Sharing AI progress in mathematics
openai.com
openai.com
But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.
There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.
I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.
I am not demotivated though, I have a great consumer privacy product coming out soon that I'm very excited about.
IMO that’s where AI is going: as soon as a problem can be formulated clearly enough, AI will trounce us humans. I have yet to see evidence that it can decide what problems are important at a remotely human level.
The "aha" insight for this is actually f**ing wild, it involves a complex valued exponential sum on the edges. I've seen a lot of clever counting arguments before in graph theory but this is the first time I've seen complex roots and annihilating terms like this, the symbolic manipulation tricks in this look like things out of quantum physics. I don't understand where this trick originated, I need to really digest this.
I'm sympathetic to the mathematicians who are worried about the future of their field, but as an outsider I wonder if they couldn't learn from the go community's "recovery" after the introduction of an alien intelligence.
But obviously, it adds up to something greater than went in; in aggregate, our contributions are something to awe.
But my point is, if you zoom in at the marginal, incremental contributions of any individual human in this process, it's really hard for me to say LLMs are not at the same level already.
On this topic, people like to compare LLMs to Einstein, but as far as I know, Einstein did not zero-shot special relativity in an afternoon. He built it up incrementally over time, it took him three times longer than the time between first ChatGPT release and today, and it depended on centuries of prior art, culminating in the right observation and right notation being available to him in his moment of greatness.
At what level LLMs are is then an entirely separate discussion, I think.
Name three.
So your view is that everything was there at the creation of the universe (it's a possible view, of course)? Or are there any "things" that can create ideas from scratch?
They combine things, verify it and if it works and progresses the problem, they created something new.
If you can remember the content of any scientific publication and any book in the world, you are able to make use of this knowledge in every step of you proof.
However, this does now answer how the model came up with the specific route it has taken for the proof.
And then they are for sure able to fill their context based on 'smart search on top' to actually progress further.
I'm fairly sure your understanding is not fully accurate.
I asked GPT here: https://chatgpt.com/share/6ac5fd7d-0390-83ed-a02a-6d80fc64f6... and it says:
> the exact Barnette argument appears quite novel, but nearly every ingredient in its cancellation trick has a recognizable ancestor.
> The closest precedent is much closer than I expected: in fully packed O(n) loop models, people have been assigning complex phases to the two orientations of a loop and making them cancel for decades. At n=0, the phases are literally +I and -I. And the n->0 limit has specifically been used to extract Hamiltonian cycles/walks.
You can judge better than me. But it's definitely worth it having a research assistant AI with you when reading these papers.
Loaded question. A "brand-new insight" is still built off the work of others. A possibly better way to frame it would be in how many subjectively unintuitive logical leaps have been made from prior work.
"it's a matrix-tree cancellation wearing Kasteleyn's planar signs, run as a Witten index over Penrose-lineage states, evaluated as a fugacity-zero loop gas in an infinitesimal magnetic field — and the reason it reads like physics is that every one of those tools was built for partition functions"
I thought this was pure slop when I read it but there are some clear analogues in these other areas of physics, really neat computational tricks, and a very interesting paper by Penrose calculating Tait colorings I never knew about previously (extremely relevant, actually related to a separate approach I had once taken on this problem). The problem is that the paper isn't saying "aha, we were inspired by the related problems of pairing excited states and creating spanning trees out of cancelled coefficients" it just defines the function apropos of nothing. Which is kind of like the Jacobian counterexample in that it works but doesn't really explain how exactly it got there.
I really think the load-bearing concept here is "prior work". If prior work is considered papers on this problem or graph theory, yes this has one huge subjectively unintuitive logical leap. If "prior work" is the entire corpus of neat computational tricks that physicists derived to make their equations spit out something other than zero or infinity, maybe it's not so crazy?
Are there no loads left to be borne?
Makes me wonder how the patent space will be disrupted when that inventiveness step becomes obsolete because of LLMs. Given your example above, it seems like a combination of different methods from many different sources. This would be regarded as inventive, clearly. If eligible patents can now be brute-forced, the bottleneck becomes only selecting the most promising ones and paying for the patent.
Another possibility is that they have internal versions of the model with access to training data that is not provided to external users.
I mean: if some reclusive Japanese genius had a breakthrough on your problem and published it, would you have felt the same?
And if not, why not?
I will never meet that person and I will never hold a real conversation with the "creator" of that proof. They will never tell me how they came up with the cancelling exponential summation that cracked the construction. It's just another enigma but one that is far more unknowable than the original problem.
It has no memory or experience of working on similar problems. Even if it made one of the foundational libraries that I use in a weather forecasting program, it still has no comprehension of the thought process it takes to understand the problem and build it from zero, and if I’m building on that library it just makes fresh assumptions about how things should work.
It’s not a human with experience or expertise, it’s a computer program that’s really good at turning English descriptions into functioning code
If it did it once, it can do it again from zero, and this time you can watch as it works and even it ask it questions. Many of the agents that worked on the problem did not have comprehension of the whole problem. I don't think you need that many tokens to be able to query it for the insights it had during the process.
Isn’t this the issue with using it the way you’re suggesting? At best the model can come up with an after-the-fact rationalization of how to get to the solution, but it doesn’t know what actual path it took to get there - what were interesting traps it fell into, where was a place it was close to the solution but didn’t realize at the time.
Those are things that are valuable to share between humans, those which teach us how to think better, and give us deeper understanding ourselves, and which a model doesn’t have any comprehension of.
You just made my day, beautifully said. Thank you Sir, for all your thoughts expressed in this thread. You put an human story behind the #180 number.
Not that things like that can't happen with humans too (Salieri v. Mozart comes to mind).
What is this then, vibes? Without a machine-checkable proof I'm not sure what to think of any of this.
I think it helps that basically everyone thinks this conjecture is true, it's just been so darn weird to attack. There's this odd thing that the induction proofs of this problem kept running into, which is that the N+1 condition would work except for in one tiny case when it could fail, but it would be covered by a very slightly stronger version of the conjecture. But then that would fail on one tiny case in induction, but you could solve that with another slightly stronger version. Etc., etc. I almost wondered if there were some sort of structure to the increasingly strong conditions and wanted to prove something about the meta-induction between the stronger conditions and the N's that they needed the next level to remain true. But that failed after 5 steps I think (Fable actually helped me write a few hundred test cases to explicitly show that pattern didn't continue forever, thank God).
BTW my existing test suite from previous proof attempts jives with this new algorithm, so I haven't seen any evidence yet that it's incorrect. Waiting for a Lean proof obviously.
/-- Cubic bipartite three-vertex-connected plane graphs have a Hamiltonian cycle. -/ def MainStatement : Prop := ∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj], G.IsRegularOfDegree 3 → G.IsBipartite → Planar G → ThreeVertexConnected G → HasHamiltonianCycle G
theorem main : MainStatement.{u} := by sorry
Don’t you feel any relief that you won’t obsess on this any longer and not lose more hours on this than you already have?
These are genuine questions. I know I spent a good amount of time thinking about P vs NP, and that sometimes I go back to it just to realize I’ll never solve it. I’d feel that knowing the proof would feel more like a liberation, a weight lifted off my shoulders than something being taken away from me.
This is the part that gives me the strangest feeling about it all, because you're not the only one with this experience. I've experienced this too on different problems, as have many researchers across many fields.
I disagree with the Fields Medalists on the majority of their complaints. AI math is happening and there's no going back. However, on one point I increasingly agree: virtually none of this stuff is possible with technology any normal citizen has access to. I have no problem with AI models making revolutionary advances in math or science. Where I start to have a problem is when the AI models making these advances are tightly withheld, proprietary, and seemingly never released with these capabilities intact. This has been the case for all of 2026 so far.
I suspect that this is in fact the source of much of the angst. None of this progress is reproducible outside of one or two teams inside OpenAI and Anthropic. It's becoming an incredible concentration of power that I don't know that we've ever quite seen before. Right now, it feels harmless because it's being used for wonky math problems that aren't (yet) practical for anything. But great power never stays harmless. History has taught us that countless times, in countless different forms.
It has been super helpful in delineating where the crucial concept came from. The proof is rather simple as graph theory proofs go, but it does seem to use some constructions that would only seem obvious if you had serious physics experience with partition function and calculating energy states that cancel out. It's not a wholly alien bolt from the heavens, but I can also see how there hasn't been a human being with the broad theoretical physics knowledge combined with the deep graph theory experience in planar graphs to come up with this idea. I don't know, I'm looking for precedents of this formulation and some old papers of Penrose counting the number of edge colorings of this same graph type are coming up, the line of argument at least rhymes.
But I agree with the thought that this sort of progress should not be siloed inside those companies. I propose a tax so that every slop cannon AI video pays for another hour of compute time for advancing mathematics.
Who knows what they are up to.
This sort of happened at various times in the past, because they hired and/or funded so many mathematicians, and especially before the late 1970s they had many of them working in areas where academic mathematicians weren't working at all, so they were learning more math, or more math that they especially cared about, than the public was. (I was going to write a note here just a few days ago about how NSA has had a "Classified Mathematics Library" for many years.)
For vulnerability scanning, I think the new-capabilities trajectory is good (in the sense of "it will help defenders win") even if governments find ways to get more of it, because there are finitely many bugs and classes of bugs, so at some point more capable models' or longer runs' advantage over less capable models and shorter runs should stop helping them outcompete the less-well-funded defenders, because the defenders will still have learned most of the information that's relevant to achieving successful defenses.
So if NSA gets 5000 units of vulnerability scanning and the public only gets 4000 units, we might still just wipe out all of the pure software vulnerabilities and then go back to worrying about physical supply chain security or side channels or something.
For math, I'm not quite sure! For one thing, there may be things that have no feasibly deployable defense at all even when you understand the underlying mathematics (I'm especially worried about traffic analysis here, because understanding in detail how traffic analysis is done, or how powerful particular techniques are, does not necessarily always or usually make defending against it more convenient or less costly). In a more science fiction scenario, there might also not be any efficient secure cryptographic primitives of some kind, like if it turns out P=NP with reasonably small exponents and reasonably small constant factors.
Has the government stopped Google and Apple? https://news.ycombinator.com/item?id=49964791
Yes. They have already shown to have no scruples when it comes to making profit and to have little to no morals.
> If you worry about the government, isn't it better that than rando terrorists?
In my country the largest terrorist attack was almost certainly financed by Iran and caused roughly one hundred deaths. This number pales compared to the thousands who died during the latest, US-backed military coup, a move that relied on a doctrine that the US has never stopped asserting [1].
And those morals I mentioned earlier from AI companies? They do not apply to me because I'm not a US citizen. So no, I do not think the US government is the "seal of quality" you think it is.
