> 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.””
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.
> 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.
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.”
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.
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.
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.
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
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?
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.
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.
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
Hi, I’d like to signal that I’m part of your in-group. One time I used AI and it sucked. Every time someone says it’s good, they are lying because they are shills. Upvotes to the left please.
> 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 claims 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.
> 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?
You know, if you attack ad personam you've got to be ready that someone will do same against you - what progress did you make in the last 10 years (or in your entire life for that matter)?
Trust me bro, just 100 more cubits, I swear we'll break everyone's encryption and cause the downfall of society, please bro just one more grant, It'll be stable this time :'(
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.
> 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.
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.
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.)
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.
> 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.
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.
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.
>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.
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.
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.
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.
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’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).
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.
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.””
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.
> 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)
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.”
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.
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.
Vibe mathing
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.
I see many parallels to genAI-assisted code development. Not surprising I think.
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
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?
[delayed]
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.
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
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
Hi, I’d like to signal that I’m part of your in-group. One time I used AI and it sucked. Every time someone says it’s good, they are lying because they are shills. Upvotes to the left please.
> 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 claims 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.
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.’
that paper isn't saying that. why are there so many single digit karma accounts misrepresenting that paper?
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.
> 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?
If you haven't made progress in Quantum Computing in the last 10 years, lecturing others can become a popular pastime.
You know, if you attack ad personam you've got to be ready that someone will do same against you - what progress did you make in the last 10 years (or in your entire life for that matter)?
He was created 14 minutes ago, give him a break!
Trust me bro, just 100 more cubits, I swear we'll break everyone's encryption and cause the downfall of society, please bro just one more grant, It'll be stable this time :'(
Having stuff explained to you in patronizing tones? How horrible Scott!
more like 3/4 of the comments but yea
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.
> 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.
I'll just leave this here https://owl.purdue.edu/owl/general_writing/academic_writing/...
Do you have any specific one in mind or did you feel one must fit and wasn’t sure which one?
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
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.)
Contact them again in 15 years and ask if they changed their mind. Could be interesting to see the results
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.
> 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.
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.
priesthood
Food and shelter /s
I think this should be called “mathematician discover vibe maths”
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?
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.
Many people thought it could never be less than 2. They proved that it can. What is the true value? Nobody knows, now.
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.
https://hideoushumpbackfreak.com/algorithms/algorithms-stras...
You just run it again and again and again
Doesn't look like these proofs are from the book.
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.
The irony. Something that is born out of a science, eats up that science.
Wasn't openai accused of stealing personal work of some mathematicians? It's going so fast I'm unable to keep up
Yes it was:
https://cepr.net/publications/ai-didnt-steal-the-mathematici...
Frontier theft is just faster.
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...>.
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.
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.
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.
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.
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’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.
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.
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.