> It feels like something written by someone who’s on psychedelics. So much unclear and doesn’t make sense. Lots of name dropping of previous work without discussing why it can be used despite impossibility results
> Basically the paper is so horribly written that it’s impossible to read it without AI help
That's interesting and haven't seen this in all the coverage of this event.
It sounds horrible to wade through - like trying to understand someone else's messy code that still produces the correct output.
Math proofs need to produce the correct output correctly, which is not quite the same thing.
This looks like an AI IPO PR powerplay, because at this point the proofs haven't been checked and it may not be possible for a human to check them - because proofs should be clear, not horribly written and noisy.
The noise is suspicious because it's the difference between brute forcing and cognition. A human proof won't just be logically correct, it will be cognitively distilled and coherent. It may still take years to understand it, but the logical flow will be straightforward, not obfuscated.
You want the path through the maze to be as short as possible and the map to be as clear as possible.
This sounds like the opposite. There may be a genuine path through the maze, but if it's too convoluted and takes too long it will be impossible to confirm.
I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
I suspect that's possible without tripping over the halting problem. (But I can't prove it.)
> “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be “natural” in Donald Newman’s sense [13]:
> This term . . . is introduced to mean not having any ad hoc constructions or brilliancies. A “natural” proof, then, is one which proves itself, one available to the “common mathematician in the streets.””
> I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
In 1976, the proof of the Four Color Theorem was controversial because it was done with a computer examining over 1000 cases by brute force and was essentially not comprehensible by humans. But mathematicians ended up accepting it. So mathematics has a 50-year precedent of not requiring human-scale proofs. How is the current situation different?
(Disclaimer: Apologies if this sounds dismissive or argumentative. I genuinely think that the Four Color Theorem should play a role in these discussions and suspect that many people are unaware of the controversy over it.)
There's also another (that I find more concerning) aspect to it.
As AIs become smarter and smarter, there will be no amount of clarity that will make more complex proofs understandable to humans - this is an inevitable effect of the cognitive capacity gap.
Complaining about bad style can make some sense now (I disagree anyway), but it's an argument that will be dead shortly.
Arguably no human understands 100% of how a smartphone is produced, and, it doesn't matter?
Maybe no human will fully understand a future proof, but they could fully understand a little piece of it. And many humans in aggregate could understand it, each with their own little piece.
If intelligence is compression, and these models are a different form of lesser intelligence than human, but being scaled up to brute force problems, then it makes sense the artifacts that produce (the proofs) would have worse compression than a human proof would.
In other domains I have seen first hand overwhelming evidence of how things that cause the AI to make mistakes also cause humans to make the same mistakes.
I wonder if the proofs being produced that are hard for humans to interpret are also hard for other LLMs to interpret.
In other words, I wonder if humans are still much better at compressing understanding into proofs than the best LLMs, and what it will take for LLMs to exceed them.
It kind of an explicit example of how the LLMs can be materially less intelligent than people, but still be more productive through scaling, and yet they also can't replace people because they are a categorically different kind of intelligence. It's like all the AI debates compressed into one example showing countwr-intuitive answers.
> I wonder if humans are still much better at compressing understanding into proofs than the best LLMs, and what it will take for LLMs to exceed them.
Isn't it fairly established that (generally [0]) manually written / optimized skill files perform a lot better than generated ones? Meaning that yes, this likely does hold.
[0] or to be specific, that the pecking order is: ai generated < human co/written < hyperoptimized for the specific model via some convergence process
idk if i'd even say they're "lesser", just very different. so they look like gods/babies depending on what they're doing because we anthropomorphise them.
In terms of synapse density, neuron diversity, energy efficiency, memory access, etc. they are orders of magnitude lesser.
My mental model - for better or worse - is that intelligence has both shape and area, and LLMs are orders of magnitude smaller area but very different shape, and they have more intelligence area in the kind that humans have lesser of.
So yes very different, more in some material ways and lesser in others, but in total intelligence are still orders of magnitude lesser.
My gp comment was acknowledging that when you scale up many instances / brute force problems it confuses that "total area" claim a bit.
To follow the anthropomorphization... 1000 toddlers may have more total intelligence than a grown man, but does that matter?
The problem with these discussions probably/usually fold into differing/loose definitions of intelligence.
perhaps the chat-based ux has sort of fooled us into comparing these things to human intellegence. we don't really do this with chess, or other forms of ai, nor computers at large.
> A human proof won't just be logically correct, it will be cognitively distilled and coherent. It may still take years to understand it, but the logical flow will be straightforward, not obfuscated.
Not a counterpoint, actually case in point, because:
>Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.
The post says there's a Lean certificate for this and other proofs ("some [...] not all of them").
> This looks like an AI IPO PR powerplay,
Interestingly, the post has actually also an argument for this:
> Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.
> So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.”
It's on OpenAI and Anthropic to prove that they obtained these results legitimately and credited all researchers who deserve credit. They do not get the benefit of the doubt.
But if you think they got them illegitimately, then how did they get them? And why are mathematicians reacting to this as a sudden explosion of new results that have resisted sustained effort? Where is the sudden productivity rise coming from?
I'd say it comes from the same mathematicians that were strongly encouraged to use the machine to solve their problems for the last 2 years or so.
There's clear benefit in a babelfish that can coordinate disparate efforts, the only problem with the current iteration is giving credit to said efforts.
Google went quite far down the road to hell, but stopped short of taking credit for websites' content since the company understood that poisoning the well only goes so far. At this point, one can safely conclude that _Chat_GPT was an intentional attempt to squeeze out more data once they mined the internet dry.
Have you actually read the article? It's been actually written, among the other things, because the author's wife has been trying to solve one of the problems for her whole life.
Surely by the time of the IPO we will know whether the main results are correct, if only because a different AI will have produced a lean proof or found a logical flaw (the second case would be hard to verify but probably not impossible).
Also from what I can tell from the few fields I understand, the proofs aren't that long or complicated they are just terribly written.
Why should that make material difference to the IPO? What is the economic value of those results?
The entire US federal budget for math research is something like $100M annually. And mathematicians in other countries are hardly making bank either. How does one reconcile how the market has historically valued mathematics with the cash-strapped frontier labs ploughing so much money into that enterprise?
They're burying any doubt that the models are capable of superhuman performance on intellectual tasks. Neural nets aren't calculators, and were notably poor at mathematical reasoning tasks for a long time. Now they're not, and the labs are proving that by chewing through what would ordinarily be decades of progress in a month. And the reason to go for math in particular is because there's no wiggle room. You can't just dismiss it as hallucination.
If the models can do this, they're almost certainly good at just about everything, because the reasoning and creativity required to solve these problems will translate. And even if they were only good at this stuff, that's still a tremendously valuable thing, because quantitative reasoning and analysis is the bedrock for many, many industries.
oAI is gunning for the largest IPO in history at this point, and they might actually get there.
> If the models can do this, they're almost certainly good at just about everything, because the reasoning and creativity required to solve these problems will translate.
The "then" in your "if-then" bears a heavy load. Why would society assign so little economic value to pure mathematics if the skills for proving math theorems translate to massive value in "just about everything"? Would you expect top mathematicians to cure cancer if you transplanted them from the math department to a medical research lab?
Because research mathematics is a subset of all quantitative work that gets done, but it's by far the most technically difficult subset. If the models can handle research grade math and produce ironclad proofs, they can probably handle the quantitative side of just about any discipline in a trustworthy fashion. Think about how many dinky spreadsheets have gone on to become critical tooling for large organizations. Even if you consider that many disciplines hide technically demanding work behind tooling (e.g. essentially no one is writing a stiffness matrix FE routine by hand), this model would be capable of writing a direct competitor from scratch to produce the same result.
All of STEM relies on mathematical analysis, and new models are now superhuman at that. And yeah, I'd go a step further and say that the reasoning and creativity required to solve cutting edge math problems probably does translate to other tasks like interpretation of the law, or medical diagnosis, or accounting, etc., for the same reasons that I think most top tier mathematicians would excel at those tasks were they so inclined.
That sort of worked for Eric Lander but as he moved higher up in the bureaucracy, he found himself suddenly having to manage people who weren't nerds like him. He was bad at that had to step down.
Do you think mathematicians are not already working on cancer research? There's quite a bit of heavy math in biostatistics, medical imaging, machine learning, etc.
I'd assume that the majority of people who study math take their skills and move onto some related STEM career that isn't pure math. Academia is incredibly small and competitive.
The market works in mysterious ways. What companies do for marketing is often irrational, what companies do to attract investors is likewise often irrational, and why investors invest in companies is also often irrational.
Why should that make a material difference to the IPO? Because of the vibes, and investors are indeed all about the vibes.
> I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
Who do we demand this from? The AI companies? Or the mathematicians who are worried they will have nothing left to do?
We should consider the possibility that at some abstraction levels, we can safely stop chasing "clarity" or "coherence" which is circularly defined in such a way that it's capped by human processing power.
Developers and people in CS in general seem to have gotten used to the idea that most productive SWEs don't need to exactly know how to produce assembly or trace every branch prediction or even most of the optimization the CPU (or even their compiler) is running. Mathematicians will get there.
But it doesn't follow that these proofs - or any proofs - are automatically on the far side of that limit.
The human usefulness of a proof depends entirely on its human legibility. Much of the value of proofs is in inventing new techniques and concepts and having new insights into relationships. Occasionally you get some game changing insight into practical physics or engineering. But that's rare.
Without that, proving or disproving a conjecture is an excuse for new and original thinking.
Compilers are not the same problem. The point of code is to produce reliable-ish consequences from various possible inputs. It's not a creative exercise in logical consistency, which is what maths proofs are, ultimately.
CS got that idea from mathematics. Theorems (with the definitions required to state them) are supposed to be self-contained units. Once the general consensus is that a theorem has been proven correct, people can use it without understanding the proof. Of course, people still want to understand how things work, and it often makes sense to understand them a couple of layers below the one you usually work at. But at some point, you should stop distracting yourself with irrelevant details and focus on your actual work.
Thing is that many of the theorems here are not useful work in and of themselves, but were posed as research problems because it wasn't clear how they could be resolved with current techniques, implying that the process of trying to find a proof might result in new techniques. It's those new techniques that are the actual goal, but if they can't be easily extracted because the proof isn't structured to enable this, that's a bit of a headache.
It also aligns with the fear that these proofs present a risk to the ecosystem by out-competing attempts at more human-readable proofs. Perhaps though we end up with more math influencers who edit and annotate these proofs to bring them back to us.
The ecosystem is (ahem) gated by hiring committees. There is no risk of AI replacement from the inside. "Replacement" is not even a possible movement. The funding for mathematics worldwide comes mostly from endowments, which are investment pools.
I think that's ultimately a good thing. As proofs weren't supposed to be the point as stated by William Thurston long ago. Maybe now the focus can be more on better explanations and creating tools for growing understanding and intuition.
It's worthy of note that most humans, do not find most mathematicians understandable. As is frequently demonstrated in Calculus classes. Therefore it is arguable that even human produced results are not generally human understandable.
Isn't that just the default experience with AI these days? In small enough scale, AI models can express their ideas clearly. But the larger and more complex the ideas are, the less suitable the outputs are for human consumption. I guess AI models think too different from humans, and nobody has trained them to communicate complex ideas in the way human experts in that particular topic expect.
Meanwhile you can use the model to help you out as Scott comments "Just now, however, Dana tells me that she’s been asking Astra all day to explain the new proof of the UGC to her and it’s been doing an amazing job and she’s starting to understand the construction."
This makes sense and looks exactly what something would look like that is smarter than us.
