theory-explanation and machinery-building for their entire career - and there are lots of these 'cause academic pipeline produces more people who can fill a job than jobs in every imaginable category and subcategory.
Academic competitiveness doesn't explain all the hysteria and why those at the top of the field (Field Medal and all) wrote that breakup letter over AI. There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence. Literally a math academic crying while being interviewed on AI impact, talking about not knowing what to do with himself.
Who knew 2-3 SD won't save you from being a pattern matcher that can't stand to see a better one.
it's not just math and CS. it's a human trait. we are seeing it in the arts and creative fields where technology and AI in particular are making their impact felt. there will be much, much more of this sort of reaction in the years ahead. we'll all have to learn how to adapt. and in the process rediscover that what makes us valuable as people is not our skills but our souls.
Do male chicken have souls? Because female chicken have the skill to lay eggs, and that's what makes the latter valuable and the former bonemeal upon birth.
Seeing work as your purpose was supposed to be the holy grail of work that most people sadly couldn't achieve. Now we just immediately toss that away like a mouldy tomato and what's left? The new highest plane of human existence is working for capitalists to make them more profitable, in exchange for 10% of their profit?
Its equally devastating when you're a renowned expert developer at a specific stack that falls out of favor and nobody cares about your skills any longer. This happens relatively often in tech.
> There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence.
I don't see this as a toxic trait at all. Some people have the fortune and luxury to do what they love - I don't think that's a bad thing.
I think this quote from Gauss puts it very well: "It is not knowledge, but the act of learning, not possession, but the act of getting there, which grants the greatest enjoyment." So tons of mathematicians feel like AI is stealing this "act of learning which grants the greatest enjoyment", and I think it's fine that they are lamenting this.
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
I don't really understand this kind of triumphalism. Would you, personally, identify as a pattern matcher who's happy to see better ones? Or do you think you have some deeper skill more robust against disruption than the math and CS folks who don't know what to do with themselves?
Note that this may include things that you don't think of as being a "skill". I've seen some people brag about being a particularly hard worker, or about a deep instinct towards Getting Stuff Done, with the clear implication that these things can't be automated the way mere mathematical analysis can.
I don't read it as triumphalism. I read it as an accurate (to my experience, as a math PhD) reflection on the existentially toxic culture of math and other intellectual disciplines, one that AI is now forcing us to reckon with.
n.b. I think math culture is probably one of the least toxic cultures that have ever existed. Its equanimity and principles are a big part of what drew me to math in the first place. But humans are human.
I really don't see how to read "hysteria", "breakup letter", "literally a math academic crying" as anything other than an attempt to dunk on mathematicians.
I suppose, maybe they are dunking on mathematicians. I can't claim to speak for them. But I at least see truth in their observations. It would be inaccurate to downplay the sentimentality of the math community right now. I don't think that sentimentality has overly clouded the community's judgment (so "hysteria" is probably overdoing it) but there is a lot of heartache and unease.
The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be indistinguishable, especially because there will very likely be “math bros” just like “ai bros” and “crypto bros” before them who want to pretend they’re mathematicians just as we see people pretending to be engineers and financial experts based on how fast they got on the hype train.
I’m no mathematician, though I do have an undergrad degree in math, and unless I had a graduate degree or made substantial contributions at that level, I would never consider myself anything but a “student of mathematics,” and that too not just because I have a degree in the subject, but because I still have a great interest in certain branches of math and I keep learning in my spare time from books or from adjacent applied fields I continue to formally study. As a kid, I grew up hating math but I became interested in it later in life because of coding, computer science and formal logic, pushing myself a lot through my college years to get some foundational understanding of the subject at a deeper level than the general population, just to get my foot in the door, so to speak.
Even if I’m an outsider from the circle of mathematicians, this whole AI takeover of mathematics leaves such a bad taste in my mouth because I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general, and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject). It feels like a great loss for humanity to lose a brunt of our engagement with it to the machines, like a whole wing of a gigantic library starting to be closed off from us indefinitely because some people decided that we don’t _need_ to access it anymore, except on an even grander scale possibly than my imaginary library metaphor.
> I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general
Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
> and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject).
There's plenty of controversy in mathematics. Controversy comes from the people, not the field of study.
I know there is some controversy, I would say that the field of study also “causes” controversy in a sense if the disagreement has to do with the subject itself, that is true for any field. For example, set theory “caused” significant controversy in the early years of its development, yes it was people in disagreement and insulting one another, but the fundamental questions were from the subject matter itself.
Regarding widespread perception of math as useless, I guess I was thinking mainly of academic circles rather than the general public in that statement. Even while I hated math, I knew it wasn’t useless because my late father (who was a civil engineer) was very good at it, and he was adamant on teaching it to us. It didn’t work on me back then though lol.
