To me, the real lead of the article is buried here:
> apparently they tried the model on about 8,000 problems. So, right now it “merely” solves ~5% of the longstanding open mathematical problems that it’s asked about, the problems that whole communities have spent years on, after a single 3-hour attempt on them.
Back-of-the-envelope cost calculation:
3 hours of GPT-Pro compute per attempt / 5% success rate → 60 hours of compute per solved open math problem
At GPT-6 Astra API long-context pricing ($75/M output tokens), assuming 50 reasoning tokens/s, that's roughly $810 per solution:
60 h × 3,600 s × 50 tokens/s × $75 / 1,000,000 = $810
Of course, actual token usage is unknown. But even if off by 10x, $8100 is cheap for solving a longstanding mathematical problem.
“On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems.”
That leaves a lot of ambiguity. Is “result” exclusively “positive result shown here” or “all results”?
Since proving infeasibility is a valuable result, the unpublished problems must not have reached a valuable end state. Thus it’s an operator decision of when to turn the machine off and try another problem. I’d thought they had a three hour time box on this, but apparently not.
As bureaucracies look at AI and staffing, this is going to be the headline in the executive summary.
Sure yeah, today it's only a 5% solution (to the hardest problems), and we know it will top out somewhere.
But the cost reduction is just too good to ignore.
Terry Tao was saying that we need more mathematicians [0]. My paraphrase of his article is that we will need them to check the AIs, essentially (I missed things didn't I?).
But with a near 10,000X reduction in cost [1] today, it's really hard to argue for that when budgets are continuously tight.
[1] assume ~$150k/year for a mathematician of the caliber that can do this work. 2x that for overhead like healthcare, 401k, etc. Do this for 35 years of work = $10,500,000. Assume about 1 result of this caliber per mathematician. So ~$10M per result. OpenAi is saying they can do it for ~$1k, a 10^4 X decrease.
If we have a jagged frontier inside the domain of math (very plausible, the jagged frontier concept seems to be quite fractal), the model solved the easier problems in his area of major strenght, some of them will be in the bottom 5% of the easiest problems in the full set, and some in remaining 95% of problems
I think it's not just to check - it's to interpret, simplify, apply taste, basically get these proofs into a form that humans can grok and build on. AI could do all of that but the grok-ing -- as we are tapping the font of cheap proofs it still behooves humans to keep track of what we find interesting, keep internalizing and learning concepts and make sure whatever we're building right now will be generationally useful. At least that's my take
>I think it's not just to check - it's to interpret, simplify, apply taste, basically get these proofs into a form that humans can grok and build on.
Sure, but we are approaching a time at which we will have to answer a "why" here. I.e. anything beyond checking and validation becomes of mostly artistic value at some point. And there is nothing wrong with that.
IMHO it's groking that allows higher level metaphors and eventually more powerful theories to be built. If all you are doing is proving everything reachable by construction, you aren't really learning, as this process won't be drawn to the more compressed expressions by necessity but only as a "please" from the humans.
I’d say the artistic value is the main value in general —- if you know mathematicians they will often refer to a ‘beautiful’ proof, aesthetics is a critical part of math.
An early (easy to understand) proof that fits the bill for me is proof that the sqrt(2) is irrational: a very natural thing to think about —- for a unit square, what is the measure of the diagonal? And people were killed in ancient Greece over the proof, but it’s comprehensible with basic algebra.
You're off by an order of magnitude at least. A mathematician does much more than just prove theorems. Let's start by teaching and administration duties.
And then you're off an order of magnitude of the efficacy of OpenAI. We don't know how much work of mathematicians is behind all these results, what's the real cost (which includes those problems they couldn't solve), etc.
Oh I entirely agree! Mathematicians are full human beings.
But a soulless admin is not.
And they care only about the budget.
And the real danger here is that these large groups of blame avoidance mechanisms take at best half a look at the cost of an AI and the cost of a pre tenure track new assistant professor (or whatever), and they decide, oh maybe next year we can hire someone.
This comment seems to be a hypercorrection, where someone attempts to demonstrate superior linguistic knowledge by correcting something that isn't actually wrong in the first place. You've likely heard somewhere before, almost certainly on HackerNews in fact, "lede" is proper and "lead" is not but don't really have a genuine understanding of it and are just repeating what you've heard before.
In actuality both "lede" and "lead" are correct. "Lede" is mostly an American and recent respelling of the word "lead" and "lead" still remains the predominantly used term (while "lede" is predominantly an American journalistic spelling).
The expression comes from journalism, hence why lede is preferred. It's something one should not do when writing an article - bury the most important point in the middle of the text.
I heard they used lede to distinguish from (hot) lead. But since linotype has been dead for 50 years this is probably moot now.
