"Early in the history of Multivac, it had become apparent that there was one big bottleneck: the questioning procedure. Multivac could answer the problems of humanity, all the problems, if -- if it were asked meaningful questions. But as knowledge accumulated at an ever-faster rate, it became ever more difficult to locate those meaningful questions."
I'd say that a more appropriate reference from that time would be "The Nine Billion Names of God" by Arthur C. Clarke [1], which actually deals with the finiteness of the list of problems that a machine successfully exhausts.
That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle.
The mathematical community was very competitive in its early years, but in the last 70 to 100 years, it has been generally less competitive and very collegial. The community was in a good place, and progress has been very good. In a few cases when competitiveness was ramped up, it lead to bad behaviour and destructive fights. Few would like to return to those competitive years.
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
-----
Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
I've always found the story of A. Wiles sad and frustrating.
He worked in secret for 7 years. He submitted a (incorrect) proof at year 4 or so. Reviewers found a problem, but he decided kept all secret for many years after.
He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I found this behavior against healthy science practices and only driven by ego. Unfortunately, I find this too often at work (working in academia).
Most probably I'm too naive...
While I can sympathize with this perspective, I don’t think it’s right to call it driven by “ego.” Sometimes one just wants to go at a problem without being second guessed on approaches or led astray with suggestions by others.
> He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I think that was Ken Ribet?
Grigori Perelman and the Poincaré Conjecture is more interesting. IIRC he turned down Millennium and was decidedly not all about the Fields Medal - mostly because Richard Hamilton didn't get credit? Anyway, I am grateful I had the opportunity to learn about Poincaré in college taking a few classes from a professor who was a key contributor to the conjecture and got a Fulbright for it when I was there
Eh, it seems like it's pretty necessary for success on such a problem (but obviously not sufficient). These problems gain a reputation, and you either get judged for it or get too much attention for it.
You miss the point. Humans don't mind competing with others. I love competition, but I don't want to compete with you and your machine. I love to play chess, I don't care if you are grand master, whoop my ass. But not if you are going to pair up with stockfish. I don't even care if you are a newbie that started playing yesterday with an ELO rating of 900. If I wanted to play the damn computer I'll do it myself. Likewise, mathematicians will not mind sharing and competing with other fellows, but if another has a billion dollars worth of GPU and you don't? Then you best be carefully what you say.
Could you tell the difference between a grandmaster and stockfish if playing them online? If not, why would you care which one you are playing against?
Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress.
(the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them).
FWIW, your post comes off as a middlebrow dismissal, surface level and not actually engaging with the substance of the comment. It's also just wrong. You claim "it’s also a misstatement of mathematical history", but don't specify which part. That there's "centuries of traditions of open science"? But your examples are from centuries (and millennia) ago, and there was never any claim that these traditions are universal.
But more fundamentally, competition doesn't mean you can't also have open science. And the very long, damaging events like the Leibniz/Newton feud are exactly what make many mathematicians work to maintain a spirit of collaboration and attribution even when they're competing on approaches.
> Nothing in their comment reads to me as "arguing against"
If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.
> Reads like nothing but historical context
They literally accuse Tao of "a misstatement of mathematical history".
> They aren't arguing against open science, they are trying to educate you on the history of science. It's always been this way.
Always been what way? And how does that contrast to what Tao said (since it was apparently "a misstatement of mathematical history")?
> Also, your third paragraph is highly ironic.
You'll have to be more specific, since I engaged with my parent's argument, while they waved away Tao's quote by suggesting he was wrong because of exactly the kind of events that helped lead to the norms and mores working mathematicians have today.
Not arguing against open science - it's super valuable. I'm saying that pearl clutching by people reading Tao isn't useful, because it misses some long history which tells us that this kind of science has been seen as fundamentally competitive for millennia.
Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?
These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from competition in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.
There's nothing that AI won't be able to mine and accomplish (aside from being literally human), it's only a matter of hardware and scale at this point. Generalized problem solving is a factor of search efficiency over the problem space. The actual software part is all figured out, the only open questions are how to do things efficiently and what the trade-offs are from a hardware perspective, but if hardware paradigms are unlocked then efficiency becomes a secondary factor for the problems we care about. Why bother making an LLM twice as fast if you can make a chip that can process 100mil TPS, for example. You're already in a ballpark where it can do anything you want, with plenty left to spare.
The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.
Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.
"In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. "
I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.
Doesn't this seem to be where all domains are headed?
I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?
Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do?
How is this any different from people in any field that are impacted by AI and lose the utility of their skills and endeavors over the past decades? Are we saying that we're running out of problems to solve because of AI and hence it should be stopped? I am not underestimating the importance of the collective knowledge of the mathematics community and the role of mathematics as the enablers of other sciences, but opposing meaningful progress in that discipline or any for that matter feels counter intuitive. I would rather have the mathematics community start collaborating closely with the this newly evolving and powerful tool to expedite humanity's progress.
Do you see an end state in this? When AI is better than humans at everything (and I used to be very sceptical of that claim but I'm getting less and less by the day), I don't see the Wall-E version of humanity as some sort of utopia, and that's the good outcome.
Seems to be it, though Im not particularly concerned about this problem personally.
The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point.
Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.
I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession.
Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes attached.
> An AI-generated solution always provides ... proof that there is a solution
This is only true in the most trivial sense. A solution is a solution, sure... but how do you know it's a solution, and not an incoherent jumble of words? A human has to review and vouch for it.
Just because the AI gives you an arxiv-worthy PDF, or a Lean proof which compiles, doesn't mean it proves what the AI says it does. The AI could give you the same PDF/Lean code and says it proves the opposite, how would anyone know the difference?
You can't advance human understanding unless you produce things that humans can understand.
Not an expert by any means but the assumption here as I understand it is that the arxiv worthy PDF would not be acceptable or meaningful for impossible to understand proofs. And the lean proof would be meaningless unless the specific expression being proven is human understandable as the direct translation of the question the human is asking in formal form. So proving the negation is not a thing but if you make a subtle mistake in translating the statement you want to prove then obviously the QI is going to be proving the wrong thing. And otherwise you're relying on the correctness of lean as a system and on identifying/preventing if the proof is adversarially exploiting bugs in lean to falsely prove things.
I might be wrong, but making an assumption that you could learn to read the mathematical output of the AI long before you could write a solution yourself. But hey, what do I know, I'm not a mathemagition.
This is definitely true in an information theory sense: having more knowledge is always better than less knowledge. However, it may not be true in math as a social human endeavor, and having answers without interesting paths to get there may not expand human mathematics in the same way.
If Fermat had a book with larger margins, would Weil have devoted so much time to proving the Taniyama-Shimura conjecture? No one can say.
Were early in this cycle, we will learn to do more, and exercise our new capabilities more fluently, which in turn will create more skilled practitioners
Consider the abacus, calculator, computer, etc, each of these enhanced mathematicians’ capabilities and thus outputs.
Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses.
Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career of the professional that takes the challenge.
In mathematics, finding novel proofs of a given result is often valuable; it may be a shorter proof (demonstrating better/expanded understanding of the problem) or a translation of the problem into a new domain, setting up more cross-domain advances.
> Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
A lot. In fields where knowledge is incrementally building on previous work the reason the whole field hasn't collapsed from the replication crisis is that usually the results that are really high impact are replicated in as an initial step in new research building on it. It's almost never the focus of the paper but you'll often find a quick mention in methods/supplemental of some previous work that was verified to be valid by a replication of a key technique etc. you'll have crisis where old tools are found to be problematic and findings end up revisited etc. Plus fields like clinical research where there's an awful lot of focus on replicating findings using staged clinical trials with increasing statistical power to determine if new interventions work - that's driven by regulatory requirements grounded in good science and a lot of people make careers in just that.
I find that to be an issue of maturity (focusing only on the climax and not the process). In Japan, where I live, the culture has a greater appreciation for the context & process, not just the moment of victory.
If you examine the consequences of the inversion of the peak, you realise the need for a balanced perspective.
>Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one’s beliefs in the way new research does. And if the goal of science is to change our beliefs and bring them closer to what is “real”, successful replications can’t be as valuable as the initial research almost by definition.
From a pure statistical perspective the first scientific paper shouldn’t update your beliefs as much as the independent replication study.
People don’t behave this way, but a high percentage of all papers have known flaws and that goes up even higher when you consider unknown flaws. Replication doesn’t own its own solve the underlying issue, but independent replication removes a huge range of potential issues on top of providing more information.
I agree, with the caveat that it probably matters how novel the paper is, the effect size, and the confidence interval. A reputed new psychological phenomenon I'd be more skeptical of than, idk, a newly detected exoplanet.
That alleged superconductor from a few years ago - everybody kind of held their breath and waited for the reproduction.
You have created a fraud machine. Why? With no answer checking then why not make up the most fraudulent crap you can get away with?
Examples: A huge portion of recent non-reproducable science papers.
---
Your thinking, along with everybody that's doing this rat race is causing the pumping out of papers with questionable data, but very little to ensure we are actually making correct science.
More of a 'its the journey' rather than the destination type of thing.Since the insights , quirks, tricks and procedures gained along the way allows insights intoother at that moment unknown problem/domains in the future.
As far as researchers sharing their data/notes with the AI hyperscalars looks like that would be coming to an end wihth a mor guild-like structure going forward to prevent their progress being fron-run by the AI labs.
I wonder if it would be possible for researchers and scientists to submit their papers to an organization which would then collect them, submit them for peer review by other experts in the field, and then release them in periodical form ONLY to individuals and organizations who pay a subscription fee in order to read them while suing those who try to redistribute them without permission?
Why would society fund mathematicians if they decided to become a guild that hides secrets? They could pursue that as a hobby, but they’d end up like the coders who refuse to use LLMs - rapidly becoming irrelevant and a bit sad from an outsider’s perspective.
> Mathematicians will be less likely to work on a problem if there is a solution
Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel.
Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore? I genuinely don't think so.
I think his point is that AI is not creating new problems. It may solve "the Riemann hypothesis" but may completely fail to posit a "Mythos hypothesis" which is vital to advance the field. In fact, achieving the former may make the latter even harder because it will disincentivize production of human mathematics which has till now been the only source of "interesting" problems.
FWIW this is my understanding of his argument and I am not a mathematician.
As Tao points out, merely suggesting new open questions isn't really sufficient. Part of what gives these problems their fame is their notoriety, their difficulty, the fact that many prodigious mathematicians have spent an evening or week or month or several years studying it.
It wouldn't be as interesting if it had just been solved by the fifth random mathematician who considered it
Notably, gardening a new field of study in math is somewhat nontrivial. You have to introduce the field, illustrate some relevance or connections, and then - and this is key - not solve all of the low-hanging fruit yourself! Because you need somebody else to become an expert in that particular field.
The analog in programming is: if a large company merely open sources a product that's decent but not great and in a language nobody wants to maintain, but they don't commit to maintaining it themselves.
Suddenly there's a bit of a vacuum because in order to provide something of value, you either need to:
1. Implement something more complete than was initially open sourced
2. Or maintain something in a horrendous language while incrementally improving it and keeping it relevant
3. Or rewrite it into a tolerable and maintainable modern language.
What the large company has done is create a vacuum in the tool space where you now require extreme motivation to get someone else to step in.
Note that in this scenario, in 2026, it's actually not such a big deal. I think several recent models could happily translate it into a more maintainable language themselves or happily maintain it in the original crufty one. And so the question is: which parts of this analogy are true in math, too?
The sphere of human comprehensible mathematics is finite. Once everything is solve it is not necessary to advance the field. The recurring error her is to say ai is not the product of human effort but another agent. Ai is human. Ai may well be speeding up human comprehension of math to its limits in which case there is no further need to advance the field and mathematicians might need to get a job. Why is this a bad thing?
> Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore?
In the case of mathematicians, I think not as researchers. What would a research mathematician do? I don't think there would be any reason to try to gain insight from proofs that AI made for the sake of understanding. I don't see what that would achieve besides just retaining extremely niche knowledge (which AI or the oracle already does). The whole point of having that knowledge was to build toward novel work which the AI/oracle does. Also, the time spent and difficulty understanding them could be very high but with no payoff besides just understanding them because the AI/oracle would be used to solve all the problems anyway.
But this oracle doesn't just say true / false. It also gives a proof. That makes it much less exciting (not to mention beneficial for your career) to find another one (or even worse, the same one).
What do u think lean is? That's like saying a program that works, is inscrutable because it appeals to the oracle of "code test cases" to prove itself correct.
You're either being intentionally obtuse, or unintentionally ignorant.
I mean, the oracle doesn't really seem so hypothetical right now. And clearly it's going to drastically change these fields, and mathematics, particularly pure mathematics, must change most of all in order to adapt to the existance of a math oracle (or something close to it).
AI companies don't share the dead ends and only sometimes a bit of the process toward success so people don't understand what was curious along the way.
This is my question too. If we are all just going “well that sucks” after AI solves this problem, why did anyone care about the problem being solved in the first place?
Is the bummer that we got a solution we didn’t want - that navier-stokes is not always applicable or something, but we hoped it was?
I think the Navier Stokes problem kind of illustrates what he’s highlighting. I think most people even before AI expected that this would resolve in the negative and that you could get finite time blow up. There wasn’t really ever going to be a situation where the resolution to this question, or really any of the other Millenium Prize problems as far as I know, gives some kind of immediate massive practical feedback.
The hope with many of these problems in math is that in trying to prove that, we get some additional insight into why it blew up that could be applied elsewhere to more general PDEs that cannot be easily controlled.
I think the observation from Tao and many others is that when humans solved these problems, the additional insights into intuition and theory building came for free since humans can give expository on what they found hard or what was their own intuition. This is much more difficult or tedious to extract from an AI model. Even when people did have access to the chain of thought, it wasn’t always very helpful to figure out what was the exact thing that made it all click. This is even more difficult how that the CoT are hidden but I would think the sort of difficulty of extracting the key ideas for a human might be worse now with more advanced models.
There’s a long term aspect to this too where we have historically used these problems as markers for the other parts of mathematics but if AI can solve it all, then suddenly this signal is not very meaningful.
Maybe to bring it closer to home. If an oracle just gave you P \neq NP, then this would be generally uninteresting since this was already expected. There’s a deeper question of why that needs to be answered. However, one would hope that creating such a separation would give us tools that allow us to create lower bounds on a lot more problems we do care about and perhaps some bigger insight onto what makes a problem intrinsically hard or easy. These long term considerations are helpful but are definitely more vague. The remarkable part is that AI is separating the part about proving theorems and the “free” insight you get.
Honestly, the attitude of the math community is a bit cringe and increasingly I think some of the elite/mystical aura is fading. Rather than a rich fertile jungle where AI can barely chomp through a fraction of the luscious terrain, one gets the sense it's a desert and all the oases are running dry.