As I was getting more interested in math while getting my degree in computer science, I realized much of my own issue with math earlier in life was because of the same thing just about everyone else that struggles with it has: the lack of any practical relevance in my own life because of how it’s generally taught. And the fact that many math teachers themselves don’t know the foundational/philosophical aspects of the subject which would go a long way I think towards explaining that relevance if introduced in some pieces earlier on rather than treating it like a rote memorization subject, and being so focused on continual practice to develop heuristics with exercises that some students couldn’t care less about. Of course, that works for some people clearly, but it didn’t for me at the time and I’m of the opinion high schools should focus more on discrete math like set theory, and formal logic rather than calculus, which is of little relevance at that stage unless you’re also invested in studying physics. Sets and logic on the other hand imo are general enough right at the beginning to be relevant to just about everything people of just about any age are concerned with.
fewer mathematicians means fewer problems to solve. All problems have human origin, right? Can AI create a problem on its own and what would be the point? Closer to can god create a stone he can't lift.
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.
"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!
Question: Have you actually used AI to do research-level math? At the current time, it is not "tackle any problem you wish and you can get headline results in ~3 hours." Their method is spraying-and-praying across the entire mathematical frontier, which is something only AI demonstrators are interested in. And as far as I know, they have never elaborated on what their "average of 3 hours" metric even means (e.g. including or excluding all unsuccessful leads?).
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.
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.
> I never had a chance against Terence Tao. Neither did you.
I'm not a mathematician at all, so yeah I certainly didn't have a chance.
> 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.
I don't see these as equivalent circumstances. For most people if they get beat by someone smarter or someone stronger, it may be disappointing but they can still respect and admire that they were beat by talent/determination/etc.
Getting beat due to access to money just feels slimy, because at some level we all realize money is not distributed on a meritocratic curve at all, at least not if you are measuring values any well-adjusted person would consider positive.
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.
For sure, he was arguably the first well known person to bring chatGPT to maths.
When his more "anti-ai side" (again, it's more nuanced but I'm calling it that) was showing, everyone was so quick to call him a Luddite. No one stopped to think why a well know ai in maths fan all of a sudden had a more negative opinion. Even reading his actual nuanced take was too much.
The best example was the following 2023 quote which has been repeatedly and wrongfully used as a criticism of Tao right up until he was proven exactly correct
>I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well. Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools? We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions. Our usual technology paradigms will not serve as an adequate guide for navigating these uncharted waters. Perhaps the greatest challenge will be transitioning to a new AI-assisted world as safely, wisely, and equitably as possible. https://unlocked.microsoft.com/ai-anthology/terence-tao/
The ever prescient Tao, over 3 years ago in 2023. You can search that quote and find lots of examples of "Lol he's so wrong" right up until very recently. Which is exactly the 2026 timeline Tao stated.
And his viewpoint really hasn't changed or needed to change. He was 100% correct.
To be fair, he’s had a bunch of guest posts on his blog that didn’t seem as reasonable to me (and some people seemed to not realize they were guest posts).
Edit: lol, this article is itself a guest post. My bad for not reading it first! But it does serve as an excellent demonstration of my point, since it seems like neither of us realized we weren’t talking about his words.
>> where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
This statement is a monstrosity, and the reason why real software engineers know vibe coding will fail. It is fundamental misunderstanding of thinking that that the intellectual work consists of knowing what you want, and coding is merely the mechanical labor of making it happen.
The coding is the intellectual work!
Its where the actual problems are discovered, where the difficult decisions are made, where vague ideas become precise, and where solutions are invented. There is not such thing as a fully formed solution waiting to be translated into code. :-)
Again just the modern variation of the idea guy who goes around saying I have a genius idea for a new startup...I just need some Developers to implement 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).
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.
They build cities around the car, they defund public transit, they close down rail lines, they build new housing a 5 km away from your closest grocery store, and 20 km away from any cafés and other cultural venues. Your job is located 50 km north of where you live, your partner’s job is 60 km in the other direction, and your child’s school is 10 km in the third direction.
They neglect to build sidewalks, and force you to walk 1-2 meters from cars racing at 70 km/h. With all the noise and high risk stressing you up to the level you just wanna scream.
stevenhuang: No one forces you to drive. You can still walk.
It's pretty well documented they deliberately made things impractically large distances apart and scrapped the idea of footpaths, to encourage car sales, paid for by car companies to your local city government several decades ago.
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 the other option is that AI does what you say. And if you think about it (and you are a mathematitian and not an engineer) it means that AI is able to read our minds and know our desires. And at that point who cares about existence anymore?
"Choosing which terrain is worth exploring" is more clear if you have an objective function (if you are doing engineering). But mathematics is closer to art in that regard. What is a terrain worth exploring in art?
You don't think OpenAI has any interest in having agents identify new interesting research terrain? They just want to solve existing important millennium problems and then they're done?