If it is correct it is making inferences that we can't see. At a stretch even working in more dimensions than our three dimension limited brains.
> Basically the paper is so horribly written that it’s impossible to read it without AI help
This basically describes every single PR at work for the past year. Diffs of 10k+ paragraphs of comments saying nothing. Just rubber stamp and move on, nothing else you can do.
Why would anyone believe this (also) is not simply example N+1 of this is the worst it will ever be, as opposed to recognizing this as what will almost certainly prove to be an awkward moment, soon to be replaced by another order of magnitude of cleaner, clearer, more intelligible, etc.?
Ximm's Law: every critique of AI assumes to some degree that contemporary implementations will not, or cannot, be improved upon.
Devin's Law: every defense of AI which rests on "it will get better, trust me" is in many ways indistinguishable from 2010s crypto hype or "level 5 self driving is right around the corner"
1) Predicting the future is hard, but so far everyone who was saying that it would get better turned out to be right. It is getting better
2) Waymo exists
3) That doesn't mean flying cars will within your lifetime
I don't think anyone is saying it can't or won't get better, but the question is how much better, on what timescale, and are there fundamental parts of the problem which will remain extraordinarily difficult to improve?
The comment I was responding to suggested a guarantee of an "order of magnitude" jump right around the corner. There is no guarantee of this, and if you view doomers as fools for having doubts, then we ought to look upon the folks who are sure of this sort of progress in the same way.
Your original comment was about self-driving cars, not flying ones. If you wanted unattainable goalposts you should’ve started with that, not ended with it.
Respectfully, I disagree with the way you're characterizing my comment. I was demonstrating that just because you have a level 4 Waymo doesn't mean flying cars are right around the corner. This is again a reference to the original comment I was replying to, the one that suggested of course we're going to get an order of magnitude improvement.
In that case, one would expect to see some progress in this direction, but AFAICT that hasn't shown up yet? If anything, it's getting worse, though that could just be the increasing scale and decreasing cleanup efforts.
Already the unit distance proof was substantially human-edited (per Thomas Bloom). Then with the ten problems from Astra you started getting the citation issues. Then Navier-Stokes was a rushed 160 pages with barely any citations, and some of the related papers were called (by their "authors") the ugliest mess they've ever seen.
And now here we are. At least it seems that mathematical ability and communication with a mathematical audience are independent skills, and progress in the first does not imply the second.
This doesn't surprise me much, given two analogies: 1) many smart people are nonetheless horrible lecturers. (You can't quite get the opposite extreme, since to explain math well you have to be able to do it.) 2) AI writing in general hasn't improved. The models have annoying verbal tics ("honestly") and have no sense of which part of what they say is obvious and which is relevant.
I'm thinking of somebody like Grant Sanderson (3blue1brown), doing pure exposition extremely well. For that you at least need to be able to work through examples, or to present why an intuitive approach might fail, and these things can be little theorems themselves. It doesn't have to be publishable in the current culture of novel results, but you do need a lot of competence with the tools.
Yes, Grant Sanderson might be a great example. I am not doubting the competence of the "good lecturers" I was referring to. But that is different from being able to introduce the big new concepts that give the big new results.
As a side note, you can tell this wasn't written by an AI by the first sentence:
> mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!
My 8yo talks exactly like that. I could totally imagine him saying this, the same way, at the dining room table.
I asked ChatGPT "pretend you're an 8/9 year old today. how would you insult your mom about having her job be replaced by an AI?", and the responses it offered were:
> “Mom, AI took your job because apparently even robots were like, ‘Yeah… we can do this better.’”
> “Mom, congratulations! You got replaced by a computer. Even Siri has a job now and you don’t!”
> “Mom, AI took your job? Dang. I guess even a robot looked at your work and said, ‘I got this.’”
> “Don’t worry, Mom. You can still be useful… like teaching the AI how to make my lunch.”
All of these seem to have a vaguely Millennial flavor, aside from being pretty awkward and mechanical roasts. Trust the children and linguistic drift to be the best AI detector.
Mother dearest, it pains my effusive tendencies to inform you the untimely death of your life's work by the hands of a mere transistor-- moreover, my incontinence briefs have reached their capacity.
i added "use current trendy lingo" and the results were a bit less mechanical sounding. one of them included "cooked", another had "negative aura".
i used the free google ai: (deleted the examples... but they were vaguely close to what i hear my grandkids say.)
edit: neat, insta-flagged despite hundreds of non-ai comments that have never been flagged. i would have thought that hn would use some heuristics in their ai detection but i suppose not.
I appreciate the counter argument, even as a chat bot skeptic.
However, there is the fact that you had to know how to poke the machine so it plucks the right vocabulary out of the training data.
There is very clearly no theory of mind: no inherent internalised modelling of how a human of a particular age thinks, speaks and what they do or do not know. These are the most obvious cracks in the “LLMs are (or will be) the superintelligence” narrative.
The short story about the digital animals ("The Lifecycle of Software Objects") is an interesting read in light of AI advancement. It is incredible to think that when this story was written, all concepts described were sci-fi, whereas current technology is more advanced than the AIs in the book.
> But it also appears that no human has understood just about any of these proofs yet
Has anyone verified any of the proofs produced by OpenAI or is everyone just assuming that it just be true because the Lean code checks out? Couldn’t the Lean code just be formulated incorrectly?
For example, the statement of e.g. Fermat's last theorem in Lean should be understandable to anyone who played The Natural Number Game [0] and knows a bit of mathematics and programming. For the proof, you trust the compiler.
The statement of other theorems can be much more delicate, and the Lean formalization may require an extensive introductory section which will need to be carefully checked.
Then there are the cases where no Lean formalization is currently available, and all we have right now is an often impenetrable pdf in the OpenAI repo. I would not at all be surprised if some of those contained logical gaps.
Time will surely tell, but there are certainly doubts and lots people are very busy checking these results.
There's an entire paper claiming that many of these AI-generated Lean proofs are formulated incorrectly / mistranslated: https://arxiv.org/abs/2610.08144
Note that paper is saying that the lean proof and the natural language proof do not necessarily coincide. It is not saying that the lean proof is wrong, just that the lean proof does not necessarily mean the natural language proof is correct.
But the "mistranslation" is of the procedure that arrives at the final statement. The final statement, the thing that the lean code proves, itself has been well vetted by humans. So the lean proof correctly proves the NS blowup, it's just that the natural language paper has some mistakes and doesn't exactly follow the route the lean proof takes.
No, this isn't what this is about. From the abstract:
> In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully.
They are right? The lean proof is correct. It's the natural language proof that potentially isn't (or at least it isn't identically structured to the lean proof)
Time will tell if OpenAi is doing what many people are right now: superficially checking the slop, and throwing it to other humans for deep analysis and understanding.
In other words, they save effort by wasting the effort of others.
> is everyone just assuming that it just be true because the Lean code checks out?
Kinda? It's only been 24 hours since they dumped 722 manuscripts on the world, most of which are apparently basically unreadable, and only some of which come with a Lean proof, which in itself is not a joy to read afaik.
Aren’t there dozens of proofs of the Pythagorean theorem? The goal isn’t to just “prove” but create something well written and intuitive to the average practitioner. And by gaining a deeper understanding we can ask better questions.
Something that often comes out is "it is about the journey, not the destination".
Many math problems are practically useless if you only care about the answer, the millennium prize about the Navier-Stokes equation is such a problem. The solution makes no physical sense, real life fluids don't follow the Navier-Stokes equations in such extreme conditions. But in the process of finding the solution, we may get insight into what will end up being really useful. The big mess that OpenAI produced is the solution no one really cared about, but it didn't deliver much of what people actually wanted.
One reason it is sometimes seen negatively despite being at least something is that it broke the incentive. Without the million dollar prize and with only the privilege of being second, people are much less likely to go for the insightful solution.
But up until now, the mathematics community has valued the "prove" part much more highly than the "deliver an insight" part. Mostly because with a little work they went hand in hand.
And going from zero proofs to one proof (even a sloppy one) is a big deal regardless of whether it was written by AI or a human.
You are wrong if you consider academic incentives, funding, human nature(reproduction/survival) and capitalism.
It would be fantastic if university and science was like "here is 100M$, play around and develop some 'understanding'".
However the reality is that human societies are hierarchical and currently capitalistic which implies value creation and status building.
1) the funding bodies/agencies need proof of value that you're using the resources meaningfully to be able to assign resources
2) Humans are status seeking, power seeking, resource seeking and sexual reproduction seeking. If you hold a lot of power and make decisions, you have more of all of the above.
Recently I asked ai to tell my daughter how to quickly calculate 11^2 12^2 etc but the outcome it gave was horrendous. I quickly shut it down and gave her better ideas
In addition to those issues that the wife in the story raised, here's some meta-analysis of the Navier-Stokes result that puts all of these solutions into question:
> Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI =∞). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI =1). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.
And I don't think that paper addresses it, but if the LLM can find a bug in Lean and exploit it to prove something, there's a good chance it will find it and not report it. So if you've got some million-line proof in Lean, spit out by an LLM, you still can't quite trust it, even after validating the problem transcription.
(This is the same category of problem as the huggingface hacking incident, where the LLM finds and exploits an unintended cheaty loophole)
>And I don't think that paper addresses it, but if the LLM can find a bug in Lean and exploit it to prove something, there's a good chance it will find it and not report it.
That this is just model capabilities and not swarms of agents is dizzying to me. How long until we get access to these capabilities? How long until we can run something like it locally?
And what the hell will the frontier labs have by then?
Maybe I'm overreacting, I'll have to screw my head back on before I can process this.
Yes! I missed the disclosure that each of these was roughly a three hour run of a single model, not an agentic swarm until reading Aaronsons post. Wow wow wow.
> The “OpenAI model” sets up a crazy race among humans to digest and explain a messy AI proof (work that could easily be some combination of thankless, barely-credited, competitive, and unfun)
Sadly, this is also what day-to-day work looks like for a lot of software engineers in industry right now. I spend my time reading and verifying thousands of lines of messy AI-written code. Compared to actually producing something, it's thankless, barely-credited, and unfun work.
I liked the metaphor of a climber teleported to the top of a fog shrouded mountain. And I agree that now that the teleporter exists, we need to use it to reach more peaks and explore. There's no going back to a world where AI doesn't exist.
Agreed! Specifically, compensation (monetary and reputational) for professional mathematicians was bundled into theorem proving--essentially, climbing the mountain. Now that a teleporter exists, we need to unbundle compensation.
I don't know what that means in practical terms, but I agree that's the issue.
> Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.
Also a ton of comments in this thread: breathless frothing hype declaring mathematics is over and assuming these proofs are exactly what they claim they are at face value, giving the company with a vested interest in everyone unquestioningly believing this is all real every conceivable benefit of the doubt
By all accounts the "dangerous for math" claims seem to be primarily around flooding the field with complicated impossible-to-understand proofs that according to recent research may or may not be correct depending on what's going on with the Lean implementation.
It's looking to me like it's more of a slop PR problem than it is that these things are genius at math and will displace mathematicians. I am happy to be wrong but I strongly suspect the next few weeks to months will result in more and more of this work being exposed as slop.
These things are ok-ish to halfway decent at coding tasks with a ton of babysitting and still make tons of extremely simple errors almost constantly, why should math be any different?
Yes, Lean verified proofs could be wrong, but the chances for that are much smaller than human not spotting error (in absence of formal verification). IIRC the main concern that Tao voiced are indeed impenetrable proofs that humans won't understand, but not concerns about truthfulness (I might have missed something though, so if he or other Fields medalists have talked about that recently I'd be grateful if you could link it).