Current career structure of mathematicians works partially by looking at whether they have solved novel and interesting problems, or at least done theory-building that can help solve such problems. Many mathematicians are also motivated by being the world's first to solve such problems
Removing this measure suddenly means that academic mathematic norms need to adapt rapidly, and, even more importantly, intrinsic motivation for many mathematicians needs to change rapidly. That is understandably a sea change for the current mathematics community.
> and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge.
You'd be more sure if you read the tweets.
Tao's point is very simple.
1. Working on problems that AI solvers can solve is a waste of human time.
2. We have no idea which problems can be solved by AI solvers...
3. ...Because the AI labs are keeping their negative results secret, and don't tell us which problems they've tried and failed to solve, and why they've failed to solve them (or succeeded at solving others).
There are additional points surrounding it, but that is the thrust of his argument. His issue is not the existence of AI, but the anti-scientific secrecy in how it is used to solve problems. All the incentives around its current use result in closed, uncollaborative work - which while very attractive to a vulture capitalist, is anathema to scientists.
---
He also posits that having a solution to a problem is a small part of the value of solving a problem. What the AI labs are doing is the equivalent of a student turning in their homework, which has 100% of the right answers, but with none of the 'show your work' steps. Those steps are a critical artifact for doing mathematics, because the process of solving a difficult problem teaches us things about other problems.
It seems relevant that Terence Tao is the author of the paper that just about convinced everyone that the Navier-Stokes equations blow up in finite time, 12 years ago: http://arxiv.org/abs/1402.0290
We're having to rediscover in real time the extremely hard way, why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system
If theft becomes more profitable than genuine creation, then nobody will create anything. Then there's nothing to steal, at which point all progress collapses
1. using copyrighted material to train LLMs is fair use, not theft
2. The topic we are dissussing concerns LLMs being trained on logs from previous LLM chats. If you're prompting a model and it spits out some unique mathematical insight, you do not have copyright on that.
> why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system
What is interesting is that LLM's do not directly violate copyright. The settlements we have seen are for how the works were acquired (that was a copyright violation) not the use of the works.
The vectors of a book, or a paper, are not the paper. They are, for all intents, facts about the work itself, and more generally writing. You can not copyright a fact.
It also means that the weights, the things that (mostly) matter can not be copyrighted either.
> Copyright maximalism is a bad look on a site called "Hacker News." Perhaps other sites beckon.
Frankly it's more of an insult to the "hacker" name to be apologising for big companies profiting off of frontrunning existing work for PR purposes, if the claims about piggybacking on human-directed efforts/prompting are true.
Being pro-copyright in order to protect the work of an individual from being reconstituted into the corporate machine is VERY hackery. Novel use for an existing tool, to fight the dominant system.
(Of course, we're on a so-called "hacker" site hosted by a company run by squarely-establishment individuals acting in an extremely un-hackery-field (investing), so the irony here has been at least one layer deep since the start.)
It is now clear to me why the AI labs are sponsoring these mathathons: https://mathathonchallenge.com/. They are basically crowdsourcing human researcher data to get access to promising directions possibly later to scoop others.
Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them?
Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource."
The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and offer a prize to solve them.
The incentives are massively skewed towards the AI labs investing their massive amounts of compute into being the first to solve an outstanding problem.
It's a marketing game for them, any societal benefits are secondary. Winning a prize is going to get headlines and feed into the "AGI soon, machine replaces another career" narrative they crave unlike coming up with some (possibly) interesting problems.
I think the problem with AI asking questions is that it will ask questions that are interesting to it but not necessarily us. AI, as a model, will never be a perfect copy of a human. It will always be a simulation, and thus to some extent, will ask questions that humans find irrelevant and solve problems that humans find irrelevant.
For anyone facing an existential crisis on AI, your ace in the hole is your humanity. Only you have it, and only you will be the best judge of what is good and interesting (to a human at least).
I didn't realize that open math problems were a finite resource.
I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.
They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community if people are disincentivized from trying to solve deep problems or study them for understanding as opposed to simply focusing on verification. It's more of a social and community problem than a fundamental problem with mathematics itself becoming "completed".
So we can let the ai generate some math problems based on the solutions found? Other fields (computer science, physics, ...) can generate math problems too.
There's an infinite number of possible math problems, but the things that make these open problems worthwhile is they're interesting to people who have worked in related areas.
They're good to give to new mathematicians, and they're good to help humans understand the shape of the problem space and relative difficulty with the tools we have.
Cheesing these problems with LLMs gets rid of both the training benefit and our ability to create good related problems. There's an aesthetic part of this, too, that LLMs do not capture.
This kinda reminds me of the guys who decided to industrialize digging up dinosaur fossils, in order to feed the dinosaur fossil collector market. They were amazed that paleontologists were so "inefficient" at finding and digging up dinosaur fossils.
But from paleontologists' perspective, they go out looking for dinosaur fossils when they have questions that digging up a fossil may answer. The metric they're focusing on isn't tons of fossil mined out of the ground, it's a developing understanding of extinct life.
These open problem solutions often reveal tighter bounds on prior conjectures. Even if the solutions produced are far from elegant and only machine verifiable, we do learn new information. But I agree that just like writing prose and code, brainstorming frontier math proofs is a perishable skill
It's easy to come up with new open problems. It's hard to come up with new open problems that seem to teach us something fundamentally new about the world. Our current batch of problems went through a complex selection process over decades (or centuries) based not purely on difficulty but also on perceived insightfulness.
I studied math, but I am not a mathematician, so I think I have a slightly different perspective on this than Tao overall. This is certainly the definitive end of an era in mathematics, but I think he's wrong that insightful new open problems are truly non-renewable. They might be non-renewable by humans at the rate at which they are being closed, but I see no reason why AI systems could not also discover insightful new open problems. In fact, once we have Riemann-capable AI mathematicians, I'd personally love to see what the next Riemann hypothesis is, which even these AI systems cannot solve with any amount of available compute.
I think we're about to find that, on the spectrum of mathematical intelligence, the best human mathematicians were only a fraction of a percent forward from the very beginning, and there's a vast universe of mathematical depth that's beyond our ability to imagine or work on directly in any way. We're used to feeling like we're able to directly perceive the Platonic realm, but we're almost certainly going to discover that our own minds, even when joined together over centuries of deliberation, can only interact with a tiny little shadow within it.
What you are saying implies that by some technique that hasn't been discovered yet, we can make the models to have the capabilities of extrapolate the information they are trained on and also interpret that what they are extrapolating are Riemann-capable hypothesis.
I do believe it will accelerate the discovery of that "vast universe of mathematical depth that's beyond our ability" but at the cost of removing the "fun part" of solving the problems. Not sure if the community is willing to do that.
I haven’t been following the AI proof stuff very closely, but the impression I got was that these models are producing massive Lean programs that prove the statement one way or another, but are quite difficult to fully understand.
Actually, I have to admit I don’t really know what math is. With physics we suspect there’s a universe, and when we study physics we’re improving our description of the behavior of that universe, right? The universe exists whether or not we know how it works.
Eventually, as you suggest, maybe we’ll hit math that won’t fit in anybody’s head at all. What is the nature of mathematics that doesn’t fit in any human’s head? Does it even exist in some sense?
I think part of mathematics is taking things that don't fit in our head and giving them human abstractions so they can.
Take infinity. Infinity can't fit in your head, hell, it can't fit anywhere, but you can abstract away the endlessness and look at infinities of different sizes, et al.
Now, is there a single formula for something actually represented in this world that would take most of a humans life just to read it, no idea.
The models produce both Lean code for formal verification and a traditional-style narrative proof. Like the general long-form output of frontier models, the math papers produced appear to be generally correct technically, but written in an ungraceful and sometimes hard-to-follow style, so they are often polished by a human mathematician as of today.
I'm surprised nobody has stated the obvious: a hard math problem that has been open for ten years (because many serious people have given it serious thought and been unable to make significant progress) is, in fact, nonrenewable.
The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a tree is twenty years ago, the next best is today: we do need to start posing some hard math problems and deciding if they are interesting merely because there are challenging or because of something else (eg busy beaver problems are arbitrarily hard, but does solving them imply anything other than "another busy beaver problem was solved"?)
Eh, if AI quickly solves most of our mathematics problems that are solvable then it might be time for us to hang up our hat as our little monkey brains aren't very good at this stuff.
Now, I think AI will solve some, but we'll find out that some are just either unsolvable or wildly huge that nothing is solving them any time soon.
And a whole lot of these problems have been around quite some time, when even knowing how to do advanced math meant you were a landed gentry or someone of high wealth. If those problems fall, they fall. They aren't pets we keep around forever. And new problems will crop up over time for both AI and men to scratch their brains over.
I think "close to completion" is not the right framing. Creating good open problems was an achievement because these problems often sit at the edge of known techniques, and solutions require inventing "new math". It's hard to find these problems, and they take decades to mature as they withstand scrutiny by many people.
In another comment below, I likened this to clear-cutting a forest. Growing the forest takes a lifetime; destroying it could happen in the next few months.
his whole point is that specifically problems that have been held as important by consensus in the field are a finite resource. obvious example being the Clay millennium prize problems. seems like they function to shape the direction of future research into useful directions. which is to say, the process of developing a solution itself generates more useful problems.
of course thrrr are tons of problems once you remove this social consensus based filter. if i’m not mistaken Ramanujan left a book of dozens of unproven theorems, for one quick example. i don’t think that that has opened up dozens of fields of mathematical research.
Deforestation might be a better metaphor than mining. Logging is renewable if for each tree you chop down you plant several more. AI companies are operating "in a non-renewable fashion" by chopping down trees without planing seeds. Open problems are a renewable resource, but only if harvested sustainably.
Relevant, interesting problems that we have some immediate hope of making genuine work on might be, if not finite, quite difficult to produce. And it's also plausible that AI will not do as good a job of producing these as it does at solving them.
The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.
> famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Web search turns up Gauss's comment, with a bit more nuance:
"I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of." (https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)
I think you can't have read the thread.
The whole point is that there is no end of mathematics, an infinite sea; but the constitution of an 'open math problem' is a delicate piece of mathematical thought, at any moment a small supply of drinking water developed by finitely many human being.
This is worsened by OAI/Ant's strategy of grabbing the glory and running rather than spending effort trying to to advance understanding of math. Tao, who is quite sophisticated in use of AI, has said a lot about this, including in meme form: https://mathstodon.xyz/@tao/117068266026618494
The AI labs' approach to math is immature in a way they can't get away with in coding. In coding, they realize that a pile of code that technically works is not enough: they need the code output to be a foundation to build on, and they need their agents to work well with humans which means explaining things in a way that makes sense.
In math, their goal seems just to be to exploit mathematics' reputation as full of hard problems with a general population that can't tell a pile of Lean from a good proof. OpenAI pretty much said this work is just to show off at the end of the post. Anthropic said their FLT formalization is a research artifact they do not intend to clean up or improve in any way.
Besides uniting mathematicians in irritation at the labs, the other flaw with this strategy is that it ignores that organizing knowledge is part of intelligence, much like not just producing a mess that runs is part of programming. You can write a proof that uses algebraic geometry because someone organized what could have been a bunch of disparate ideas (or fragments of a Lean repo no one will read) into a toolbox where an expert can find the tool they need.
I hope they change tack. Perhaps instead of making an explicit strategy of taking the credit from mathematicians but doing little for actual understanding, they could let some math departments at their swarms or best models, ask for a bit of acknowledgement, and hopefully they approach it by trying to write good papers, simplify, etc. rather than just rushing for headlines. (Tao's post about digesting an LLM-generated proof https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the... is an interesting read for a sense of what he means by 'digestion'.)
On that last note, it's also important (Tao's also noted) for the mathematical community to properly value digestion and organization of results, so that given the incentives of mathematics and availability of new tools you end up with good papers and textbooks and so on, not just mathematicians taking the labs' current role of pushing incomprehensible-even-to-specialists proof code to repos.
Besides Sendov’s Conjecture, Terence Tao has also shared his AI-assisted digestion of the counterexample to the Jacobian conjecture. In simple words, digestion just means full human understanding of the results. It could come from understanding the result from a different perspective, or perturbing the result and seeing what breaks.
The Jacobian one is great, and fits here -- 1) the counterexample came out as a tweet of a single expression which seems exceptionally hard to turn into something sensible, which crystallizes the lab's approach; 2) it links (like the Sendov post) to a chat transcript showing a little of what was involved in untangling it (though of course all that went into asking the right questions is invisible!).
Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.
Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).
It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.
This reminds me of the time an AI was taught how to play a racing sim game (Gran Turismo if I remember correctly). The AI was able to race its car very well, but it took a lot of risks that a human player probably would not. A human player might be able to copy the approach the AI took, but they would probably crash.
Going back to chess, I think the situation is similar where you can’t expect an amateur player to get better by trying to play like a strong engine. I think even professional chess players mainly use engines to prepare or memorize variations that are counterintuitive for their opponent. In other words, getting into situations that look wild, but that part of one player’s preparation.
I’m not sure how it is in math, but in chess, it seems like top players can play just like engines when they are in “normal” positions, so that is where I get a bit confused as to where the direction of insight is coming from because it’s been my view that AI is able to make leaps that we would never think of taking and I’m not sure that anyone could actually learn how to do that on their own unless they were willing to keep failing over and over.
> The AI was able to race its car very well, but it took a lot of risks that a human player probably would not.
There is a parallel with autonomous vehicles in real life. On northbound 1 in SF going through GG Park, the left turn lane onto Crossover Drive is always backed up. Waymos often do a very late merge into that turn lane in order to jump the queue and save time. With 360 degree sensing they can do this safely in real time but it feels too risky for most humans to attempt.
Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?
This is the direction of experimental particle physics and observational astronomy, where the budgetary scale at which progress occurs means we now fund these efforts at a societal level. These fields have graduated beyond "tabletop science".
For 3000 years mathematics has only been a "tabletop science". Even big programs like the classification of finite simple groups have been comprised of small teams chipping away at different (publishable) parts of an overall program.
This latest Navier-Stokes advance cost something like $22m in tokens, already well beyond what a mathematician's research grant can fund. As the easy open problems get mined, the cost of frontier progress will continue to climb. Some part of mathematics as a field will need to transition from tabletop science to big science: Coordinated top-down programs addressing high-priority objectives.
TBD is what the role of individual mathematicians will look like in a "big science" paradigm, but we could look to experimental high energy physics for ideas. For all practical purposes, AI converts math from a theoretical field into an experimental/observational one.
Yes. But this is hard for mathematicians to stomach, because like everyone else, deep down in a place where they don't like to talk about at parties, they have egos and a sense of purpose based in part on demonstrating mastery of a technically difficult field, as well as social connections based on their participation in it, and taking all that away from them probably feels like a kind of death.
The situation is not that different from John Henry competing against the machine. The question is really: Do people deserve to be allowed to continue doing what they have always done, when doing it is no longer necessary to advance the greater good?
I totally understand and agree, I just feel like with the progress that we have in automating informational work, you are going to have an exponential amount of these "deaths" as new fields where humans can provide any kind of value get more and more short-lived.