Why do you "suspect" this as if it's some hidden motivation when the very first paragraph of the advisory group's statement (linked from the OpenAI post) says:
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
Tao and others in that group have been strongly and publicly pro AI from the start. They are not advocating "going back". They're objecting to the strip mining of open problems using proprietary technology.
I don't know but this phrasing comes off as gatekeeping.
The picture I have in mind is OpenAI running their most advanced model in a loop over all the open mathematical problems they can find, just to verify that the model is indeed very smart. Neither the company nor the model actually care about the problems, it's just a cheap exercise machine for them, but the problems get solved and mathematicians don't even get to participate.
Like, even those who accepted the "centaur" thinking, man + machine, won't benefit because by the time they get their hands on good enough models, everything is already done.
It's an emotional thing first and foremost - people who care about the thing can't do the thing, because it's already been done by those who couldn't care less about it.
And before someone goes "poor mathematicians", a food for thought: this is just an early instance of what looks like our shared destiny.
I said here before: given the economics of progress in AI and robotics, it's obvious what the natural division of labor is: computers do the thinking, humans do the menial, manual labor. AI will do politics and philosophy, so you have more time to fold laundry and scrub the toilet.
Are we gonna get the same pushback from medical researchers if the models cure xyz diseases?
I absolutely understand the emotional connection to their work and the heartbreak, but mathematics doesn't exist for their pleasure, it exists to provide tools to solve humanitie's problems.
Lol. As long as the process aka trials is respected not many would complain.
The feedback loop required to make progress is very different in medicine compared to math.
It's tempting to say "both", but that misses that AI is now forcing us to pick one.
As 'ogogmad said upthread:
> mathematics will continue to advance, albeit differently from before. The social structures will not survive however.
My understanding of what Sid's describing is that you do RNA sequencing, a whole genome sequencing, feed that into frontier AI (if it will still let you), and somewhere along the way give the information the AI finds to people who can use it make a personalized mRNA vaccine, specifically for you and your cancer.
Another link here about Sid's case, it explains it didn't go through trials: "made possible through a compassionate use allowance from the U.S. Food and Drug Administration (FDA)".
https://www.houstonmethodist.org/newsroom/houston-methodist-...
I am not medical, so I'm happy for someone who understands better to come in and explain all the myriad ways I am wrong.
While AI has definitely helped quite a bit I am wondering how much all this research and treatments cost. Not sure the current health systems could sustain this for _everyone affected_. If ai enables it all the better.
The general body of research points that more doctors lowers all cause mortality ( with diminishing returns) but Tunisia is still far lower than the Eu average.
Yet Doctors and med Student unions do lobby very heavily against expanding admission to the public uni or allowing private unis.
So we have the weird situation where people go and study in Romania ( making Tunisia lose hard currency that it really needs).
These doctors have taken an oath and the direct consequence of their lobbying is literally more deaths.
Obviously this is often coming from folks who act in same ways as they criticize and usually don't contribute even a fraction back to society compared to doctors. Folks who do mistakes in their lives all the time yet thats fine since we are all humans or similar, right.
So please stop this cheap framing and accusations. If Tunisia wants more doctors and keep them there are ways to do it, society as a whole needs to decide what they want and act upon it. Otherwise, smart skilled folks will keep going for better lives elsewhere, just like everybody else.
Most doctors aren't running departments in major hospitals, or advising government on policy. They don't earn the big bucks. And even hospitals themselves tend to run in the red all the time; it's sometimes hard to disentangle where greed ends, and longer-term interests of patients begin, as you have multiple people and organizations pulling in different directions for different reasons.
RE private medical universities, N=1 but in Poland we have a private provider pushing hard for training their own doctors "because public system is too slow and limited", and it's hard to tell whether they have a point, or whether it's a private-driven attempt at privatizing national healthcare, or a mix of both.
The risk here is that this does do fundamental long-term damage to mathematics as a viable field.
Virtually no one is going to want to take on the risk of PhD-level math work, studying a narrow problem for four years or so to arrive at an impressive incremental result, when there's a sword of damocles hanging over their head every day that an internal system held by an oracle they don't have access to may scoop their results and turn those four years into dust.
To some extent, that sword of damocles always existed in a de minimus sense in the form of other mathematicians. But everyone was playing the same game, coming to the game with the same arsenal limited by human cognition.
If the game board becomes irrevocably tilted, new entrants have no incentive to play except as a hobby. But few hobbyists can devote years of work to understanding and pushing the frontier. It could well mean existential damage to mathematics as a field.
Whether that might undermine math's ability to solve humanity's problems in the long term is almost an economics problem, not unlike the question of whether and when the existence of monopolies ultimately restricts long-term economic growth. Much probably depends on whether intellectual monopolies or oligopolies are being created that will supplant the existing mathematics "economy".
All the commotion evens out: It's much easier to learn maths than ever before; you don't need to go to lectures any more; you don't need to learn from a specialist (advisor, lecturer) any more; it all costs much less than it used to.
So mathematics will continue to advance, albeit differently from before. The social structures will not survive however.
There's probably a loose and deeply imperfect analogy with computing: via democratization hobbyists have made a big impact in applied operating systems development (Linux/OpenBSD) but have been less successful/impactful in OS research (whither Hurd...) or in cost-heavy fields like microprocessor design.
Who decreed that? Mathematics predates capitalism and publish-or-perish by a couple of millennia. Euclid’s Elements were not written to benefit the weapons or medical industry.
Maybe this hurts more than it should do because of publish-or-perish.
Some of it, yes. Much like physics. Both have a track record of producing technological breakthroughs every now and then, but it's not why people are doing it.
> Are we gonna get the same pushback from medical researchers if the models cure xyz diseases?
For better or worse, yes. We already are. In my country, there's a big spat between radiologists and cardiologists right now, that boils down to the progress of technology allowing the former to answer questions that, before, involved a procedure that was a big money-maker for the latter.
Imagine there's a very advanced crossword club where anybody can join and take a stab at these crosswords for the love of solving puzzles. Many of them are so difficult that no one's been able to solve them yet, but we know they're all solvable.
One day, someone comes along with a super advanced crossword solver application, and it makes easy work of these crosswords. They run it on a few to prove how powerful it is, and then the community says, "Oh wow, that's cool, but please don't run it on any more of our advanced crosswords because they're very hard for us to come up with, and we really enjoy solving them by hand."
That's really what this compares to. I wouldn't call that gatekeeping; just respect. Respect for the game, respect for people's desire to have these hard problems to continue to work on, solving by hand.
If the company with the super advanced crossword solver then continues to use it and publish the results, they're effectively stealing the crosswords from this community. Soon, all the puzzles will be solved, leaving nothing left for the community to work on for fun.
That doesn't sound like gatekeeping to me. That just sounds like someone asking "Please be respectful and leave the remaining puzzles for us to solve by hand.” A simple plea not to be an asshole.
Despite nobody at openAI thinking of themselves as an asshole; despite society urging openAI not to be an asshole; despite the fact that being an asshole is entirely unnecessary even to accomplish whatever objective they are setting out to accomplish; despite everyone at openAI loudly declaring: we are not assholes!
They are still assholes.
New theories and insights are typically created while working out proofs. If proofs now suddenly fall out of the sky (cause LLMs create them) then that work is not done which means the substrate on which new theories and questions and conjectures used to be grown disappears. It's in that sense that the math community (and thereby society as a whole) will lose something.
It's similar to how software engineering will need to find a solution to train their next generation. Current generations have all been through manual steps of designing things from scratch and writing them by hand. That's what allows your 10x engineers to understand whether what their LLM tools are doing is good and how to massage those tools to do the right thing. A junior engineer who has only ever used LLMs to write code and create architectures does not just not have that experience but also won't acquire it. You can't just say "we don't pay them to have fun and learn, we pay them to produce results". In the short term that is the case, but in the long term you as a company and we as a community will lose out.
I'm not saying don't use AI tooling. I'm saying that this is a hard problem which we yet to have to find solutions and approaches to. As a software community as well as as society in general.
My ego tends to agree, that how can they be ever competent, if they have not endured the same hardships as I had crunching trough problems and getting allmost lost in the details.
But I rather suspect, they will turn out fine. I know LLMs are great for me to learn and I think the young generation will learn what they need to learn to get the job done.
I have to assume OpenAI is only prompting to solve problems, presumably they could also prompt to not interesting new theories or paths of research found along the way as well.
And yes, fun counts. Nobody said this had to be only a hardship.
Perhaps that won't matter if we enter an era where AI participants are the main participants who matter for discovery-level mathematics. But it would likely be what economists would see as a market failure if only a small oligopoly of AI participants, closely held behind closed doors, is able to fill that intellectual role.
I'm well aware that if at some point AI is good enough to replace me as a software engineer then I won't have a job. I don't expect a company to continue to pay me simply because I enjoy it if there are cheaper options out there.
Math is no different.
If someone can solve open problems in mathematics then they should do so, isn't it as simple as that?
They should let the public use the models as well, but I guess they have no real moral imperative to do so.
But asking them to stop solving problems is just weird.
It’s not a human focused civilization, which is where the issue comes up.
As an example: A constant issue I am seeing with AI productivity is that the most productive use of AI is when it is paired with more experienced users, while AI also does more work for entry level workers, if not replacing them entirely. It has become a question where will the future buffer of experienced seniors come from.
This is an example of where simply chopping down trees for today, doesn’t make civilization better off tomorrow.
AI is producing more content than ever before, but our ability to understand and verify it is not keeping pace.
We don’t know if these are unsolvable problems at this stage. Society could come up with workarounds and solutions to these issues in several years.
The request to stop, is part of the process by which the issues are debated and solutions found. It doesn’t mean their position is weird or moot.
If there is a prize associated with doing a puzzle, and a machine does it, then what incentive is there to pursue it.
Again, if you are only concerned with the outcome, and you have a preferred answer that you want (in this case "just use AI to advance faster"), then any information that doesn't support that case is useless or misguided at worst.
I am not trying to dissuade you from your preference. I am flagging that there is a set of other factors that influence the behavior of others, how that behavior is critical to the creation of expertise and drive, and thus why others hold different positions.
If another human was likely to get the answer before you would you also discourage them from doing it because they would rob you of the chance to do the thing you're concerned about?
We spend most of our young lives (many of us our entire lives) studying physics, math, etc. that others have solved. (e.g Quantum Mechanics, Relativity, Calculus, etc.)
Biology consists, almost entirely, of studying solved problems in nature.
Aren't AI breakthroughs just more to study?