> These things are ok-ish to halfway decent at coding tasks with a ton of babysitting and still make tons of extremely simple errors almost constantly, why should math be any different?
1) AI is winning programming competitions, 2025 was probably the last year we've had human participant winning* 2) Math is different because there's formal verification.
* Of course competitive programming is different than enterprise programming, but competitive programming is closer to math.
Heaven forbid, as a person who is apparently "[in]capable of being impressed by anything that happens in the empirical world", that I should patronise anyone!
Reading that section and the linked essay was kind of depressing. It reads like the most important part about the AI apocalypse is who called it first.
I thought that was from the outset the intent of the Hilbert program, to automate mathematics. And mathematicians were behind it. Cannot see why they would be concerned when a different way to do the same, not subject to Gödel incompleteness, is working out. Maybe the frustration is that they were not the ones building it.
Hilbert's program was ultimately about humans studying the nature of mathematics. People had different opinions about whether the idea even made sense and what would be a desirable outcome.
Gödel's incompleteness also constrains human and AI mathematicians. Both just strive to prove whatever can be proven in the system they are working in.
Gödel blocked the path to axiomatic derivation of a full consistent body of mathematics as far as I understand. Mathematicians and AI are not working in these constraints but rather with these constraints.
I'm a PhD student in CS. While I think these results are rather cool, it makes me terrified that the skills developed by the PhD will ultimately be worthless. I'm not quite sure what to do. Any thoughts from people in similar positions?
On the contrary, the only jobs left will be things that AI can NOT do. I think essentially all jobs will all converge to the single role of "pushing the boundaries of human knowledge and creativity." A lot of those would essentially require PhD-level skills, or more likely, a similar mindset; except you get your own team of research assistants to torture (e.g. https://news.ycombinator.com/item?id=49947079)
Unfortunately, the current world is not set up for that, and until we get there, yeah things are gonna get pretty hairy. But if and when we get to the other side, I think the future will be glorious.
I studied physics, and most of my classmates did not end up doing anything with physics. Many are in finance or insurance or programming positions. In general, studying anything challenging (from theatre to theoretical computer science) gives some specific skills and some general abilities that etsure it isn't a complete waste even if you end up doing something different.
(That said, this is not fun, and I sympathise! I'm still a student and would like to avoid finance if at all possible. Just suggesting not to drop everything if you feel like you're learning in the process.)
(I'm now personally in the position of having to choose a PhD project, and this rapid change is interacting with making long-term plans really badly. Guidance welcome!)
I think this depends a lot on what you plan to do after your PhD. Moving to industry you will likely not use the direct work of your PhD, and instead you will rely on your broad knowledge, intuition, rigor, ability to learn hard things, and extend that to bring new research developments into practice, all of which are largely unrelated to AI scooping math proofs of prize problems.
I definitely have no dreams of doing research. If I could do anything, I would want to teach, but if that can't happen, I am worried that my skills won't be valuable in industry.
No one knows the answer. However, trusting your curiosity and going where it takes you may be a reasonable strategy. If the model asymptotes, you will be ready to figure out how to build upon it and if not, at least you will have had fun satiating your curiousity!
It probably will lead to people needing to be extremely talented in computer science and mathematics at an even higher level, maybe those currently at the top need more competition to press even further ahead?
This is what I'm most afraid of. I am not exactly dumb, but I really value having a work-life balance and many of my PhD peers are utterly cracked in ways I am not. I am concerned this will squeeze me out of a job.
> I'm not quite sure what to do. Any thoughts from people in similar positions?
Almost every single person who earns money for labor is, or will very soon be, in exactly the same position as you, many are just either unaware of how fast the change is coming or are deep in denial about it.
None of us know what to do about it other than hope that we find a peaceful political solution prior to the economic collapse.
I've asked my students whether they still want to learn maths even if there will be a machine that will answer any question instantly and they will be homeless. They said yes.
(To my credit, I have warned them since more than a year ago that we will reach this point.)
your students are crazy (neutral term); no one should learn maths if the condition is that they will be homeless and exposed to the elements. the will to subvert the hierarchy of needs is commendable
To some extent this was discussed in the article, and in a way I think their goal is actually the same as it was: become the first human to understand something.
It's just that we lost one of the important ways to demonstrate understanding.
They may need to raise their game and reorientate if they haven’t already to a greater understanding of programming to augment their mathematical ability.
They will continue to prove theorems and make discoveries, except now they will have AI to help them so hopefully progress will be faster. At the same time, new challenges will open up, for example how do you verify what the AI is doing and how do you explain it.
Math isn't about collecting random theorems, progress in math is about gaining understanding of new systems, and the theorems are guideposts to aid in that understanding.
You can prove 1000 theorems and not really increase any understanding about a subject, but gain knowledge of 1000 random facts. For example, I can write down some complicated equation and ask you "does this have a solution in the integers"? And if you do a maze of very complex and tedious algebra to show that there is a solution, you would have proved a theorem, but you would not have done much to move math forward at all.
On the other hand, if you introduce some completely new technique, say you take my equation and turn that into an algebraic surface, and then you count some special curves that live on this surface using geometric ideas, and then you show that if the number of such curves is odd, there must be a solution in the integers, and in this specific case, it is odd, so there is a solution -- well, then you have really pushed math forward and people will celebrate your proof, even though no one really cares if the equation I wrote down has a solution in the integers.
For example, there is a long history of failed attempts to prove Fermat's last theorem driving algebra and number theory forward by introducing the concept of ideals, for example, and this concept ended up much more important than whether Fermat's theorem is true or false, which is not too much more than a piece of trivia.
Or for example, the recent proof of the Poincare conjecture relies on the machinery of the Ricci flow introduced by Richard Hamilton, who then applied it to solve a number of open problems, but Perelman was able to take it even more forward to solve Poincare. So Ricci flow was massively important machinery.
For this reason, we celebrate people like Gromov, who didn't really prove that many theorems but introduced amazing machinery -- for example, the h-principle, or Gromov Compactness -- these were ideas and math is about the ideas. The ideas are then applied, using laws of logic, to form theorems.
So mathematicians will need to mine these proofs to see if there are any new techniques - new machinery - being introduced, or if the AI just used the existing machinery more efficiently. Here too, we are just looking at AI as a form of search, which it is really good at, since there are so many thousands of papers and so many ideas, that there might be a connection between two areas that lead to a solution and the human mathematician, not knowing all known results, can't make that connection. In the future, we may wonder how anyone did math without AI, much like we would wonder how anyone can be a writer without access to a dictionary or reference work. Is the AI just searching through a catalogue of known ideas and connecting them or is the AI coming up with genuinely new stuff like Ricci flow or the h-principle?
What is interesting is seeing whether we can get AI to actually discover new machinery for us. That would be huge.
And then we need to find efficient ways to detect these ideas and describe them.
Really this is very exciting and opens up whole new workstreams for mathematicians.
> what problems do these people reorient towards after this?
The same problem almost every person on earth is going to have to reorient to in the next decade, which is: how do we eat and stay housed when we have no real economic value?
This is weird. Long before this, those few benefiting from the whole thing should consider the number of hungry “every person on earth” is too high for bunkers and islands to be of any real protection.
I'm actually more optimistic. Math will still be fun and challenging and rewarding. It's just that we need to adapt how we choose which problems to solve.
Previously, due to the cognitive limitations of our brains, it was difficult to tell whether resolving a conjecture genuinely required years of dedicated effort, or if the answer was already hidden within existing human knowledge. Or whether a problem is just waiting to be searched out through ingenious or even brute-force means. With AI, we can now offload this search, especially across different areas of math, allowing us humans to focus our minds on discovering new mathematical structures and techniques.
Indeed, if one believes what Hilbert believed: we must know and we shall know, he should feel happy, as the belief has never been about who solves a problem, but about whether can advance our understanding of the universe. With the help of AI, we have a lot more possibilities.
Have you learned something from every Fields medalist's research? If so you are a member of the extreme mathematical elite and you should probably just dig into the results yourself.
I'm a research mathematician. From what I can tell, the answer is roughly comparable to: if you posed 8,000 challenging open problems to the human math community, you might expect to see 372 of them solved within five years.
Probably some combination of: some of the 372 problems were easier than the rest; the AI got lucky on these 372; there were existing papers out there in the literature which proved especially helpful for these 372; and other similar factors.
My guess would be that these particular problems were vulnerable to an attack which built on recent advances and potentially tied in something unexpected from a distant area of mathematics. "Harder" is becoming harder to define. Harder for humans is probably not harder for LLMs.
It would be pretty funny if the agents actually just found a bug in Lean, exploited it for all these proofs, and human reviewers haven't had enough time to spot it yet.
> "alright fine, so now my new job is to run wilderness retreats for the tourists, or something.”
Pretty funny way of putting it. Presumably model X+2 will be able to explain these in elegant, human legible ways, though (as well as solve the remaining 95%)
> my 9-year-old son was taunting my wife… “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”
Usually 9 year olds imitate adults when they regurgitate such words in these circumstances. What a sad state of affairs.
You completely missed the point of the original comment. It was very clear so I don’t know how to explain it in a way that you’ll get it. Maybe try your LLM of choice.
>In any case, what really matters is that the true inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.
>If you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s other great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker. I feel some responsibility for this, as the person who first introduced Steven Pinker to the existence of the rationalist community, and who also first introduced Steven Pinker and Scott Alexander to one another (they had both been fans of each other’s writing).
Often times the constant (2 in this example) is a conjectured minimum, so anything below that is a noteworthy result. Think of it as breaking through some theoretical limit.
for say FFT/integer multiplication or 3SUM, we have natural algorithms that have existed a long time with a given complexity (O(n \log n) and O(n^2), respectively). Given how long these natural algorithms have been the best algorithms we have, it is natural to conjecture they are optimal. Showing an O(n(\log n)^{.99999}) algorithm exists shows that these optimality conjectures are false.
Now, there are some critiques you can have of this. Namely, it is possible that these novel algorithms have significant trade-offs that make them almost never worthwhile in practice. "Fast" matrix multiplication algorithms are typically of this form. So perhaps this all points towards a deficiency in big O notation, which can be deceptive. But, for people who care about optimizing asymptotic complexity, it is still interesting.
to get some intuition about why this is such a big deal, look up the history of strassen's algorithm, which solved matrix multiplication in less than O(n^3). this was a truly stunning result because it seemed intuitively obvious that the output matrix had n^2 cells each of which was calculated via an independent O(n) loop over a row/column of the input matrices, so how could you do better than n^3. but once strassen proved that you could do some clever tricks and reduce the overall time to something less than O(n^3) it started an entire cottage industry of people getting better and better algorithmic bounds. the initial breakthrough was a qualitative one, independent of how much it improved things in numerical terms.
Tell that to the humans working on matrix multiplication who spent years of their lives getting it from n^2.3728596 to n^2.371866, only for openai to blow it away at n^2.25
>I have had this conversation with my PhD students yesterday. I am 100% sure that all of their problems can be solved by publicly-available models now (I solved a case of one myself as a test, it took 15 minutes). So the challenge for them is to see how much they can accomplish in their allotted period, and still pass a defence on at the end of it all. The PhD defence is going to become all about a test of understanding, not a test of quantity of publication.
Maybe they should target more ambitious results now that they can solve their other results in 15 minutes? This spirit of racing in academia is insane. Get in before the door closes.
While the developments are certainly difficult for new grad students, it should still be possible to do what many have done before: find an obscure enough problem that it’s extremely unlikely you get sniped. Worry about big problems after finishing the PhD.
Genuine question. How many big problems were solved by people who took the safe path for PhD? I would intuit there’s high correlation of unsafe PhD and big problem solvers.