At some point in my suggestion the machine will ask better questions than you, and that will be pointless as well, and you keep doing what you like doing, or you move on to something new. But if you keep tying your value to outcome and recognition instead of process you are going to have some incredibly depressing years ahead, and every time will just be as hard to stomach because of your ego.
So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before?
Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
If this is V0.5 of AGI/ASI then by V1 the only system that will be understanding any of this is the AI itself. If AI creates a new field of mathematics month 1, then solutions to new problems in month 2, then another field of mathematics on top of that at month 3 there's no human who will ever keep up with that.
Or the alternative is a flattening of abilities, the AI cannot proceed further than the collective intelligence of humans and in that case this is correct. We'd be in a future where nobody wants to work in a field with an AI dominating it and when AI hits the limit of no useful training data input we'd have this giant gap of nobody know wtf it's done for years and nobody willing to figure it out and advance it.
Ooo here's a dytopian story:
- AI gets better at everything humans do
- humans stop trying
- AI cannot improve anymore than its input data + human support
- AI slowly degrades itself (model collapse) for decades, it slowly hallucinates little by little until its hallucinating entire scientific fields losing quality over time
- there's a mass population of people in the future who never learned to do anything and now have to relearn and figure out the equivalent of 100k years of AI work in order to prevent its slow degredation while all the systems they've come to rely on start failing around them. The AI has solved every problem but every real solution is saturated with 1000 false ones.
- humanity starts from scratch?
I love the idea of an archive of every solution to every problem existing but it's impossible to figure out the correct one. Infinite library like!
Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.
It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.
If said human is kicked to the street with thousands of other homeless people that can't get jobs because AI then robots replaced them, then those fast math problems sound like a pretty bad trade off.
Now, if there's some future where AI leads to abundance and we can all live off UBI, well, probably a worthwhile trade.
The biggest issue I see is the more controversial people leading the AI race at the moment are not the kind of people I'd hand kids safety scissors much less the future of the human race.
there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.
> Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.
IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.
> Unless [...] mathematicians are effectively useless?
It's always been a bit bizarre that this isn't the case. Mathematicians are almost always working on problems that there is no good reason to expect to have utility in the real world... problems they selected because of their elegance or whatever... yet there is a strong historical trend of their work having huge importance after the fact. Sometimes in fields that weren't even invented yet at the time of the work.
There's something to be said for the idea that disrupting a system that is working well for no apparent reason is a bad idea.
Think of it like software going from programmers understanding every instruction, knowing where every byte of memory was being used and why, and using this knowledge to build optimised systems
During the process of optimising and understanding the programmer might learn something new or have some kind of "aha" moment of insight that might lead them down a new path of study where fantastic new technologies and capabilities can be realised
Fast forward to 2026
Most web pages take several seconds to load
Applications crash often for no apparent reason
A vast majority of programmers have no idea what their applications are actually really even doing anymore, so they stack bloat on top of bloat and if something breaks, well I guess that's someone elses problem cos I have no idea what's going on anymore
There's something to be said about levels of abstraction being useful, but abstracting away understanding of the task itself is not the path to generating useful knowledge or applications for humanity
We might be gaining the "what" but we are losing the "why" and the "how" and these are generally fundamentally more important
Mathematicians provide two complementary services bundled together.
1. Proving theorems - what AI can apparently replicate faster and better.
2. Creating definitions and new theorems from those definitions to prove, selecting which of the possible statements to work on. I.e. developing the "language" of mathematics. So far there is no evidence that LLM can do this at all well. And there's some reason to think that mathematicians won't be as good at this if they aren't also doing the first part.
The value to society only comes when they do both "well", and it's 2 which is really the black magic where we don't understand why they've been so useful to us.
So I totally agree if AI also cannot do the second part better than a person.
Honestly though, I wouldn't want to take that bet. I never thought that the first thing AI would become super human AGI like is math.
You ask me 10years ago and I'd think the opposite. I think we all would have said we'd have super human HR employees before a super human mathematician.
Because there's lots and lots of money in that and there's not in funding pure math. It sounds like your problem is with the people holding the purse strings.
Perhaps an AI could! Today they do not, because the people driving them understand constructing the proof rather than understanding the proof to be "the problem".
(I suppose it's possible that in some distant AI future there might be no value in people understanding theoretical math, but I'm pretty skeptical of that; to me it seems like the same error as thinking nobody needs to understand multiplication because you can ask the computer to solve any multiplication problem.)
It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.
People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.
Knowing the answer has value. But, often in math the best thing was how someone got there.
Pure math is practiced mostly for the intellectual thrills and recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.
This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.
There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.
> Now, Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
This is an absurd thing to say. Hacker news is not the only place that knows about the most famous mathematician in the world. Glancing at Google Trends he seems to be roughly as famous as Linus Torvalds. Not exactly a household name but by no means obscure.
I'm going to charitably assume that you forgot to include quotes around the names in your query, because that's absolutely not what Google Trends shows.
Stop 100 people on the street in NYC or Berlin or Tokyo and I bet none of them will be able to name any living mathematician. A few of them might know Linus, though.
Yes, it absolutely is what it shows[0]. I have no idea what data you're looking at that suggests otherwise. Worldwide or in the US, as public individuals or as search terms with quotes or without quotes, over any reasonable timespan, they get roughly the same amount of searches.
This series of posts by Terry Tao is a direct response to the Navier-Stokes results (multiple results!) from the last 24 hours. The question is what is left after the levelling of mathematics, in all its senses, occurs? How can you protect a field that's under this much pressure in the next 6 months?
> [I]t is now the identification of a promising problem which is the scarce and precious resource. We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.
yes but the tweets don't really address why the field needs to be protected instead of adapting and evolving.
And to reply to the sibling since I hit my comment limit and I'm going to probably forget about this conversation until tomorrow:
But our situation before 2023 was one in which we had an endless abundance of solutions and ideas. I understand that AI can generate bad ideas faster than we can discern them, but we already have tried and true mechanisms to filter good ideas from bad (e.g. the scientific process), why can't they be adapted?
The reason is because the entirety of society, historically, has been based upon humans using their differential skills to further it, which in turn promotes societal cohesion. If most human endeavours are solved, then we will enter a period of abundance that paradoxically will erode the glue holding society together. In short, endless abundance of solutions and ideas cannot coexist with a healthy society. Only those who are priveleged and have a naive belief in a Star Trek utopia think otherwise.
>endless abundance of solutions and ideas cannot coexist with a healthy society.
This really cuts to the heart of the problem with AI. Not only does AI undermine the monetary economy, it undermines the intellectual economy. What is humanity without the need for collaboration for survival or for intellectual progress, ultimately providing the impetus to build something greater as a result? I don't know, and I'm not looking forward to finding out.
This far into history I don't think many are going to buy a "the next technology is to be the undoing of society itself" pitch until after society is already undone by something. They're as easy to make and hard to concretely evaluate ahead of time as the utopian predictions while offering little in the way of practical approach to preventing the same result from occurring anyways.
I mean the past 10 to 20 years of social media have shown a lot of societies glue already breaking.
A society can live just fine in a period of abundance. The societies that we currently have on earth do make it questionable of 'we' can right now. I mean I see people posting stuff like "I'd rather burn it all to the ground rather than see one cent more tax" kind of stuff when they have millions. That kind of person doesn't want more people uplifted and it takes away from their idea of being special.
>naive belief in a Star Trek
The naive ones don't read into ST lore to know it comes after WWIII.
Aren't we in a similar position to what chess went through in the 2000s when Deep Fritz came out, and a desktop PC was able to defeat a reigning World Chess Champion? Did chess players just give up and stop playing? No, they didn't. They used these new chess engines to become better players. Computer programmers and mathematicians will probably go through something analogous.
Presumably it is only a matter of time until these frontier models are used to create new interesting conjectures. I don't get Tao's line of reasoning.
Perhaps his takes are evidence of just usual human fear to new things. I see he relies quite much on the "community" or "social" aspects of the discussion.
I probably have delusional expectation of what a mathematician of his level should be talking about, but I expected from him a pure objective analysis on what to do with this new AI thing , what are its limitations, how it can improve the field and the creation of human knowledge, etc.
Seems like in current cultural and economic context, short term extraction is what we’re going to do
> In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.
Is there an analogy here to the phenomenon that senior {engineers, designers, PMs} are now able to be insanely productive with AI, but it's also very hard to train junior folks to develop the sense of judgment that senior folks have?
I think this highlights one of the fundamental differences between humans and our current AI systems. They can still only try to solve problems in the given well defined parameters they are given (with some exceptions). The human is able in the effort to solve problems to intuit where there may be new interesting problems adjacent to the current problem.
I don't think we can confidently say, yet, that LLMs can't discover those connections. We just haven't explored them yet, because 99.99% of the prestige is locked up in proving hard results, which also happen to be easier to assess objectively (thanks to automated proof checkers), and so that's where all the effort has thus far been exerted.
No matter what happened this must be a wake up call for all of us. We’re basically sharing everything we have with these companies/AI systems. This is wildly different than a human wiretapping our private messages. Because it is systematic and automated in an astronomical scale. There is no real privacy in this new world. Law? I think “National Security” is a good enough excuse to screen anything constantly, including foreign researchers in case they are close to a breakthrough.
>We’re basically sharing everything we have with these companies/AI systems.
My sense of the word 'share' is that it traditionally involves agency by all parties involved. There are a lot of words in English for describing taking things without permission and profiting thereby - words like piracy, banditry, and larceny.
Next you will try to tell me that these companies built upon nothing but piracy, banditry and larceny would continue to commit piracy, banditry or larceny.
I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.
Probably an AI-written Lean proof is very different to how a human would write it, and some may say it's more like mathy neuralese. For sure it works but it is not human-friendly and needs to be transformed into something more readable and digestible to be able to extract insights from it.
Not that different from when trying to read an out-of-control vibe coded codebases, or an sloppy AI long email that someone may send you at 9 AM.
Tao has a spiel in his recent interview with Dwarkesh where he says that AIs are very good at explaining things - so just have the AI explain the proof in a human-friendly way.
For sure, but this was supposedly ~18 million dollars of compute, afaik 100 pages paper / lean proof and only god knows how many bytes of chat interactions + thought traces. Scale matters.
I bet it can be decomposed quite nicely though. At the top level, there are probably only like five steps. Dig as deep as you want into any of those steps (i.e. engineering).
Because the problem has almost no value unto itself. The clay statement of navier stokes is not relevant to how CFD is done in practice.
It's about what is non verifiable versus verifiable. The same way it produces "slop" code (which, if you give it test cases, will be 100% correct), it also produces "slop" math.
Code that serves a business function, it's ok if its slop. Math that serves directly a business function also can be slop.
But most open problems are not directly for a particular usecase. People agree widely to attack it due to the perceived possibility of encountering useful mathematical objects along the way, that will then expand the world's mathematical toolset. This is not something that you can easily express in a verifier, and is thus something that is hard to force an LLM system to do.
You are right in that understanding it retrospectively is possible, but that is not going to be as useful as the desired "elegant" objects that expand and unify mathematics. You can't represent these concepts in verifiers.
Again, if you let AI rip at something like say "beat shannon capacity" and suppose it comes up with MIMO as paulraj did, great! It's useful and you can retrospectively understand it, say by expanding shannon to multiple dimensions, as foschini and telatar did. But most math problems are not in that category.
The question then is, if AI is really good at this type of math, how much of the existing mathematical community+process is necessary? I think it will still be necessary, just maybe in fewer cases. Wherever the primary purpose of the math is in a domain and that domain has a verifiable target, we can directly optimise it to that verifiable target in-domain rather than reach for the mathematical community. How well will this work? We'll see. It's not clear if it's even possible to represent most problems this way.
My model of mathematical intelligence for a little while now has been 3 levels:
1. I give you a proof, you tell me if it's correct
2. I give you a theorem, you give me a correct proof
3. I give you nothing, you give me a theorem
1. is largely solved by modern LLMs and they took a big step toward 2. today with the Navier-Stokes proof. But they're definitely not there yet. It's unclear what progress is being made toward 3. for the time being that remains the realm of humans.
>"While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions without such analysis would be of negligible or even negative value for these purposes."
Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...
The point of mathematics is not to prove results. It is to build conceptual thinking about mathematics. Important problems are important because in order to solve them we have to build concepts tying different things together.
We're not searching for answers. We're searching for insights. Trying to understand the problem causes us to draw the connections and find those insights.
AI gives us answers. But it doesn't help us build those insights. AI has a complete mastery of existing human insights. But doesn't build new ones from its own experience. In a real way, it does not find the opportunity to really learn.
So it tackles problems and either solves them or not. If solved, we now have an answer. If not, it's too hard for humans.
The issue I see with a handed-over proof is tunnel-vision: you explore only the understanding of the proof.
Without a proof, your exploration branches out much further, in directions that could seem fruitless, but may uncover new understandings that are now "hidden" because the handed-over proof drastically lowered the incentives to find them.
- If you have only a fuzzy idea of how to get to your travel destination, wrong turns and alternate routes may reveal sights and places you'd never have encountered without that wandering.
- If your GPS directs you straight to your travel destination, you are now where you wanted to be but missed out on the exploration. This is the sort of consequences the AI math proofs have.
STEM research thrives on that side exploration and unearthing unexpected things along the way. James Burke's famous documentary Connections spends the middle episodes talking about the unexpected directions that exploration has taken science. It's very hard to credibly make the case that this sort of meandering exploration is not valuable.
I think you're misunderstanding the point of math problems. Mathematics is as much a process as it is a result. This is why even from early on, relatively rudimentary mathematics questions you are graded by your capacity to correctly achieve the desired process to the answer than getting the answer correct. The risk here is that AI generated proofs removes the process part of mathematics, where actually interesting concepts live (because then you can apply novel concepts to other unsolved problems and then thereby unlock new concepts that way...) Sure you can kind of try to reverse-engineer it but you lose the entire intuition and "we tried applying it in X, Y, Z ways and it didn't work" intuition, because even the non-working process can teach you about how not to apply the working process to novel problem spaces.
Basically: Tasting a delicious soup doesn't tell you how to layer the flavors, but if you want to be a good chef, you better be learning flavors more than you learn dishes!
He's saying that in such a scenario, almost all of the value is located in the analysis and just dumping the proof has "negligible or even negative value". (The negative value would occur in the cases where the proof doesn't contain enough information to reconstruct what insights would have led a person to it.)
> Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.
Perhaps the pace of applied mathematics would rise sharply, given cheap intelligence. And this* may end up being the forefront driving progress in mathematics.
*Or maybe a split between the human domain and the practical real world. Where the human domain might end up with a variation of a "No machine contributions" policy. Sorta like the recent gcc policy.
While AI companies have almost infinite money, they still don’t want to blow million dollar budgets on problems if there isn’t high likelihood that it will be successful.