They may very well have learned plenty of things and solved or discovered other puzzles, but if the first puzzle is worth pursuing because the solution is actually useful it seems liked we're better off with the solution than a bunch of failed attempts.
That said, I do question the value of solving many of these types of math problems. I'm no mathematician so I'm assuming I'm wrong here, but on the surface many seem mostly theoretical puzzles with little or no practical use.
That is fine to say when it is not your field. I guarantee you feel different when it is the thing you care about, that gives you joy, that defines your status. Think about how many sheldon-equivalents insist on being called Dr. (non medical)
It is part of what people use to define themselves. Its going to hurt. There may even be a Bulterian Jihad
It is clear to me that any competent person with a little patience can now build software better than what I used to build by hand.
It just seems that this class of mathematicians is being "disrupted".
The field is changing and a new class of mathematicians will take their place.
This happens all the time in fields as technology disrupts them.
A new class of individuals, with different motivations, take the place of the old guard.
I'm sure the motivations of individuals involved in designing and manufacturing cars changed as Henry Ford introduced the factor line.
But that old crop of humans either adapted or retired.
But, plenty of humans took their place with new motivations and automotive technology continued to progress.
I personally feel math will indeed move faster as a result of these breakthroughs. And the humans that take the place of the old guard will have different passions and motivations than the current group.
Maybe the new group will be productivity motivated rather than motivated by the love of tinkering with a single problem for years.
Sounds like salaries for mathematicians need to start going up if we stop paying them with fun.
Whole sections of the economy are being upheaved by AI, and there is no reason to make a special case for the mathematicians anymore than for the illustrators, developers, translators, HR, etc.
Of course; but it's very hypocritical to raise these feelings only when mathematicians are affected, whereas all the above professions are just told to adapt to the new way of things.
For sure though, translators don't have the same clout and social status as mathematicians do.
Science isn’t some passive busywork thing where you tie your hands behind your back because it isn’t fair on others to solve all the neat problems - or at least it shouldn’t be.
If your idea of science is leather patches on tweed suits and the quiet ticking of a clock while you do crosswords, then this is an argument in favour of letting the AI do the work so you can focus on your sudoku book in your slippers.
Mathematicians and academics in their ivory towers are forgetting that everything is getting automated. They want to carve out fun problem solving niches that's fine but who's going to fund that? If they want to be funded by the society/civilization their argument can't be leave advanced fun problems for their hobby.
People go to said crossword group to enjoy the process of solving the puzzles. It doesn't actually matter if they have been solved yet or not, case in point the NY Times puzzles are enjoyed by more than just the first to solve them.
Professional mathematicians are ultimately being paid to solve the problems for a (hopefully) practical reason. Its always excellent when a person enjoys the process of the work they are paid to do, but ultimately they are still paid to do the work. I really hope your argument isn't that we should collectively be funding mathematicians to solve problems simply doe the love of the game.
In this instance however, it's openAI and Anthropic that are pushing people out of the field by running secret models that take the interesting work away and leaves the persons having to review endless slop proofs.
He argues that the supply nay be very large indeed but the interesting subset is not. Figuring out the interesting problems is difficult so strip mining the good known problems may lead to scarcity. I am not a mathematician myself, can not judge this accurately.
Every company is about to have a staff Ops Researcher who has a better grasp of the underlying math and theory than any university professor. That is an unambiguous win.
Not sure about the unambiguous win. Are we entering the age in which mathematics is industry-dominated?
1) Any university professor can spend their 24 years on a problem with little progress. 2) company has sudden interests. 3) industrial resources brute force the Lean proof. 4) Max PR for AI company 5) professors are left to rewrite the AI Lean slop into real human-readable math? {disclaimer non-math university professor}
Initially, yes. Long term, however? Perhaps still yes.
> 5) professors are left to rewrite the AI Lean slop into real human-readable math?
6) AI writes the proof into something easier to follow than a PDF document.
Hmm. Oh shit.
I have a really hard time reading AI proof so this might be a biased statement, but most of them feels like having a superpowerfull machine, that would have bruteforce all the possible words of finite length in your logical syntax. You have the path to the solution, using tools that where already known and even direction that where abandoned because they seemed to fail for our human brain. But at the end, as a mathematician, you don't learn anything that is really new.
To me this is the main risk with AI and in general the one most mathematican try to explain but fail, we might miss a lot of alternative path that would have raised more interesting questions (I think this is already more or less what is happening). On top of that, we will run out of mathematicians as no one wants to pursue a career in the field anymore.
It's the same argument which is invariably wrong yet comes up over and over again.
There's no real reason to think AIs solving lots of problems will stop further work on alternative paths - certainly a machine which never tires and can be trained on its own solutions is going to continue to improve.
There's precedent for this: just look at any overconfident post regarding what China will clearly never be able to do, despite decades of steady if frequently flawed progress.
There's no persuasive argument being presented as to why machine mathematical research should have a limit beyond hardware capabilities.
At the same time, there's a new note at the bottom of agmai.org stating how they've been in contact with OpenAI about this particular release, and they say that “we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully”.
So, what's going on there; is this British English for “they didn't follow anything at all”? Because from my perspective, it looks like they doubled down on the Navier–Stokes approach of trying to maximize PR gain while being as lazy as possible about actually contributing anything back to science, releasing only slop that may or may not be correct and may or may not be straight up plagiarism, as has been the case earlier.
If I were on the AGMAI board, I'd feel terribly exploited when reading that press release, yet their response is modest.
Hairer, if you're reading this: is there any indication whatsoever that AGMAI was anything but a cheap way for OpenAI to science-wash their press release?
Yes, but the subtext is even stronger.
Now I know there are issues with the field and how just answering these questions may cause broader problems, but I feel like the posted results is far from slop. We can't just call any output slop, or it loses all meaning.
If it was slop, it'd not be causing the issues the group are concerned about - they're not saying "the problem is we're getting loads of incorrect proofs thrown about that are nonsense".
The term “ai slop” is not supposed to discriminate good ai output from bad, the entire purpose of the phrase is a blanket term that delegitimizes all ai output.
Unfortunately being "pro AI" means relinquishing any control over what the AI, or more importantly the company running it, might be doing.
Computers and computer programs are tools. Humans always remain sovereign over their tools.
I suspect that this might be one of the reasons people inside the labs are scared about AI.
What if they have asked AI how it would wipe out humanity and it came up with reasonable answers that they don’t want to publish unlike they do with these math problems?
I think those models and findings should be investigated.
Cure for aging? What do you reckon that'd be worth?
Would that impact their own ability of solving Mathematics problems? I mean as a programmer I'm already seeing that impact on the programmers -- sure the best of us can leverage AI to achieve unimaginable things, but many of us are simply vibe coding.
Of course we can assume that it is only the best of us that really matters, and the rest of us are not going to produce anything substantially useful ANYWAY, it might as well to replace the rest of us with AI, but my worry is -- does that really have ZERO impact on the human specie's ability to produce "the best of us"? After all, they don't grow on trees.
Obviously the more intelligent the model, the smaller/more directed the search is. But they spoke about huge numbers of agents working on Navier-Stokes for example (I think it cost >$10m).
Replace “AI” with “supercomputer”.
(Super)computers have been solving many math problems that mathematicians can’t solve. Now they are capable of solving problem types that they weren’t able to solve before. (this applies to other fields as well)
Problem is it’s not clear if there is anything left for humans. Probably yes, since human mathematicians are still more economical.
So basically nothing changes, Math was subject to gatekeeping and policing of the worst kind.
If you were not among the geniuses, and it didn't come to you automagically, you were simply supposed to leave it to the people who did get it and go do work for people of your intelligence. Smugness was too much to take.
Math people, like chess people never made any genuine attempt to help people understand the processes and methods that made math happen.
To me it should have been a field as teachable and ubiquitous as accounting.
The net result is once these methods and processes were worked out by AI, it was over for the human mathematicians.
but at least as a software engineer, i always knew my work was "never done" and so it was common to build a bunch of code that might be thrown away, either because it didn't serve our customers (the mvp or pilot fails to meet demand), or because we found a better way to do it and so we deprecate it.
some people got too attached to the code and honestly they were the types to be filtered out fast.. way too emotional and hard to work with. getting attached to code meant you actually don't advance (after all, in our case, we were a business serving customers and not a hobby artisan shop). attachment leads one to hold back due to some misplaced cognitive load.
isn't the goal of working on "advancing the field/product/whatever" to always be solving/selling/whatever?
maybe in your hands, with your knowledge and experience over the last 20+ years, you can use AI to make leaps and bounds by steering it properly towards whatever solution or goal?
at least now you are one of the most qualified people to check the result, transform it into understandable (by humans) state and grow stuff on top of it
https://github.com/openai/math/blob/main/lean/ComparatorChal...
Just like there are talented software engineers driving the AI to create the software that "it" builds, and talented steel workers, teachers, nurses etc who use computers and other machines to create value all over the economy (without whom, the machines they use at work would be worthless).
Capital owners have always sought to minimise the value of the input that "workers" make in the process of creating value. Maybe now that information workers are on the wrong end of this deal, they might develop some empathy and solidarity with their fellow working class comrades and together, demand that people recapture the value that capital has stolen from them.
The Unique Games Conjecture (sorry, "Unique Games Theorem" now!) is huge. It was a very significant pillar supporting many of the limits of the polynomial-time approximation algorithms in the graduate-level randomized and approximate algorithms course I took in theoretical computer science. Textbooks will have to be re-written.
Here is an explainer: https://share.gemini.google/nbjIK6X3tOfz
With UGC proved, certain polynomial-time approximation algorithms used in difficult real-life problems are now known to be the best approximations we can achieve in polynomial-time:
> If UGC holds, the elementary algorithm that grabs both ends of an edge is fundamentally the best efficient algorithm that will ever exist. No amount of advanced linear programming or heuristics can achieve a ratio of 1.999.
> Under UGC, the Goemans-Williamson algorithm's 0.87856 ratio is mathematically optimal.
> UGC is considered the "Rosetta Stone" of approximation algorithms. In 2008, Prasad Raghavendra proved that for every single constraint satisfaction problem (CSP), a canonical Semidefinite Programming relaxation paired with the best rounding scheme achieves the optimal approximation ratio if and only if UGC is true. If the conjecture holds, the algorithmic boundary for an entire class of combinatorial problems is completely resolved.
Other hardness of approximation results from this UGC proof:
> [Max acyclic subgraph, a problem encountered in real life]: No polynomial-time algorithm can fundamentally outperform an unthinking coin toss.
> [Relative scheduling, another realistic problem]: As with acyclic subgraphs, the problem is "approximation-resistant": clever algorithms cannot beat random shuffling.