Which is not the same thing that was written in informal prose & their proof has 8k non-computable axioms so you can't even write a program to verify their blowup example is actually a valid instance of a smooth flow that blows up.
I see, so you have no idea if the proof is correct or not & you have no clue how many non-computable axioms are in the lean certificate. Seems like I have wasted enough time here so good luck to you.
You also have to count the amount of tokens and human time spent afterward to verify these proofs. There's already been some papers retracted and it seems likely to me that there will be many more as people continue digging into the more dubious ones.
$8100 doesn't include the negative value of all the knowledge not gained along the way, and perhaps never to be gained in the future, since there is "no reason" to take the road to a known solution.
Many discoveries were made in search of something else.
I'll leave you with this:
"When I woke up just after dawn on September 28, 1928, I certainly didn't plan to revolutionize all medicine by discovering the world's first antibiotic, or bacteria killer. But I suppose that was exactly what I did."
> 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.
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.
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.
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.
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.
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.)
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?
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.
The first paragraph is describing a general trend over multiple events across multiple agents. Between zero and three of these can be true for any transaction across every transaction.
The second paragraph is specific to OpenAIs behavior.
> what companies do to attract investors is likewise often irrational, and why investors invest in companies is also often irrational.
Note that the cool thing about rationality is that it does not depend on transitivity. If what companies do to attract investors works, then it is in fact rational of them, regardless of the rationally of investors.
And neither is rationality absolute nor consistent. One company can exhibit rational behavior while attracting investors, while a different company does not. A single companies can be rational at times and irrational at other times. And quite often companies will behave in extremely dumb ways. Especially in our current economic system where the cost of being dumb is negatively correlated with the amount of money available to you, and that cost goes down on a sigmoid curve.
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?
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.
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.
American mathematician Jim Simons was worth some $31 billion at the time of his death in 2024. The lack of economic value in pure math doesn't mean that applied math is of little value. Physicists and mathematicians with PhDs "sell out" to join Wall Street as quants and make a killing there, Jane Street is full of them.
> If the models can do this, they're almost certainly good
I would argue that "good at math" was a short hand for "good at X" because mathematicians were historically good at engaging with very complex ideas, distilling them and coming up with precise and concise answers they could validate by themselves.
Given that AI solutions are described as "psychedelical" and they rely on the outside source to validate the result, I don't think the same logic could apply to them.
> 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.
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.
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.
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.
> 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.
That's because there's a bunch of people who work on that stuff. Just because web developers don't care about it doesn't mean it doesn't exist. Utter magical thinking.
You seem to be confusing your quantifiers a little. (Not all SWEs must understand all CS) does not imply (there can exist parts of CS that no SWE understands).
> “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.””
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.
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.
> 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
> 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?
> 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.
The point isn't that 1 person can understand an entire smartphone silicon up. It's that 100% of a smartphone can be understood by people. A maths theorem that can only be partially understood/proven only has value as marketing collateral.
>A maths theorem that can only be partially understood/proven only has value as marketing collateral.
There are many problems that aren't "interesting" in a mathematical sense but can still have valuable proofs. The entire field of formal methods in software development is mostly about this kind of problem. If I want to be sure that no input can cause out-of-bounds memory access, or that my superoptimizer found the lowest possible cycle count, I don't care if it's elegant, I just need some machine-readable proof I can run through my trusted verifier.
When we use verification to build software. We have an actual software as an output and hopefully some sort of problem solvsd the verification is also not in an of itself valuable.
The verification itself is valuable because it gives us confidence that the software is correct. This knowledge changes what we do with the software. E.g. do we need to sandbox it before processing untrusted inputs? Is it safe to put it on a hostile network? Can we use it to control an airplane or a nuclear reactor? Even if the code remains identical, formally verifying software improves it because it lets us use it in situations where unverified software would be too risky.
For one person that understands a proof of some matrix multiplication algorithm lowering its current complexity bound in an useful way (e.g. not with absurdly gigantic <= 2nd order polynomial term), there's likely thousands of companies willing to pay for the improved efficiency
And if anything that useful does come out of this output/"slop dump" then that might begin to make a strong argument for it. But every other time it was "that's the theorem done for the headline figuring out everything important and how to use it is an exercise for the reader".
> A maths theorem that can only be partially understood/proven only has value as marketing collateral.
If this is true, then what's really the point of math? A lot of math is actually useful. A proof regarding cryptography, for example, would have practical application even if not understandable by humans.
Cryptography is 100% understandable by humans so I'm not sure what the point is. You do understand that the argument isn't everything has to be understandable by every human right just to check?
While any logical proof written in lean can be broken down into things a human can follow, there is no requirement that the whole proof of something is ever simple enough that all humans combined could follow it.