But within next 10 years as costs drop significantly and even more improvements are made, yes it is very likely that almost every single existing math problem will get a serious AI cracking done on it
A "massive amount of AI-powered effort" costs a ton of money. What's the return on investment for these frontier labs? Do headline-grabbing successes in mathematics translate to expected *profitability* in disciplines with more immediately quantifiable economic value.
The way to tame a profit-maximizer is to make the most profitable choice the one that creates the most societal good.
I would like to see the Clay Institute give zero recognition for formalizations without human-readable proofs. That would incentivize OpenAI to scram or create something that's actually useful.
It would be trivially easy to convert Lean into English, so I’m not sure what the human-readable criteria gets you. There are also human written proofs that are considered not human-readable by most of the mathematics community (ABC conjecture).
Human-readable means multiple humans can read and understand it in full. A human-readable proof is more worth more than one that is not, because mathematicians can read the proof and extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. A proof that only a few humans can read is more useful than one that no human can read because the few mathematicians that can read the proof can still extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. That's what the human-readable criteria gets you.
Mochizuki's claimed proof of the ABC conjecture is not unintelligable; it has errors. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
The four color theorem states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. It was the first theorem proved with substantial computer assistance. The theorem was proved by showing there could not be a counterexample. The authors made a list of maps where if a minimal counterexample existed, it would be one of these maps. There were 1,834 maps in that list, and each one was checked by computer. You could turn each of those cases into a picture or paragraph, but the resulting artefact would not meet my definition of human-readable.
Human-readable does not just mean in English. Humans can only hold a few objects in their short-term memory at once, not hundreds. Though some proofs require significant background knowlege, any proof written by a human will respect the fundemental limits of the human mind. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
I suspect large lean proofs generated by LLMs do not respect the fundemental limits of the human mind. If no human can read and understand them, no one can extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. If LLM proof generators can be made to write proofs with the same value as humans, that would be great! OpenAI would be a celebrated collaborator if they created as much value as a human does.
OpenAI has already said they aren't going to claim the $1,000,000. If this proof claim holds up, then the Millennium prizes will be 2 for 2 for rejections of the prize money for valid solutions. Maybe that will be the precedent for other AI labs as well.
What is this man talking about. You can speak physics into existance now, yet it still has to be proven with math. Until we are walking through worm holes and driving around in spaceships that travel in a warp drive could he even begin to say there is non-renewable. But even then..
There is also a large incentive for OpenAI to fold user conversations into the training process. Proving this happened is unduly hard, and given that professionals are using frontier LLMs for daily work, such training would make it easier for labs to scoop said professionals.
I have seen private correspondence between one mathematician working on Navier-Stokes and OpenAI that makes it sound like OpenAI deliberately scooped this Navier-Stokes result. The alleged correspondence also contained veiled threats if said mathematician went public.
Well, we name conjectures after the conjecturer not the (dis)prover so there is some incentive to be the guy who comes up with a hard problems. It is curious that we haven’t had something like this improve OR etc. problems. Perhaps not glorious enough.
> Well, we name conjectures after the conjecturer not the (dis)prover
That’s not universally true. Some conjectures are renamed after being proven. For example, Fermat’s Last Theorem is now sometimes called the Fermat-Wiles Theorem, the Taniyama-Shimura Conjecture is often referred to as the Modularity Theorem now, etc.
I think this is where people will have to let go of the ego of being the "sole author" of a solution for us to move into the next era of human flourishing.
Actually there will because if you've actually ever spent time doing math, you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI. The analogy is interesting but incomplete and misleading.
I've actually spent time doing math and literally everyone I know who knows math learnt it by reproving things that people proved before them. I don't see why AI proving things makes this form of learning any more economically infeasible than the field of mathematics already is - and since it was apparently economically feasible before AI I expect it to stay that way.
There are two stages to learning research level math. First you learn to reprove other people work. This is relatively easy because the language, notion and prior explanations have already been optimized to help prove the next thing, and you know a solution exists. Then in your PhD you try to solve problems you don't really know a solution exists, and if it does, how long and complicated it is, what techniques will be used, etc.
You need to solve the second to get a PhD. For good reason. The second is way harder than the first. And now AI is making second way obsolete. It's not the end of the world, but it is the end of how things have been done for centuries.
you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI
The thing is, nobody has time for that. Look at Mochizuki's work. It takes years of hard labor by high-level mathematicians to come up with stuff like that, and years of hard labor on the part of other mathematicians to validate it. The low-hanging fruit in math has all been picked, AI or no AI, and Tao doesn't seem to acknowledge that.
The mathematics community needs better tools or they're out of business anyway. Now they're getting those tools... and bickering and complaining about it?
I agree that the low-hanging fruit has largely been picked. But then I think we should acknowledge that and stop innovating, or just work less on new solutions and more on clarifying old ones, and start to work on degrowth rather than useless problem solving. Because I really don't feel that any of this is really necessary or beneficial for the human race in the long-term.
See also his previous thread from before the result was published (and before he knew it was coming [1]) on how a to this problem seemed increasingly likely to be solved by AI in a way that caused us to miss the insights that would traditionally be associated with solving it: https://mathstodon.xyz/@tao/117207849921390904
There seems to be a sense wher mathematicians are gamifying math, but are frustrated that AI labs are better at gamifying math.
If an AI solves a problem in an unenlightening way, then there's no reason for mathematicians to stop studying it. Pythagoream Theorem has hundreds of different proofs!
If an AI solves a problem in an enlightening way, mathematicians should study it and propose extensions.
It's the same pipeline problem coders have been talking about; once AI does all the work, how are people going to get the experience necessary to take part productively?
Plenty of new open problems will come from applications, and applications are what actually matters. Pure mathematicians are wrapped up in math for the sake of math which is a fun academic game but not something the rest of us should care about.
The lack of reasoning traces in frontier model output is hurting science and progress... thats my take away from reading that, and why open source models are so critical and so needed, because they actually do expose the chain-of-thought reasoning traces recently missing from the frontier models (openAI, anthropic, etc). By encrypting and purposely hiding this important information from public inspection, it makes for a world where people lack the true understanding of how a problem gets solved.
I think he's suffering from a sort of static-universe fallacy. People aren't going to keep doing what they are doing, but secretly.
What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.
Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.
In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.
Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
> If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)
People who work hard to build intuition and also use AI effectively will dominate the field.
This will be literally every field, sooner or later, at least until the human and their intuition is just slowing the AI down. There’s no scenario where humans without AI beat humans with AI in the long run, unless there are fields where the “alien intelligence” somehow hurts more than it helps (like artistic pursuits perhaps?)
This to me is another level of dishonesty. Imagine if a company producing math software starts to hear "rumors" someone is using their software to prove an important result and start massive runs of that software to beat the team. That may not be illegal per se but it is incredibly deceitful. I wonder if the original mathematicians should start a lawsuit for theft of intelectual property.
I would expect this for service providers that provide their "free" services but the mathematicians said they paid OpenAI to help generate this result. This is a clear conflict of interest. Similar to inside trading. Or the same lawyer being hired from two opposing sides which is a big no no. There are rules and laws that already deal with this in other industries and I expect that there will have to be regulation developed for cases where there is a conflict between AI customers and AI provider interest.
This is just the age of slop mathematics, if it doesnt lead to our lives neing improved none of this matters. Math peeps are being nerd sniped by AI in the same way SWEs (the worst ones) got sniped by claude code. Building solutions to problems that dont matter for the sake of doing it just because you can.
You'll ultimately waste a ton of time and get lapped by people doing real world work that actually improves the lives of regular people.
alright, i'll bite: what real work did you put out there that has improved the lives of regular people in the past one year? (without the use of ai, needless to say)
The difference is no one writes punchcards so the comparison is inapt. It is more like supposing AI eats everything humans attempt to do. They will take your work as you are working on it, any interaction at all. It is meta-plagiarism, on another level.
No evidence until there's an answer to the question of if prompt data from the other researchers was used to train the internal model.
I think the reasonable ethical question to consider when a company has an incentive to prove its value, and learns of the state of the art of a flashy research topic being close to a solution, and then throws loads of resources at trying to solve it first. I agree that "AI" didn't "take" but it's not an incomprehensible shortcut of criticism of the aforementioned behavior, not that precision should be avoided here.
Ok so if no plagarism has occured here in any sense then there's no discussion here?
I agree though if they did plagarism it's really really bad for OpenAI. They would instantly have evaporated all remaining good will to gloat over a stolen discovery.
I think there's still an ethical question of a business trying to scoop a researcher solely because the business learned of promising progress. It's not a very big leap to consider "multiple discovery" (a la calculus by Leibniz and Newton) and chance that the model that is good at math proofs might have enough ambient context to work faster from a similar knowledge base on the problem for which promising progress was discovered. I think that's not nice at the very least. Surely the above-board way to approach that is "hey researcher, we learned, indirectly, that you might have made significant progress on flashy research topic and would like the chance to throw our vast computational resources in to try to help work toward the conclusion of that effort." But, maybe that's asking too much?
I think there's an important part which is that the nature of the beast implies there's effectively zero way to do that.
They have no idea what context, triggering this pathway, was from this guys reddit post 4 years ago.
I also assume that if any AI lab could actually do that it'd be an instant pissing context over who gives who credit the mostest. "This software engineer wrote a sick sorting algorthm everyone now uses, thank you random dude for that training contribution"
Maybe we will in the future, maybe we'll learn that the solution to this prize is from 3 peoples schizo rant on reddit a decade ago. Would be kinda sick actually.
It sounds like a fancy way of saying kids will forget how to do math if they use calculators. The flattening of the space makes it hard to find new problems deemed significant? Maybe that went over my head, I’ll concede that point.
What? No, it's nothing like that. It means that if you have a critical insight and you share it out loud, somebody with no particular mathematical inclination can just spend lots of money to steal the credit from you.
It shifts power further from the worker toward those with capital.
> We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.
In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?
Struggling to understand how a solution to a millennium problem like this isn't a net positive. Presumably Open AI employs mathematicians in these efforts anyway. And I can think of far worse uses of the AI compute resources.
If there are no incentives for mathematicians to work on and share progress in tough problems because they will get scooped, then they will end up quitting and in net we will see less progress overall.
From "Jokester" by Isaac Asimov 1956:
"Early in the history of Multivac, it had become apparent that there was one big bottleneck: the questioning procedure. Multivac could answer the problems of humanity, all the problems, if -- if it were asked meaningful questions. But as knowledge accumulated at an ever-faster rate, it became ever more difficult to locate those meaningful questions."
[0] https://web.archive.org/web/20150118004835/http://www.sffaud...
I'd say that a more appropriate reference from that time would be "The Nine Billion Names of God" by Arthur C. Clarke [1], which actually deals with the finiteness of the list of problems that a machine successfully exhausts.
[1]: https://hex.ooo/library/nine_billion_names_of_god.html
I don't see why that should be a problem, as we already know the answer is 42 in any case.
There is as yet insufficient Data for a meaningful answer.
[1]: https://www.imdb.com/title/tt0708807
??? http://www.thelastquestion.net/
The findings of the present study suggest that more funding is necessary, roughly the amount of money necessary for a trip to Burning Man
That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle.
Mathematics has always been highly competitive.
The mathematical community was very competitive in its early years, but in the last 70 to 100 years, it has been generally less competitive and very collegial. The community was in a good place, and progress has been very good. In a few cases when competitiveness was ramped up, it lead to bad behaviour and destructive fights. Few would like to return to those competitive years.
The story of the cubic equations is another great example: https://en.wikipedia.org/wiki/Cubic_equation
Dudes straight up used to hoard solutions to equations and use them in math battles.
Right, the point is we're trying to avoid reverting back to such practices.
Showing my ignorance, but the only thing I can picture when I hear 'math battles' is akin to the 'street Countdown' scene from the IT Crowd
My computer contains the prime factorization of probably several dozen (if not more) large integers, and I refuse to share them with anyone!
(Because they are my private RSA keys)
Why are you still using RSA?
Also Andrew Wiles working in secret for 7 years out of fear of someone scooping him.
Partially; but also in order to be able to focus, as stated by himself in https://www.pbs.org/wgbh/nova/transcripts/2414proof.html:
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
-----
Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
I've always found the story of A. Wiles sad and frustrating. He worked in secret for 7 years. He submitted a (incorrect) proof at year 4 or so. Reviewers found a problem, but he decided kept all secret for many years after. He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I found this behavior against healthy science practices and only driven by ego. Unfortunately, I find this too often at work (working in academia). Most probably I'm too naive...
Obviously, he was trying to avoid being labeled as a crank for working on a famous problem like that for so long.
While I can sympathize with this perspective, I don’t think it’s right to call it driven by “ego.” Sometimes one just wants to go at a problem without being second guessed on approaches or led astray with suggestions by others.
> He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I think that was Ken Ribet?
Grigori Perelman and the Poincaré Conjecture is more interesting. IIRC he turned down Millennium and was decidedly not all about the Fields Medal - mostly because Richard Hamilton didn't get credit? Anyway, I am grateful I had the opportunity to learn about Poincaré in college taking a few classes from a professor who was a key contributor to the conjecture and got a Fulbright for it when I was there
Eh, it seems like it's pretty necessary for success on such a problem (but obviously not sufficient). These problems gain a reputation, and you either get judged for it or get too much attention for it.
Yeah but a highly productive last two decades of math research from https://en.wikipedia.org/wiki/Polymath_Project has come from collaboration.
You miss the point. Humans don't mind competing with others. I love competition, but I don't want to compete with you and your machine. I love to play chess, I don't care if you are grand master, whoop my ass. But not if you are going to pair up with stockfish. I don't even care if you are a newbie that started playing yesterday with an ELO rating of 900. If I wanted to play the damn computer I'll do it myself. Likewise, mathematicians will not mind sharing and competing with other fellows, but if another has a billion dollars worth of GPU and you don't? Then you best be carefully what you say.
Could you tell the difference between a grandmaster and stockfish if playing them online? If not, why would you care which one you are playing against?
Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress.
(the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them).
FWIW, your post comes off as a middlebrow dismissal, surface level and not actually engaging with the substance of the comment. It's also just wrong. You claim "it’s also a misstatement of mathematical history", but don't specify which part. That there's "centuries of traditions of open science"? But your examples are from centuries (and millennia) ago, and there was never any claim that these traditions are universal.
But more fundamentally, competition doesn't mean you can't also have open science. And the very long, damaging events like the Leibniz/Newton feud are exactly what make many mathematicians work to maintain a spirit of collaboration and attribution even when they're competing on approaches.
Nothing in their comment reads to me as "arguing against"
Reads like nothing but historical context
> Nothing in their comment reads to me as "arguing against"
If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.
> Reads like nothing but historical context
They literally accuse Tao of "a misstatement of mathematical history".
They aren't arguing against open science, they are trying to educate you on the history of science. It's always been this way.
Also, your third paragraph is highly ironic.
> They aren't arguing against open science, they are trying to educate you on the history of science. It's always been this way.