Yes? I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
Of course, general improvements may be sufficient to continue to drive math improvements even without specific attention - we’ll see.
Not a chance. They'll go for room temperature superconductors and other stuff that is currently in the quite possible but not yet proven and patent the hell out of it and charge rent to the rest of humanity.
Simulation to the point that the accuracy is large enough to extrapolate to real materials. You can do a lot of stuff when you have access to a million GPUs that would otherwise be much harder.
A lot of materials science is already done through simulation, same with a large chunk of chemistry and biology. That fraction will only increase over time.
I am aware of advances, but I think a large chunk is a stretch in various ways. We'll just see what happens - simulating even a glass of water fully is tough.
That depends on the granularity of your simulation. For most - not all - purposes a glass of water can be simulated coarsely or by simulating a far smaller amount with greater fidelity and then to extrapolate.
For useful materials science the numbers are still large but far lower than the numbers involved in 'a glass of water'.
I really depends on what is needed. If you know that something quite "coarse" is enough to create/uncover room temperature superconductors with high critical current - fair enough. Again, we'll see what happens.
With the kind of compute they have at their disposal 'quite coarse' may actually be 'impressively detailed' and that is why I think they have an edge on this kind of stuff. It's like being able to borrow a trillion and then to go play research lab, it would be highly surprising if they weren't able to monetize such a head start.
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"
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.
> 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.
I wish mathematicians would just focus on this. This really is the problem, at the end of the day.
Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.
We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.
But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
Not only are the proofs long and unintelligent, but they don't cite previous work.
In fact, they completely disregard all standards of mathematical dissemination. They just post a bunch of 100+ page papers online and call it the day.
A real mathematician polishes and polishes for months until it is understandable by the community. Then they communicate the results to a journal where there are editors and peer reviewers who come with detailed feedback which is then implemented. And just then it is ready to publish.
"Polish this paper. Check all proofs with Lean. Try to find shorter proofs of lemmas where possible. Write for a undergraduate math student. Add brief explanatory material as needed. Cite previous work where relevant. Check all citations. Convert to LaTeX."
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.
He did attach his name to it by posting it. I'd imagine if he disagreed strongly he wouldn't do that, or he would at least add a footnote.
What you could argue though is that he is muddying the waters with all these guest posts on a domain that is as personal as it gets. I appreciate he's trying to be the 'Carlos Santana of mathematics' by sharing the spotlight but it might be good to highlight the distance a bit more, for instance by adding his own views.
Well, you're wrong, given how many of the guest posts featured by Tao are inherently contradictory and so it would be strange for him to be personally endorsing each one! He also posts his own views elsewhere: see https://teorth.github.io/tao-web/ai-views.html.
The obvious goal is to highlight differing attitudes within the mathematics community so the mathematics community can have a dialogue and see what other people are thinking. Luckily, most mathematicians seem to be able to actually read and understand not all posts are from Tao. It is unfortunate it seems many people on this site cannot.
Posting something on a domain with your name attached to it is an endorsement of sorts. You can argue with that all you want but that's just how it works. Having a dialogue happens on neutral ground and Tao's blog is not neutral ground, even if that might be his intention.
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.
> 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
If rich people want to use their money to fund math with the goal of NOT funding math with any known useful or practical purpose, that sounds like a hobby to me.
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 mean Lean has two things: a full programming language, and proof checker. The programming language is used to create either code, or proofs. But the "created proofs" are using a very simple subset (a bit like an "assembly language"), and only need somewhat simple rules to be checked
So as long as the "somewhat simple" core that checks to proof is correct, it doesn't matter if the "programming language" has bugs: either in the end the "proof assembly code" generated by the program is right and is validated by the core, or it's not, and you need to revise your proof, or to check for either a bug in the programming language (it didn't generate the proof you were expecting), or in very rare case detect a bug in the simple "core"
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.
We invested at least a couple trillion of speculative value into making ugly cartoon apes the future of art, finance and property. I'm not sure it's always possible to overwhelm reality with sufficient money.
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?
The whole situation in mathematics reminds me of two years ago when models started coding well and all the developers were really sad. Now we use it as a tool and, as a whole, are able to make much more.
It has changed from elegance and human understanding to brute force. But that trend was started a while ago, before AI became really useful.
From TFA:
"I’m ready to board the plane, because I want to see what’s there. At the same time, I would add that people who previously have been walking around on earth with their eyes closed are not going to recognize anything however high they fly; I still believe that many of the skills that mathematicians have now will not only be transferable but vital to the new world order."
> gulf between ... novel proofs and ... understanding
Machine proofs are useful machine understanding. You don't read the proofs, that's holding it wrong. You point another agent at the proof and prompt "use these ideas to prove this lemma that's useful to me". And it works, agents are faster given access to the proofs than without.