Given the LLMs are struggling to explain themselves clearly, but also that these explanations cleared up somewhat by having an expert using one to query some of this research, but also the LLMs can solve problems faster than humans can read the proofs, it is possible for this category to have both examples of things that can be rendered in a human-comprehensible form, and also examples where it is not.
As an analogy: any human can check any single arithmetical calculation from a computer, that's not even particularly difficult. But a Raspberry Pi Zero can do those arithmetical calculations so fast that even if literally every single human was trained to do this at the level of the current world record holder, humanity as a whole could not keep up.
Exactly. We need "observability tools" that allow us to keep a handle on things we can't see or comprehend innately.
We have such tools for a large number of events that we're incapable of witnessing or understanding in "raw form".
We can't possibly read through thousands of raw records of recordings of individual's heights and make sense of it. But we can use Excel to calculate the average in seconds. We can chart the results and the image is perfectly comprehensible. We can trust the average calculation and the image because we trust the mechanism for transforming the data. We have what I would call a trusted path of provenance. We don't need to nor do we want to read the raw data.
Now we need things like this of the 2nd order. We need mechanisms of transforming trusted paths of provenance into things we can look at and easily verify.
We can do formal verification of code that would be hard to do by hand. We need formal verification tools for formally verifying formal verification tools. Eventually we will need multiple layers of this. As long as the chain and reasoning is intact, we should be alright.
It's kind of like with a horse. A horse is much stronger than us, but we can control it by pulling on two strategically connected pieces of rope.
I don't know much about this controversy, but looking at the Wikipedia page for the Four Color Theorem, it looks like since the initial Appel-Haken result, mathematicians have continued to work on the Four Color Theorem to try to find a simpler proof. So did they really accept it?
There are a lot of proofs for Pythagoras' Theorem and other theorems.
There are mappings between complex numbers and 2D matrices allowing problems to be solved in either domain.
There's research into The Langlands Program looking to connect number theory and harmonic analysis.
There's research into Category Theory looking to define core concepts and relate them do different fields so that results in one field can be applied to another due to equivalence.
It was also not some random small pc or minicomputer. It required 1200 hours on a then mainframe computer. Anyone reading this may estimate the rates to rent an IBM 360 in 1976 and use an inflation calculator if you wish.
> 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.)
I doubt that it is. For the language of proofs to be powerful enough to be able to
express an arbitrary proof it would have be Turing Complete. Proving that a proof
is the smallest proof of a given concept (i.e. there is no smaller human understandable proof)
would then be proving the minimality of a program in a Turing Complete language.
> 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.
This is basically how all AI approaches appear to work. They solve the problem you give them (in many cases) but in a very over-complicated way.
It's like they can't step back from the problem and realise that they need to simplify to make it work better. Nope, just keep hammering more code/proofs against the problem and eventually you'll hit the goal.
> It's like they can't step back from the problem and realise that they need to simplify to make it work better. Nope, just keep hammering more code/proofs against the problem and eventually you'll hit the goal.
There is nothing wrong with that approach if it accomplishes the goal faster, i.e. you can work at the speed of an AI.
> There is nothing wrong with that approach if it accomplishes the goal faster, i.e. you can work at the speed of an AI.
Sure, and god knows I've done the same myself when under time pressure (so similar to an AI in RL training, just get the goal).
The part that's lacking (that humans are much better at right now) is the meta-cognition around this works, but it's far more complicated than it needs to be.
Potentially because LLMs have larger "working memory" than humans, they can avoid this for a while, but once it blows up the context they fail in much worse ways than a human would.
Hmmm, this feels like a good metaphor, I need to write about this at some point.
> the difference between brute forcing and cognition
There’s not really a clear distinction between these things, in my opinion. Problem solving (and intelligence?) is a mix of search and compression. We like solutions that are elegant (high compression, simple search). But often what appears elegant to some is harder to appreciate for those without the same background knowledge or even the same amount of mental bandwidth (if you’ve ever worked with someone simply much, much smarter than you, you may intuit this!).
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.
> 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.
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.
If it were submitted to a journal, it would be outright rejected. We should give the peer review process -however flawed it is- some credit here. Dumping hundreds of AI-generated preprint articles on the field should not be allowed.
No. Letting some tiny handful of individuals selected by a tiny set of review-delegators review this is immeasurably worse than just releasing it publicly and letting anyone with the sufficient expertise review it. You know nothing of how peer review actually works, or have not thought about why that process would be only harmful in this case.
Normal peer review works by submitting the article to the appropriate journal, suggesting some preferred reviewers that have expertise in the subject, after which the rest of the process ideally is then orchestrated by an experienced editor.
If a journal receives a paper that is unreadable, it is outright rejected with the comment that it should be made readable, regardless the results in that paper. What is the point of having results if you are unable to convince your audience of those results? You could just as well just sit on them and never communicate them. What is harmful about this process?