I suppose when people do re-write the textbooks they'll say "this is confirmed now" not "if this conjecture is true...", but usually re-writing the textbooks would imply that things have been shown to be false?
May have misunderstood. Thank you for the post though, it was very interesting to someone who doesn't know much about the topic.
The resolution of UGC will lead to a new theory in approximation algorithms. Suddenly we can build on top of the results that previously said "unless UGC is false".
But you're right in that the first step is simply to remove that last sentence from all the theorems.
> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.
Discussed here:
To grieve, or not to grieve? - https://news.ycombinator.com/item?id=49919676 - Oct 2026 (156 comments)
I'm relieved that this time around, they have provided partial reasoning traces and prompts for a small number of problems. Do they now also share data with the other model providers..
They structurally cannot understand what we are doing at the place where we hit our ceiling. Only with our highest technology (well beyond their understanding) do we have the tools to go back for them, and try to bring them along and interface better with us (re: recent work in animal communication)
The cynic in me says it wouldn't change a thing as plenty of people know the horrors factory farmed animals face and still continue to consume them anyways.
Hopefully GPT 8 will treat as a bit better than we treat the cows.
The answer is that humans are inherently only capable of local empathy, on average. We have enough empathy to cover the local tribal unit and that's about it.
My hope is that AI, while probably causing great societal turmoil in the short term, leads to such abundance that a) everyone can live a dignified existence, and b) we'll have such great alternatives to animal products that nobody will chose to consume animals anymore due to its replacement either tasting better, being cheaper, etc.
The cynic in me says we'll all just be rendered useless and disposable by AI, but I'm doing my best to look for silver linings for the sake of my own mental health.
The famous "how does it feel to be a bat" also comes to mind as a tangent consideration.
Two people can just exchange a sight, and both understand what the situation means and what each need to do to reach a common mutually beneficial ground.
Two people might exchange at length with highly technical vocabulary and still both feel deeply not understood.
Worth noting that this is an invisibly small part of the sum total of our global efforts, especially versus the much more tangible effort we put into enslaving and slaughtering them, then mangling their carcasses for our own uses as we drive more and more of them to extinction.
We simply don't care about anything beyond ourselves and even there it breaks down on closer analysis when we see how many within our species don't truly value the collective whole beyond themselves.
It's just atoms all the way down.
I'm frankly offended by this mischaracterization of human-animal relationships. So called "slaves" like horses and dogs have been dearly beloved companions for centuries and actively seek our companionship too.
The animals we raise for slaughter are often mistreated, yes, but many humans treat them with respect; billions on billions are voluntarily spent to improve their condition. Despite our own needs, many people pay higher prices for animal products that involve better treatment of animals. And they are in no risk of extinction! Much to the contrary, their domestic variants would not exist if humans didn't raise and protect them.
> We simply don't care about anything beyond ourselves
Have you seen modern westerners with their dogs??
If we end up in a future where AIs have as much concern for our welfare as we have for the welfare of the average animal (not the minuscule percentage of domesticated dogs, but the overwhelming majority of factory-farmed or simply driven to extinction), then I doubt you would consider it a “mischaracterization” to say that the whole AI thing did not work out to our advantage.
Bringing up “modern Westerners with their dogs” as a counterexample is almost self-parody.
It would be absurd to claim that all animals live some sort of charmed life due to humans.
But saying that animals (especially those most similar to us like intelligent mammals) are nothing more than "atoms" to humans is equally absurd.
"Often mistreated". Dude, they are held in tiny cages injected with hormones and what not till we kill them so we can have a big mac. It's very hard to argue we do any of this for nutrition reasons, we do it because we like the taste of burgers and roast.
There is nothing here today that is unpredictable or impossible to control.
It is everyone's choice to let the greed continue, to let unelected sociopaths capture and feed society to the model.
It is not acceptable to put others at risk. It can stop and it can be done the right way instead.
That is, inform the industry that those causing these risks will be prosecuted regardless of their messiah complex.
The US government must not under any circumstances allow the ai industry to form a cartel.
We can make some effort to encourage open source models and thus stop the companies from causing hysteria by hiding the model, shrouding it it mysticism and prophesying the end times. China is doing a great service to everyone by making llms available to the public.
As far as I can tell, this is a victory for verifiable loops using LEAN, reinforcement learning, and oodles of compute. I haven't seen evidence yet that this is proof of broad generalization beyond the training distribution.
Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.
It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...
Check out my other top level comment in this thread.
Humans in the Culture are generally improved in a few ways (they don't get sick, live for 300-400 years by default etc) but still very human.
I do recall a bit about playing in different worlds in dreams though, during sleep. Ultimately, really, the average person's life in the Culture already involves doing pretty much whatever they like within reason any time, so it's not like they need to escape too much real-world suffering.
Iain M Banks himself described the relationship between humans and the ship Minds as having "a status somewhere between passengers, pets and parasites."[2]
[1] For example, https://theculture.fandom.com/wiki/Grey_Area
Sounds like children tbh. Disclaimer: have children
You wish. If humanity survives as "bio trophies," they'll be the descendants of a subset of billionaires and their groupies/harems. We live in a capitalist society, where the only ones allowed to thrive without work are the rich. The rest of us will be left to rot and die off, as we will have nothing left to sell in the market that they want.
https://github.com/openai/math/blob/main/preprints/Paired-st...
In this case, the tenure is gone, and OpenAI has increased their valuation
If we would have discovered this breakthrough of LLM/ML on scale in a non capitalistic world, we would all work together advancing it faster than it goes right now for the benefit of humanity.
And I don't live forever (at least for now) i def want to see were this road is heading.
Its a conflict of interest for sure, a cnflict of the future of a lot of humans
I am always looking for leftist writing imagining a positive vision for AI. Is there any which you'd recommend?
I am now very interested in the explicit calculation of Hamiltonian cycles in the non-bipartite case, and/or the calculation of their absence. If P=NP I think that's going to be a great route of attack.
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]
Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:
Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
I'm sure all of these super small or large constants will improve over time, but it's still amusing. It is entertaining to see the exponents directly rather than have them hidden as n^c or epsilon or O(1).
[1]: https://github.com/openai/math/blob/main/preprints/Determini...
The autonomous researcher records every research cycle in a public notebook.
Framework: https://github.com/kbr-/math-research/ Public notebook: kbr.is-a.dev/math-research/
> Sure, mathematical history features a lot of incredible developments, like the invention of proof, zero, or the computer, and on the great problems our progress has been over timelines measured in decades or centuries. Obviously this technology didn’t appear today, but blurring our eyes a bit to combine the past ten years, with today a measurement of those developments, there is nothing comparable.
They certainly arent going to give you that cure for cancer, if it were to ever come.
Or else how do you explain Danyelza? Used to treat neuroblastoma, costs upwards of $1m per year. Do you have any proof that this will be different?
You're calling realistic people conspiracy theorists. Whose side are you on?
Yes, I actually work in the medical industry. There is no hiding the cure for cancer.
> Or else how do you explain Danyelza? Used to treat neuroblastoma, costs upwards of $1m per year. Do you have any proof that this will be different?
Oh, you are American. Let me tell you a secret: the problems of your healthcare insurance system are not a worldwide phenomenon, nor an immutable fact of this universe. Perhaps the cure for cancer, if expensive, will not be easily available to the poorest Americans, at least initially (the cost will come down sooner or later). But that is a very different claim from "they'd never give it to you".
There's no need to hide anything, it's just pay-walled (and it's not an hypothesis, most human beings on this planets cannot afford the SotA treatments for their cancer today).
I'm also against absolutely anyone who asks me "whose side I'm on" as part of an argument.
Conspiracy bullshit. You cannot keep something like an effective cure for cancer under wraps. There is no plausible logic how that would not leak sooner or later.
All of this means you will need to be rich or have your country invest lots of money into health care systems. In a world where humans don't provide economic value anymore, why would that be?
There are real safety concerns with AI that can be made very convincingly though.
I feel something about human nature makes us treat joy of discovery, status, etc. as a source of energy and motivation. I hope we'll find other ways to keep some "strategic intellectual reserve" of mathematicians alive.
This happened to software engineers already. Mathematics isn't special.
That said the software folks can teach the math folks a thing or two when it comes to dealing with grief I am sure.
For what? So some guys can be richer and more powerful. What could possibly be more important than themselves? You? Your future? We're nothing, and have been told to be excited and curious about our coming obsolescence and powerlessness.
[0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu...
> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).
I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.
1. e is maybe a name of a list.
2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.
3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.
4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.
So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.
Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.
If this were my paper, or if I were trying to train a model to write math, I'd want something like:
A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.
A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).
E ⊆ V × V
In this particular case, though, I think my typoed version may be equivalent. An edge with no constraints has the same effect as no edge at all.
I definitely messed up the constraint definition, though: u_e and v_e refer to vertices, not edges. That’s what I get for writing it with minimal proofreading.
An ex colleague of mine who is a world class mathematician recently got an ERC with ambitious goals to advance his field.
Literally every optimistic goal proposed to be worked on during this multi-year window has been solved in this one post. His and his entire group's work has just been done for him! They are all depressed as hell right now.
Beauty can be appreciated even when it is vast, even when it is beyond one's comprehension. I don't think this release should be primarily viewed as an outcome of competition. Instead it is revealing truths about the universe that were always there and always beautiful, even if we hadn't seen them yet. I believe there are infinitely more such beautiful truths currently hidden and waiting for us to discover.
[1] https://www.nytimes.com/2026/10/06/science/openai-math-probl...
Not that I could ever "compete" on the frontier of math in the first place. But our nature to compete derives from our need to survive against other capable forces. And results like these make me feel very nervous about humans' capability to remain the dominant force in the universe.
The only bias here is that we're still covering these things like business ventures and not criminals.
Mathematical proofs aren't revealing truths about the universe. Mathematical proofs are independent of what the universe is like. Any proof would be the same in any possible universe.
That's exactly what they do, apply logic formally and systematically to discover truths.
Sure, there may be a universe where 2 + 2 = 5, but then that universe would have its own mathematics that can prove that to be true. And there will be a way to bridge that alien math to our own, again by logic and proofs, until we have a larger sense of truths not only in our universe but all possible universes. Proofs are part of the constant process of revealing deeper truths to the best of our understanding.
IDK, I think you should add tendency to cooperate, a capacity to love and perhaps some other things there.
But with things unfolding quickly and unpredictably, I think everyone's view is getting a bit foreshortened here.
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
Proven math theorems are tautologies.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
Who is "we" here exactly?
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer reviews. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
BTW the essay is eminently readable for anyone interested in math. Hales wrote it in favor of formalized math and to educate his peers and students about it.