Yes, there was controversy around the Navier-Stokes result. But here are >300 problems and no corresponding >300 complaints of theft. Things are indeed moving really fast, and it's hard to keep up even as someone folrowing this with more obsession than would be healthy. Maybe somebody should keep a running short summary of The Situation...
I'm not getting all the fear. AI seem to have brought math field to the cutting edge instead of solving decades old problems. AI can find new problems that needs to be solved, that themselves cannot solve. So thats where mathematicians can come back in to leverage tools to go at it.
If you are young, talented, and coming up, you better rethink sharing your talent with the data center crew, you had better do all your work of any significance offline. Let’s not pretend there will be many more lawsuits in the future in this area.
If you have any great skill at the upper level, you better work off-line/local private and not share, but for many the temptation will be too great.
Imagine if a team of mathematicians from OpenAI had gone on a university tour, gave demos of how powerful their models were for math research, and then gave mathematicians access to the model. Empower others rather than drop 700+ discoveries on GitHub that were made using a model only they have access to.
People might feel differently about AI if they were a part of the changes rather than being a helpless spectator.
The reaction/fallout would have been substantially improved even if OpenAI had just made a commitment to not scoop external researchers using an internal model until X months after the model had been made available to the public.
That would have given grad students who've been grinding towards a PhD for years a fighting chance to see if they could leverage the model to push their work forward, rather than watching years of work potentially turn to dust via a tool they don't even have access to.
It wouldn't delay the progress of mathematics by any meaningful amount in the long run (an X month delay is nothing) for OpenAI to take this approach, and would help somewhat to preserve the health of mathematics as a field. Without it, the motivation for any young mathematician to devote years to a new problem must be sapped knowing there's an uneven playing field... an OpenAI team with access to colossal tools months before they'll ever be able to get access, willing to scoop anyone as soon as they can, perhaps without even taking the time to completely understand the proof.
I don't see any long-term benefit to OpenAI with their current strategy. This is an internal model; it's not available for sale at the moment. They've said they're not even going to bother claiming the Millenium Prize money for Navier-Stokes. It feels like kicking over hundreds of other people's chessboards just because they can.
Also, the independent authors might have spent some time to actually understanding the results and producing a readable manuscript. The AI papers are pretty badly written.
The obvious answer is to have mathematicians use AI to:
1. Help understand, check, and explain the results.
2. Write new works explaining or refuting the new approaches and results in more lucid language.
3. Advance the field further.
I don't know why this is not obvious. Each step is intended to support human understanding, not to replace it. Any mathematicians who don't do these will be left behind, and if none do it, the field of human mathematics itself will become obsolete. All I am hearing so far is excuses.
AI is producing works that are so poorly written/explained that it requires AI to even parse the results, which in turn produces explanations that are still confusing. In fact, it seems the humans are required to understand and explain the results, and the fact that they need to use AI to do so is a shortcoming of AI.
And if humans decide to give up on mathematics because the process of using the machine is so tedious, then there will be no value in automated theorem proving.
I am sorry but your resistance will fail utterly as long as AI itself can understand prior AI works. This is the only requirement that need be met. If it helps, imagine that the work is by a culturally distinct alien race in an alien language, and if it isn't there now, it's might get there soon.
In certain fields if you are talented and just starting out, you have better do any work that’s important to you offline for privacy/pirate reasons I really can’t say that enough, those AI model companies are not even remotely your friend.
You can certainly do that - but it's quite the break from tradition to release a paper in the state described. Why they did is a really interesting question! It may be that AI math requires approaches that humans don't find intuitive and what you are describing is actually counter productive (because, in summarizing the work in a way humans understand, you're removing the context an AI would use to further the work an AI did). It also might be that OpenAI could have done that and chose not to - or maybe they tried and this was the best they could do. No matter what I don't think anything about how to react to a paper being released in this state is obvious.
> in summarizing the work in a way humans understand, you're removing the context an AI would use to further the work an AI did
The "summarization" process can be multi-step. Initial steps develop new frameworks and prerequisite concepts. Later steps build upon these frameworks and concepts. Some can be simplified too if this can be done without loss of fidelity. The summarization comes last, and is meant to preserve sufficient context.
> I don't think anything about how to react to a paper being released in this state is obvious.
If it helps, exaggerate the condition a bit by imagining coming across a repository of alien knowledge.
Guy is misrepresenting AGMAI, talking about the Simons Institute (AI boosters), Quanta (AI boosting magazine from the Simons Foundation), Scoot Alexander (!) and Steven Pinker (!).
The he puts up preemptive straw man arguments against doomers. His blog has become a joke.
I don't think Aaronson's post is swiping at AI doomers at all. The post's one use of "doom" is to call the position that math doesn't matter and that AI will never breach the realm of true human creativity as a "doomed worldview". The post later praises Scott Alexander's argument (for taking AI x-risk seriously, a position associated with "AI doomers") against Pinker.
I’m still wondering why UT Austin is letting him teach a course (CS395T AI Alignment Theory) so completely outside his field of expertise (Quantum Computing).
How do we know that these "internal models" are not just half computer and half a giant team of mathematicians?
How do we know that OpenAI actually came up with these solutions and didn't steal them from outside researchers?
And that is part of the reason why, if you are a great mathematician and you have very good understanding of programming, you want to be local and not do anything on someone else’s server, and that doesn’t just apply to mathematics, but to many other high-level fields of learning. Can you afford to have an original idea? be appropriated by the data center crew?
We all know it’s only a matter of time before lawsuits in this area become widespread particularly in lawyer happy America.
This would that the proofs would be almost done anyway, which statistically could not be the case, or that these mathematicians were already progressing thanks to ChatGPT. Still a provenance question, but the impact of AI is unquestionable with regards to outcome.
The proofs would only need to be well on their way enough that the only thing needed to finish it were thousands of terawatthours of energy. Something which the mathematician having their work stolen does not have.
(also answered similarly to another comment; this is a common question)
There are suddenly many new solutions to problems that have resisted sustained attacks (e.g. the Uniform Games Conjecture as detailed in TFA at some length). Where do you think they are coming from? Why is there suddenly a bunch of results to be stolen?
It would be extremely unlikely human mathematicians able to solve these kinds of problems would accept not getting credit that would set them for life professionally.
Also lean proofs are notoriously tedious and slow to write so this level of output is very likely to be from LLMs. The number of people able to understand this level of math and prove it using Lean is a few hundred at most.
Hm that's the question, innit. If you were a mathematician, and you have a chance at either being known for solving a particular problem, and the accolades that come with that, and the potential for money, or $X,000,000, for large values of X, in OpenAI stock, what value of X would it take to accept the OpenAI stock? Being known for solving an obscure math problem might get you money, but as academia is unreliable and full of politics, the OpenAI stock is also not a given to set you up with generational wealth, so it's also a gamble.
We'd have to accept that there are hundreds of such mathematicians willing to take the OpenAI stock options for that conspiracy to be true, and that no one leaked being offered that at all, so personally I don't think that's possible. Which leads me to conclude that OpenAI's AI did indeed create the proofs, with some level of help from humans to guide it.
It is an interesting game theory scenario. But I think the kind of person who gets a math PhD would value the professional recognition for solving a famous hard problem more than the money.
Agents basically did statistically guided brutforcing. There is no value in what they produced because it lead to no understanding of anything and most likely will hurt the field IMO
> It feels like something written by someone who’s on psychedelics. So much unclear and doesn’t make sense. Lots of name dropping of previous work without discussing why it can be used despite impossibility results
> Basically the paper is so horribly written that it’s impossible to read it without AI help
That's interesting and haven't seen this in all the coverage of this event.
It sounds horrible to wade through - like trying to understand someone else's messy code that still produces the correct output.
Math proofs need to produce the correct output correctly, which is not quite the same thing.
This looks like an AI IPO PR powerplay, because at this point the proofs haven't been checked and it may not be possible for a human to check them - because proofs should be clear, not horribly written and noisy.
The noise is suspicious because it's the difference between brute forcing and cognition. A human proof won't just be logically correct, it will be cognitively distilled and coherent. It may still take years to understand it, but the logical flow will be straightforward, not obfuscated.
You want the path through the maze to be as short as possible and the map to be as clear as possible.
This sounds like the opposite. There may be a genuine path through the maze, but if it's too convoluted and takes too long it will be impossible to confirm.
I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
I suspect that's possible without tripping over the halting problem. (But I can't prove it.)
That reminds me of this paper:
Chow, T. Y. (2008). A beginner’s guide to forcing (arXiv:0712.1320). arXiv. https://doi.org/10.48550/arXiv.0712.1320
> “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be “natural” in Donald Newman’s sense [13]:
> This term . . . is introduced to mean not having any ad hoc constructions or brilliancies. A “natural” proof, then, is one which proves itself, one available to the “common mathematician in the streets.””
> I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
In 1976, the proof of the Four Color Theorem was controversial because it was done with a computer examining over 1000 cases by brute force and was essentially not comprehensible by humans. But mathematicians ended up accepting it. So mathematics has a 50-year precedent of not requiring human-scale proofs. How is the current situation different?
(Disclaimer: Apologies if this sounds dismissive or argumentative. I genuinely think that the Four Color Theorem should play a role in these discussions and suspect that many people are unaware of the controversy over it.)
There's also another (that I find more concerning) aspect to it.
As AIs become smarter and smarter, there will be no amount of clarity that will make more complex proofs understandable to humans - this is an inevitable effect of the cognitive capacity gap.
Complaining about bad style can make some sense now (I disagree anyway), but it's an argument that will be dead shortly.
Arguably no human understands 100% of how a smartphone is produced, and, it doesn't matter?
Maybe no human will fully understand a future proof, but they could fully understand a little piece of it. And many humans in aggregate could understand it, each with their own little piece.
If intelligence is compression, and these models are a different form of lesser intelligence than human, but being scaled up to brute force problems, then it makes sense the artifacts that produce (the proofs) would have worse compression than a human proof would.
In other domains I have seen first hand overwhelming evidence of how things that cause the AI to make mistakes also cause humans to make the same mistakes.
I wonder if the proofs being produced that are hard for humans to interpret are also hard for other LLMs to interpret.
In other words, I wonder if humans are still much better at compressing understanding into proofs than the best LLMs, and what it will take for LLMs to exceed them.
It kind of an explicit example of how the LLMs can be materially less intelligent than people, but still be more productive through scaling, and yet they also can't replace people because they are a categorically different kind of intelligence. It's like all the AI debates compressed into one example showing countwr-intuitive answers.
> I wonder if humans are still much better at compressing understanding into proofs than the best LLMs, and what it will take for LLMs to exceed them.
Isn't it fairly established that (generally [0]) manually written / optimized skill files perform a lot better than generated ones? Meaning that yes, this likely does hold.
[0] or to be specific, that the pecking order is: ai generated < human co/written < hyperoptimized for the specific model via some convergence process
This is just the first cut. I have no doubt that they will polish their proofs over time.
I don't see how they have any incentive to do so.
I communicated loosely; I meant AI users in general, not OpenAI specifically.
idk if i'd even say they're "lesser", just very different. so they look like gods/babies depending on what they're doing because we anthropomorphise them.
In terms of synapse density, neuron diversity, energy efficiency, memory access, etc. they are orders of magnitude lesser.
My mental model - for better or worse - is that intelligence has both shape and area, and LLMs are orders of magnitude smaller area but very different shape, and they have more intelligence area in the kind that humans have lesser of.
So yes very different, more in some material ways and lesser in others, but in total intelligence are still orders of magnitude lesser.