Try thinking for just two seconds about what your process would result in, given the scale and amount of results here. You are basically asking for these results to become available to all experts only months / years from now, only after some select experts are chosen and first get to read these results. If there is any error or bias in that selection process, the whole thing is delayed even further or lost to the file drawer entirely. Or, since you say these are all unreadable, and should be rejected, you are in fact arguing these results should never be released at all.
Which is why it is obvious post-publication peer review is the only sane way to handle this. Formal peer review is not even remotely up to the task here.
For what it’s worth I’ve heard from several different ex mathematician colleagues that found papers in the stack that was adjacent to their work or solved a problem they were acquainted with in their career, and they all said the results are actually quite readable.
Maybe there’s some variance or it depends on the reader. That said, like with a lot of other complaints about AI: human papers can be poorly written and poorly explained too. That’s always been the case. And sometimes a proof is just complicated and hard to expose nicely.
> It sounds horrible to wade through - like trying to understand someone else's messy code that still produces the correct output.
That's what a lot of us do nowadays, but instead of maths, it's code that looks sensible on the surface, but when you try to understand it's some kind of "alien" logic , names don't really make sense, etc.
It’s been said every time about these LLM math results, but of course you don’t see it in press releases or articles blindly written based on said press releases.
But if you train the next generation of models on it, will AI be able to use it? How much larger will these models be, etc.
Also, AI has limited context. At a certain point proofs may become so complicated that an AI cannot keep most of it in memory and will not reuse it for new proofs.
As mentioned in the first paragraph, the proofs come with Lean certificates. Lean is a theorem-proving language. If Lean accepts the proof, it’s legit.
They already had to withdraw three of the manuscripts and correct several others. Who knows how much broken stuff there's in there; unfortunately they were too lazy to check themselves.
What I don’t get: if someone that’s apparently not the average mathematician has so much trouble reading and understanding the paper, then why is everyone so sure that GPT really delivered a genuine proof? This is an honest question; I really don’t understand that.
they are Lean certified, so in theory they should work. It's automated programming language for proving math, but it doesn't mean that you can't make mistakes there, although way less likely
It's a well-established fact that Llama simply can't write comprehensible maths, and it's one of the major reasons that OpenAI are catching the flak they are. Dumping the manuscripts in their current form is a sign of laziness and incompetence.
And you're right, it's not really a theme around here, but there aren't that many mathematicians around HN. Check one of the maths forums and it'll be a common point of complaint.
That's what Buckmaster and Alpöge said about the Navier–Stokes equations paper: they had the meat of their own paper ready by the beginning of August thanka to LLM assistance, but it was so messy they spent the whole month turning that into a paper worth publishing. Then OpenAI heard about their work, spent a few millions of dollar of compute to beat them (likely exploiting their own work with ChatGPT in the process) and released something.
When all you value is being the first, you have not time producing useful papers.
The proofs could be optimized for human consumption.
OpenAI could optimize each proof for fewest theorems, low branching, or short dependency chains.
OpenAI could have an adversarial network enforce that proofs look human.
OpenAI could spend a couple more machine hours per paper with the prompt "edit for clarity".
I think OpenAI wants the outputs to be unreadable.
If outputs "too advanced for human comprehension" become the norm, then every interaction requires tokens.
If someone can buy a day pass, get what they need, and leave, then there's no recurring revenue stream.
> 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.
Is anybody getting good code out of these things reliably without a bunch of additional legwork? Like, what a 2020 mid-level software engineer would write? I know it's old fashioned to read code these days.
To be clear, SotA models/harnesses are really good at making things that work one-shot, and their code golf and debugging game is insane.
When I go to implement, even with a good spec, I end up with code that's 80% of the way there in a fractal manner. The modular decomposition is 80% of the way to good code. The function decomposition is 80% of the way there. The computation structure and variable naming within functions is 80% of the way there. I can walk the AI through it and address each level of issues, but it's tedious as hell, and not clearly faster than doing it myself in some cases.
This is with omp/claude/codex, with Fable 5.1/Opus 5.5/6 Astra. The Chinese models do better at staying coherent, but they're a little less smart IME. I've tried many permutations of "fable planning, opus sub-agent implementation"; "pass a branch back and forth between opus and astra, as reviewers and feedback implementers". I haven't tried the "software factory with an architect and 2 juniors" thing. I haven't tried AGENTS.md beyond the /i-have-adhd, iso-24495 (lol at the Opus 5 induced PTSD), "No cleft constructions or Latinate absolutes" and general project orientation. Model character seems to change too fast to make agents worth it, and I see stuff about skills and excessive AGENTS reducing model capabilities.
actually Matt mentions the same problems as Scott Aronson. If an LLM works on a codebase with lots of tiny modules, or a code base that has ten data layers added by previous runs, then poor LLM will make bad decisions.