> 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.
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.
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.
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.
These fall under the class of problems that have solutions which can be inferred more easily from applying massive computation, which is a subset of problems that human minds are trying to solve.
It's like being a CIA analyst, maybe. You have an overwhelming number of information sources of varying quality and non-obvious relationships. Applying massive computation will definitely allow you to draw new insights and identify trends that would have been difficult, or nearly impossible for unassisted human reasoning.
From what I understand, this is the sort of thing that the LLMs have been particularly effective at in mathematics, applying massive computation to the existing field and with a combination of impressive vector space search and runtime evaluation, deriving plausible solutions to certain narrowly defined problems.
Here's a harder one: you're in a bar, sitting having drinks with friends. There's an attractive girl sitting with some of her friends that has glanced at you a couple of times. What should you do?
Taking into account, of course, the entire human experience from marriage status, age, rejection likelihood, your friends opinions, her friends opinions, your family, your hormones, everyone's intoxication level, how well you slept, the noise level, how long her glance lingered, and the billions of other things that go into it - entire lifetimes of experience, genetic behavior predisposition, etc...
This is just a small example of a brief problem that a normal human has to solve hundreds or thousands of times per day that don't fit into the class of solutions that LLMs would be particularly adept at. It's easy to come up with a lot more.
So yes, I'd argue that those neatly formulated, clean problems with clearly provable solutions are low hanging fruit.
> But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.
Low hanging fruits for the agents that solved them. It's very possible that there are whole classes of problems that AI won't be able to solve or will require another few orders of magnitude of compute to solve. A human might even be required to first make a breakthrough on some field before AI can re-apply it exhaustively and find new discoveries on top of that.
It's amazing to still see posts like this in late 2026. I suggest you recalibrate your ideas on what both AI and human intelligence is and can do. My take: both are trained systems. Both are as "statistical" as the other. Any idea that "because statistics" means that humans are different or better is bogus. Both can do things based on their training data. You are totally correct that a human has access to training data that AI doesn't (the smell of new mown hay, for example, or the fear of being crushed by the garbage compactor room in Star Wars). But there's no "humans have more training data therefore better" argument to be made about fields like mathematics. In fact the opposite argument works quite well: AI can have all mathematics ever invented in its training data, something that's pretty hard to achieve as a human.
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.
I would really like to see an AI invent a new set of (consistent) axioms from which a whole slew of new problems and solutions arise. Something akin to Euclidean geometry, but for a whole new landscape of theorems.
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?
> 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.
There’s an elegance to how the Feds just got out of funding this stuff and within a year all proofs were solved by the private sector
I know, unrelated truths and coincidental timelines, part of me giggles nonetheless since the whole belief system behind the market relies on infinite coincidences
> 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.
> 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.
This echoes SICP's admonition that "Programs must be written for people to read, and only incidentally for machines to execute."
Which is absurd on it's face for most day to day programmers, but a critical distinction for theoretical computer scientists.
One of the best comments I read this month and in October
I feel like the hyper-competitiveness of academia generally is the elephant in the room.
The disaster for a problem-solution-oriented mathematician isn't that once problem-solution is easy they'll just switch to theory-explanation and machinery-building. It is that they will have to compete (and won't be able to compete) with students and post-docs who've been doing theory-explanation and machinery-building for their entire career - and there are lots of these 'cause academic pipeline produces more people who can fill a job than jobs in every imaginable category and subcategory.
Academic competitiveness doesn't explain all the hysteria and why those at the top of the field (Field Medal and all) wrote that breakup letter over AI. There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence. Literally a math academic crying while being interviewed on AI impact, talking about not knowing what to do with himself.
Who knew 2-3 SD won't save you from being a pattern matcher that can't stand to see a better one.
it's not just math and CS. it's a human trait. we are seeing it in the arts and creative fields where technology and AI in particular are making their impact felt. there will be much, much more of this sort of reaction in the years ahead. we'll all have to learn how to adapt. and in the process rediscover that what makes us valuable as people is not our skills but our souls.
Do male chicken have souls? Because female chicken have the skill to lay eggs, and that's what makes the latter valuable and the former bonemeal upon birth.
What benefit is a human soul to a feudal lord?
Seeing work as your purpose was supposed to be the holy grail of work that most people sadly couldn't achieve. Now we just immediately toss that away like a mouldy tomato and what's left? The new highest plane of human existence is working for capitalists to make them more profitable, in exchange for 10% of their profit?
Its equally devastating when you're a renowned expert developer at a specific stack that falls out of favor and nobody cares about your skills any longer. This happens relatively often in tech.
> There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence.
I don't see this as a toxic trait at all. Some people have the fortune and luxury to do what they love - I don't think that's a bad thing.