Always been what way? And how does that contrast to what Tao said (since it was apparently "a misstatement of mathematical history")?
> Also, your third paragraph is highly ironic.
You'll have to be more specific, since I engaged with my parent's argument, while they waved away Tao's quote by suggesting he was wrong because of exactly the kind of events that helped lead to the norms and mores working mathematicians have today.
Not arguing against open science - it's super valuable. I'm saying that pearl clutching by people reading Tao isn't useful, because it misses some long history which tells us that this kind of science has been seen as fundamentally competitive for millennia.
Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?
These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from competition in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.
this is not untrue but it's a pendulum swinging back to ancient times man
Between people.
There's nothing that AI won't be able to mine and accomplish (aside from being literally human), it's only a matter of hardware and scale at this point. Generalized problem solving is a factor of search efficiency over the problem space. The actual software part is all figured out, the only open questions are how to do things efficiently and what the trade-offs are from a hardware perspective, but if hardware paradigms are unlocked then efficiency becomes a secondary factor for the problems we care about. Why bother making an LLM twice as fast if you can make a chip that can process 100mil TPS, for example. You're already in a ballpark where it can do anything you want, with plenty left to spare.
The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.
Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.
Tao's central point seems to be:
"In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. "
I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.
Doesn't this seem to be where all domains are headed?
I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?
Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do?
How is this any different from people in any field that are impacted by AI and lose the utility of their skills and endeavors over the past decades? Are we saying that we're running out of problems to solve because of AI and hence it should be stopped? I am not underestimating the importance of the collective knowledge of the mathematics community and the role of mathematics as the enablers of other sciences, but opposing meaningful progress in that discipline or any for that matter feels counter intuitive. I would rather have the mathematics community start collaborating closely with the this newly evolving and powerful tool to expedite humanity's progress.
Do you see an end state in this? When AI is better than humans at everything (and I used to be very sceptical of that claim but I'm getting less and less by the day), I don't see the Wall-E version of humanity as some sort of utopia, and that's the good outcome.
Seems to be it, though Im not particularly concerned about this problem personally.
The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point.
Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.
I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession.
Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes attached.
An AI-generated solution always provides two pieces of info:
Maybe the solution is pretty inscrutable, but it's almost always better than nothing.So, both of these pieces of info would be at least marginally useful for advancing human knowledge.
> An AI-generated solution always provides ... proof that there is a solution
This is only true in the most trivial sense. A solution is a solution, sure... but how do you know it's a solution, and not an incoherent jumble of words? A human has to review and vouch for it.
Just because the AI gives you an arxiv-worthy PDF, or a Lean proof which compiles, doesn't mean it proves what the AI says it does. The AI could give you the same PDF/Lean code and says it proves the opposite, how would anyone know the difference?
You can't advance human understanding unless you produce things that humans can understand.
Not an expert by any means but the assumption here as I understand it is that the arxiv worthy PDF would not be acceptable or meaningful for impossible to understand proofs. And the lean proof would be meaningless unless the specific expression being proven is human understandable as the direct translation of the question the human is asking in formal form. So proving the negation is not a thing but if you make a subtle mistake in translating the statement you want to prove then obviously the QI is going to be proving the wrong thing. And otherwise you're relying on the correctness of lean as a system and on identifying/preventing if the proof is adversarially exploiting bugs in lean to falsely prove things.
That's why the solution should be presented in a verifiable formal language, such as Lean. Which is the case with the Navier-Stokes problem.
I might be wrong, but making an assumption that you could learn to read the mathematical output of the AI long before you could write a solution yourself. But hey, what do I know, I'm not a mathemagition.
What does "mathematical output of the AI" even mean? A proof? Intermediate tokens?
It's a Lean program that proves the theorem.
This is definitely true in an information theory sense: having more knowledge is always better than less knowledge. However, it may not be true in math as a social human endeavor, and having answers without interesting paths to get there may not expand human mathematics in the same way.
If Fermat had a book with larger margins, would Weil have devoted so much time to proving the Taniyama-Shimura conjecture? No one can say.
It demotivates mathematicians. That’s a pretty large negative!
* current mathematicians
Were early in this cycle, we will learn to do more, and exercise our new capabilities more fluently, which in turn will create more skilled practitioners
Consider the abacus, calculator, computer, etc, each of these enhanced mathematicians’ capabilities and thus outputs.
This feels a lot like drafters complaining that nothing will get designed when CAD starts being used.
That’s a skill issue.
Will somebody please let Professor Tao know that he's simply experiencing a skill issue?
Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses.
Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career of the professional that takes the challenge.
In mathematics, finding novel proofs of a given result is often valuable; it may be a shorter proof (demonstrating better/expanded understanding of the problem) or a translation of the problem into a new domain, setting up more cross-domain advances.
> Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
A lot. In fields where knowledge is incrementally building on previous work the reason the whole field hasn't collapsed from the replication crisis is that usually the results that are really high impact are replicated in as an initial step in new research building on it. It's almost never the focus of the paper but you'll often find a quick mention in methods/supplemental of some previous work that was verified to be valid by a replication of a key technique etc. you'll have crisis where old tools are found to be problematic and findings end up revisited etc. Plus fields like clinical research where there's an awful lot of focus on replicating findings using staged clinical trials with increasing statistical power to determine if new interventions work - that's driven by regulatory requirements grounded in good science and a lot of people make careers in just that.
I find that to be an issue of maturity (focusing only on the climax and not the process). In Japan, where I live, the culture has a greater appreciation for the context & process, not just the moment of victory.
If you examine the consequences of the inversion of the peak, you realise the need for a balanced perspective.
>Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one’s beliefs in the way new research does. And if the goal of science is to change our beliefs and bring them closer to what is “real”, successful replications can’t be as valuable as the initial research almost by definition.
From a pure statistical perspective the first scientific paper shouldn’t update your beliefs as much as the independent replication study.
People don’t behave this way, but a high percentage of all papers have known flaws and that goes up even higher when you consider unknown flaws. Replication doesn’t own its own solve the underlying issue, but independent replication removes a huge range of potential issues on top of providing more information.
I agree, with the caveat that it probably matters how novel the paper is, the effect size, and the confidence interval. A reputed new psychological phenomenon I'd be more skeptical of than, idk, a newly detected exoplanet.
That alleged superconductor from a few years ago - everybody kind of held their breath and waited for the reproduction.
Hello.
You have created a fraud machine. Why? With no answer checking then why not make up the most fraudulent crap you can get away with?
Examples: A huge portion of recent non-reproducable science papers.
---
Your thinking, along with everybody that's doing this rat race is causing the pumping out of papers with questionable data, but very little to ensure we are actually making correct science.
It is true for mathematics certainly. I would guess it is less true for science per se.
I think successful replications are as valuable as the original research because they're not unsuccessful replications
More of a 'its the journey' rather than the destination type of thing.Since the insights , quirks, tricks and procedures gained along the way allows insights intoother at that moment unknown problem/domains in the future.
As far as researchers sharing their data/notes with the AI hyperscalars looks like that would be coming to an end wihth a mor guild-like structure going forward to prevent their progress being fron-run by the AI labs.
I wonder if it would be possible for researchers and scientists to submit their papers to an organization which would then collect them, submit them for peer review by other experts in the field, and then release them in periodical form ONLY to individuals and organizations who pay a subscription fee in order to read them while suing those who try to redistribute them without permission?
To what end?
Why would society fund mathematicians if they decided to become a guild that hides secrets? They could pursue that as a hobby, but they’d end up like the coders who refuse to use LLMs - rapidly becoming irrelevant and a bit sad from an outsider’s perspective.
Mathematicians will be less likely to work on a problem if there is a solution - even an incomprehensible one.
> Mathematicians will be less likely to work on a problem if there is a solution
Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel.
Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore? I genuinely don't think so.
I think his point is that AI is not creating new problems. It may solve "the Riemann hypothesis" but may completely fail to posit a "Mythos hypothesis" which is vital to advance the field. In fact, achieving the former may make the latter even harder because it will disincentivize production of human mathematics which has till now been the only source of "interesting" problems.
FWIW this is my understanding of his argument and I am not a mathematician.
Have we asked AI to create new interesting math problems? XD
Yes, many mathematicians have.
As Tao points out, merely suggesting new open questions isn't really sufficient. Part of what gives these problems their fame is their notoriety, their difficulty, the fact that many prodigious mathematicians have spent an evening or week or month or several years studying it.
It wouldn't be as interesting if it had just been solved by the fifth random mathematician who considered it
Notably, gardening a new field of study in math is somewhat nontrivial. You have to introduce the field, illustrate some relevance or connections, and then - and this is key - not solve all of the low-hanging fruit yourself! Because you need somebody else to become an expert in that particular field.
The analog in programming is: if a large company merely open sources a product that's decent but not great and in a language nobody wants to maintain, but they don't commit to maintaining it themselves.
Suddenly there's a bit of a vacuum because in order to provide something of value, you either need to:
1. Implement something more complete than was initially open sourced
2. Or maintain something in a horrendous language while incrementally improving it and keeping it relevant
3. Or rewrite it into a tolerable and maintainable modern language.
What the large company has done is create a vacuum in the tool space where you now require extreme motivation to get someone else to step in.
Note that in this scenario, in 2026, it's actually not such a big deal. I think several recent models could happily translate it into a more maintainable language themselves or happily maintain it in the original crufty one. And so the question is: which parts of this analogy are true in math, too?
Mathematics isn't art. It doesn't gain its value in human affairs from being interesting to study. I fail to see why we should cater to that.
What is it, if not an art?
A science?
Have we asked it new interesting math problems, after studing some space for an evening or a day?
The sphere of human comprehensible mathematics is finite. Once everything is solve it is not necessary to advance the field. The recurring error her is to say ai is not the product of human effort but another agent. Ai is human. Ai may well be speeding up human comprehension of math to its limits in which case there is no further need to advance the field and mathematicians might need to get a job. Why is this a bad thing?
> Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore?
In the case of mathematicians, I think not as researchers. What would a research mathematician do? I don't think there would be any reason to try to gain insight from proofs that AI made for the sake of understanding. I don't see what that would achieve besides just retaining extremely niche knowledge (which AI or the oracle already does). The whole point of having that knowledge was to build toward novel work which the AI/oracle does. Also, the time spent and difficulty understanding them could be very high but with no payoff besides just understanding them because the AI/oracle would be used to solve all the problems anyway.
But this oracle doesn't just say true / false. It also gives a proof. That makes it much less exciting (not to mention beneficial for your career) to find another one (or even worse, the same one).
The "proof" is merely an appeal (unreadable program) submitted to a different oracle (Lean).
What do u think lean is? That's like saying a program that works, is inscrutable because it appeals to the oracle of "code test cases" to prove itself correct.
You're either being intentionally obtuse, or unintentionally ignorant.
Yes, the present developments, and the present approach, mean we will not have mathematicians any more.
Your confident re-assertion still doesn't convince me, why do you think so?
I mean, the oracle doesn't really seem so hypothetical right now. And clearly it's going to drastically change these fields, and mathematics, particularly pure mathematics, must change most of all in order to adapt to the existance of a math oracle (or something close to it).
AI companies don't share the dead ends and only sometimes a bit of the process toward success so people don't understand what was curious along the way.
Why was there a prize attached to this problem then? What does humanity get out of this being proved?
The millennium problems is something done by a single institute to motivate progress on known open problems: https://en.wikipedia.org/wiki/Millennium_Prize_Problems
This is my question too. If we are all just going “well that sucks” after AI solves this problem, why did anyone care about the problem being solved in the first place?
Is the bummer that we got a solution we didn’t want - that navier-stokes is not always applicable or something, but we hoped it was?
I think the Navier Stokes problem kind of illustrates what he’s highlighting. I think most people even before AI expected that this would resolve in the negative and that you could get finite time blow up. There wasn’t really ever going to be a situation where the resolution to this question, or really any of the other Millenium Prize problems as far as I know, gives some kind of immediate massive practical feedback.
The hope with many of these problems in math is that in trying to prove that, we get some additional insight into why it blew up that could be applied elsewhere to more general PDEs that cannot be easily controlled.
I think the observation from Tao and many others is that when humans solved these problems, the additional insights into intuition and theory building came for free since humans can give expository on what they found hard or what was their own intuition. This is much more difficult or tedious to extract from an AI model. Even when people did have access to the chain of thought, it wasn’t always very helpful to figure out what was the exact thing that made it all click. This is even more difficult how that the CoT are hidden but I would think the sort of difficulty of extracting the key ideas for a human might be worse now with more advanced models.
There’s a long term aspect to this too where we have historically used these problems as markers for the other parts of mathematics but if AI can solve it all, then suddenly this signal is not very meaningful.
Maybe to bring it closer to home. If an oracle just gave you P \neq NP, then this would be generally uninteresting since this was already expected. There’s a deeper question of why that needs to be answered. However, one would hope that creating such a separation would give us tools that allow us to create lower bounds on a lot more problems we do care about and perhaps some bigger insight onto what makes a problem intrinsically hard or easy. These long term considerations are helpful but are definitely more vague. The remarkable part is that AI is separating the part about proving theorems and the “free” insight you get.
Honestly, the attitude of the math community is a bit cringe and increasingly I think some of the elite/mystical aura is fading. Rather than a rich fertile jungle where AI can barely chomp through a fraction of the luscious terrain, one gets the sense it's a desert and all the oases are running dry.
Current career structure of mathematicians works partially by looking at whether they have solved novel and interesting problems, or at least done theory-building that can help solve such problems. Many mathematicians are also motivated by being the world's first to solve such problems
Removing this measure suddenly means that academic mathematic norms need to adapt rapidly, and, even more importantly, intrinsic motivation for many mathematicians needs to change rapidly. That is understandably a sea change for the current mathematics community.
> and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge.
You'd be more sure if you read the tweets.
Tao's point is very simple.
1. Working on problems that AI solvers can solve is a waste of human time.
2. We have no idea which problems can be solved by AI solvers...
3. ...Because the AI labs are keeping their negative results secret, and don't tell us which problems they've tried and failed to solve, and why they've failed to solve them (or succeeded at solving others).
There are additional points surrounding it, but that is the thrust of his argument. His issue is not the existence of AI, but the anti-scientific secrecy in how it is used to solve problems. All the incentives around its current use result in closed, uncollaborative work - which while very attractive to a vulture capitalist, is anathema to scientists.
---
He also posits that having a solution to a problem is a small part of the value of solving a problem. What the AI labs are doing is the equivalent of a student turning in their homework, which has 100% of the right answers, but with none of the 'show your work' steps. Those steps are a critical artifact for doing mathematics, because the process of solving a difficult problem teaches us things about other problems.
It seems relevant that Terence Tao is the author of the paper that just about convinced everyone that the Navier-Stokes equations blow up in finite time, 12 years ago: http://arxiv.org/abs/1402.0290
We're having to rediscover in real time the extremely hard way, why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system
If theft becomes more profitable than genuine creation, then nobody will create anything. Then there's nothing to steal, at which point all progress collapses
What theft? LLM output has never been copyrightable.