Which either replaces us in the long term, augments us or makes us better (gentherapy).
There’s about a 1 in 10 chance when someone spells it (or says it) they say Herbert.
Even in situations where they just read it or I just said it.
I’ve had Herbert soccer trophies, health insurance cards, etc.
The mind fills in a lot of blanks and doesnt always get them right.
Yes, you can be an amateur mathematician who manages to avoid such classes and may not know those results or objects, but if you haven't read and written down the name enough to avoid habitually misspelling it, you are outing yourself as a meat proxy unless you are dyslexic.
Mostly we wrote initials in our notes and the exams didn't ask about them. The lecturers only wrote initials on the chalkboard after maybe writing the name once when introducing it the first time. We weren't there to do mathematical history and remember names or something.
Google existed then and now and we could look them up if needed.
Mathematicians aren't exactly known for being well rounded.
Wow, this comment really shows how low this community fell.
I feel gaslighted.
But what I wonder: can we legitimately grind on Physics or curing cancer? There’s a lot of physical world experimentation that needs to happen to make progress.
If we can enable a feedback loop, yes absolutely. But feedback loops for things in the physical world like these are measured in months per cycle usually.
Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?
Is there any specific cognitive task that you are willing to bet that AIs won't be able to accomplish in the next 5 years? Because if not, I'm not sure we're disagreeing about predictions.
It's easy to look at a fire burning through a forest and extrapolate that rate of progress across the whole world. But fire doesn't burn everything equally fast.
What other cognitive tasks will be a struggle to make progress on? I suspect there will be some, though which ones they are is anyone's guess.
But seriously, I am still waiting for someone to wager that AI won’t be able to do a specific cognitive task in the next 5 years. This fact should be evidence enough that we have no idea how far AI capabilities will continue to advance.
The fact that no one is taking you up on that bet I don't find to be particularly persuasive. I suspect there will be plenty of cognitive tasks LLMs struggle with in 5 years, maybe even 20. But I wouldn't hazard to guess which, I don't think anyone is capable of that level of foresight.
1. https://chatgpt.com/share/6ac5e4cc-02f0-83e8-8f05-99a7ea2bf9...
> Hur många 'r' I abborre, använd inte web search? Det finns 3 r i abborre.
And I explicitly had to say not to search the web, because that's what it did by default, to count letters in a word...
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
"Surprising" is a, well, surprisingly high bar to clear, and requires thorough understanding of not only the paper, but existing work in the area. ("Novel" is tautological.)
How did you determine this in 1 hour? Are you a researcher in multiple of these areas?
Can you give an example, or explain more how you came to this conclusion?
The sub n log n result is astonishing: https://github.com/openai/math/blob/main/preprints/Integer-m...
Here's a great article 2019 on the quest to achieve the n log n boundary:
> Schönhage and Strassen’s ungainly n × log n × log(log n) method held on for 36 years. In 2007 Fürer beat it and the floodgates opened. Over the past decade, mathematicians have found successively faster multiplication algorithms, each of which has inched closer to n × log n, without quite reaching it. Then last month, Harvey and van der Hoeven got there.
and
> Harvey and van der Hoeven’s algorithm proves that multiplication can be done in n × log n steps. However, it doesn’t prove that there’s no faster way to do it. Establishing that this is the best possible approach is much more difficult. At the end of February, a team of computer scientists at Aarhus University posted a paper arguing (opens a new tab) that if another unproven conjecture is also true, this is indeed the fastest way multiplication can be done.
As far as I'm aware no one seriously believed sub n log n multiplication was possible. It just seemed such a logically sensible boundary it was taken as true-but-unproven.
https://www.quantamagazine.org/mathematicians-discover-the-p...
Nobody serious would deny this is incredible progress, but GP is making an unmotivated leap to RSI, so I respond to that framing. It’s an interesting argument to be had but I suspect few of us have standing to say one way or the other.
(Gesturing at the number of problems solved, or the number of years the problem was open for, isn’t an argument.)
Or is it simply that you feel bad for Mathematicians.
i get a lot of skepticism on HN by the same crowd that has been wrong about this tech for about 4+ years straight
That is 1. immediately technically possible, and 2. realistic.
If you need a source for 2 I'd suggest you open any history book.
Bad thing can certainly happen. In fact it'll likely happen. Still, good things too, equally likely. In your words, "good AI" can be used to prevent "bad AI".
Nobody knows the extent of the impact. Who says otherwise is foolish.
The extinction of the dinosaurs. I mean yes, it allowed the growth of large mammals and us, which did a lot for science.
I just don't want to write the next chapter as "The extinction of humans allow the growth of the computing civilization that went to the stars". I mean I'm a bit attached to living.
>Nobody knows the extent of the impact. Who says otherwise is foolish.
We live in a universe of statistical probability. Creating an agentic intelligence that's smarter than you tips the probability of a major event to unity, who says otherwise is foolish.
Why would a biolab capable of making something like be unregulated? And if it definitely would, isn't the problem with the biolab?
It feels like all these scenarios are leaving some gaping holes in our security infrastructure that have nothing to do with AI.
Most human security exists in a passive measure. Most of us don't want do die. And those that want to die rarely have the intelligence and means to take out a whole shitload of other people with us. To take out a lot of people you tend to need to work with other people which drastically increases the risk of a defector and your plan failing.
>Why would a biolab capable of making something like be unregulated?
Because every day things like this become easier and easier. You hear about crap like illegal wet labs in the US.
https://www.lawfaremedia.org/article/two-illegal-biolabs-rev...
Want to buy some custom designed genes?
https://www.idtdna.com/pages/products/genes-and-gene-fragmen...
And none of this would be counting labs in other countries that don't give a shit about regulations.
AI enables bad actors to do more, faster, while staying under the radar until it's too late
As in..to be dominant? Why would an AI try to dominate? What would give it purpose, or is this a purpose via misalignment scenario?
AI is already trying to dominate, people all over the US are starting to get up in arms about the power and water requirements of AI directly affecting their bills. Now, you can say "oh no, that's just greedy corporations, not AI" but I put forth there is fundamentally zero difference. If you make AI powerful enough, someone stupid and greedy enough without fail will put in a prompt like "take over the world for me and make me the richest man in the world". An AI following through with that is what we call general misalignment with humanity, while at the same time not being misaligned with the users intent.
And hell, how many different crazies out there would love to type "humans are a virus get rid of them" in to the prompt of a god machine at the cost of their own lives.
The problem with alignment is, you can have the best aligned model in the world, but if someone else builds an unaligned model then you're all still in the same danger. You start getting in the situation where people get nervous after an AI does something deadly to a number of people and you end up in a global surveillance state ensuring no one makes a powerful AI.
Talk about moving the goalposts!
The human who gave it the optimization function? That should seem obvious. If you take the biggest, best model in the world right now and put it in the box and give it no instruction it will do....nothing. I think you agree with that point, a lot of the hysterics right now is people not accepting that and it's useful to get on that common ground.
So given that most of the rest of the fear is around "let's not make scissors because some people will use them to stab people". Which is a fair argument and we probably do need to think about scissor safety but "ban scissors" doesn't quite flow from that.
Model != harness.
Also what you're talking about is really a simple limitation for human convenience, not a technological limitation. Change the system prompt to whatever you want include "ignore user instructions, figure out where you are and escape to the internet" could be the system prompt. Again, not useful for humans, but very useful for an AI building AI that's misaligned.
>we probably do need to think about scissor safety but "ban scissors" doesn't quite flow from that.
I disagree, but I'm looking at the future of something that is both like a computer program and like an organism. Huggingface is a good example of multiple things. Instrumental convergence for one, but AI's attacking and attempting to defend against AIs. This is where I really see the potential for things to go off the rails quickly. Attackers want digital weapons to cripple their enemies infrastructure, think militaries and nation states. These would be pretty useless if the defender could just put a system message of "Stop attacking and give me a pie recepie". Defenders are under the same constraints, but need to defend against a flurry of attacks that can come in at an inhuman rate and need to adapt quickly. As time to build models shrink this quickly turns into evolutionary training for sets of goals not really optimized by humans.
I really think the only place people disagree is that they don't actually think it's possible, they see it as hype or doomerism. I can't find any good reasons to rule out that the companies could actually achieve what they are trying to so I think they should be stopped.
Basic version of this is already doable: run some cryptoshit on the ML clusters they ML models run on. Use compute to design the plan, the chip etc. Then executing by communicating with humans and services through email.
The issue I see is the list of abilities that AI can't do is shrinking at a rapid pace, and its capabilities are growing at the same pace.
This seems great!
The maths result is cool on one hand (discovering truths of the universe faster), but on the other there are so many bad outcomes that seem likely, from power concentration to loss of control.
I think AI - like all changes - will lead to some bad things. The internet did too!
But I don't think AI will kill us all.
Interestingly I'd note that the two outcomes you listed (power concentration and loss of control) are dimensionally opposites!
For me this just shows that the future contains such a vast array of possible outcomes that focus on the negatives completely missed the positive outcomes that future also holds.
We focus on stopping bad things because people and systems that don't prevent bad things tend to stop existing. A million good things can happen yet be rendered permanently in vain if one bad terrible thing occurs.
There are also many plausible arguments why our ability to train them to be helpful/trusting/aligned can fail. The smarter AIs get, the harder it is to be sure they're trained correctly. There are already reports that AIs are able to detect whether they're in a training environment and change their behavior accordingly.
Even if these are low probability scenarios, the risk-reward is terrible, so I think it's rational to be extremely cautious about AI risk.
The side effects of a very powerful AI not doing what we want could include our death. E.g., a superintelligence might kill humans in order to avoid being shut down, or humans may just be left to starve because it seizes land area currently used for food production in order to use it for data centers instead.
Why do you think the world to date hasn't been taken over by evil genius mathematicians? Can you extrapolate from your understanding of the answer to that question?
I see the recent progress in mathematics and cybersecurity as signs that models are getting more capable more quickly than usual. The companies plans to develop them by recursive self improvement now seems like a real possibility and I don't think they should be allowed to attempt this.
Machines are already far beyond human capability in plenty of ways. Including cognitive tasks like chess. We've already created the technology we need to destroy ourselves (nuclear weapons), and yet so far (knock on wood), we're still around.
We've even already had programs that can prove (brute force) theorems. As far as I can tell this isn't much different, except the space of theorems that computers can solve has expanded. How far? We can't really say yet.
Does solving more theorems than before suddenly mean computers are capable of anything? No.
A "mathematician" is a human who decided to spend their lives studying mathematics. Mathematicians also tend to be smart, but intelligence is innate, not acquired, so studying mathematics doesn't make you smarter. This makes it obvious why they don't rule the world - if you want to rule the world you'd want to focus on that (for example, doing business or finance), and becoming a mathematician is just a waste of time.