I think this quote from Gauss puts it very well: "It is not knowledge, but the act of learning, not possession, but the act of getting there, which grants the greatest enjoyment." So tons of mathematicians feel like AI is stealing this "act of learning which grants the greatest enjoyment", and I think it's fine that they are lamenting this.
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
Given the pay of a typical career in math it would seem logical that the typical mathematician's self worth is associated with the work.
I don't really understand this kind of triumphalism. Would you, personally, identify as a pattern matcher who's happy to see better ones? Or do you think you have some deeper skill more robust against disruption than the math and CS folks who don't know what to do with themselves?
Note that this may include things that you don't think of as being a "skill". I've seen some people brag about being a particularly hard worker, or about a deep instinct towards Getting Stuff Done, with the clear implication that these things can't be automated the way mere mathematical analysis can.
I don't read it as triumphalism. I read it as an accurate (to my experience, as a math PhD) reflection on the existentially toxic culture of math and other intellectual disciplines, one that AI is now forcing us to reckon with.
n.b. I think math culture is probably one of the least toxic cultures that have ever existed. Its equanimity and principles are a big part of what drew me to math in the first place. But humans are human.
I really don't see how to read "hysteria", "breakup letter", "literally a math academic crying" as anything other than an attempt to dunk on mathematicians.
I suppose, maybe they are dunking on mathematicians. I can't claim to speak for them. But I at least see truth in their observations. It would be inaccurate to downplay the sentimentality of the math community right now. I don't think that sentimentality has overly clouded the community's judgment (so "hysteria" is probably overdoing it) but there is a lot of heartache and unease.
The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be indistinguishable, especially because there will very likely be “math bros” just like “ai bros” and “crypto bros” before them who want to pretend they’re mathematicians just as we see people pretending to be engineers and financial experts based on how fast they got on the hype train.
I’m no mathematician, though I do have an undergrad degree in math, and unless I had a graduate degree or made substantial contributions at that level, I would never consider myself anything but a “student of mathematics,” and that too not just because I have a degree in the subject, but because I still have a great interest in certain branches of math and I keep learning in my spare time from books or from adjacent applied fields I continue to formally study. As a kid, I grew up hating math but I became interested in it later in life because of coding, computer science and formal logic, pushing myself a lot through my college years to get some foundational understanding of the subject at a deeper level than the general population, just to get my foot in the door, so to speak.
Even if I’m an outsider from the circle of mathematicians, this whole AI takeover of mathematics leaves such a bad taste in my mouth because I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general, and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject). It feels like a great loss for humanity to lose a brunt of our engagement with it to the machines, like a whole wing of a gigantic library starting to be closed off from us indefinitely because some people decided that we don’t _need_ to access it anymore, except on an even grander scale possibly than my imaginary library metaphor.
> I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general
Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
> and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject).
There's plenty of controversy in mathematics. Controversy comes from the people, not the field of study.
I know there is some controversy, I would say that the field of study also “causes” controversy in a sense if the disagreement has to do with the subject itself, that is true for any field. For example, set theory “caused” significant controversy in the early years of its development, yes it was people in disagreement and insulting one another, but the fundamental questions were from the subject matter itself.
Regarding widespread perception of math as useless, I guess I was thinking mainly of academic circles rather than the general public in that statement. Even while I hated math, I knew it wasn’t useless because my late father (who was a civil engineer) was very good at it, and he was adamant on teaching it to us. It didn’t work on me back then though lol.
As I was getting more interested in math while getting my degree in computer science, I realized much of my own issue with math earlier in life was because of the same thing just about everyone else that struggles with it has: the lack of any practical relevance in my own life because of how it’s generally taught. And the fact that many math teachers themselves don’t know the foundational/philosophical aspects of the subject which would go a long way I think towards explaining that relevance if introduced in some pieces earlier on rather than treating it like a rote memorization subject, and being so focused on continual practice to develop heuristics with exercises that some students couldn’t care less about. Of course, that works for some people clearly, but it didn’t for me at the time and I’m of the opinion high schools should focus more on discrete math like set theory, and formal logic rather than calculus, which is of little relevance at that stage unless you’re also invested in studying physics. Sets and logic on the other hand imo are general enough right at the beginning to be relevant to just about everything people of just about any age are concerned with.
fewer mathematicians means fewer problems to solve. All problems have human origin, right? Can AI create a problem on its own and what would be the point? Closer to can god create a stone he can't lift.
It's still confusing to call that a "change". Apart from the way the media covers the topic, I've never known mathematics to be about proofs.
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.
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!
Question: Have you actually used AI to do research-level math? At the current time, it is not "tackle any problem you wish and you can get headline results in ~3 hours." Their method is spraying-and-praying across the entire mathematical frontier, which is something only AI demonstrators are interested in. And as far as I know, they have never elaborated on what their "average of 3 hours" metric even means (e.g. including or excluding all unsuccessful leads?).