Now what happens to mathematics, if prior art gets filled up with spaghetti code like proofs that no one can verify? Maybe there is some hallucination somewhere in the middle that has been left unchallenged by the other agents? Nobody can tell.
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.
Math is, of course, an ideal field for AI to work in. However, I find this quote revealing:
"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."
This makes one wonder how many of the proofs are actually valid, and how many are just impenetrable hallucinations.
> Math is, of course, an ideal field for AI to work in.
I find it interesting that you state this as obvious. When ChatGPT was introduced four years ago, the argument was that language was fuzzy and probabilistic and that LLMs would thereby never work in math.
It's not hard to tie a deterministic process to any stream of output. The thing "hand waves" a proof and some other process exhaustively searches for either lean operations or some other math-manipulator operations that may create a real proof. The processes go back and forth. A proof is a verifiable string at the end of the day and this is better-than-pure-brute-strategy search strategy guided by heuristics from math literature.
I think someone asked here a bit ago whether other search strategies using OpenAI levels of compute have been tried and think the answer is no (even 3 hours of present GPU computer I think is vast compared to anything available 10 years ago, say).
The proofs could be optimized for human consumption.
OpenAI could optimize each proof for fewest theorems, low branching, or short dependency chains.
OpenAI could have an adversarial network enforce that proofs look human.
OpenAI could spend a couple more machine hours per paper with the prompt "edit for clarity".
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.
None of the soundness bugs found in lean so far could have been plausibly exploited by accident. For example you might have to set up weird recursive types that never come up in ordinary mathematics. Of course, past performance is not indicative of future results.
On the other hand, if a proof exploits a bug in Lean then it's ~trivial to prove a contradiction, so you can just check all the proof steps to see if it also allows doing that.
Similarly if the formal Lean problem statements (human generated) are correct translations into Lean (and the original NL statements are sound, which one would hope after decades), and no Lean bugs are abused by the proof (as defined above), then the proof is valid.
The NL/Lean discrepancies are super annoying and will make human analysis hard and fraught, but as many posters have found out the models themselves will gladly pick apart the NL-Lean translation for errors, and so my guess is that finding the discrepancies will not take too long. OpenAI really should have done a dynamic workflow over every lemma and step to ensure pointwise accuracy in the translation.
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.
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.
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.
Eh I don't think the content is out of line with what an 8yo understands. Mine tells stories about bombs flying in the Russia/Ukraine war, which required a lesson about tact because some of his classmates are both Russian and Ukrainian refugees and this subject may be a little close to home (literally, his best friend's grandma lives in Ukraine right now).
You're probably still stuck with old beliefs from when all available AI checkers were snake oil. Pangram is, to my knowledge, the only AI checker today that actually passes accuracy assessments. See e.g. on their own site: https://www.pangram.com/blog/third-party-pangram-evals
(Incidentally, if you asked me a few years ago I'd have predicted that reliable AI detection was impossible. I still suspect that it's impossible in general and that Pangram would break as soon as companies figure out how to make LLMs stop all sounding the same - but until then, it'll work fine.)
> 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?
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.
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)
By "it" people in such cases are usually referring to the NL proof. It seems nobody has any hope of undersanding AI-generated Lean code any more, if only due to volume.
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.
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.
> 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.
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.
The others are not in any way compelled to do that deep analysis and understanding. They're doing it because they find real value in the material produced.
If the problem statements in lean are correctly formalized, I think we can be pretty confident the proofs are correct, which does not mean their natural language counterparts are faithful representation of the proofs though.
I'm however puzzled by the number of proof claims without lean proofs. How does OpenAI have confidence in those, especially if, as noted in the blog posts, the papers are very hard to read?
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.
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.
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
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.
Getting recognition is a fundamental human motivation to work on hard problems. Being first is a really important part of getting recognition, measuring by the last few hundred years.
You might as well say "the obsession with sex has always held humanity back". Maybe, but it's complicated...
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.
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.
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> apparently they tried the model on about 8,000 problems. So, right now it “merely” solves ~5% of the longstanding open mathematical problems that it’s asked about, the problems that whole communities have spent years on, after a single 3-hour attempt on them.
Back-of-the-envelope cost calculation:
3 hours of GPT-Pro compute per attempt / 5% success rate → 60 hours of compute per solved open math problem
At GPT-6 Astra API long-context pricing ($75/M output tokens), assuming 50 reasoning tokens/s, that's roughly $810 per solution:
60 h × 3,600 s × 50 tokens/s × $75 / 1,000,000 = $810
Of course, actual token usage is unknown. But even if off by 10x, $8100 is cheap for solving a longstanding mathematical problem.
“On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems.”