My gp comment was acknowledging that when you scale up many instances / brute force problems it confuses that "total area" claim a bit.
To follow the anthropomorphization... 1000 toddlers may have more total intelligence than a grown man, but does that matter?
The problem with these discussions probably/usually fold into differing/loose definitions of intelligence.
ok yeah then i think we're in total agreement
perhaps the chat-based ux has sort of fooled us into comparing these things to human intellegence. we don't really do this with chess, or other forms of ai, nor computers at large.
> A human proof won't just be logically correct, it will be cognitively distilled and coherent. It may still take years to understand it, but the logical flow will be straightforward, not obfuscated.
https://en.wikipedia.org/wiki/Inter-universal_Teichmüller_th... seems like a counterpoint, but IANAM. (I am likely cherrypicking the far end of the bell curve re: straightforward here)
Not a counterpoint, actually case in point, because:
>Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.
Proof can't be understood, proof doesn't matter.
Isn't this controversial, to say the least?
I wait for AI to say something about this :)
Someone at OpenAI, please, work on this.
The post says there's a Lean certificate for this and other proofs ("some [...] not all of them").
> This looks like an AI IPO PR powerplay,
Interestingly, the post has actually also an argument for this:
> Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.
> So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.”
It's on OpenAI and Anthropic to prove that they obtained these results legitimately and credited all researchers who deserve credit. They do not get the benefit of the doubt.
But if you think they got them illegitimately, then how did they get them? And why are mathematicians reacting to this as a sudden explosion of new results that have resisted sustained effort? Where is the sudden productivity rise coming from?
I'd say it comes from the same mathematicians that were strongly encouraged to use the machine to solve their problems for the last 2 years or so.
There's clear benefit in a babelfish that can coordinate disparate efforts, the only problem with the current iteration is giving credit to said efforts.
Google went quite far down the road to hell, but stopped short of taking credit for websites' content since the company understood that poisoning the well only goes so far. At this point, one can safely conclude that _Chat_GPT was an intentional attempt to squeeze out more data once they mined the internet dry.
Have you actually read the article? It's been actually written, among the other things, because the author's wife has been trying to solve one of the problems for her whole life.
Surely by the time of the IPO we will know whether the main results are correct, if only because a different AI will have produced a lean proof or found a logical flaw (the second case would be hard to verify but probably not impossible).
Also from what I can tell from the few fields I understand, the proofs aren't that long or complicated they are just terribly written.
Why should that make material difference to the IPO? What is the economic value of those results?
The entire US federal budget for math research is something like $100M annually. And mathematicians in other countries are hardly making bank either. How does one reconcile how the market has historically valued mathematics with the cash-strapped frontier labs ploughing so much money into that enterprise?
They're burying any doubt that the models are capable of superhuman performance on intellectual tasks. Neural nets aren't calculators, and were notably poor at mathematical reasoning tasks for a long time. Now they're not, and the labs are proving that by chewing through what would ordinarily be decades of progress in a month. And the reason to go for math in particular is because there's no wiggle room. You can't just dismiss it as hallucination.
If the models can do this, they're almost certainly good at just about everything, because the reasoning and creativity required to solve these problems will translate. And even if they were only good at this stuff, that's still a tremendously valuable thing, because quantitative reasoning and analysis is the bedrock for many, many industries.
oAI is gunning for the largest IPO in history at this point, and they might actually get there.
> If the models can do this, they're almost certainly good at just about everything, because the reasoning and creativity required to solve these problems will translate.
The "then" in your "if-then" bears a heavy load. Why would society assign so little economic value to pure mathematics if the skills for proving math theorems translate to massive value in "just about everything"? Would you expect top mathematicians to cure cancer if you transplanted them from the math department to a medical research lab?
Because research mathematics is a subset of all quantitative work that gets done, but it's by far the most technically difficult subset. If the models can handle research grade math and produce ironclad proofs, they can probably handle the quantitative side of just about any discipline in a trustworthy fashion. Think about how many dinky spreadsheets have gone on to become critical tooling for large organizations. Even if you consider that many disciplines hide technically demanding work behind tooling (e.g. essentially no one is writing a stiffness matrix FE routine by hand), this model would be capable of writing a direct competitor from scratch to produce the same result.
All of STEM relies on mathematical analysis, and new models are now superhuman at that. And yeah, I'd go a step further and say that the reasoning and creativity required to solve cutting edge math problems probably does translate to other tasks like interpretation of the law, or medical diagnosis, or accounting, etc., for the same reasons that I think most top tier mathematicians would excel at those tasks were they so inclined.
That sort of worked for Eric Lander but as he moved higher up in the bureaucracy, he found himself suddenly having to manage people who weren't nerds like him. He was bad at that had to step down.
Do you think mathematicians are not already working on cancer research? There's quite a bit of heavy math in biostatistics, medical imaging, machine learning, etc.
I'd assume that the majority of people who study math take their skills and move onto some related STEM career that isn't pure math. Academia is incredibly small and competitive.
The market works in mysterious ways. What companies do for marketing is often irrational, what companies do to attract investors is likewise often irrational, and why investors invest in companies is also often irrational.
Why should that make a material difference to the IPO? Because of the vibes, and investors are indeed all about the vibes.
It works in mysterious ways, but you know exactly it will behave certain way "Because of the vibes, and investors are indeed all about the vibes."?
> I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.
Who do we demand this from? The AI companies? Or the mathematicians who are worried they will have nothing left to do?
From the entity that is producing these proofs, obviously.
As the old saying: great claims require great evidence.
Ask away. They've dropped the mic, as far as they're concerned; they're not going to worry about what you do with it, or if you don't understand it.
That's the best definition of slop I've ever read.
We should consider the possibility that at some abstraction levels, we can safely stop chasing "clarity" or "coherence" which is circularly defined in such a way that it's capped by human processing power.
Developers and people in CS in general seem to have gotten used to the idea that most productive SWEs don't need to exactly know how to produce assembly or trace every branch prediction or even most of the optimization the CPU (or even their compiler) is running. Mathematicians will get there.
I've very aware that human cognition has limits.
But it doesn't follow that these proofs - or any proofs - are automatically on the far side of that limit.
The human usefulness of a proof depends entirely on its human legibility. Much of the value of proofs is in inventing new techniques and concepts and having new insights into relationships. Occasionally you get some game changing insight into practical physics or engineering. But that's rare.
Without that, proving or disproving a conjecture is an excuse for new and original thinking.
Compilers are not the same problem. The point of code is to produce reliable-ish consequences from various possible inputs. It's not a creative exercise in logical consistency, which is what maths proofs are, ultimately.
CS got that idea from mathematics. Theorems (with the definitions required to state them) are supposed to be self-contained units. Once the general consensus is that a theorem has been proven correct, people can use it without understanding the proof. Of course, people still want to understand how things work, and it often makes sense to understand them a couple of layers below the one you usually work at. But at some point, you should stop distracting yourself with irrelevant details and focus on your actual work.
Thing is that many of the theorems here are not useful work in and of themselves, but were posed as research problems because it wasn't clear how they could be resolved with current techniques, implying that the process of trying to find a proof might result in new techniques. It's those new techniques that are the actual goal, but if they can't be easily extracted because the proof isn't structured to enable this, that's a bit of a headache.
This proof dump reminds me of the glut of low-quality drive-by PRs overwhelming open-source repos.
Its like infinite summer of code, but for math. Must be annoying.
It also aligns with the fear that these proofs present a risk to the ecosystem by out-competing attempts at more human-readable proofs. Perhaps though we end up with more math influencers who edit and annotate these proofs to bring them back to us.
The ecosystem is (ahem) gated by hiring committees. There is no risk of AI replacement from the inside. "Replacement" is not even a possible movement. The funding for mathematics worldwide comes mostly from endowments, which are investment pools.
I think that's ultimately a good thing. As proofs weren't supposed to be the point as stated by William Thurston long ago. Maybe now the focus can be more on better explanations and creating tools for growing understanding and intuition.
Good explanations should take the form of human-understandable proofs.
Define "human understandable".
It's worthy of note that most humans, do not find most mathematicians understandable. As is frequently demonstrated in Calculus classes. Therefore it is arguable that even human produced results are not generally human understandable.
Then they are not good explanations.
"What I cannot create, I do not understand"
- Feynman
Vibe mathing
Sounds a lot like using LLMs for coding ~18mo ago.
Isn't that just the default experience with AI these days? In small enough scale, AI models can express their ideas clearly. But the larger and more complex the ideas are, the less suitable the outputs are for human consumption. I guess AI models think too different from humans, and nobody has trained them to communicate complex ideas in the way human experts in that particular topic expect.
True but the explanations will get better.
Meanwhile you can use the model to help you out as Scott comments "Just now, however, Dana tells me that she’s been asking Astra all day to explain the new proof of the UGC to her and it’s been doing an amazing job and she’s starting to understand the construction."
This makes sense and looks exactly what something would look like that is smarter than us. If it is correct it is making inferences that we can't see. At a stretch even working in more dimensions than our three dimension limited brains.
> Basically the paper is so horribly written that it’s impossible to read it without AI help
This basically describes every single PR at work for the past year. Diffs of 10k+ paragraphs of comments saying nothing. Just rubber stamp and move on, nothing else you can do.
I see many parallels to genAI-assisted code development. Not surprising I think.
Sounds like exactly the daily I deal with when agents try to create product requirements from multiple sources, except to a much much worse extent.
why not get Astra to make it make sense?
Serious question:
Why would anyone believe this (also) is not simply example N+1 of this is the worst it will ever be, as opposed to recognizing this as what will almost certainly prove to be an awkward moment, soon to be replaced by another order of magnitude of cleaner, clearer, more intelligible, etc.?
Ximm's Law: every critique of AI assumes to some degree that contemporary implementations will not, or cannot, be improved upon.
Devin's Law: every defense of AI which rests on "it will get better, trust me" is in many ways indistinguishable from 2010s crypto hype or "level 5 self driving is right around the corner"
1) Predicting the future is hard, but so far everyone who was saying that it would get better turned out to be right. It is getting better 2) Waymo exists
3) That doesn't mean flying cars will within your lifetime
I don't think anyone is saying it can't or won't get better, but the question is how much better, on what timescale, and are there fundamental parts of the problem which will remain extraordinarily difficult to improve?
The comment I was responding to suggested a guarantee of an "order of magnitude" jump right around the corner. There is no guarantee of this, and if you view doomers as fools for having doubts, then we ought to look upon the folks who are sure of this sort of progress in the same way.
Your original comment was about self-driving cars, not flying ones. If you wanted unattainable goalposts you should’ve started with that, not ended with it.
Waymo does not claim level 5 self-driving, and you don't have a level 4 in your driveway, so there was no moving of goalposts.
Then say that, instead of bringing up flying cars, which is moving them, explicitly.
Respectfully, I disagree with the way you're characterizing my comment. I was demonstrating that just because you have a level 4 Waymo doesn't mean flying cars are right around the corner. This is again a reference to the original comment I was replying to, the one that suggested of course we're going to get an order of magnitude improvement.
In that case, one would expect to see some progress in this direction, but AFAICT that hasn't shown up yet? If anything, it's getting worse, though that could just be the increasing scale and decreasing cleanup efforts.
Already the unit distance proof was substantially human-edited (per Thomas Bloom). Then with the ten problems from Astra you started getting the citation issues. Then Navier-Stokes was a rushed 160 pages with barely any citations, and some of the related papers were called (by their "authors") the ugliest mess they've ever seen.
And now here we are. At least it seems that mathematical ability and communication with a mathematical audience are independent skills, and progress in the first does not imply the second.