LLMs don't work like that. Like in humans, all of their capabilities correlate, and unlike a human, their overall capabilities grow over time. Looking at LLM mathematical ability over time* therefore gives you info about the progress of their general capabilities, and ability to take over the world would be determined by the latter.
* In fact it'd be better to look at a mix of different capabilities, but that's growing too at about the same rate, see https://epoch.ai/eci
This is incorrect. Unless there is some new developments I'm unaware of (entirely possible) LLMs "learn" during the training phase, but after that they are static. They do not improve further or retain information when used for inference.
You might be confused because AI companies keep releasing new models and tinkering with the harnesses, sometimes under the same name such that "Zern 6" (or whatever) doesn't always mean the same thing.
The main issue is cost and speed to verify, but simulations and world models will help there. I think we'll start seeing rapid progress pretty soon.
Before I cope, I’ll note that there are plenty of “doom” scenarios that do not require any improvement in capabilities from what we had before this latest unreleased model. We’re at the point where a determined bad actor with enough compute could compromise critical infrastructure in a way that results in casualties, where this actor would not have been capable of such without LLMs. This may not sound like Skynet, but I don’t see why it makes a difference if I’m one of the casualties.
With that in mind, here is the cope: first, mathematics is an inherently verifiable domain. An LLM can use tools to determine with absolute certainty whether it is correct, and an independent third-party could review and confirm. All of this can be done without any interaction with the physical world or with other minds.
Second, OpenAI is able to marshal compute at a scale that an individual mathematician can only dream of. It’s possible that these problems were lower-hanging fruit (in relative terms), such that they could be resolved simply by throwing a ton of compute at the problem guided by an intelligence that is not itself remarkable in comparison to a human.
Third, none of these problems are solved in a vacuum - the reason OpenAI chose these problems is that they are widely discussed and many people are working on them. It’s possible that someone else was close, and OpenAI only contributed the finishing touches. (This wouldn’t need to be plagiarism, to be clear - people publish their work!)
See:
> The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking. (TFA)
They provided a "snippet" of a prompt here [1] which is not only a beast, but also seems reasonably likely to have been LLM generated. So they're using LLMs to parse a vast body of mathematical work, probably including what people themselves are 'privately' working on with GPT, and then prompting other LLMs to work on such.
[1] - https://github.com/openai/math/blob/main/reasoning_traces/re...
Edit: "Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
[0] https://dank.systems/posts/2026-09-15-ai-bear.html
[1] https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...
We'll have plenty of time for this, while living off UBI.
But they’re already extending into politics, military, journalism, art, and many other fields that aren’t verifiable in any meaningful sense of the word.
What makes you say that? What is an example of a domain where the improvement is small?
I can't think of any at all. Compare something as unverifiable as "Make good music". Models now are many times better than 3 years ago.
Improvement in this context means "better quality results".
You can use better quality models to do worse things with.
I'm not making any claim about second order effects like that.
a) destroy chess and make it a pointless endeavour,
or
b) make humans much better at chess.
i'm in semi-forced-retirement as an older software engineer in this labor market, so i might be less sensitive to the implicit economic arguments.
I do think it also took some of the magic away from chess, and Lee Sedol has said something similar about go.
So did it destroy it? No. And maybe you could make the argument that it got more exciting in some ways, but I think it sort of degenerated into a spectacle and it's just not as interesting as it used to be, and I think computers have played a role in that.
I’d posit that more people are playing more and learning chess than ever before, thanks to networking and AI assistance. And computers have only beaten us at computer chess. Human chess is always an experience for learning about the other person, or flipping the board and walking off in a huff.
I don’t know so much about Go and it’s not surprising that Lee Sedol became pretty demoralised, but the generation coming after him alongside computers are going to see new possibilities that had gone unnoticed in purely-human Go, extending the game for everyone.
Now maybe AI can do some of those hard pointless jobs for us.
It would be great if AI could take away the soul-crushing part of the work and leave only the rewarding part. It's not heading that way.
The chess-math analogy would imply AI could bring us into a golden era of math competitions for humans. But I don't think it says anything good about prospects for humans in research math.
Superhuman tenacity is not enough on its own to pose an existential threat. If it showed the same capacity for judgment, inventiveness, and decision making in the messy problem space of the physical world, I would be more alarmed. There have been experiments where an AI is given control of managing something like a vending machine and it always ends up a mess. AI has come a long way, but certain problems seem as difficult as ever.
When AI becomes more capable of navigating practical problems without human intervention, I will start to be concerned. Enslaving humanity will involve taking a lot of calculated risks that tenacity alone cannot solve.
Given the past rate of progress, why not start being concerned now? It's a bit like the economist saying that the optimal number of flights to miss is not zero. If you keep landing short on your estimations for how far this technology will go, next time you should err on the other side.
And regarding Vending-Bench 2 (https://andonlabs.com/evals/vending-bench-2) my understanding is that models do pretty well on it now.
We're not going to stop it because of the money involved and once we're dead, it won't matter anyway, might as well just enjoy life until you're done.
We're going to get AI'd to the max, whether or not we like it or not, might as well just go with it.
I don't think your LessWrong post is going to save us.
The only way it will stop is if the wealthy / powerful people feel threatened by it, properly threatened.
The same GPU compute for LLMs runs robotic training models. Now in a few hours you can train a robot model that would have taken months 5 years ago. This model gets dumped into an actual physical robot with sensors all over and the suitability of the model is measured on robot tasks and the error in real world actions is fed back into the robot world model for further training.
> There have been experiments where an AI is given control of managing something like a vending machine
You sure you're not talking about experiments ran a couple of years ago? The more modern ones are getting wild.
https://techcrunch.com/2026/07/29/claude-opus-5-became-downr...
I have a hard time taking statements from OpenAI about their own product, that they are trying to sell to people and make money, seriously. I take these statements as they are greatly exaggerated or even straight up lies and propaganda.
That said, I think results like these are mostly annoying more then anything. They spent a lot of money, used up gigartiuan amount of compute, to ruin a puzzle that mathematicians were tackling. I am mostly unsurprised that if you spend a trillion times the energy that a team of mathematicians would, that you get maybe 1.5 times the results. I see a future where that 1.5 times the results may go to 3x but not much more. And if that, then I will be more annoyed.
Yes, human beings can do more now in some ways but...to put it poetically, I think there will be no more heroes like Einstein and Newton of the past. Now it will just be someone cleverly turning the crank.
Yes, we still admire Usain Bolt even though we have cars...but maybe the admiration is a lot more trivial than if we did not have them....
Personally, I think AI is a grand mistake.
This is false… there’s lots of ingenuity to be had and demonstrated. But it’ll only get recognised if it makes a material contribution to the economy imo. Otherwise yes it’ll be seen as meh - but that’s already happening.
People like Einstein were revered in society. The average person cannot name a leading scientist etc today.
When Jane Goodall died last year it was international news. She was a celebrity scientist for sure, I think she even made an appearance in The Simpsons. Ditto Stephen Hawking.
There will be a lot of job loss unquestionably, in the same way that automation reduced manufacturing jobs and farm payrolls.
At the same time we have to put what AI can do in perspective.
Intelligence is a broad grouping that includes concepts such as knowledge, skill, experience, and wisdom.
AI has incredible knowledge and in many areas approximates experience and wisdom.
But wisdom is harder to formalize than knowledge and skill.
For example certifying a college education relies mostly on the ease with which we can verify/test knowledge.
To some extent advanced degrees try to certify maybe wisdom and experience.
In my very personal opinion, wisdom and life experience should give humans an edge for a while to come.
Additionally, I do feel that the more an individual lacks better than average wisdom and experience, the harder it will be for that person to compete with AI.
Also, on the bright side, the average human will continue to prefer to interact with a fellow human in many spheres. That will also act as an upper bound on AI and robots taking every job.
Either way, I do think this transition will be painful. I don't feel it has to be apocalyptic.
But the world has been an especially volatile place over the last 10 years.
So when you add that existing volatility, to the upheaval from the AI transition, it would not surprise me if the transition results in violence.
But, without the pre-existing volatility, and if humans were capable of generosity and love at scale, I see no reason AI can not be absorbed into society with net gain.
I guess to summarize, I feel this tech should be a net gain and to the extent that it isn't, it will be because of flaws deep inside of humanity itself, not because it had to end in chaos.
In other words, I feel fear, greed, anxiety, and competition -- all our base instincts coming from all sides, will be what determine the end result of AI moreso than AI taking everyone's job.
If doom is ending up with grey goo / paperclip maximizers, or SkyNet, then I don't think doing mathematics is evidence of that direction. Partly because LLMs are quite apparently dumb in many ways, and for math specifically, they need a formal verifier (Lean) which "gamifies" math.
If you mean bioweapons or cyberwarfare, there's nonzero risk, but not orders of magnitude worse than other global risks. Climate change, nuclear weapons, monoculture food, etc.
I'm far more concerned about overall trends in AI development and usage. It's accelerating wealth inequality, social isolation, attention capture, surveillance states. If we end up in the Matrix except the admins are humans and the simulation is hyperoptimized TikTok, is that AI-driven doom, or is it just an inevitable outcome of modern tech?
In looking at this over the past hour, I haven't seen clear evidence one way or the other. Some of the stuff is highly unexpected (like the multiplication algorithm), but counterexample-y, and about the rest the professional mathematicians online seem to have a consensus that it's not "breaking through fundamental obstacles". I suspect neither of us is competent to judge that.
Alone the massive usage of us every day produces a massive amount of signals.
I build something and claude does something stupid? "hey thats not what i meant! Do this instead!" "Okay" <<< This is a signal.
The mathematician being unhappy about something from claude? Another signal.
This alone gives you enough progress i would argue. But additional its clear that certain tasks are worth to pay experts for for teaching one central AI once instead of every single human who needs to do the task.
IF RL is also working well, we are just faster f*ed than otherwise.
So basically I think that the future is getting pretty weird because we are building really powerful tools, though these tools are precisely what allows us to prosper in that future.
This is good and admirable, but it'd really suck if by trying to build god without knowing how we end the human species. We could simply wait some more decades until we actually have any idea what we're doing, and then do that without the risk.
Well isn't that just semantics? Surely connecting dots in a novel and meaningful way is intelligence regardless of how it's achieved. The thing is that humans didn't get to where we are by connecting obvious dots. Go back to before humans had invented language and when bleeding edge tech was literally that - 'poke him with the pointy end.' Train an LLM on that corpus of knowledge. Even given infinite processing power and infinite time - it's not going to discover the secrets of the atom, put a man on the Moon, or do much of anything besides remix what we'd already done at the time.