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.
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.
> I never had a chance against Terence Tao. Neither did you.
I'm not a mathematician at all, so yeah I certainly didn't have a chance.
> 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.
I don't see these as equivalent circumstances. For most people if they get beat by someone smarter or someone stronger, it may be disappointing but they can still respect and admire that they were beat by talent/determination/etc.
Getting beat due to access to money just feels slimy, because at some level we all realize money is not distributed on a meritocratic curve at all, at least not if you are measuring values any well-adjusted person would consider positive.
This has unfortunately been the case in experimental academia forever.
Mathematics ignored meritocracy for at least a decade. It is almost impossible to get tenured as a white male!
And you do not need millions of dollars to make a dent with LLMs. Very basic subscription can get you a long way.
For now. It once took Deep Blue hardware to beat Kasparov. Now most laptops can do it.
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.
yeah, Tao has been extremely reasonable throughout this, while the coverage has been... maybe 'motivated' is the right word?
For sure, he was arguably the first well known person to bring chatGPT to maths.
When his more "anti-ai side" (again, it's more nuanced but I'm calling it that) was showing, everyone was so quick to call him a Luddite. No one stopped to think why a well know ai in maths fan all of a sudden had a more negative opinion. Even reading his actual nuanced take was too much.
The best example was the following 2023 quote which has been repeatedly and wrongfully used as a criticism of Tao right up until he was proven exactly correct
>I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well. Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools? We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions. Our usual technology paradigms will not serve as an adequate guide for navigating these uncharted waters. Perhaps the greatest challenge will be transitioning to a new AI-assisted world as safely, wisely, and equitably as possible. https://unlocked.microsoft.com/ai-anthology/terence-tao/
The ever prescient Tao, over 3 years ago in 2023. You can search that quote and find lots of examples of "Lol he's so wrong" right up until very recently. Which is exactly the 2026 timeline Tao stated.
And his viewpoint really hasn't changed or needed to change. He was 100% correct.
To be fair, he’s had a bunch of guest posts on his blog that didn’t seem as reasonable to me (and some people seemed to not realize they were guest posts).
Edit: lol, this article is itself a guest post. My bad for not reading it first! But it does serve as an excellent demonstration of my point, since it seems like neither of us realized we weren’t talking about his words.
He's not out of the woods yet.
"This is a guest post by Frank Calegari."
>> where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
This statement is a monstrosity, and the reason why real software engineers know vibe coding will fail. It is fundamental misunderstanding of thinking that that the intellectual work consists of knowing what you want, and coding is merely the mechanical labor of making it happen.
The coding is the intellectual work!
Its where the actual problems are discovered, where the difficult decisions are made, where vague ideas become precise, and where solutions are invented. There is not such thing as a fully formed solution waiting to be translated into code. :-)
Again just the modern variation of the idea guy who goes around saying I have a genius idea for a new startup...I just need some Developers to implement it ....
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.
That is a great example, car engines (and derivatives) did create tremendous economical value but are in the process of making our planet unhabitable.
Maybe we should listen to people that care about negative externalities more often, and mitigate them.
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).
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.
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.
Do you feel robbed of your scenic expeditions because vehicles and highways exist?
Not really. I feel sad for the people who miss them.
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.
It's a pretty straightforward geometric argument, from your own analogy
I'm sure there's a coherent rebuttal but "That is plainly rubbish" isn't the reply I'd expect in good-faith discussion
It's well in the realm of obvious, but possibly outside the realm of personal experience on your part.
I think pointing at Greenland for a start and asking you to think about the problem for a bit will lead to the same coherent view on the matter.
But how would you get to Greenland?
No one forces you to drive. You can still walk.
They build cities around the car, they defund public transit, they close down rail lines, they build new housing a 5 km away from your closest grocery store, and 20 km away from any cafés and other cultural venues. Your job is located 50 km north of where you live, your partner’s job is 60 km in the other direction, and your child’s school is 10 km in the third direction.
They neglect to build sidewalks, and force you to walk 1-2 meters from cars racing at 70 km/h. With all the noise and high risk stressing you up to the level you just wanna scream.
stevenhuang: No one forces you to drive. You can still walk.
It's pretty well documented they deliberately made things impractically large distances apart and scrapped the idea of footpaths, to encourage car sales, paid for by car companies to your local city government several decades ago.
But Terry T said I have to drive or die.
Or has he changed his mind again.
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 the other option is that AI does what you say. And if you think about it (and you are a mathematitian and not an engineer) it means that AI is able to read our minds and know our desires. And at that point who cares about existence anymore?
Sorry, why would it need to read our minds exactly?
"Choosing which terrain is worth exploring" is more clear if you have an objective function (if you are doing engineering). But mathematics is closer to art in that regard. What is a terrain worth exploring in art?