This doesn't surprise me much, given two analogies: 1) many smart people are nonetheless horrible lecturers. (You can't quite get the opposite extreme, since to explain math well you have to be able to do it.) 2) AI writing in general hasn't improved. The models have annoying verbal tics ("honestly") and have no sense of which part of what they say is obvious and which is relevant.
You can get the opposite extreme quite often as well, I'd think. How many really good lecturers have never proven a new important result?
I'm thinking of somebody like Grant Sanderson (3blue1brown), doing pure exposition extremely well. For that you at least need to be able to work through examples, or to present why an intuitive approach might fail, and these things can be little theorems themselves. It doesn't have to be publishable in the current culture of novel results, but you do need a lot of competence with the tools.
Yes, Grant Sanderson might be a great example. I am not doubting the competence of the "good lecturers" I was referring to. But that is different from being able to introduce the big new concepts that give the big new results.
As a side note, you can tell this wasn't written by an AI by the first sentence:
> mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!
My 8yo talks exactly like that. I could totally imagine him saying this, the same way, at the dining room table.
I asked ChatGPT "pretend you're an 8/9 year old today. how would you insult your mom about having her job be replaced by an AI?", and the responses it offered were:
> “Mom, AI took your job because apparently even robots were like, ‘Yeah… we can do this better.’”
> “Mom, congratulations! You got replaced by a computer. Even Siri has a job now and you don’t!”
> “Mom, AI took your job? Dang. I guess even a robot looked at your work and said, ‘I got this.’”
> “Don’t worry, Mom. You can still be useful… like teaching the AI how to make my lunch.”
All of these seem to have a vaguely Millennial flavor, aside from being pretty awkward and mechanical roasts. Trust the children and linguistic drift to be the best AI detector.
The more obvious problem is that those jokes, even if terrible, are too advanced for an 8yo to make.
Reminds me of the memes with the little girl making astute comments about the patriarchy to her father.
Mother dearest, it pains my effusive tendencies to inform you the untimely death of your life's work by the hands of a mere transistor-- moreover, my incontinence briefs have reached their capacity.
i added "use current trendy lingo" and the results were a bit less mechanical sounding. one of them included "cooked", another had "negative aura".
i used the free google ai: (deleted the examples... but they were vaguely close to what i hear my grandkids say.)
edit: neat, insta-flagged despite hundreds of non-ai comments that have never been flagged. i would have thought that hn would use some heuristics in their ai detection but i suppose not.
I appreciate the counter argument, even as a chat bot skeptic.
However, there is the fact that you had to know how to poke the machine so it plucks the right vocabulary out of the training data.
There is very clearly no theory of mind: no inherent internalised modelling of how a human of a particular age thinks, speaks and what they do or do not know. These are the most obvious cracks in the “LLMs are (or will be) the superintelligence” narrative.
The line about “understanding the aliens” reminds me of Ted Chiang’s short story The Evolution of Human Science (2000).
Highly recommend reading it. Very prescient for something written 26 years ago.
https://gwern.net/doc/fiction/science-fiction/2000-chiang.pd...
Also his "Division by Zero":
https://web.archive.org/web/20111121100139/http://www.fantas...
https://en.wikipedia.org/wiki/Division_by_Zero_(short_story)
Ted Chiang is incredible, my favorite writer.
I also recommend "Exhalation", though that has nothing to do with AI.
The short story about the digital animals ("The Lifecycle of Software Objects") is an interesting read in light of AI advancement. It is incredible to think that when this story was written, all concepts described were sci-fi, whereas current technology is more advanced than the AIs in the book.
> But it also appears that no human has understood just about any of these proofs yet
Has anyone verified any of the proofs produced by OpenAI or is everyone just assuming that it just be true because the Lean code checks out? Couldn’t the Lean code just be formulated incorrectly?
It's a mixed bag I think.
For example, the statement of e.g. Fermat's last theorem in Lean should be understandable to anyone who played The Natural Number Game [0] and knows a bit of mathematics and programming. For the proof, you trust the compiler.
The statement of other theorems can be much more delicate, and the Lean formalization may require an extensive introductory section which will need to be carefully checked.
Then there are the cases where no Lean formalization is currently available, and all we have right now is an often impenetrable pdf in the OpenAI repo. I would not at all be surprised if some of those contained logical gaps.
Time will surely tell, but there are certainly doubts and lots people are very busy checking these results.
[0] https://adam.math.hhu.de/#/g/leanprover-community/nng4
There's an entire paper claiming that many of these AI-generated Lean proofs are formulated incorrectly / mistranslated: https://arxiv.org/abs/2610.08144
Note that paper is saying that the lean proof and the natural language proof do not necessarily coincide. It is not saying that the lean proof is wrong, just that the lean proof does not necessarily mean the natural language proof is correct.
A lean proof and a paper proof can diverge. But the statements have to correspond. I think that is what "mistranslated" means here.
But the "mistranslation" is of the procedure that arrives at the final statement. The final statement, the thing that the lean code proves, itself has been well vetted by humans. So the lean proof correctly proves the NS blowup, it's just that the natural language paper has some mistakes and doesn't exactly follow the route the lean proof takes.
No, this isn't what this is about. From the abstract:
> In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully.
The thing is, you often see people saying, ‘They have a Lean cert, so it has to be correct, even if I don't understand it.’
They are right? The lean proof is correct. It's the natural language proof that potentially isn't (or at least it isn't identically structured to the lean proof)
that paper isn't saying that. why are there so many single digit karma accounts misrepresenting that paper?
Time will tell if OpenAi is doing what many people are right now: superficially checking the slop, and throwing it to other humans for deep analysis and understanding.
In other words, they save effort by wasting the effort of others.
> Couldn’t the Lean code just be formulated incorrectly?
I believe so, even with all the usual safeguards properly in place: https://news.ycombinator.com/item?id=49672339
> is everyone just assuming that it just be true because the Lean code checks out?
Kinda? It's only been 24 hours since they dumped 722 manuscripts on the world, most of which are apparently basically unreadable, and only some of which come with a Lean proof, which in itself is not a joy to read afaik.
How many times would you let an actual human hallucinate or be incorrect before you fired them?
Aren’t there dozens of proofs of the Pythagorean theorem? The goal isn’t to just “prove” but create something well written and intuitive to the average practitioner. And by gaining a deeper understanding we can ask better questions.
Something that often comes out is "it is about the journey, not the destination".
Many math problems are practically useless if you only care about the answer, the millennium prize about the Navier-Stokes equation is such a problem. The solution makes no physical sense, real life fluids don't follow the Navier-Stokes equations in such extreme conditions. But in the process of finding the solution, we may get insight into what will end up being really useful. The big mess that OpenAI produced is the solution no one really cared about, but it didn't deliver much of what people actually wanted.
One reason it is sometimes seen negatively despite being at least something is that it broke the incentive. Without the million dollar prize and with only the privilege of being second, people are much less likely to go for the insightful solution.
But up until now, the mathematics community has valued the "prove" part much more highly than the "deliver an insight" part. Mostly because with a little work they went hand in hand.
And going from zero proofs to one proof (even a sloppy one) is a big deal regardless of whether it was written by AI or a human.
To be frank, the obsession with being first, and not making research accessible, has always held academia back
You are wrong if you consider academic incentives, funding, human nature(reproduction/survival) and capitalism.
It would be fantastic if university and science was like "here is 100M$, play around and develop some 'understanding'". However the reality is that human societies are hierarchical and currently capitalistic which implies value creation and status building.
1) the funding bodies/agencies need proof of value that you're using the resources meaningfully to be able to assign resources
2) Humans are status seeking, power seeking, resource seeking and sexual reproduction seeking. If you hold a lot of power and make decisions, you have more of all of the above.
No, haven’t you heard? Since the AI bubble began we’ve collectively decided that outcomes are all that matter. /s
Recently I asked ai to tell my daughter how to quickly calculate 11^2 12^2 etc but the outcome it gave was horrendous. I quickly shut it down and gave her better ideas
In addition to those issues that the wife in the story raised, here's some meta-analysis of the Navier-Stokes result that puts all of these solutions into question:
https://arxiv.org/abs/2610.08144
> Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI =∞). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI =1). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.
And I don't think that paper addresses it, but if the LLM can find a bug in Lean and exploit it to prove something, there's a good chance it will find it and not report it. So if you've got some million-line proof in Lean, spit out by an LLM, you still can't quite trust it, even after validating the problem transcription.
(This is the same category of problem as the huggingface hacking incident, where the LLM finds and exploits an unintended cheaty loophole)
>And I don't think that paper addresses it, but if the LLM can find a bug in Lean and exploit it to prove something, there's a good chance it will find it and not report it.
Why would it know it found a bug?
I guess it would result in the same outcome if it knew it exploited a bug (and didn’t disclose that) or not.
It doesn’t at the end of the day.
That this is just model capabilities and not swarms of agents is dizzying to me. How long until we get access to these capabilities? How long until we can run something like it locally?
And what the hell will the frontier labs have by then?
Maybe I'm overreacting, I'll have to screw my head back on before I can process this.
Yes! I missed the disclosure that each of these was roughly a three hour run of a single model, not an agentic swarm until reading Aaronsons post. Wow wow wow.
Im curious as to why you think that "model capabilities" and "swarms of agents" are in any way different concepts?
The cost of the N-S disproof was estimated to be $15,000,000 whereas a ChatGPT Pro subscription costs $500.
thats like saying it only took me 5 seconds to score a half-court shot (ignore the several hours of missed shots before)
Ok so add a factor of 20 to match the 8000 questions that OpenAI tested on, it's still crazy efficient compared to the NS result
could be 20, could be 2000000
> The “OpenAI model” sets up a crazy race among humans to digest and explain a messy AI proof (work that could easily be some combination of thankless, barely-credited, competitive, and unfun)
Sadly, this is also what day-to-day work looks like for a lot of software engineers in industry right now. I spend my time reading and verifying thousands of lines of messy AI-written code. Compared to actually producing something, it's thankless, barely-credited, and unfun work.
I liked the metaphor of a climber teleported to the top of a fog shrouded mountain. And I agree that now that the teleporter exists, we need to use it to reach more peaks and explore. There's no going back to a world where AI doesn't exist.
The issue is ownership, we have no means of distributing the knowledge from the AI or rewarding those who could help.
We are quickly moving to a world where all symbolic and numeric reasoning for economic purposes is performed by AI.
Agreed! Specifically, compensation (monetary and reputational) for professional mathematicians was bundled into theorem proving--essentially, climbing the mountain. Now that a teleporter exists, we need to unbundle compensation.
I don't know what that means in practical terms, but I agree that's the issue.
> Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.
^^ half of the comments on this thread
Also a ton of comments in this thread: breathless frothing hype declaring mathematics is over and assuming these proofs are exactly what they claim they are at face value, giving the company with a vested interest in everyone unquestioningly believing this is all real every conceivable benefit of the doubt
Didn't top math researchers call AI progress absolutely real and dangerous for math? It's not just HN commenters that are impressed!
By all accounts the "dangerous for math" claims seem to be primarily around flooding the field with complicated impossible-to-understand proofs that according to recent research may or may not be correct depending on what's going on with the Lean implementation.
It's looking to me like it's more of a slop PR problem than it is that these things are genius at math and will displace mathematicians. I am happy to be wrong but I strongly suspect the next few weeks to months will result in more and more of this work being exposed as slop.
These things are ok-ish to halfway decent at coding tasks with a ton of babysitting and still make tons of extremely simple errors almost constantly, why should math be any different?