That leaves a lot of ambiguity. Is “result” exclusively “positive result shown here” or “all results”?
Since proving infeasibility is a valuable result, the unpublished problems must not have reached a valuable end state. Thus it’s an operator decision of when to turn the machine off and try another problem. I’d thought they had a three hour time box on this, but apparently not.
Sure yeah, today it's only a 5% solution (to the hardest problems), and we know it will top out somewhere.
But the cost reduction is just too good to ignore.
Terry Tao was saying that we need more mathematicians [0]. My paraphrase of his article is that we will need them to check the AIs, essentially (I missed things didn't I?).
But with a near 10,000X reduction in cost [1] today, it's really hard to argue for that when budgets are continuously tight.
[0] https://terrytao.wordpress.com/2026/09/24/were-gonna-need-a-...
[1] assume ~$150k/year for a mathematician of the caliber that can do this work. 2x that for overhead like healthcare, 401k, etc. Do this for 35 years of work = $10,500,000. Assume about 1 result of this caliber per mathematician. So ~$10M per result. OpenAi is saying they can do it for ~$1k, a 10^4 X decrease.
Isn't that more likely the easiest problems in the set? I suppose the hardest problems compared to those that have been solved.
Otherwise your points stand.
Sure, but we are approaching a time at which we will have to answer a "why" here. I.e. anything beyond checking and validation becomes of mostly artistic value at some point. And there is nothing wrong with that.
An early (easy to understand) proof that fits the bill for me is proof that the sqrt(2) is irrational: a very natural thing to think about —- for a unit square, what is the measure of the diagonal? And people were killed in ancient Greece over the proof, but it’s comprehensible with basic algebra.
The problem is getting the funding orgs to care.
And then you're off an order of magnitude of the efficacy of OpenAI. We don't know how much work of mathematicians is behind all these results, what's the real cost (which includes those problems they couldn't solve), etc.
But a soulless admin is not.
And they care only about the budget.
And the real danger here is that these large groups of blame avoidance mechanisms take at best half a look at the cost of an AI and the cost of a pre tenure track new assistant professor (or whatever), and they decide, oh maybe next year we can hire someone.
And then they do that again next year.
And then the next
https://www.merriam-webster.com/wordplay/bury-the-lede-versu...
In actuality both "lede" and "lead" are correct. "Lede" is mostly an American and recent respelling of the word "lead" and "lead" still remains the predominantly used term (while "lede" is predominantly an American journalistic spelling).
I heard they used lede to distinguish from (hot) lead. But since linotype has been dead for 50 years this is probably moot now.
How much are you willing to bet on this?
https://polymarket.com/event/openai-retracts-navier-stokes-s...
https://polymarket.com/event/openai-retracts-navier-stokes-s...
i hope to hear all about your riches in 2027
> I am certain their Navier-Stokes proof is incorrect.
i at no point said anything so stupid so i'm not betting anything.
> I am certain their Navier-Stokes proof is incorrect.
Many discoveries were made in search of something else.
I'll leave you with this:
"When I woke up just after dawn on September 28, 1928, I certainly didn't plan to revolutionize all medicine by discovering the world's first antibiotic, or bacteria killer. But I suppose that was exactly what I did."
What might we miss out on?
> 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.
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.
- Feynman
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.
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.
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.
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.)
> 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.”
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.
Obviously they stole them from the Proof Fairy.
It's grapes so sour they could etch metal.
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.
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?
Why should that make a material difference to the IPO? Because of the vibes, and investors are indeed all about the vibes.
The second paragraph is specific to OpenAIs behavior.
Note that the cool thing about rationality is that it does not depend on transitivity. If what companies do to attract investors works, then it is in fact rational of them, regardless of the rationally of investors.
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.
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?
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.
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.
I would argue that "good at math" was a short hand for "good at X" because mathematicians were historically good at engaging with very complex ideas, distilling them and coming up with precise and concise answers they could validate by themselves.
Given that AI solutions are described as "psychedelical" and they rely on the outside source to validate the result, I don't think the same logic could apply to them.
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)
>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.
Someone at OpenAI, please, work on this.
There is ongoing AI assisted work on this, and precisely the problematic gap in the theory (around Corollary 3.12).
As of now, a required step appears to definitely be missing, but it's not clear if this is a genuine / unrepairable defect.
https://zen.ac.jp/news/zmcpostevent0331e
https://github.com/lana-agents/iut
https://github.com/LANA-Project/genl
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.
That's because there's a bunch of people who work on that stuff. Just because web developers don't care about it doesn't mean it doesn't exist. Utter magical thinking.
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.””
https://en.wikipedia.org/wiki/Nothing-up-my-sleeve_number
?
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.
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.
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
Who do we demand this from? The AI companies? Or the mathematicians who are worried they will have nothing left to do?