Sometimes I wonder if "art" is also just some form of pattern matching on a very meta level.
Just because they could, doesn't mean anyone would bother having the AI agents do so.
You don't think OpenAI has any interest in having agents identify new interesting research terrain? They just want to solve existing important millennium problems and then they're done?
Yes? I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
Of course, general improvements may be sufficient to continue to drive math improvements even without specific attention - we’ll see.
Not a chance. They'll go for room temperature superconductors and other stuff that is currently in the quite possible but not yet proven and patent the hell out of it and charge rent to the rest of humanity.
Why would that necessarily be accessible by thinking only? Or do you see them running or directing large laboratories of various types?
Simulation to the point that the accuracy is large enough to extrapolate to real materials. You can do a lot of stuff when you have access to a million GPUs that would otherwise be much harder.
I have my doubts there, but we will see.
A lot of materials science is already done through simulation, same with a large chunk of chemistry and biology. That fraction will only increase over time.
https://en.wikipedia.org/wiki/Computational_chemistry
https://en.wikipedia.org/wiki/Computational_biology
https://en.wikipedia.org/wiki/Computational_materials_scienc...
These fields are all going through a tremendous acceleration right now.
I am aware of advances, but I think a large chunk is a stretch in various ways. We'll just see what happens - simulating even a glass of water fully is tough.
That depends on the granularity of your simulation. For most - not all - purposes a glass of water can be simulated coarsely or by simulating a far smaller amount with greater fidelity and then to extrapolate.
For useful materials science the numbers are still large but far lower than the numbers involved in 'a glass of water'.
I really depends on what is needed. If you know that something quite "coarse" is enough to create/uncover room temperature superconductors with high critical current - fair enough. Again, we'll see what happens.
With the kind of compute they have at their disposal 'quite coarse' may actually be 'impressively detailed' and that is why I think they have an edge on this kind of stuff. It's like being able to borrow a trillion and then to go play research lab, it would be highly surprising if they weren't able to monetize such a head start.
If they did that and were able to find someone to pay them, that'd be a pretty good contribution, no?
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"
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...
> 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.
I wish mathematicians would just focus on this. This really is the problem, at the end of the day.
Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.
We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.
But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
> Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
I don't see it. What I see is more wealth and power concentration and with the worst group of people possible.
Not only are the proofs long and unintelligent, but they don't cite previous work.
In fact, they completely disregard all standards of mathematical dissemination. They just post a bunch of 100+ page papers online and call it the day.
A real mathematician polishes and polishes for months until it is understandable by the community. Then they communicate the results to a journal where there are editors and peer reviewers who come with detailed feedback which is then implemented. And just then it is ready to publish.
"Polish this paper. Check all proofs with Lean. Try to find shorter proofs of lemmas where possible. Write for a undergraduate math student. Add brief explanatory material as needed. Cite previous work where relevant. Check all citations. Convert to LaTeX."
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.
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.
He did attach his name to it by posting it. I'd imagine if he disagreed strongly he wouldn't do that, or he would at least add a footnote.
What you could argue though is that he is muddying the waters with all these guest posts on a domain that is as personal as it gets. I appreciate he's trying to be the 'Carlos Santana of mathematics' by sharing the spotlight but it might be good to highlight the distance a bit more, for instance by adding his own views.
Well, you're wrong, given how many of the guest posts featured by Tao are inherently contradictory and so it would be strange for him to be personally endorsing each one! He also posts his own views elsewhere: see https://teorth.github.io/tao-web/ai-views.html.
The obvious goal is to highlight differing attitudes within the mathematics community so the mathematics community can have a dialogue and see what other people are thinking. Luckily, most mathematicians seem to be able to actually read and understand not all posts are from Tao. It is unfortunate it seems many people on this site cannot.
Posting something on a domain with your name attached to it is an endorsement of sorts. You can argue with that all you want but that's just how it works. Having a dialogue happens on neutral ground and Tao's blog is not neutral ground, even if that might be his intention.
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.
> 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
If rich people want to use their money to fund math with the goal of NOT funding math with any known useful or practical purpose, that sounds like a hobby to me.
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.
Haha, the breaks from reality there are two, one professionals do not spend most of their time inventing proofs, and two, practical use? :-)
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.
I mean Lean has two things: a full programming language, and proof checker. The programming language is used to create either code, or proofs. But the "created proofs" are using a very simple subset (a bit like an "assembly language"), and only need somewhat simple rules to be checked
So as long as the "somewhat simple" core that checks to proof is correct, it doesn't matter if the "programming language" has bugs: either in the end the "proof assembly code" generated by the program is right and is validated by the core, or it's not, and you need to revise your proof, or to check for either a bug in the programming language (it didn't generate the proof you were expecting), or in very rare case detect a bug in the simple "core"
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
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.