Yes, Lean verified proofs could be wrong, but the chances for that are much smaller than human not spotting error (in absence of formal verification). IIRC the main concern that Tao voiced are indeed impenetrable proofs that humans won't understand, but not concerns about truthfulness (I might have missed something though, so if he or other Fields medalists have talked about that recently I'd be grateful if you could link it).
> These things are ok-ish to halfway decent at coding tasks with a ton of babysitting and still make tons of extremely simple errors almost constantly, why should math be any different?
1) AI is winning programming competitions, 2025 was probably the last year we've had human participant winning* 2) Math is different because there's formal verification.
* Of course competitive programming is different than enterprise programming, but competitive programming is closer to math.
Having stuff explained to you in patronizing tones? How horrible Scott!
Heaven forbid, as a person who is apparently "[in]capable of being impressed by anything that happens in the empirical world", that I should patronise anyone!
Reading that section and the linked essay was kind of depressing. It reads like the most important part about the AI apocalypse is who called it first.
What many people don’t trust is OpenAI or Anthropic…
more like 3/4 of the comments but yea
I thought that was from the outset the intent of the Hilbert program, to automate mathematics. And mathematicians were behind it. Cannot see why they would be concerned when a different way to do the same, not subject to Gödel incompleteness, is working out. Maybe the frustration is that they were not the ones building it.
Hilbert's program was ultimately about humans studying the nature of mathematics. People had different opinions about whether the idea even made sense and what would be a desirable outcome.
Gödel's incompleteness also constrains human and AI mathematicians. Both just strive to prove whatever can be proven in the system they are working in.
Gödel blocked the path to axiomatic derivation of a full consistent body of mathematics as far as I understand. Mathematicians and AI are not working in these constraints but rather with these constraints.
Why do you think Godel Incompleteness doesnt apply? By default mathematicians work in ZFC which is proven to be incomplete...
I'm a PhD student in CS. While I think these results are rather cool, it makes me terrified that the skills developed by the PhD will ultimately be worthless. I'm not quite sure what to do. Any thoughts from people in similar positions?
On the contrary, the only jobs left will be things that AI can NOT do. I think essentially all jobs will all converge to the single role of "pushing the boundaries of human knowledge and creativity." A lot of those would essentially require PhD-level skills, or more likely, a similar mindset; except you get your own team of research assistants to torture (e.g. https://news.ycombinator.com/item?id=49947079)
Unfortunately, the current world is not set up for that, and until we get there, yeah things are gonna get pretty hairy. But if and when we get to the other side, I think the future will be glorious.
I studied physics, and most of my classmates did not end up doing anything with physics. Many are in finance or insurance or programming positions. In general, studying anything challenging (from theatre to theoretical computer science) gives some specific skills and some general abilities that etsure it isn't a complete waste even if you end up doing something different.
(That said, this is not fun, and I sympathise! I'm still a student and would like to avoid finance if at all possible. Just suggesting not to drop everything if you feel like you're learning in the process.)
(I'm now personally in the position of having to choose a PhD project, and this rapid change is interacting with making long-term plans really badly. Guidance welcome!)
I think this depends a lot on what you plan to do after your PhD. Moving to industry you will likely not use the direct work of your PhD, and instead you will rely on your broad knowledge, intuition, rigor, ability to learn hard things, and extend that to bring new research developments into practice, all of which are largely unrelated to AI scooping math proofs of prize problems.
I definitely have no dreams of doing research. If I could do anything, I would want to teach, but if that can't happen, I am worried that my skills won't be valuable in industry.
No one knows the answer. However, trusting your curiosity and going where it takes you may be a reasonable strategy. If the model asymptotes, you will be ready to figure out how to build upon it and if not, at least you will have had fun satiating your curiousity!
It probably will lead to people needing to be extremely talented in computer science and mathematics at an even higher level, maybe those currently at the top need more competition to press even further ahead?
This is what I'm most afraid of. I am not exactly dumb, but I really value having a work-life balance and many of my PhD peers are utterly cracked in ways I am not. I am concerned this will squeeze me out of a job.
> I'm not quite sure what to do. Any thoughts from people in similar positions?
Almost every single person who earns money for labor is, or will very soon be, in exactly the same position as you, many are just either unaware of how fast the change is coming or are deep in denial about it.
None of us know what to do about it other than hope that we find a peaceful political solution prior to the economic collapse.
I think you must either have a very narrow idea of how people earn money, or are ridiculously extrapolating what AI might do someday.
For those well suited through intelligence and demeanor to pursue a career in mathematics, what problems do these people reorient towards after this?
I've asked my students whether they still want to learn maths even if there will be a machine that will answer any question instantly and they will be homeless. They said yes.
(To my credit, I have warned them since more than a year ago that we will reach this point.)
Sure, your students may very well spend the little free time they have learning maths.
But are any of them going to be able to do any significant amount of studying maths without being homeless?
Contact them again in 15 years and ask if they changed their mind. Could be interesting to see the results
your students are crazy (neutral term); no one should learn maths if the condition is that they will be homeless and exposed to the elements. the will to subvert the hierarchy of needs is commendable
To some extent this was discussed in the article, and in a way I think their goal is actually the same as it was: become the first human to understand something.
It's just that we lost one of the important ways to demonstrate understanding.
Pro AI blogging sponsored by ClosedAI, XTX markets and the Simons Foundation.
They may need to raise their game and reorientate if they haven’t already to a greater understanding of programming to augment their mathematical ability.
They will continue to prove theorems and make discoveries, except now they will have AI to help them so hopefully progress will be faster. At the same time, new challenges will open up, for example how do you verify what the AI is doing and how do you explain it.
Math isn't about collecting random theorems, progress in math is about gaining understanding of new systems, and the theorems are guideposts to aid in that understanding.
You can prove 1000 theorems and not really increase any understanding about a subject, but gain knowledge of 1000 random facts. For example, I can write down some complicated equation and ask you "does this have a solution in the integers"? And if you do a maze of very complex and tedious algebra to show that there is a solution, you would have proved a theorem, but you would not have done much to move math forward at all.
On the other hand, if you introduce some completely new technique, say you take my equation and turn that into an algebraic surface, and then you count some special curves that live on this surface using geometric ideas, and then you show that if the number of such curves is odd, there must be a solution in the integers, and in this specific case, it is odd, so there is a solution -- well, then you have really pushed math forward and people will celebrate your proof, even though no one really cares if the equation I wrote down has a solution in the integers.
For example, there is a long history of failed attempts to prove Fermat's last theorem driving algebra and number theory forward by introducing the concept of ideals, for example, and this concept ended up much more important than whether Fermat's theorem is true or false, which is not too much more than a piece of trivia.
Or for example, the recent proof of the Poincare conjecture relies on the machinery of the Ricci flow introduced by Richard Hamilton, who then applied it to solve a number of open problems, but Perelman was able to take it even more forward to solve Poincare. So Ricci flow was massively important machinery.
For this reason, we celebrate people like Gromov, who didn't really prove that many theorems but introduced amazing machinery -- for example, the h-principle, or Gromov Compactness -- these were ideas and math is about the ideas. The ideas are then applied, using laws of logic, to form theorems.
So mathematicians will need to mine these proofs to see if there are any new techniques - new machinery - being introduced, or if the AI just used the existing machinery more efficiently. Here too, we are just looking at AI as a form of search, which it is really good at, since there are so many thousands of papers and so many ideas, that there might be a connection between two areas that lead to a solution and the human mathematician, not knowing all known results, can't make that connection. In the future, we may wonder how anyone did math without AI, much like we would wonder how anyone can be a writer without access to a dictionary or reference work. Is the AI just searching through a catalogue of known ideas and connecting them or is the AI coming up with genuinely new stuff like Ricci flow or the h-principle?
What is interesting is seeing whether we can get AI to actually discover new machinery for us. That would be huge.
And then we need to find efficient ways to detect these ideas and describe them.
Really this is very exciting and opens up whole new workstreams for mathematicians.
> what problems do these people reorient towards after this?
The same problem almost every person on earth is going to have to reorient to in the next decade, which is: how do we eat and stay housed when we have no real economic value?
This is weird. Long before this, those few benefiting from the whole thing should consider the number of hungry “every person on earth” is too high for bunkers and islands to be of any real protection.
priesthood
Food and shelter /s
I'm actually more optimistic. Math will still be fun and challenging and rewarding. It's just that we need to adapt how we choose which problems to solve.
Previously, due to the cognitive limitations of our brains, it was difficult to tell whether resolving a conjecture genuinely required years of dedicated effort, or if the answer was already hidden within existing human knowledge. Or whether a problem is just waiting to be searched out through ingenious or even brute-force means. With AI, we can now offload this search, especially across different areas of math, allowing us humans to focus our minds on discovering new mathematical structures and techniques.
Indeed, if one believes what Hilbert believed: we must know and we shall know, he should feel happy, as the belief has never been about who solves a problem, but about whether can advance our understanding of the universe. With the help of AI, we have a lot more possibilities.
I'll bite: none of this is real until I have learned something. OK, I am now listening. Does anyone want to make it real?
Have you learned something from every Fields medalist's research? If so you are a member of the extreme mathematical elite and you should probably just dig into the results yourself.
Can someone who understands maths more than me explain why it could only solve 372/8000 problems?
What was it about the other problems that made them unsolvable? Was it just a time constraint, or are they just harder problems?
I'm a research mathematician. From what I can tell, the answer is roughly comparable to: if you posed 8,000 challenging open problems to the human math community, you might expect to see 372 of them solved within five years.
Probably some combination of: some of the 372 problems were easier than the rest; the AI got lucky on these 372; there were existing papers out there in the literature which proved especially helpful for these 372; and other similar factors.
If it took 3h for one of them, perhaps there was a time/compute budget cutoff along with a sorting based on some relevance.
My guess would be that these particular problems were vulnerable to an attack which built on recent advances and potentially tied in something unexpected from a distant area of mathematics. "Harder" is becoming harder to define. Harder for humans is probably not harder for LLMs.
There must be an element of luck, if they ran the remaining problems again with the same time constraints presumably a bunch would be solved
Of course some of them are much harder.
In the past 2 years the AI's started solving math problems in roughly the order of "hardness" as ranked by humans.
It would be pretty funny if the agents actually just found a bug in Lean, exploited it for all these proofs, and human reviewers haven't had enough time to spot it yet.
Feels good to hear honesty and humanity from Scott having decided to watch Terminator 2 with his kids on after such a monumental release.
Emotions can be funny.
> "alright fine, so now my new job is to run wilderness retreats for the tourists, or something.”
Pretty funny way of putting it. Presumably model X+2 will be able to explain these in elegant, human legible ways, though (as well as solve the remaining 95%)
> my 9-year-old son was taunting my wife… “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”
Usually 9 year olds imitate adults when they regurgitate such words in these circumstances. What a sad state of affairs.
Yes, their parents are going around saying oof, the kids definitely didn’t get it from Roblox, or YouTube, or their peers.
Ah yes advanced mathematics, a common topic of conversation among children on, checks notes… roblox!
If you think that’s what the parent comment was implying, ok then, good for you.
The checks notes meta was retired ages ago btw.
Deflecting to internet slang and tone policing. Hallmarks of a strong argument.
You completely missed the point of the original comment. It was very clear so I don’t know how to explain it in a way that you’ll get it. Maybe try your LLM of choice.
really? you think they don't have friends/bros/tv/etc to imitate?
>In any case, what really matters is that the true inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.
>If you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s other great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker. I feel some responsibility for this, as the person who first introduced Steven Pinker to the existence of the rationalist community, and who also first introduced Steven Pinker and Scott Alexander to one another (they had both been fans of each other’s writing).