As the old saying: great claims require great evidence.
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.)
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.
Maybe what we need is an “AGI Academy” that helps humans climb that same ladder and level up alongside AI.
(in other words - we should not give up Yet)
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.
There are many problems that aren't "interesting" in a mathematical sense but can still have valuable proofs. The entire field of formal methods in software development is mostly about this kind of problem. If I want to be sure that no input can cause out-of-bounds memory access, or that my superoptimizer found the lowest possible cycle count, I don't care if it's elegant, I just need some machine-readable proof I can run through my trusted verifier.
If this is true, then what's really the point of math? A lot of math is actually useful. A proof regarding cryptography, for example, would have practical application even if not understandable by humans.
Given the LLMs are struggling to explain themselves clearly, but also that these explanations cleared up somewhat by having an expert using one to query some of this research, but also the LLMs can solve problems faster than humans can read the proofs, it is possible for this category to have both examples of things that can be rendered in a human-comprehensible form, and also examples where it is not.
As an analogy: any human can check any single arithmetical calculation from a computer, that's not even particularly difficult. But a Raspberry Pi Zero can do those arithmetical calculations so fast that even if literally every single human was trained to do this at the level of the current world record holder, humanity as a whole could not keep up.
We have such tools for a large number of events that we're incapable of witnessing or understanding in "raw form".
We can't possibly read through thousands of raw records of recordings of individual's heights and make sense of it. But we can use Excel to calculate the average in seconds. We can chart the results and the image is perfectly comprehensible. We can trust the average calculation and the image because we trust the mechanism for transforming the data. We have what I would call a trusted path of provenance. We don't need to nor do we want to read the raw data.
Now we need things like this of the 2nd order. We need mechanisms of transforming trusted paths of provenance into things we can look at and easily verify.
We can do formal verification of code that would be hard to do by hand. We need formal verification tools for formally verifying formal verification tools. Eventually we will need multiple layers of this. As long as the chain and reasoning is intact, we should be alright.
It's kind of like with a horse. A horse is much stronger than us, but we can control it by pulling on two strategically connected pieces of rope.
There are mappings between complex numbers and 2D matrices allowing problems to be solved in either domain.
There's research into The Langlands Program looking to connect number theory and harmonic analysis.
There's research into Category Theory looking to define core concepts and relate them do different fields so that results in one field can be applied to another due to equivalence.
> I suspect that's possible without tripping over the halting problem. (But I can't prove it.)
I doubt that it is. For the language of proofs to be powerful enough to be able to express an arbitrary proof it would have be Turing Complete. Proving that a proof is the smallest proof of a given concept (i.e. there is no smaller human understandable proof) would then be proving the minimality of a program in a Turing Complete language.
This is basically how all AI approaches appear to work. They solve the problem you give them (in many cases) but in a very over-complicated way.
It's like they can't step back from the problem and realise that they need to simplify to make it work better. Nope, just keep hammering more code/proofs against the problem and eventually you'll hit the goal.
RL has a lot to answer for, I guess.
There is nothing wrong with that approach if it accomplishes the goal faster, i.e. you can work at the speed of an AI.
Sure, and god knows I've done the same myself when under time pressure (so similar to an AI in RL training, just get the goal).
The part that's lacking (that humans are much better at right now) is the meta-cognition around this works, but it's far more complicated than it needs to be.
Potentially because LLMs have larger "working memory" than humans, they can avoid this for a while, but once it blows up the context they fail in much worse ways than a human would.
Hmmm, this feels like a good metaphor, I need to write about this at some point.
There’s not really a clear distinction between these things, in my opinion. Problem solving (and intelligence?) is a mix of search and compression. We like solutions that are elegant (high compression, simple search). But often what appears elegant to some is harder to appreciate for those without the same background knowledge or even the same amount of mental bandwidth (if you’ve ever worked with someone simply much, much smarter than you, you may intuit this!).
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.
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."
If a journal receives a paper that is unreadable, it is outright rejected with the comment that it should be made readable, regardless the results in that paper. What is the point of having results if you are unable to convince your audience of those results? You could just as well just sit on them and never communicate them. What is harmful about this process?
Which is why it is obvious post-publication peer review is the only sane way to handle this. Formal peer review is not even remotely up to the task here.
And yes, I am arguing that they should not be released if they cannot be verified within a reasonable amount of time.
Also, how will this impact PhD student who might need this research to get their career started. How will it affect the inflow of junior researchers?
By the way, your tone is very unpleasant. Go outside and touch grass, or have a wank for all I care. You are insufferable.
Maybe there’s some variance or it depends on the reader. That said, like with a lot of other complaints about AI: human papers can be poorly written and poorly explained too. That’s always been the case. And sometimes a proof is just complicated and hard to expose nicely.