We invested at least a couple trillion of speculative value into making ugly cartoon apes the future of art, finance and property. I'm not sure it's always possible to overwhelm reality with sufficient money.
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
Did any ai made any new interesting question or conjecture?
No.
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?
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?
> 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?
The whole situation in mathematics reminds me of two years ago when models started coding well and all the developers were really sad. Now we use it as a tool and, as a whole, are able to make much more.
It has changed from elegance and human understanding to brute force. But that trend was started a while ago, before AI became really useful.
From TFA:
"I’m ready to board the plane, because I want to see what’s there. At the same time, I would add that people who previously have been walking around on earth with their eyes closed are not going to recognize anything however high they fly; I still believe that many of the skills that mathematicians have now will not only be transferable but vital to the new world order."
That sounds a bit disconnected and rude.
> gulf between ... novel proofs and ... understanding
Machine proofs are useful machine understanding. You don't read the proofs, that's holding it wrong. You point another agent at the proof and prompt "use these ideas to prove this lemma that's useful to me". And it works, agents are faster given access to the proofs than without.
Useful reference to "Colossus - the Forbin Project".[1] Watch starting at 00:30:00, and see this scenario predicted in detail, half a century ago.
[1] https://archive.org/details/colossus-the-forbin-project-1970...
https://archive.org/details/colossus-the-forbin-project-1970...
here is the direct link of the video which starts at 1800 seconds.
Offtopic: Fun fact but aside from archive.org there is also a way to have a link which links to any direct video at any time.
Someone made a good HN article about it that I vaguely remembered[0]
[0]: https://www.autodidacts.io/how-to-deeplink-to-specific-part-...
> 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.
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.
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.
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?
GP missed the most important part of TFM's introduction:
> AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics.
In the "theory of mind" sense, yes, probably.
These fall under the class of problems that have solutions which can be inferred more easily from applying massive computation, which is a subset of problems that human minds are trying to solve.
It's like being a CIA analyst, maybe. You have an overwhelming number of information sources of varying quality and non-obvious relationships. Applying massive computation will definitely allow you to draw new insights and identify trends that would have been difficult, or nearly impossible for unassisted human reasoning.
From what I understand, this is the sort of thing that the LLMs have been particularly effective at in mathematics, applying massive computation to the existing field and with a combination of impressive vector space search and runtime evaluation, deriving plausible solutions to certain narrowly defined problems.
Here's a harder one: you're in a bar, sitting having drinks with friends. There's an attractive girl sitting with some of her friends that has glanced at you a couple of times. What should you do?
Taking into account, of course, the entire human experience from marriage status, age, rejection likelihood, your friends opinions, her friends opinions, your family, your hormones, everyone's intoxication level, how well you slept, the noise level, how long her glance lingered, and the billions of other things that go into it - entire lifetimes of experience, genetic behavior predisposition, etc...
This is just a small example of a brief problem that a normal human has to solve hundreds or thousands of times per day that don't fit into the class of solutions that LLMs would be particularly adept at. It's easy to come up with a lot more.
So yes, I'd argue that those neatly formulated, clean problems with clearly provable solutions are low hanging fruit.
Wait until you find about RizzGPT.
> But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.
Low hanging fruits for the agents that solved them. It's very possible that there are whole classes of problems that AI won't be able to solve or will require another few orders of magnitude of compute to solve. A human might even be required to first make a breakthrough on some field before AI can re-apply it exhaustively and find new discoveries on top of that.
It's amazing to still see posts like this in late 2026. I suggest you recalibrate your ideas on what both AI and human intelligence is and can do. My take: both are trained systems. Both are as "statistical" as the other. Any idea that "because statistics" means that humans are different or better is bogus. Both can do things based on their training data. You are totally correct that a human has access to training data that AI doesn't (the smell of new mown hay, for example, or the fear of being crushed by the garbage compactor room in Star Wars). But there's no "humans have more training data therefore better" argument to be made about fields like mathematics. In fact the opposite argument works quite well: AI can have all mathematics ever invented in its training data, something that's pretty hard to achieve as a human.
I won't even start on "AI is unconscious"...
I think Sutton agrees [1].
[1] https://youtu.be/ThFq87Rp21s?is=xvgU-cKCpK0KwG_M
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.
I would really like to see an AI invent a new set of (consistent) axioms from which a whole slew of new problems and solutions arise. Something akin to Euclidean geometry, but for a whole new landscape of theorems.
> 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?
> 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.
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?
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
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.
There’s an elegance to how the Feds just got out of funding this stuff and within a year all proofs were solved by the private sector
I know, unrelated truths and coincidental timelines, part of me giggles nonetheless since the whole belief system behind the market relies on infinite coincidences
> 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.
Wake me up when AI solves Collatz.
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> 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"?
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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?
The original OpenAI blog post.
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.