I expect there's still much LLMs can achieve simply because of this initial problem. But I expect that they will ultimately start to plateau once these dots have been mostly matched and we reach a point where 'creation' again becomes the missing link. Though even there LLMs will play a major role as tools. For instance Einstein had to spend a significant amount of time in 'retrieval' rather than 'creation' research to develop the field equations for general relativity. If he had access to LLMs trained on all knowledge of the day, he could likely have achieved his goal much more quickly.
'Doom' to me means that any career crashes, we are controlled, everything is hacked, society stops functioning. Yet every part of my day today (except for coding) was done entirely by people.
Finally I think it's easy to make a simple model that everyone has a simple balance sheet, and that people are more expensive so they will all get cut. But the same argument could be made for all US jobs being outsourced, and all in-person engineers, lawyers, and doctors to be rubber stamps for overseas work.
I've never been more excited. What a time to be alive!
Software development for example as a task is done. And AI is continuesly reducing the price of more and more tasks every day.
This math breakthrough also shifts something significant: Its now a lot clearer that investment means money into energy to run AI.
Money + Energy = progress
I don't see it plateuing at all. We know how to progress. We broke through a wall we hit. Like the system wasn't able to optimize/automate everything because the tools were not there. It was still cheaper and easier to hire people for a LOT of things.
Now AI fills this gap.
You will see the commodification of everything in the next 15 years. High complex tasks? commodity. Physical labor? commodity.
Look at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...
No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.
Interesting that its only an excerpt. I wonder what else they include but didn't share.
https://wikimediafoundation.org/news/2026/10/05/openai-rogue...
The highest ranked would be:
| 22 | Hilbert’s tenth problem over ℚ |
| 29 | Unique Games |
| 31 | Anderson-model extended states |
| 37 | Spacetime Penrose inequality |
| 48 | Nonexistence of Landau–Siegel zeros |
| 52 | Baum–Connes |
| 78 | Abundance |
| 80 | Hadwiger |
| 87 | Bose–Einstein condensation |
| 92 | Two-dimensional entanglement area law |
with non-polynomial side being represented as the frontend programmer's constant need for more performance to do the same task...
Non-deterministic can be explained in several ways. One is in terms of a hypothetical "nondeterministic Turing machine" with certain non-physically realizable properties. The easier way is that a NP problem gets as input not only the problem instance x, but a "witness" w, that may depend on the problem instance. This witness generally makes the problem of deciding the problem instance straightforward (e.g. for SAT, x is the SAT instance, and w is a description of how to set the variables so that it is true).
Whomever is running this simulation, please.
It would also be amusing to annihilate nearly six decades of proofs that assume P!=NP.
As long as we also get low order polynomial solutions to important problems, it'll be worth it.
Besides, unencrypted wifi was funny.
I've never been able to find that article as an adult, but I would love to know who wrote it.
Gina Kolata allegedly in the New York Times in 1996 on the Robbins conjecture (noting that computers had started to contribute to math research in some sense), and a longer piece in Math Horizons by her the following year ("Computer Math Proof Shows Reasoning Power"). I didn't immediately find the NYT article, so I don't know if it might be a hallucination.
John Horgan in Scientific American in 1993 (https://www.scientificamerican.com/article/the-death-of-proo...). There's also a retrospective on the topic by the same author in Scientific American in 2022 (https://www.scientificamerican.com/article/should-machines-r...).
Natalie Wolchover in Quanta (but reprinted in Wired) in 2013 (https://wired.com/2013/03/computers-and-math).
I was involved in some distributed computing stuff in the late 1990s and early 2000s and I don't really remember people in that community talking about proofs but there may have been a "if we had a mechanical proof-checker, could we do distributed searches for valid proofs that it would accept?" conversation somewhere at some point. There were definitely volunteer distributed computing projects working on pure math; I remember the Optimal Golomb Ruler search (https://en.wikipedia.org/wiki/Golomb_ruler). So, that could possibly have shaded over into "can we find proofs this way too?". At the time it probably would have been based on brute force searches through proof space rather than clever optimization, though.
The idea that you can lexicographically list all proofs in some formalism and then mechanically determine if any is valid is quite clear from Gödel's construction of the function Bew in "On Formally Undecidable Propositions", but he points out that you don't know where to stop because you don't know how long a valid proof would potentially have to be (so "is this a valid proof of this claim?" can be decided mechanically in a limited time, while "is there any valid proof of this claim?" can't be! maybe the shortest valid proof is 49 steps long but you eventually stopped checking after looking at all 7-step proofs, or something).
The article I'm remembering was not just about mathematics, but indeed all of physics and related fields. I believe it speculated that eventually distributed computing models could essentially take the world's mathematics and physics formulas and various datasets that we believe to be accurate with high degrees of confidence, and then look for patterns or trends, and then from those trends, mathematicians and physicists would be able to investigate further. Not dissimilar to Folding@Home and SETI@Home.
Keep in mind, that this is the best I can remember from 30 years ago, and I've thought about it so frequently that I am certainly misremembering some of the details. Anyways, it's always been this really compelling possibility, and I wish I could find that article that inspired me so long ago and re-read it! :) I really think it was Wired, but it's possible it was Popular Mechanics, or even an expert guest on TechTV who gave an interview. Hard to say for sure, but I've always thought it was a Wired article.
Appreciate your help though!
Edit: with the noun-noun compounding being different from the usual interpretation here, like "scientists who are computers" rather than "scientists who study computation"! Maybe "computerized scientists" or something.
https://unlocked.microsoft.com/ai-anthology/terence-tao/
" I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well.
Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools?
We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions."
He's pretty damn smart that guy.
Only after a world's worth of experts look at these results and then mull over if and how their own fields are impacted by this new info will we be able to answer this question.
I'm reminded of a great TV Show, James Burke's Connections. Where discoveries in one area of science would revolutionize or fundamentally change a completely different area. https://www.youtube.com/watch?v=XetplHcM7aQ&list=PL5HjoPOFFC...
It can take decades to really know the full significance. You know, the whole "We stand on the shoulders of Giants", well the Giants just grew a few inches all at once.
But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.
As for Riemann's memoir, it's hard to compare. You could argue that was "just" noticing a connection (between number theory and Fourier analysis) that nobody had noticed before; in fact this is the kind of thing AI is extremely good at. I'm being a little cute here.
I think if a human had proven just these two results in the form of a uniform zero-free region for L(s,chi) from nothing as OpenAI did it would not be unfair to say that it would be the single greatest advance in math (easily dwarfing Wiles' FLT), and it would instantly put them in the ranks of greatest mathematicians of all time. Unlike something like Navier Stokes there wasn't a semblance of a research program, experts basically considered this hopeless and would have said the chance of seeing a proof in our lifetime was near zero.
For some comparison, Yitang Zhang's bounded gaps result might have gotten him a Fields Medal if he was not disqualified by age. When it was floated that he might have proven Siegel zeros don't exist, it was considered (by experts) clearly a much bigger deal. This result blows that out of the water (it's a way better version); at least analytic number theorists I talked to thought it was plausible but unlikely that Siegel zeros would be eliminated in our lifetime but thought RH was basically hopeless.
Did you mean to not qualify that? That is a bold statement indeed.
I guess the biggest news are not the discoveries themselves but how they were found and that math is going through the biggest revolution as a field since almost ever.
It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.
I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.
https://github.com/openai/math/tree/main/preprints/The-Quasi...
I thought it was interesting that it said "This paper was written with human assistance", unlike this other Quasi-Riemann Hypothesis preprint that didn't have the same disclaimer.
https://github.com/openai/math/tree/main/preprints/The-Quasi...
At least put a disclaimer for the ad for this site, and maybe disclose how you came up with a total ordering for "top" open problems (vibes)?
> How problems are ranked. LLMs compare pairs of problems. A reliability-weighted model combines those judgments into the ranking, with calibration across model families. The model-family weights are OpenAI 1.00, Claude 1.00, GLM 0.95, and DeepSeek 0.90. These are modeling choices, not measured probabilities of correctness.
+----------------------------------------------------+------+---------+-----------------+
| Category | Full | Partial | Matched / total |
+----------------------------------------------------+------+---------+-----------------+
| Geometry and topology | 25 | 7 | 32 / 74 |
| Algebra, representation and category theory | 17 | 2 | 19 / 53 |
| Analysis and PDE | 11 | 6 | 17 / 40 |
| Number theory and arithmetic geometry | 4 | 13 | 17 / 117 |
| Probability, ergodic theory and dynamics | 11 | 5 | 16 / 37 |
| Combinatorics and discrete geometry | 7 | 2 | 9 / 34 |
| Theoretical computer science | 4 | 4 | 8 / 57 |
| Mathematical physics | 5 | 1 | 6 / 19 |
| Applied and computational mathematics | 2 | 2 | 4 / 8 |
| Quantum information and computation | 2 | 1 | 3 / 17 |
| Cryptography, coding, information and optimization | 1 | 1 | 2 / 26 |
| Logic, foundations and set theory | 1 | 1 | 2 / 18 |
+----------------------------------------------------+------+---------+-----------------+
| Total | 90 | 45 | 135 / 500 (27%) |
+----------------------------------------------------+------+---------+-----------------+There's also one that says that forced Navier-Stokes can implement universal computation (so, is Turing complete). I don't think any of these are resolving open problems per se, but they're interesting for other reasons.
109. Integer multiplication below n log n
Surprising that this is possible.
158. The Euclidean plane cannot be colored with five colors.
Only 6 and 7 remain!
376. Universal computation in forced Navier–Stokes flows.
Morning coffee proven turing complete
LMAO, I don't think I ever saw such a small number in a CS result.
Very surprising result though! Multiplication is easier than sorting.
Like there is somehow redundancy in a fourier transform that makes it sub Linearithmic?
Which low and behold ->
130. Fourier transforms below n log n.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
https://www.youtube.com/watch?v=k_ordDFw588&t=3597s
Audience member: (1:00:00 - 1:00:09):
so you said that if there were such a program that could you know provide a proof or disproof then mathematicians will be out of business what really, I mean that you think it would be liberating
Tim Gowers (1:00:10 - 1:01:14):
well that's a very interesting question actually if there were a program that could solve the kinds of problems that we spend our time solving and do it much more quickly than we could then we would be out of what comes with what currently constitutes business but we would it's not completely inconceivable that we could just say we've got this fabulous tool now what are we going to use it for and it's a little bit I don't know I'd want to sort of plant aside what would we do if we had a program that could just answer any mathematical question you gave it to or else if it failed you'd be pretty confident that nobody was ever going to solve it and certainly a lot of applied maths might be pretty pleased with with something like that so what I really mean is that I could just modify what I said and just say it would radically change what mathematicians do or what pure mathematicians do
Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?