I'll give you the benefit of the doubt. GP was referring to the use of copyrighted material to train LLMs.
Which is not "theft" according to current legal precedent, and also common sense.
It is theft under common sense.
Only in the sense that you reading a book from the library also constitutes theft of knowledge. Does it?
1. using copyrighted material to train LLMs is fair use, not theft
2. The topic we are dissussing concerns LLMs being trained on logs from previous LLM chats. If you're prompting a model and it spits out some unique mathematical insight, you do not have copyright on that.
Nobody owns math and it is ridiculous to suggest someone should.
> why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system
What is interesting is that LLM's do not directly violate copyright. The settlements we have seen are for how the works were acquired (that was a copyright violation) not the use of the works.
The vectors of a book, or a paper, are not the paper. They are, for all intents, facts about the work itself, and more generally writing. You can not copyright a fact.
It also means that the weights, the things that (mostly) matter can not be copyrighted either.
This is literally why we need a functional copyright system
To block progress. Got it.
This is the literal opposite of progress: stealing from people genuinely creating, and stealing the money they should earn
Funny, the stuff that was stolen is still there. A strange kind of theft.
Copyright maximalism is a bad look on a site called "Hacker News." Perhaps other sites beckon.
> Copyright maximalism is a bad look on a site called "Hacker News." Perhaps other sites beckon.
Frankly it's more of an insult to the "hacker" name to be apologising for big companies profiting off of frontrunning existing work for PR purposes, if the claims about piggybacking on human-directed efforts/prompting are true.
Being pro-copyright in order to protect the work of an individual from being reconstituted into the corporate machine is VERY hackery. Novel use for an existing tool, to fight the dominant system.
(Of course, we're on a so-called "hacker" site hosted by a company run by squarely-establishment individuals acting in an extremely un-hackery-field (investing), so the irony here has been at least one layer deep since the start.)
It is now clear to me why the AI labs are sponsoring these mathathons: https://mathathonchallenge.com/. They are basically crowdsourcing human researcher data to get access to promising directions possibly later to scoop others.
Not so dissimilar to my first comment on this site (which I got piled on): https://news.ycombinator.com/item?id=48959395
Except way more nefarious than I expected
Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them?
Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource."
The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and offer a prize to solve them.
The incentives are massively skewed towards the AI labs investing their massive amounts of compute into being the first to solve an outstanding problem.
It's a marketing game for them, any societal benefits are secondary. Winning a prize is going to get headlines and feed into the "AGI soon, machine replaces another career" narrative they crave unlike coming up with some (possibly) interesting problems.
I think the problem with AI asking questions is that it will ask questions that are interesting to it but not necessarily us. AI, as a model, will never be a perfect copy of a human. It will always be a simulation, and thus to some extent, will ask questions that humans find irrelevant and solve problems that humans find irrelevant.
For anyone facing an existential crisis on AI, your ace in the hole is your humanity. Only you have it, and only you will be the best judge of what is good and interesting (to a human at least).
>your ace in the hole is your humanity
Average HN Poster: [nervous sweating]
If AI can generate questions and then answer them, what are the people for?
If humans can shovel dirt, then what are the ants for?
https://en.wikipedia.org/wiki/Decline_in_insect_populations
I didn't realize that open math problems were a finite resource.
I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.
They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community if people are disincentivized from trying to solve deep problems or study them for understanding as opposed to simply focusing on verification. It's more of a social and community problem than a fundamental problem with mathematics itself becoming "completed".
That explains a lot on why his arguments always focus on the "social part"
So we can let the ai generate some math problems based on the solutions found? Other fields (computer science, physics, ...) can generate math problems too.
There's an infinite number of possible math problems, but the things that make these open problems worthwhile is they're interesting to people who have worked in related areas.
They're good to give to new mathematicians, and they're good to help humans understand the shape of the problem space and relative difficulty with the tools we have.
Cheesing these problems with LLMs gets rid of both the training benefit and our ability to create good related problems. There's an aesthetic part of this, too, that LLMs do not capture.
This kinda reminds me of the guys who decided to industrialize digging up dinosaur fossils, in order to feed the dinosaur fossil collector market. They were amazed that paleontologists were so "inefficient" at finding and digging up dinosaur fossils.
But from paleontologists' perspective, they go out looking for dinosaur fossils when they have questions that digging up a fossil may answer. The metric they're focusing on isn't tons of fossil mined out of the ground, it's a developing understanding of extinct life.
These open problem solutions often reveal tighter bounds on prior conjectures. Even if the solutions produced are far from elegant and only machine verifiable, we do learn new information. But I agree that just like writing prose and code, brainstorming frontier math proofs is a perishable skill
tl;dr - it's content creation rather than process and understanding
It's easy to come up with new open problems. It's hard to come up with new open problems that seem to teach us something fundamentally new about the world. Our current batch of problems went through a complex selection process over decades (or centuries) based not purely on difficulty but also on perceived insightfulness.
I studied math, but I am not a mathematician, so I think I have a slightly different perspective on this than Tao overall. This is certainly the definitive end of an era in mathematics, but I think he's wrong that insightful new open problems are truly non-renewable. They might be non-renewable by humans at the rate at which they are being closed, but I see no reason why AI systems could not also discover insightful new open problems. In fact, once we have Riemann-capable AI mathematicians, I'd personally love to see what the next Riemann hypothesis is, which even these AI systems cannot solve with any amount of available compute.
I think we're about to find that, on the spectrum of mathematical intelligence, the best human mathematicians were only a fraction of a percent forward from the very beginning, and there's a vast universe of mathematical depth that's beyond our ability to imagine or work on directly in any way. We're used to feeling like we're able to directly perceive the Platonic realm, but we're almost certainly going to discover that our own minds, even when joined together over centuries of deliberation, can only interact with a tiny little shadow within it.
What you are saying implies that by some technique that hasn't been discovered yet, we can make the models to have the capabilities of extrapolate the information they are trained on and also interpret that what they are extrapolating are Riemann-capable hypothesis. I do believe it will accelerate the discovery of that "vast universe of mathematical depth that's beyond our ability" but at the cost of removing the "fun part" of solving the problems. Not sure if the community is willing to do that.
I haven’t been following the AI proof stuff very closely, but the impression I got was that these models are producing massive Lean programs that prove the statement one way or another, but are quite difficult to fully understand.
Actually, I have to admit I don’t really know what math is. With physics we suspect there’s a universe, and when we study physics we’re improving our description of the behavior of that universe, right? The universe exists whether or not we know how it works.
Eventually, as you suggest, maybe we’ll hit math that won’t fit in anybody’s head at all. What is the nature of mathematics that doesn’t fit in any human’s head? Does it even exist in some sense?
I think part of mathematics is taking things that don't fit in our head and giving them human abstractions so they can.
Take infinity. Infinity can't fit in your head, hell, it can't fit anywhere, but you can abstract away the endlessness and look at infinities of different sizes, et al.
Now, is there a single formula for something actually represented in this world that would take most of a humans life just to read it, no idea.
The models produce both Lean code for formal verification and a traditional-style narrative proof. Like the general long-form output of frontier models, the math papers produced appear to be generally correct technically, but written in an ungraceful and sometimes hard-to-follow style, so they are often polished by a human mathematician as of today.
I'm surprised nobody has stated the obvious: a hard math problem that has been open for ten years (because many serious people have given it serious thought and been unable to make significant progress) is, in fact, nonrenewable.
The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a tree is twenty years ago, the next best is today: we do need to start posing some hard math problems and deciding if they are interesting merely because there are challenging or because of something else (eg busy beaver problems are arbitrarily hard, but does solving them imply anything other than "another busy beaver problem was solved"?)
Eh, if AI quickly solves most of our mathematics problems that are solvable then it might be time for us to hang up our hat as our little monkey brains aren't very good at this stuff.
Now, I think AI will solve some, but we'll find out that some are just either unsolvable or wildly huge that nothing is solving them any time soon.
And a whole lot of these problems have been around quite some time, when even knowing how to do advanced math meant you were a landed gentry or someone of high wealth. If those problems fall, they fall. They aren't pets we keep around forever. And new problems will crop up over time for both AI and men to scratch their brains over.
I think "close to completion" is not the right framing. Creating good open problems was an achievement because these problems often sit at the edge of known techniques, and solutions require inventing "new math". It's hard to find these problems, and they take decades to mature as they withstand scrutiny by many people.
In another comment below, I likened this to clear-cutting a forest. Growing the forest takes a lifetime; destroying it could happen in the next few months.
his whole point is that specifically problems that have been held as important by consensus in the field are a finite resource. obvious example being the Clay millennium prize problems. seems like they function to shape the direction of future research into useful directions. which is to say, the process of developing a solution itself generates more useful problems.
of course thrrr are tons of problems once you remove this social consensus based filter. if i’m not mistaken Ramanujan left a book of dozens of unproven theorems, for one quick example. i don’t think that that has opened up dozens of fields of mathematical research.
> the Clay millennium prize problems
augmented Hilbert's problems of 1900.
Surely mathematicians are creative enough to ask new questions?
If not, then the next set of challenges will be to find questions to ask!
Did you read Tao‘s tweets? That’s what he addresses
Deforestation might be a better metaphor than mining. Logging is renewable if for each tree you chop down you plant several more. AI companies are operating "in a non-renewable fashion" by chopping down trees without planing seeds. Open problems are a renewable resource, but only if harvested sustainably.
Relevant, interesting problems that we have some immediate hope of making genuine work on might be, if not finite, quite difficult to produce. And it's also plausible that AI will not do as good a job of producing these as it does at solving them.
The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.
> I didn't realize that open math problems were a finite resource.
That is exactly what Tao is explaining in that tweet.
TLDR: Open Problems are infinite, but those which are at the boundary of easy and hard problems and are interesting are far more scarce
He addresses your point in the first paragraph.
> I didn't realize that open math problems were a finite resource.
There's an interesting commentary about this: https://mathstodon.xyz/@tao/117237320796901560
> famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Web search turns up Gauss's comment, with a bit more nuance: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of." (https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)
You can indeed generate many nonsensical problems. Generating ones which require interesting and non-trivial mathematics is much more difficult.
I think you can't have read the thread. The whole point is that there is no end of mathematics, an infinite sea; but the constitution of an 'open math problem' is a delicate piece of mathematical thought, at any moment a small supply of drinking water developed by finitely many human being.
This is worsened by OAI/Ant's strategy of grabbing the glory and running rather than spending effort trying to to advance understanding of math. Tao, who is quite sophisticated in use of AI, has said a lot about this, including in meme form: https://mathstodon.xyz/@tao/117068266026618494
The AI labs' approach to math is immature in a way they can't get away with in coding. In coding, they realize that a pile of code that technically works is not enough: they need the code output to be a foundation to build on, and they need their agents to work well with humans which means explaining things in a way that makes sense.
In math, their goal seems just to be to exploit mathematics' reputation as full of hard problems with a general population that can't tell a pile of Lean from a good proof. OpenAI pretty much said this work is just to show off at the end of the post. Anthropic said their FLT formalization is a research artifact they do not intend to clean up or improve in any way.
Besides uniting mathematicians in irritation at the labs, the other flaw with this strategy is that it ignores that organizing knowledge is part of intelligence, much like not just producing a mess that runs is part of programming. You can write a proof that uses algebraic geometry because someone organized what could have been a bunch of disparate ideas (or fragments of a Lean repo no one will read) into a toolbox where an expert can find the tool they need.
I hope they change tack. Perhaps instead of making an explicit strategy of taking the credit from mathematicians but doing little for actual understanding, they could let some math departments at their swarms or best models, ask for a bit of acknowledgement, and hopefully they approach it by trying to write good papers, simplify, etc. rather than just rushing for headlines. (Tao's post about digesting an LLM-generated proof https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the... is an interesting read for a sense of what he means by 'digestion'.)
On that last note, it's also important (Tao's also noted) for the mathematical community to properly value digestion and organization of results, so that given the incentives of mathematics and availability of new tools you end up with good papers and textbooks and so on, not just mathematicians taking the labs' current role of pushing incomprehensible-even-to-specialists proof code to repos.
Besides Sendov’s Conjecture, Terence Tao has also shared his AI-assisted digestion of the counterexample to the Jacobian conjecture. In simple words, digestion just means full human understanding of the results. It could come from understanding the result from a different perspective, or perturbing the result and seeing what breaks.
The Jacobian one is great, and fits here -- 1) the counterexample came out as a tweet of a single expression which seems exceptionally hard to turn into something sensible, which crystallizes the lab's approach; 2) it links (like the Sendov post) to a chat transcript showing a little of what was involved in untangling it (though of course all that went into asking the right questions is invisible!).
For other folks, the post: https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
The transcript: https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.
Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).
It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.
This reminds me of the time an AI was taught how to play a racing sim game (Gran Turismo if I remember correctly). The AI was able to race its car very well, but it took a lot of risks that a human player probably would not. A human player might be able to copy the approach the AI took, but they would probably crash.
Going back to chess, I think the situation is similar where you can’t expect an amateur player to get better by trying to play like a strong engine. I think even professional chess players mainly use engines to prepare or memorize variations that are counterintuitive for their opponent. In other words, getting into situations that look wild, but that part of one player’s preparation.
I’m not sure how it is in math, but in chess, it seems like top players can play just like engines when they are in “normal” positions, so that is where I get a bit confused as to where the direction of insight is coming from because it’s been my view that AI is able to make leaps that we would never think of taking and I’m not sure that anyone could actually learn how to do that on their own unless they were willing to keep failing over and over.
> The AI was able to race its car very well, but it took a lot of risks that a human player probably would not.
There is a parallel with autonomous vehicles in real life. On northbound 1 in SF going through GG Park, the left turn lane onto Crossover Drive is always backed up. Waymos often do a very late merge into that turn lane in order to jump the queue and save time. With 360 degree sensing they can do this safely in real time but it feels too risky for most humans to attempt.
Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?
This is the direction of experimental particle physics and observational astronomy, where the budgetary scale at which progress occurs means we now fund these efforts at a societal level. These fields have graduated beyond "tabletop science".
For 3000 years mathematics has only been a "tabletop science". Even big programs like the classification of finite simple groups have been comprised of small teams chipping away at different (publishable) parts of an overall program.
This latest Navier-Stokes advance cost something like $22m in tokens, already well beyond what a mathematician's research grant can fund. As the easy open problems get mined, the cost of frontier progress will continue to climb. Some part of mathematics as a field will need to transition from tabletop science to big science: Coordinated top-down programs addressing high-priority objectives.
TBD is what the role of individual mathematicians will look like in a "big science" paradigm, but we could look to experimental high energy physics for ideas. For all practical purposes, AI converts math from a theoretical field into an experimental/observational one.