That's what a lot of us do nowadays, but instead of maths, it's code that looks sensible on the surface, but when you try to understand it's some kind of "alien" logic , names don't really make sense, etc.
Makes you wonder if AI can build upon such proofs.
If the AI cannot create proper abstractions, then how can it build a tower of abstractions?
Also, AI has limited context. At a certain point proofs may become so complicated that an AI cannot keep most of it in memory and will not reuse it for new proofs.
so then, if you don't understand the proof, how do you know it's a proof?
Welcome to lots of (most?) code PR's in the last year. Though for software at the PR level it has gotten better with the latest models.
And you're right, it's not really a theme around here, but there aren't that many mathematicians around HN. Check one of the maths forums and it'll be a common point of complaint.
When all you value is being the first, you have not time producing useful papers.
I think OpenAI wants the outputs to be unreadable. If outputs "too advanced for human comprehension" become the norm, then every interaction requires tokens. If someone can buy a day pass, get what they need, and leave, then there's no recurring revenue stream.
I have read many research papers in computer science, and the code there is written in math symbols... and my sentiment aligns exactly with yours
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.
To be clear, SotA models/harnesses are really good at making things that work one-shot, and their code golf and debugging game is insane.
When I go to implement, even with a good spec, I end up with code that's 80% of the way there in a fractal manner. The modular decomposition is 80% of the way to good code. The function decomposition is 80% of the way there. The computation structure and variable naming within functions is 80% of the way there. I can walk the AI through it and address each level of issues, but it's tedious as hell, and not clearly faster than doing it myself in some cases.
This is with omp/claude/codex, with Fable 5.1/Opus 5.5/6 Astra. The Chinese models do better at staying coherent, but they're a little less smart IME. I've tried many permutations of "fable planning, opus sub-agent implementation"; "pass a branch back and forth between opus and astra, as reviewers and feedback implementers". I haven't tried the "software factory with an architect and 2 juniors" thing. I haven't tried AGENTS.md beyond the /i-have-adhd, iso-24495 (lol at the Opus 5 induced PTSD), "No cleft constructions or Latinate absolutes" and general project orientation. Model character seems to change too fast to make agents worth it, and I see stuff about skills and excessive AGENTS reducing model capabilities.
Am I holding it wrong?
Matt Pocock says that he knows how, but it comes with a lot of context fiddling.
https://www.youtube.com/watch?v=v4F1gFy-hqg
"Software Fundamentals Matter More Than Ever"
This lecture is very impressive. However I haven't managed to reach AI enlightenment, yet.
Now what happens to mathematics, if prior art gets filled up with spaghetti code like proofs that no one can verify? Maybe there is some hallucination somewhere in the middle that has been left unchallenged by the other agents? Nobody can tell.
Highly recommend reading it. Very prescient for something written 26 years ago.
https://gwern.net/doc/fiction/science-fiction/2000-chiang.pd...
I also recommend "Exhalation", though that has nothing to do with AI.
https://web.archive.org/web/20111121100139/http://www.fantas...
https://en.wikipedia.org/wiki/Division_by_Zero_(short_story)
"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."
This makes one wonder how many of the proofs are actually valid, and how many are just impenetrable hallucinations.
I find it interesting that you state this as obvious. When ChatGPT was introduced four years ago, the argument was that language was fuzzy and probabilistic and that LLMs would thereby never work in math.
I think someone asked here a bit ago whether other search strategies using OpenAI levels of compute have been tried and think the answer is no (even 3 hours of present GPU computer I think is vast compared to anything available 10 years ago, say).
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)
Why would it know it found a bug?
Similarly if the formal Lean problem statements (human generated) are correct translations into Lean (and the original NL statements are sound, which one would hope after decades), and no Lean bugs are abused by the proof (as defined above), then the proof is valid.
The NL/Lean discrepancies are super annoying and will make human analysis hard and fraught, but as many posters have found out the models themselves will gladly pick apart the NL-Lean translation for errors, and so my guess is that finding the discrepancies will not take too long. OpenAI really should have done a dynamic workflow over every lemma and step to ensure pointwise accuracy in the translation.
> 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.
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.
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.
Reminds me of the memes with the little girl making astute comments about the patriarchy to her father.
(Incidentally, if you asked me a few years ago I'd have predicted that reliable AI detection was impossible. I still suspect that it's impossible in general and that Pangram would break as soon as companies figure out how to make LLMs stop all sounding the same - but until then, it'll work fine.)
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?
> 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.
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
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.
In other words, they save effort by wasting the effort of others.
I'm however puzzled by the number of proof claims without lean proofs. How does OpenAI have confidence in those, especially if, as noted in the blog posts, the papers are very hard to read?
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 might as well say "the obsession with sex has always held humanity back". Maybe, but it's complicated...
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.
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.