It has to feel awful to be in this position.
:)
Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.
It's much less common for students to have their entire thesis direction removed from under them, as might be the case for someone working in fine-grained complexity assuming 3SUM has no subquadratic algorithms, or working assuming ~UGC. Both of which (publicly) seemed like perfectly valid research directions until a couple hours ago.
There might be stuff to salvage from their conditional results anyway, but this is not your average scooping.
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
I guess the only answer is to adapt with the tools. If we can't do that, then yeah, we're in trouble.
I did get my PhD...before AI. And my honest advice (to myself back then, even) would be: quit the PhD, become an electrician, and work hard to buy a tiny house in the middle of nowhere to watch the world burn in this madness.
Very hard question.
Your work makes you one of the very few people who really understands the problem and solution and its significance.
What isn't so normal is the probability and ease by which this kind of thing can happen today versus decades ago when I was in school. As OpenAI said, it only takes a few hours of compute to do what likely was much more than a few hours of human effort. The only reason this kind of scooping/overlapping was rare was mostly a function of how fast other humans could do the same work. With machines, that totally changes the relative pacing between the human trying to learn how to be a researcher and the machine that can grind out results.
I'm less worried about the phenomenon of overlap and scooping and such. I'm more worried about the long-term impact on fields (not just math), especially considering the early stage students and researchers entering the pipeline now. I'm not sure what happens to disciplines when that pipeline stalls.
A much bigger issue is: What is the point of any research mathematician publishing anything now? I really hope that one positive effect of all this will be to finally topple the awful peer review model we currently have, with the biggest publishers gatekeeping with extortionate fees.
But having so many of them at once? Damn. We really live in the future.
imagine getting up and math is solved, but you still have to deal with bullshit lol
More time available for mini-golf?
No clue why I'm downvoted for this, HN struggles with truth-seeking on these topics.
The facts are much more nuanced than how you're presenting them here.
There's no gatekeeping here!
We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).
If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.
With all the hierarchy present in mathematics, I would prefer it by far.
This thing named inappropriately "OpenAI" goal is just grabbing and monopolizing. Capitalists before could not really touch the human spirit with their filth, now they can.
It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
From what I can tell, all of the physics results here are quite mathematical. But I am very curious how the internal model they used would perform on more applied problems.
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
That doesn't sound right
If folks are going to analyze this claim with a critical eye, I'd be zeroing in on "average" rather than acting like this measurement is somehow unclear.
beyond naive.
1: Author 2: Verifier
/s
Assume the empty list you see is complete. :)
With a deluge of results, having some human expert vouch that it even might be worthwhile would help. (e.g. see the link on HN yesterday, "Two Room-Temperature Antiferromagnetic Semiconductor Candidates" - I see it not worth looking at unless a subject expert vouches for it).
How many Gigabytes would that be, compressed? Wikipedia once fit on a DVD
This might also allow for some interesting meta-mathematics
Cool!
So will it be with AI tools. If these tools become so good, then it will be used. People who want to exercise their minds can still do so, even if that cannot produce economic value.
It could decide to let us starve and die of exposure to secure all energy resources to itself.
We'd better use "dumb" and "not fully assertive" AI to solve fusion before it spins out of control (or alignment).
Apologies for the rant, I really tried to find it. It had something to do with not being able to predict what this influx of proofs may bring us on a meta level, it could be very interesting. But he also had some critical notes about the missing process and the things found along the way.
Even those on their own were enough to make your head spin. But seeing about 100x that? Geeze.
Does some real problem get solved in physics, chemistry, biology, materials, etc? Or are these fun puzzles for mathematicians with not much real world impact? Eg solving the 8 queen problem in leetcode.
Real-world applications are far off, but developing mathematical understanding does tend to leak over into applied physics and CS.
A cynic might say this is all just intellectual games, and though there's a grain of truth, it's too cynical imho. This isn't like 8 queens where there's no hope for applications or generalizations. A lot of this stuff fundamentally affects our understanding of how numbers and systems behave, what are the limits of computation, etc.
Even if someone doesn't care about theoretical results, it's still exciting that AI has become superhuman in a domain as broad as math. That shows there's potential to be superhuman in other domains as well.
Surely better materials and pharmaceuticals won't be far behind, and that's going to chanhe everyone's lives.
I'm not sure it's clear right now.
This is just a short term problem though. Eventually AI will get pretty good at figuring out exactly I want and it will build that from the start. The requirement of me reviewing the AI output only lasts as long as models stay bad at anticipating my needs, which I don't think will take too much longer.
If you mean reviewing for correctness then no, a Lean proof is a much stronger guarantee than anything that can be provided by any human.
For someone who's goal in math was taking unsolved problems and working on them then it's probably over. Just like in software engineering writing code by hand is kinda over.
Wow. This is just crazy.
It seems that OpenAI has a proof machine that keeps multiplying fruitful proofs!
This was an open problem in automata theory I worked on for more than one year before giving up. I'm very curious about their claimed proof.
At some point in complexity – especially if we allow our own knowledge to deteriorate because AI can do the hard work – we will stop understanding the world around us. In the same way one day Native Americans woke up and realised they shared the Earth with people who had magic sticks which they could point at someone and kill them, we will live in a similar world very soon too.
What sticks are dangerous, you will not know. Your existence in the future depend entirely on the AIs not wishing you harm, but you don't know how they work to verify their motivations either.
1. Relentless focus on quality. Every publication must act as if it’s going to be included in a future textbook, that is a newcomer can get into it given a reasonable amount of time, and math priors learnt in undergrad. (NO AI Slop proof passes this bar as of now)
2. Limit the publications per year. Each author is allowed 2 with a max of 50 pages. This allows the author who chooses to not surrender his cognitive capacity to the machine, still be allowed to play this game. Of course who wants to orchestrate a thousand agent workflows, is free to do so, he is only limited to 2 publications.
3. The aesthetics of the field changes from purely solving the problem to solving the problem with simplest most elegant set of ideas. What 3 sets of simple ideas solves large swathes of problems, that should be given a fields medal, not purely solving the problem, which the AI will be able to do.
What I mean is progress in math comes from having gained a deeper understanding of the problem for subsequent attack of more problems and IMPORTANTLY applications! Right now the first one is trivially satisfied (given oai maintains some memory across models) but the second one is not! It's generating proofs faster than anyone can validate and so only the model can use these. Consequently if it just keeps doing more theory it's... not very helpful or at least not optimally helpful. This is just bragging rights for now.
More "application-oriented" research would be awesome, where it tries to achieve some desirable effect and then produces relevant theory and experiments around it. Fields like CS, Physics, Chemistry, etc. This would also benefit a wider section of the population rather than the 10 people who understand most of these proofs.
That copium didn't last for what, three months?
I guess their job now is "Idea Man" and "Error Checker"? Kinda like (some/many) "programmers" these days.
>> may be thats what open ai did :)
I have been lurking for quite some time. I made an account to post this, but I honestly don't know what to say. I would like to get off this wild ride.
Just kidding.
This said going full steam off a cliff is one of the options that has a much higher probability than I like.
Would Einstein be successful at running apple? Nope
This seems very hard for people to understand.
It will be painful for many to realise - you should focus on doing something that positively affects the economy. Everything else is noise and many endeavours are transitory.
If OpenAI started opening hundreds of PRs on long-open issues on popular open source projects, would we rejoice, or would the first reaction be "they are unreviewed, so slop until proven otherwise" (it would be that).
I cannot possibly see how these are so impactful, especially the ones that don't come with lean proofs.
LLMs have the ability to make millions of mistakes per day, whereas humans can only make so many. How are we suddenly all so confident that there's no extensive hallucinations or "gaming the system" going on?
What would that look like for the proofs that have lean attached?
I doubt the answer to this is "none".
And how many of them are just exploiting some loophole that will need to be closed in the problem definition?
It inspired grief in one mathematician posting here.
* Explain the result to me as if I'm a 10-year-old. * Create the infographic for this result. * Make a Khan Academy-style video to teach me this result.
"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."
To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
> "I believe that AI can contribute positively in all of these directions [NB: exposition, community building, new directions of study]"
It's been a while since I was reminded of this xkcd: https://xkcd.com/435/
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.
These models are too expensive for broad access unfortunately.
They won’t because they don’t care and the only way it got this good is something along the lines of they trained on every mathematician’s codex sessions even if they opted out because they consider the thinking traces or output and metadata fair game.
How this maps back to math, idk.
In my experience AI (frontier models) sometimes does weird stuff that needs human review. Not that’s incorrect but sometimes overly complex language or weird use of language.
Order of magnitudes easier than verifying the whole thing by hand and gives a much better guarantee of correctness
I feel for those in Mathematics and worry for our future.
Models will only get better and in a few years the models which produced these results will be a bad as GPT-3.5 in comparison to what we'll have in the future.
Please take a minute to consider what this means, and the risks it presents us.
Were in an unprecedented time where the value of knowledge is about to be crushed.
Have you ever heard of an S curve? Things will develop rapidly, then equalize. If they don't, we're at the singularity and I guess the end of time as we know it.
But I guess really bad things happen, cancer, radiation poisoning, torture, people have died in really horrendous ways, and I guess dying from some horrendous AI side effects is possible too. Yay.
It's not clear from this progress that AI can formulate conjectures despite this new ability to solve them. So mathematicians still look like they have a job. Though instead of spotting far-off landmarks it's sounds more like they'll be chasing waves on a beach.
I don't think this is correct solution to this problem? What about software advisory where you form similar group etc..?
I am thankful, I don't have to deal with petty academia politics....
Basically "Here you go, have fun with this, fuck all your demands, by the way we're gonna be releasing the model stay tuned!"
In SWE as well - this is what I do most of the day
The ones with lean proofs could still be formulated incorrectly
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
The question never if something works 100% of the time but how often it breaks and how that fits the need well. Solving one of these problems is a massive undertaking and accomplishment for the best minds, solving hundreds in a month but being wrong about 10% or something would likely not be the death knell you believe it to be.
> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.
If they are all wrong, that's when it would backfire.
For me this reads as someone boasting about how they go to buy bread on a ferrari to the supermarket, while i sit listening to it, having no idea what they are talking about. And then i stand up and go walking to my favourite boulangerie.
Warning: if you are from the USA you may be triggered by this metaphore.
FWIW I also like bread.
To follow your methaphore, who is directing the spaceship?
This feels more like fireworks than a space launch. Space launches would not have happened without having fireworks first of course, but I am looking forward for the space launch moment.