Yes. But this is hard for mathematicians to stomach, because like everyone else, deep down in a place where they don't like to talk about at parties, they have egos and a sense of purpose based in part on demonstrating mastery of a technically difficult field, as well as social connections based on their participation in it, and taking all that away from them probably feels like a kind of death.
The situation is not that different from John Henry competing against the machine. The question is really: Do people deserve to be allowed to continue doing what they have always done, when doing it is no longer necessary to advance the greater good?
I totally understand and agree, I just feel like with the progress that we have in automating informational work, you are going to have an exponential amount of these "deaths" as new fields where humans can provide any kind of value get more and more short-lived.
At some point in my suggestion the machine will ask better questions than you, and that will be pointless as well, and you keep doing what you like doing, or you move on to something new. But if you keep tying your value to outcome and recognition instead of process you are going to have some incredibly depressing years ahead, and every time will just be as hard to stomach because of your ego.
So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before?
Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
If this is V0.5 of AGI/ASI then by V1 the only system that will be understanding any of this is the AI itself. If AI creates a new field of mathematics month 1, then solutions to new problems in month 2, then another field of mathematics on top of that at month 3 there's no human who will ever keep up with that.
Or the alternative is a flattening of abilities, the AI cannot proceed further than the collective intelligence of humans and in that case this is correct. We'd be in a future where nobody wants to work in a field with an AI dominating it and when AI hits the limit of no useful training data input we'd have this giant gap of nobody know wtf it's done for years and nobody willing to figure it out and advance it.
Ooo here's a dytopian story: - AI gets better at everything humans do - humans stop trying - AI cannot improve anymore than its input data + human support - AI slowly degrades itself (model collapse) for decades, it slowly hallucinates little by little until its hallucinating entire scientific fields losing quality over time - there's a mass population of people in the future who never learned to do anything and now have to relearn and figure out the equivalent of 100k years of AI work in order to prevent its slow degredation while all the systems they've come to rely on start failing around them. The AI has solved every problem but every real solution is saturated with 1000 false ones. - humanity starts from scratch?
I love the idea of an archive of every solution to every problem existing but it's impossible to figure out the correct one. Infinite library like!
Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.
It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.
In that case isn't it actually more valuable to have an AI do this? It works faster and solves more math.
The random engineer looking at a funny problem 10 years later now has the literal author of the math to talk to about it and implement it.
I have never even spoken to a world class mathematician and now I can have them design with me?
How is this not better in almost everyway?
Guess it depends...
If said human is kicked to the street with thousands of other homeless people that can't get jobs because AI then robots replaced them, then those fast math problems sound like a pretty bad trade off.
Now, if there's some future where AI leads to abundance and we can all live off UBI, well, probably a worthwhile trade.
The biggest issue I see is the more controversial people leading the AI race at the moment are not the kind of people I'd hand kids safety scissors much less the future of the human race.
there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.
Fun suggestion, thanks!
> Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.
IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.
So...why can't an AI do the exact same thing? Make AI so it understands math better for future math to understand more math.
Unless your argument is that mathematicians are effectively useless?
I am assuming that's not your point though.
> Unless [...] mathematicians are effectively useless?
It's always been a bit bizarre that this isn't the case. Mathematicians are almost always working on problems that there is no good reason to expect to have utility in the real world... problems they selected because of their elegance or whatever... yet there is a strong historical trend of their work having huge importance after the fact. Sometimes in fields that weren't even invented yet at the time of the work.
There's something to be said for the idea that disrupting a system that is working well for no apparent reason is a bad idea.
So solving and discovering math is or is not the core value a mathematician provides?
If AI can perfectly replicate their work but faster and better then what?
SWE have nobody crying for them as they've been massively disrupted.
Think of it like software going from programmers understanding every instruction, knowing where every byte of memory was being used and why, and using this knowledge to build optimised systems
During the process of optimising and understanding the programmer might learn something new or have some kind of "aha" moment of insight that might lead them down a new path of study where fantastic new technologies and capabilities can be realised
Fast forward to 2026
Most web pages take several seconds to load
Applications crash often for no apparent reason
A vast majority of programmers have no idea what their applications are actually really even doing anymore, so they stack bloat on top of bloat and if something breaks, well I guess that's someone elses problem cos I have no idea what's going on anymore
There's something to be said about levels of abstraction being useful, but abstracting away understanding of the task itself is not the path to generating useful knowledge or applications for humanity
We might be gaining the "what" but we are losing the "why" and the "how" and these are generally fundamentally more important
The answer is 42 but what is the question?
Mathematicians provide two complementary services bundled together.
1. Proving theorems - what AI can apparently replicate faster and better.
2. Creating definitions and new theorems from those definitions to prove, selecting which of the possible statements to work on. I.e. developing the "language" of mathematics. So far there is no evidence that LLM can do this at all well. And there's some reason to think that mathematicians won't be as good at this if they aren't also doing the first part.
The value to society only comes when they do both "well", and it's 2 which is really the black magic where we don't understand why they've been so useful to us.
Fair take.
So I totally agree if AI also cannot do the second part better than a person.
Honestly though, I wouldn't want to take that bet. I never thought that the first thing AI would become super human AGI like is math.
You ask me 10years ago and I'd think the opposite. I think we all would have said we'd have super human HR employees before a super human mathematician.
But here we are.
We don't understand why? Reality is mathematical. As evidenced by lawful physics.
Math academia was not working well at all. Almost every single graduated from my PhD program wound up working in ads or finance.
The gatekeeping in math academia is extremely unfair, or should I say objectively fair but personally unfair. I won’t cry crocodile tears.
> wound up working in ads or finance
Because there's lots and lots of money in that and there's not in funding pure math. It sounds like your problem is with the people holding the purse strings.
Perhaps an AI could! Today they do not, because the people driving them understand constructing the proof rather than understanding the proof to be "the problem".
(I suppose it's possible that in some distant AI future there might be no value in people understanding theoretical math, but I'm pretty skeptical of that; to me it seems like the same error as thinking nobody needs to understand multiplication because you can ask the computer to solve any multiplication problem.)
It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.
People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.
Knowing the answer has value. But, often in math the best thing was how someone got there.
Pure math is practiced mostly for the intellectual thrills and recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.
This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.
There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.
> Now, Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
This is an absurd thing to say. Hacker news is not the only place that knows about the most famous mathematician in the world. Glancing at Google Trends he seems to be roughly as famous as Linus Torvalds. Not exactly a household name but by no means obscure.
I'm going to charitably assume that you forgot to include quotes around the names in your query, because that's absolutely not what Google Trends shows.
Stop 100 people on the street in NYC or Berlin or Tokyo and I bet none of them will be able to name any living mathematician. A few of them might know Linus, though.
Yes, it absolutely is what it shows[0]. I have no idea what data you're looking at that suggests otherwise. Worldwide or in the US, as public individuals or as search terms with quotes or without quotes, over any reasonable timespan, they get roughly the same amount of searches.
[0]: https://trends.google.com/explore?geo=US&q=%2Fm%2F047mjr%2C%...
Well no because it works by joining together existing novel insights.
Today, maybe. Where's the law of nature that says it won't be generating novel insights in two years? Five? Ten?
This series of posts by Terry Tao is a direct response to the Navier-Stokes results (multiple results!) from the last 24 hours. The question is what is left after the levelling of mathematics, in all its senses, occurs? How can you protect a field that's under this much pressure in the next 6 months?
> [I]t is now the identification of a promising problem which is the scarce and precious resource. We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.
If math is solved then move to an area that’s not. Why do fields need “protecting” from ai?
Have you read his tweets?
yes but the tweets don't really address why the field needs to be protected instead of adapting and evolving.
And to reply to the sibling since I hit my comment limit and I'm going to probably forget about this conversation until tomorrow:
But our situation before 2023 was one in which we had an endless abundance of solutions and ideas. I understand that AI can generate bad ideas faster than we can discern them, but we already have tried and true mechanisms to filter good ideas from bad (e.g. the scientific process), why can't they be adapted?
Math can never be fully solved by a computer, if only for lack of computational resources.
Math can never be solved, depending on exactly what set of axioms you're using. Certain problems can be though.
The reason is because the entirety of society, historically, has been based upon humans using their differential skills to further it, which in turn promotes societal cohesion. If most human endeavours are solved, then we will enter a period of abundance that paradoxically will erode the glue holding society together. In short, endless abundance of solutions and ideas cannot coexist with a healthy society. Only those who are priveleged and have a naive belief in a Star Trek utopia think otherwise.
>endless abundance of solutions and ideas cannot coexist with a healthy society.
This really cuts to the heart of the problem with AI. Not only does AI undermine the monetary economy, it undermines the intellectual economy. What is humanity without the need for collaboration for survival or for intellectual progress, ultimately providing the impetus to build something greater as a result? I don't know, and I'm not looking forward to finding out.
This far into history I don't think many are going to buy a "the next technology is to be the undoing of society itself" pitch until after society is already undone by something. They're as easy to make and hard to concretely evaluate ahead of time as the utopian predictions while offering little in the way of practical approach to preventing the same result from occurring anyways.
I mean the past 10 to 20 years of social media have shown a lot of societies glue already breaking.
A society can live just fine in a period of abundance. The societies that we currently have on earth do make it questionable of 'we' can right now. I mean I see people posting stuff like "I'd rather burn it all to the ground rather than see one cent more tax" kind of stuff when they have millions. That kind of person doesn't want more people uplifted and it takes away from their idea of being special.
>naive belief in a Star Trek
The naive ones don't read into ST lore to know it comes after WWIII.
The dangerous ones do.
https://en.wikipedia.org/wiki/Commodity_fetishism
> How can you protect a field that's under this much pressure in the next 6 months?
Sounds like they're going the way of the DoDo. better take that PhD, migrate to the new world and become a tuktuk driver.
Aren't we in a similar position to what chess went through in the 2000s when Deep Fritz came out, and a desktop PC was able to defeat a reigning World Chess Champion? Did chess players just give up and stop playing? No, they didn't. They used these new chess engines to become better players. Computer programmers and mathematicians will probably go through something analogous.
Presumably it is only a matter of time until these frontier models are used to create new interesting conjectures. I don't get Tao's line of reasoning.
Perhaps his takes are evidence of just usual human fear to new things. I see he relies quite much on the "community" or "social" aspects of the discussion.
I probably have delusional expectation of what a mathematician of his level should be talking about, but I expected from him a pure objective analysis on what to do with this new AI thing , what are its limitations, how it can improve the field and the creation of human knowledge, etc.
Seems like in current cultural and economic context, short term extraction is what we’re going to do
> In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.
Is there an analogy here to the phenomenon that senior {engineers, designers, PMs} are now able to be insanely productive with AI, but it's also very hard to train junior folks to develop the sense of judgment that senior folks have?
The "ecosystem" is dead. Tao should be thinking about what will replace it. I don't understand why he's taking this tack.
I think this highlights one of the fundamental differences between humans and our current AI systems. They can still only try to solve problems in the given well defined parameters they are given (with some exceptions). The human is able in the effort to solve problems to intuit where there may be new interesting problems adjacent to the current problem.
I don't think we can confidently say, yet, that LLMs can't discover those connections. We just haven't explored them yet, because 99.99% of the prestige is locked up in proving hard results, which also happen to be easier to assess objectively (thanks to automated proof checkers), and so that's where all the effort has thus far been exerted.
No matter what happened this must be a wake up call for all of us. We’re basically sharing everything we have with these companies/AI systems. This is wildly different than a human wiretapping our private messages. Because it is systematic and automated in an astronomical scale. There is no real privacy in this new world. Law? I think “National Security” is a good enough excuse to screen anything constantly, including foreign researchers in case they are close to a breakthrough.
>We’re basically sharing everything we have with these companies/AI systems.
My sense of the word 'share' is that it traditionally involves agency by all parties involved. There are a lot of words in English for describing taking things without permission and profiting thereby - words like piracy, banditry, and larceny.
Next you will try to tell me that these companies built upon nothing but piracy, banditry and larceny would continue to commit piracy, banditry or larceny.
I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.
Probably an AI-written Lean proof is very different to how a human would write it, and some may say it's more like mathy neuralese. For sure it works but it is not human-friendly and needs to be transformed into something more readable and digestible to be able to extract insights from it.
Not that different from when trying to read an out-of-control vibe coded codebases, or an sloppy AI long email that someone may send you at 9 AM.
Tao has a spiel in his recent interview with Dwarkesh where he says that AIs are very good at explaining things - so just have the AI explain the proof in a human-friendly way.
For sure, but this was supposedly ~18 million dollars of compute, afaik 100 pages paper / lean proof and only god knows how many bytes of chat interactions + thought traces. Scale matters.
I bet it can be decomposed quite nicely though. At the top level, there are probably only like five steps. Dig as deep as you want into any of those steps (i.e. engineering).
Hopefully. We'll need to wait until mathematicians confirm how easy it is to digest whatever GPT did.
Because the problem has almost no value unto itself. The clay statement of navier stokes is not relevant to how CFD is done in practice.
It's about what is non verifiable versus verifiable. The same way it produces "slop" code (which, if you give it test cases, will be 100% correct), it also produces "slop" math.
Code that serves a business function, it's ok if its slop. Math that serves directly a business function also can be slop.
But most open problems are not directly for a particular usecase. People agree widely to attack it due to the perceived possibility of encountering useful mathematical objects along the way, that will then expand the world's mathematical toolset. This is not something that you can easily express in a verifier, and is thus something that is hard to force an LLM system to do.
You are right in that understanding it retrospectively is possible, but that is not going to be as useful as the desired "elegant" objects that expand and unify mathematics. You can't represent these concepts in verifiers.
Again, if you let AI rip at something like say "beat shannon capacity" and suppose it comes up with MIMO as paulraj did, great! It's useful and you can retrospectively understand it, say by expanding shannon to multiple dimensions, as foschini and telatar did. But most math problems are not in that category.
The question then is, if AI is really good at this type of math, how much of the existing mathematical community+process is necessary? I think it will still be necessary, just maybe in fewer cases. Wherever the primary purpose of the math is in a domain and that domain has a verifiable target, we can directly optimise it to that verifiable target in-domain rather than reach for the mathematical community. How well will this work? We'll see. It's not clear if it's even possible to represent most problems this way.
> Code that serves a business function might as well be slop. Math that serves directly a business function also can be slop.
Both of these are simply incorrect - serving a business function means it's valuable to that function.
I phrased it badly just out of bed.
I meant what you're saying. That it's OK if it's slop if it serves a business function.
Edited
We're running out of math. Maybe the president needs to establish a Strategic Math Reserve.
My model of mathematical intelligence for a little while now has been 3 levels:
1. I give you a proof, you tell me if it's correct
2. I give you a theorem, you give me a correct proof
3. I give you nothing, you give me a theorem
1. is largely solved by modern LLMs and they took a big step toward 2. today with the Navier-Stokes proof. But they're definitely not there yet. It's unclear what progress is being made toward 3. for the time being that remains the realm of humans.