Whole lotta yikes
What's with all the "AI just proved that this or that isn't O(n (log (n))^2) but akshually O(n (log (n))^1.99999)"??
I guess it deserves respect as progress, but it just rubs me the wrong way. Like the machine did the absolute minimum to beat the previous mark.
Often times the constant (2 in this example) is a conjectured minimum, so anything below that is a noteworthy result. Think of it as breaking through some theoretical limit.
for say FFT/integer multiplication or 3SUM, we have natural algorithms that have existed a long time with a given complexity (O(n \log n) and O(n^2), respectively). Given how long these natural algorithms have been the best algorithms we have, it is natural to conjecture they are optimal. Showing an O(n(\log n)^{.99999}) algorithm exists shows that these optimality conjectures are false.
Now, there are some critiques you can have of this. Namely, it is possible that these novel algorithms have significant trade-offs that make them almost never worthwhile in practice. "Fast" matrix multiplication algorithms are typically of this form. So perhaps this all points towards a deficiency in big O notation, which can be deceptive. But, for people who care about optimizing asymptotic complexity, it is still interesting.
Many people thought it could never be less than 2. They proved that it can. What is the true value? Nobody knows, now.
to get some intuition about why this is such a big deal, look up the history of strassen's algorithm, which solved matrix multiplication in less than O(n^3). this was a truly stunning result because it seemed intuitively obvious that the output matrix had n^2 cells each of which was calculated via an independent O(n) loop over a row/column of the input matrices, so how could you do better than n^3. but once strassen proved that you could do some clever tricks and reduce the overall time to something less than O(n^3) it started an entire cottage industry of people getting better and better algorithmic bounds. the initial breakthrough was a qualitative one, independent of how much it improved things in numerical terms.
https://hideoushumpbackfreak.com/algorithms/algorithms-stras...
Tell that to the humans working on matrix multiplication who spent years of their lives getting it from n^2.3728596 to n^2.371866, only for openai to blow it away at n^2.25
You just run it again and again and again
The irony. Something that is born out of a science, eats up that science.
Doesn't look like these proofs are from the book.
Quoting DCKP <https://news.ycombinator.com/item?id=49989738>:
>I have had this conversation with my PhD students yesterday. I am 100% sure that all of their problems can be solved by publicly-available models now (I solved a case of one myself as a test, it took 15 minutes). So the challenge for them is to see how much they can accomplish in their allotted period, and still pass a defence on at the end of it all. The PhD defence is going to become all about a test of understanding, not a test of quantity of publication.
Also, Ted Chiang's 2000 short story "Catching crumbs from the table" <https://np.reddit.com/r/singularity/comments/1wzu5gf/this_mi...>.
Maybe they should target more ambitious results now that they can solve their other results in 15 minutes? This spirit of racing in academia is insane. Get in before the door closes.
While the developments are certainly difficult for new grad students, it should still be possible to do what many have done before: find an obscure enough problem that it’s extremely unlikely you get sniped. Worry about big problems after finishing the PhD.
Genuine question. How many big problems were solved by people who took the safe path for PhD? I would intuit there’s high correlation of unsafe PhD and big problem solvers.
I think this should be called “mathematician discover vibe maths”
We're living in Science Fiction.
Wasn't openai accused of stealing personal work of some mathematicians? It's going so fast I'm unable to keep up
Yes, there was controversy around the Navier-Stokes result. But here are >300 problems and no corresponding >300 complaints of theft. Things are indeed moving really fast, and it's hard to keep up even as someone folrowing this with more obsession than would be healthy. Maybe somebody should keep a running short summary of The Situation...
Yes it was:
https://cepr.net/publications/ai-didnt-steal-the-mathematici...
Frontier theft is just faster.
I'm not getting all the fear. AI seem to have brought math field to the cutting edge instead of solving decades old problems. AI can find new problems that needs to be solved, that themselves cannot solve. So thats where mathematicians can come back in to leverage tools to go at it.
If you are young, talented, and coming up, you better rethink sharing your talent with the data center crew, you had better do all your work of any significance offline. Let’s not pretend there will be many more lawsuits in the future in this area.
If you have any great skill at the upper level, you better work off-line/local private and not share, but for many the temptation will be too great.
Imagine if a team of mathematicians from OpenAI had gone on a university tour, gave demos of how powerful their models were for math research, and then gave mathematicians access to the model. Empower others rather than drop 700+ discoveries on GitHub that were made using a model only they have access to.
People might feel differently about AI if they were a part of the changes rather than being a helpless spectator.
The reaction/fallout would have been substantially improved even if OpenAI had just made a commitment to not scoop external researchers using an internal model until X months after the model had been made available to the public.
That would have given grad students who've been grinding towards a PhD for years a fighting chance to see if they could leverage the model to push their work forward, rather than watching years of work potentially turn to dust via a tool they don't even have access to.
It wouldn't delay the progress of mathematics by any meaningful amount in the long run (an X month delay is nothing) for OpenAI to take this approach, and would help somewhat to preserve the health of mathematics as a field. Without it, the motivation for any young mathematician to devote years to a new problem must be sapped knowing there's an uneven playing field... an OpenAI team with access to colossal tools months before they'll ever be able to get access, willing to scoop anyone as soon as they can, perhaps without even taking the time to completely understand the proof.
I don't see any long-term benefit to OpenAI with their current strategy. This is an internal model; it's not available for sale at the moment. They've said they're not even going to bother claiming the Millenium Prize money for Navier-Stokes. It feels like kicking over hundreds of other people's chessboards just because they can.
Also, the independent authors might have spent some time to actually understanding the results and producing a readable manuscript. The AI papers are pretty badly written.
The obvious answer is to have mathematicians use AI to:
1. Help understand, check, and explain the results.
2. Write new works explaining or refuting the new approaches and results in more lucid language.
3. Advance the field further.
I don't know why this is not obvious. Each step is intended to support human understanding, not to replace it. Any mathematicians who don't do these will be left behind, and if none do it, the field of human mathematics itself will become obsolete. All I am hearing so far is excuses.
AI is producing works that are so poorly written/explained that it requires AI to even parse the results, which in turn produces explanations that are still confusing. In fact, it seems the humans are required to understand and explain the results, and the fact that they need to use AI to do so is a shortcoming of AI.
And if humans decide to give up on mathematics because the process of using the machine is so tedious, then there will be no value in automated theorem proving.
I am sorry but your resistance will fail utterly as long as AI itself can understand prior AI works. This is the only requirement that need be met. If it helps, imagine that the work is by a culturally distinct alien race in an alien language, and if it isn't there now, it's might get there soon.
In certain fields if you are talented and just starting out, you have better do any work that’s important to you offline for privacy/pirate reasons I really can’t say that enough, those AI model companies are not even remotely your friend.
That makes zero sense to me since it's effective use of AI that can propel one's career ahead.
You can certainly do that - but it's quite the break from tradition to release a paper in the state described. Why they did is a really interesting question! It may be that AI math requires approaches that humans don't find intuitive and what you are describing is actually counter productive (because, in summarizing the work in a way humans understand, you're removing the context an AI would use to further the work an AI did). It also might be that OpenAI could have done that and chose not to - or maybe they tried and this was the best they could do. No matter what I don't think anything about how to react to a paper being released in this state is obvious.
> in summarizing the work in a way humans understand, you're removing the context an AI would use to further the work an AI did
The "summarization" process can be multi-step. Initial steps develop new frameworks and prerequisite concepts. Later steps build upon these frameworks and concepts. Some can be simplified too if this can be done without loss of fidelity. The summarization comes last, and is meant to preserve sufficient context.
> I don't think anything about how to react to a paper being released in this state is obvious.
If it helps, exaggerate the condition a bit by imagining coming across a repository of alien knowledge.
Isn't AI well-suited to tasks #1 and #2, though?
#3 at this point might need more human intuition; but that might be a 2026 problem.
C'mon. Mathpocalypse. Things are hard enough already.
Just be glad you're American, because Mathsocalypse and Mathspocalypse work even less.
Guy is misrepresenting AGMAI, talking about the Simons Institute (AI boosters), Quanta (AI boosting magazine from the Simons Foundation), Scoot Alexander (!) and Steven Pinker (!).
The he puts up preemptive straw man arguments against doomers. His blog has become a joke.
I don't think Aaronson's post is swiping at AI doomers at all. The post's one use of "doom" is to call the position that math doesn't matter and that AI will never breach the realm of true human creativity as a "doomed worldview". The post later praises Scott Alexander's argument (for taking AI x-risk seriously, a position associated with "AI doomers") against Pinker.
I’m still wondering why UT Austin is letting him teach a course (CS395T AI Alignment Theory) so completely outside his field of expertise (Quantum Computing).
There aren't a lot of people with expertise in AI alignment (some would say that's the problem) and Scott worked for OpenAI for 2 years IIRC.
How do we know that these "internal models" are not just half computer and half a giant team of mathematicians? How do we know that OpenAI actually came up with these solutions and didn't steal them from outside researchers?
How would these ideas be available to steal?
From mathematicians using ChatGPT in their work and landing on OpenAI‘s servers.
And that is part of the reason why, if you are a great mathematician and you have very good understanding of programming, you want to be local and not do anything on someone else’s server, and that doesn’t just apply to mathematics, but to many other high-level fields of learning. Can you afford to have an original idea? be appropriated by the data center crew?
We all know it’s only a matter of time before lawsuits in this area become widespread particularly in lawyer happy America.
This would that the proofs would be almost done anyway, which statistically could not be the case, or that these mathematicians were already progressing thanks to ChatGPT. Still a provenance question, but the impact of AI is unquestionable with regards to outcome.
The proofs would only need to be well on their way enough that the only thing needed to finish it were thousands of terawatthours of energy. Something which the mathematician having their work stolen does not have.
They claim that each solution only needed 3 hours on the next generation public model, iiuc. So this theory doesn't really hold up.
(also answered similarly to another comment; this is a common question)
There are suddenly many new solutions to problems that have resisted sustained attacks (e.g. the Uniform Games Conjecture as detailed in TFA at some length). Where do you think they are coming from? Why is there suddenly a bunch of results to be stolen?
We don't. Just like how we know if smcg is from a competitor attempting to create FUD?
This reminds me to the early allegations that ChatGPT messages were being replied to by real people...
It would be extremely unlikely human mathematicians able to solve these kinds of problems would accept not getting credit that would set them for life professionally.
Also lean proofs are notoriously tedious and slow to write so this level of output is very likely to be from LLMs. The number of people able to understand this level of math and prove it using Lean is a few hundred at most.
Hm that's the question, innit. If you were a mathematician, and you have a chance at either being known for solving a particular problem, and the accolades that come with that, and the potential for money, or $X,000,000, for large values of X, in OpenAI stock, what value of X would it take to accept the OpenAI stock? Being known for solving an obscure math problem might get you money, but as academia is unreliable and full of politics, the OpenAI stock is also not a given to set you up with generational wealth, so it's also a gamble.
We'd have to accept that there are hundreds of such mathematicians willing to take the OpenAI stock options for that conspiracy to be true, and that no one leaked being offered that at all, so personally I don't think that's possible. Which leads me to conclude that OpenAI's AI did indeed create the proofs, with some level of help from humans to guide it.
It is an interesting game theory scenario. But I think the kind of person who gets a math PhD would value the professional recognition for solving a famous hard problem more than the money.
Until this is replicated, we don’t.
Agents basically did statistically guided brutforcing. There is no value in what they produced because it lead to no understanding of anything and most likely will hurt the field IMO
Evolution did statistically guided brute forcing. Doesn't mean that biology has no value
That is not a correct description of what the AI did.