>"While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions without such analysis would be of negligible or even negative value for these purposes."
Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...
Perhaps reading https://www.math.toronto.edu/mccann/199/thurston.pdf will help.
The point of mathematics is not to prove results. It is to build conceptual thinking about mathematics. Important problems are important because in order to solve them we have to build concepts tying different things together.
We're not searching for answers. We're searching for insights. Trying to understand the problem causes us to draw the connections and find those insights.
AI gives us answers. But it doesn't help us build those insights. AI has a complete mastery of existing human insights. But doesn't build new ones from its own experience. In a real way, it does not find the opportunity to really learn.
So it tackles problems and either solves them or not. If solved, we now have an answer. If not, it's too hard for humans.
Not a mathematician.
The issue I see with a handed-over proof is tunnel-vision: you explore only the understanding of the proof.
Without a proof, your exploration branches out much further, in directions that could seem fruitless, but may uncover new understandings that are now "hidden" because the handed-over proof drastically lowered the incentives to find them.
- If you have only a fuzzy idea of how to get to your travel destination, wrong turns and alternate routes may reveal sights and places you'd never have encountered without that wandering.
- If your GPS directs you straight to your travel destination, you are now where you wanted to be but missed out on the exploration. This is the sort of consequences the AI math proofs have.
STEM research thrives on that side exploration and unearthing unexpected things along the way. James Burke's famous documentary Connections spends the middle episodes talking about the unexpected directions that exploration has taken science. It's very hard to credibly make the case that this sort of meandering exploration is not valuable.
I think you're misunderstanding the point of math problems. Mathematics is as much a process as it is a result. This is why even from early on, relatively rudimentary mathematics questions you are graded by your capacity to correctly achieve the desired process to the answer than getting the answer correct. The risk here is that AI generated proofs removes the process part of mathematics, where actually interesting concepts live (because then you can apply novel concepts to other unsolved problems and then thereby unlock new concepts that way...) Sure you can kind of try to reverse-engineer it but you lose the entire intuition and "we tried applying it in X, Y, Z ways and it didn't work" intuition, because even the non-working process can teach you about how not to apply the working process to novel problem spaces.
Basically: Tasting a delicious soup doesn't tell you how to layer the flavors, but if you want to be a good chef, you better be learning flavors more than you learn dishes!
He's saying that in such a scenario, almost all of the value is located in the analysis and just dumping the proof has "negligible or even negative value". (The negative value would occur in the cases where the proof doesn't contain enough information to reconstruct what insights would have led a person to it.)
@Practal's comment is interesting:
> Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.
Perhaps the pace of applied mathematics would rise sharply, given cheap intelligence. And this* may end up being the forefront driving progress in mathematics.
*Or maybe a split between the human domain and the practical real world. Where the human domain might end up with a variation of a "No machine contributions" policy. Sorta like the recent gcc policy.
Isn't it safe to say that all famous unsolved math problems will get a "massive amount of AI-powered effort" pointed at them regardless?
While AI companies have almost infinite money, they still don’t want to blow million dollar budgets on problems if there isn’t high likelihood that it will be successful.
But within next 10 years as costs drop significantly and even more improvements are made, yes it is very likely that almost every single existing math problem will get a serious AI cracking done on it
A "massive amount of AI-powered effort" costs a ton of money. What's the return on investment for these frontier labs? Do headline-grabbing successes in mathematics translate to expected *profitability* in disciplines with more immediately quantifiable economic value.
Open math problems, yes; also open source code, art, literature, and everything else as well. AI is a machine for turning commons into tragedies.
The way to tame a profit-maximizer is to make the most profitable choice the one that creates the most societal good.
I would like to see the Clay Institute give zero recognition for formalizations without human-readable proofs. That would incentivize OpenAI to scram or create something that's actually useful.
It would be trivially easy to convert Lean into English, so I’m not sure what the human-readable criteria gets you. There are also human written proofs that are considered not human-readable by most of the mathematics community (ABC conjecture).
Human-readable means multiple humans can read and understand it in full. A human-readable proof is more worth more than one that is not, because mathematicians can read the proof and extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. A proof that only a few humans can read is more useful than one that no human can read because the few mathematicians that can read the proof can still extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. That's what the human-readable criteria gets you.
Mochizuki's claimed proof of the ABC conjecture is not unintelligable; it has errors. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
The four color theorem states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. It was the first theorem proved with substantial computer assistance. The theorem was proved by showing there could not be a counterexample. The authors made a list of maps where if a minimal counterexample existed, it would be one of these maps. There were 1,834 maps in that list, and each one was checked by computer. You could turn each of those cases into a picture or paragraph, but the resulting artefact would not meet my definition of human-readable.
Human-readable does not just mean in English. Humans can only hold a few objects in their short-term memory at once, not hundreds. Though some proofs require significant background knowlege, any proof written by a human will respect the fundemental limits of the human mind. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
I suspect large lean proofs generated by LLMs do not respect the fundemental limits of the human mind. If no human can read and understand them, no one can extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. If LLM proof generators can be made to write proofs with the same value as humans, that would be great! OpenAI would be a celebrated collaborator if they created as much value as a human does.
OpenAI has already said they aren't going to claim the $1,000,000. If this proof claim holds up, then the Millennium prizes will be 2 for 2 for rejections of the prize money for valid solutions. Maybe that will be the precedent for other AI labs as well.
"In a world where the cost of answers is dropping to zero, the value of the question becomes everything"
https://www.youtube.com/watch?v=dcolM6W5Odc
What is this man talking about. You can speak physics into existance now, yet it still has to be proven with math. Until we are walking through worm holes and driving around in spaceships that travel in a warp drive could he even begin to say there is non-renewable. But even then..
Oh man am I totally going to determine the 10^10^10th digit of π and cement my name in the annals of history.
There is also a large incentive for OpenAI to fold user conversations into the training process. Proving this happened is unduly hard, and given that professionals are using frontier LLMs for daily work, such training would make it easier for labs to scoop said professionals.
I have seen private correspondence between one mathematician working on Navier-Stokes and OpenAI that makes it sound like OpenAI deliberately scooped this Navier-Stokes result. The alleged correspondence also contained veiled threats if said mathematician went public.
AIs are putting humans out of work.
Yes. We already knew this. Are we actually surprised it's happening?
I guess we are.
Well, we name conjectures after the conjecturer not the (dis)prover so there is some incentive to be the guy who comes up with a hard problems. It is curious that we haven’t had something like this improve OR etc. problems. Perhaps not glorious enough.
> Well, we name conjectures after the conjecturer not the (dis)prover
That’s not universally true. Some conjectures are renamed after being proven. For example, Fermat’s Last Theorem is now sometimes called the Fermat-Wiles Theorem, the Taniyama-Shimura Conjecture is often referred to as the Modularity Theorem now, etc.
Tao seems to be stuck in a pure-math-for-the-benefit-of-pure-mathematicians mindset. The rest of us care about applications of math, not math itself.
Timing is everything https://nonlineartransform.substack.com/p/ai-swarms-timing-i...
Article arguing math is the next "human calculator".
I think this is where people will have to let go of the ego of being the "sole author" of a solution for us to move into the next era of human flourishing.
The future sole owners of half of San Francisco’s mansions thank you for your enlightened stance.
The future I'm after involves Open models controlled by the people, not a few SC venture capitalists
If proofs are tropes, explanations are stories. There won't be an end to stories.
Actually there will because if you've actually ever spent time doing math, you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI. The analogy is interesting but incomplete and misleading.
I've actually spent time doing math and literally everyone I know who knows math learnt it by reproving things that people proved before them. I don't see why AI proving things makes this form of learning any more economically infeasible than the field of mathematics already is - and since it was apparently economically feasible before AI I expect it to stay that way.
There are two stages to learning research level math. First you learn to reprove other people work. This is relatively easy because the language, notion and prior explanations have already been optimized to help prove the next thing, and you know a solution exists. Then in your PhD you try to solve problems you don't really know a solution exists, and if it does, how long and complicated it is, what techniques will be used, etc.
You need to solve the second to get a PhD. For good reason. The second is way harder than the first. And now AI is making second way obsolete. It's not the end of the world, but it is the end of how things have been done for centuries.
you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI
The thing is, nobody has time for that. Look at Mochizuki's work. It takes years of hard labor by high-level mathematicians to come up with stuff like that, and years of hard labor on the part of other mathematicians to validate it. The low-hanging fruit in math has all been picked, AI or no AI, and Tao doesn't seem to acknowledge that.
The mathematics community needs better tools or they're out of business anyway. Now they're getting those tools... and bickering and complaining about it?
I agree that the low-hanging fruit has largely been picked. But then I think we should acknowledge that and stop innovating, or just work less on new solutions and more on clarifying old ones, and start to work on degrowth rather than useless problem solving. Because I really don't feel that any of this is really necessary or beneficial for the human race in the long-term.
Why not let AI proof or disproof this Tao - PI - Riemann zeta hypothesis
See also his previous thread from before the result was published (and before he knew it was coming [1]) on how a to this problem seemed increasingly likely to be solved by AI in a way that caused us to miss the insights that would traditionally be associated with solving it: https://mathstodon.xyz/@tao/117207849921390904
[1] https://mathstodon.xyz/@tao/117219101339291693
There seems to be a sense wher mathematicians are gamifying math, but are frustrated that AI labs are better at gamifying math.
If an AI solves a problem in an unenlightening way, then there's no reason for mathematicians to stop studying it. Pythagoream Theorem has hundreds of different proofs!
If an AI solves a problem in an enlightening way, mathematicians should study it and propose extensions.
It's the same pipeline problem coders have been talking about; once AI does all the work, how are people going to get the experience necessary to take part productively?
Plenty of new open problems will come from applications, and applications are what actually matters. Pure mathematicians are wrapped up in math for the sake of math which is a fun academic game but not something the rest of us should care about.
The lack of reasoning traces in frontier model output is hurting science and progress... thats my take away from reading that, and why open source models are so critical and so needed, because they actually do expose the chain-of-thought reasoning traces recently missing from the frontier models (openAI, anthropic, etc). By encrypting and purposely hiding this important information from public inspection, it makes for a world where people lack the true understanding of how a problem gets solved.
I think he's suffering from a sort of static-universe fallacy. People aren't going to keep doing what they are doing, but secretly.
What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.
Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.
In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.
Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
> If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)
People who work hard to build intuition and also use AI effectively will dominate the field.
This will be literally every field, sooner or later, at least until the human and their intuition is just slowing the AI down. There’s no scenario where humans without AI beat humans with AI in the long run, unless there are fields where the “alien intelligence” somehow hurts more than it helps (like artistic pursuits perhaps?)
This to me is another level of dishonesty. Imagine if a company producing math software starts to hear "rumors" someone is using their software to prove an important result and start massive runs of that software to beat the team. That may not be illegal per se but it is incredibly deceitful. I wonder if the original mathematicians should start a lawsuit for theft of intelectual property.
I would expect this for service providers that provide their "free" services but the mathematicians said they paid OpenAI to help generate this result. This is a clear conflict of interest. Similar to inside trading. Or the same lawyer being hired from two opposing sides which is a big no no. There are rules and laws that already deal with this in other industries and I expect that there will have to be regulation developed for cases where there is a conflict between AI customers and AI provider interest.
This is just the age of slop mathematics, if it doesnt lead to our lives neing improved none of this matters. Math peeps are being nerd sniped by AI in the same way SWEs (the worst ones) got sniped by claude code. Building solutions to problems that dont matter for the sake of doing it just because you can.
You'll ultimately waste a ton of time and get lapped by people doing real world work that actually improves the lives of regular people.
alright, i'll bite: what real work did you put out there that has improved the lives of regular people in the past one year? (without the use of ai, needless to say)
I mean, hasn't it always been this way? If something valuable is within reach and you disclose it, someone else might reach for it?
Just because the length of the arm is longer with an AI org doesn't mean it's somehow fundamentally a different system.
The future is that if you don't use AI your work is a lot easer to reach against someone else who has it.
That dude hand writing code with punchcards can be lapped by a 20yo with python, what's different?
The difference is no one writes punchcards so the comparison is inapt. It is more like supposing AI eats everything humans attempt to do. They will take your work as you are working on it, any interaction at all. It is meta-plagiarism, on another level.
AI didn't "take" anything. An openai researcher did?
The solution is also different to theirs? Literally no evidence of plagarism?
No evidence until there's an answer to the question of if prompt data from the other researchers was used to train the internal model.
I think the reasonable ethical question to consider when a company has an incentive to prove its value, and learns of the state of the art of a flashy research topic being close to a solution, and then throws loads of resources at trying to solve it first. I agree that "AI" didn't "take" but it's not an incomprehensible shortcut of criticism of the aforementioned behavior, not that precision should be avoided here.
Ok so if no plagarism has occured here in any sense then there's no discussion here?
I agree though if they did plagarism it's really really bad for OpenAI. They would instantly have evaporated all remaining good will to gloat over a stolen discovery.
I think there's still an ethical question of a business trying to scoop a researcher solely because the business learned of promising progress. It's not a very big leap to consider "multiple discovery" (a la calculus by Leibniz and Newton) and chance that the model that is good at math proofs might have enough ambient context to work faster from a similar knowledge base on the problem for which promising progress was discovered. I think that's not nice at the very least. Surely the above-board way to approach that is "hey researcher, we learned, indirectly, that you might have made significant progress on flashy research topic and would like the chance to throw our vast computational resources in to try to help work toward the conclusion of that effort." But, maybe that's asking too much?
It's a reasonable take.
I think there's an important part which is that the nature of the beast implies there's effectively zero way to do that.
They have no idea what context, triggering this pathway, was from this guys reddit post 4 years ago.
I also assume that if any AI lab could actually do that it'd be an instant pissing context over who gives who credit the mostest. "This software engineer wrote a sick sorting algorthm everyone now uses, thank you random dude for that training contribution"
Maybe we will in the future, maybe we'll learn that the solution to this prize is from 3 peoples schizo rant on reddit a decade ago. Would be kinda sick actually.
It sounds like a fancy way of saying kids will forget how to do math if they use calculators. The flattening of the space makes it hard to find new problems deemed significant? Maybe that went over my head, I’ll concede that point.
What? No, it's nothing like that. It means that if you have a critical insight and you share it out loud, somebody with no particular mathematical inclination can just spend lots of money to steal the credit from you.
It shifts power further from the worker toward those with capital.
> We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.
In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?
Struggling to understand how a solution to a millennium problem like this isn't a net positive. Presumably Open AI employs mathematicians in these efforts anyway. And I can think of far worse uses of the AI compute resources.
If there are no incentives for mathematicians to work on and share progress in tough problems because they will get scooped, then they will end up quitting and in net we will see less progress overall.
Mathematicians can join the club like the rest of us.
Time to consider re-training to become a nurse, electrician, auto mechanic, or a plumber.
> In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?
This is not at all what he said? I'm not sure how you got this from anything he wrote, actually.