AHM Statement on OpenAI's October 6 Release of Mathematical Documents
https://news.ycombinator.com/item?id=50000421 / https://news.ycombinator.com/item?id=49999159
Mathematics is an academy, and academies are human assemblages for producing truth (and the tools therein); they will stop producing if we forget to repair and refine them. It's just undeniable in the abstract.
(Sorry for the length, cut it as much as I could; mod(s) remove if you'd like. Talking to myself in the shadow of giants is how I'm coping with the ennui, I think.) That said, four philosophy nits on paradigms, scope, motivation, and pride:
1. Paradigms | The 'Math 1.0' rhetoric is undeniably powerful, but it makes it seem like he's unaware of his standpoint[1] by lumping all of "traditional mathematics" together. At the very least we've gone through four methodological revolutions in math, each one changing how the field is done on a fundamental level: ??? => Euclidean Certainty => Aristotlean Computation (~800s) => ~Newtonian Calculation (1600s) => ~Gaussian Systems (~1850s), and perhaps one in the 20th c. I lack the expertise to even gesture at. We also have clear analogues from parts of the other two acadamies in the 20th century alone: physics becoming an arcane, inelegant group effort in the ~1920s, and mainstream philosophy adopting a cloud of Kiki ideas vaguely revolving around Wittgeinstein & Chomsky in the ~1960s.
I totally understand this being distressing, especially when it's happening quickly. They, too, had people decrying the future of their fields. But we wouldn't obviously wouldn't change it, in hindsight; much of modern physics would be completely intractable without those strange, boring, unnerving methods, for example. More than intractable: unthinkable.
2. Scope | This all seems overly focused on Autumn 2026. Most egregiously, this is all built on the premise that RSI never happens, and we never acheive ASI. If we do, mathematics is almost assuredly A) the first academy to be completely outmoded, and B) the least of our problems. I cut a long thing about the caveats and effects here; at this point... if you know, you know.
3. Motivation | Ultimately this thread is focusing on human motivation throughout, a fact that would be more forgivable if acknowledged as an intentional tradeoff. Speculating that it'll be harder to have interest in math is just not worth withholding truth; for one thing, knowing that computers could solve a problem but it's banned to try would ruin motivation anyway, and worse. It's up to us to be motivated, and if I know us, we'll have no problem doing so as long as there's any utility there at all.
In more stark terms: trading progress in the fundamental academy for the sake of its current methods of recruitment and motivation seems like something posterity will almost definitely frown upon.
4. Pride | This is the common thread that weaves through all three preceeding points, I think, and is even stated in pretty blatant terms (that's Tao -- always a clear writer!):
...promising open directions are now being withheld from the public in fear that this will cause their own research to be "scooped"... "Math 1.0" placed a premium on being the first to solve an open problem, even if the solution was not initially well understood.
Doing science (mathematic or otherwise) in this competitive, individualistic way is just clearly counterintuitive to me, even if it weren't a recent development. Imagine taking it to its conclusion and applying some kind of patent system to mathematics -- or even worse, copyright to combinations of symbols! Perhaps more riches would motivate some mathematicians, but it would so obviously eat away at the democratic principles that have brought us unimaginably far over the past 406 years.
---
TL;DR: What worked well for the past ~century is not particularly relevant, and I think Tao is missing the forest here, despite one of the best sylvan trailblazers around. On his side practically-speaking for heuristic and contingent reasons, regardless.
Of course it's good to have the discussion... So maybe, we listen to the nay-sayers, but defer judgement on the matter... That's wisdom.
Edit, to be clear, I consider Tao to be the wisdom provider, not an early nay-sayer!
The less said about Gary Marcus the better.
This is a technology not like prior technologies. Are we okay if the technology discourages a whole generation of Mathematicians? If the technology leads to 10x fewer mathematicians -- what impact does that have on the field? These are the questions Tao is asking. And I don't think he himself claims to have all the answers, he just doesn't wanna see the math _community_ die.
Almost certainly not. It's just going to jump to a higher level of abstraction.
There seems to be more interest in hitting some arbitrary benchmark (we proved X unsolved problems) than in genuinely contributing to mathematics. But what else is to be expected? It's become a maniacal race with too much money. Too much effort is being invested in proving that the exponential curve is still holding.
One is that AI will continue hallucinating in a manner that is not easy to verify, second is that AI will not be enhanced to produced more simplified amd robust outputs, and third that a human will be required to do that. What humans in the loop are doing now is verify the process, propose shortcuts and add legitimacy, through the verification process, if that ends up being succesful its highly likely a lot less mathematicians will be required in the future.
The conclusion that this is not productive focuses on the mathematicians, but it is very productive in terms of hundreds of proofs being produced that had previously consumed uncountable hours of the brightest minds. Unless it ends up being the greatest hallucination ever ofcourse
Hold up, that's an even bigger assumption in the opposite direction, and I don't see anything to support it.
At least in terms LLMs getting all the "AI" hype these days, there is no structural/mathematical reason to believe they won't continue to have the same problem they've always had of generating plausible text over rational text, and I don't think anybody even has a clear idea how it could eventually be accomplished.
I've seen "then the magic singularity occurs and somehow it solves the problem for itself", but I would classify that more as mysticism than engineering.
There is literally not a single shred of evidence to indicate either of your supposed eventualities. The core technology of an LLM is sampling from a distribution so there is literally no way to make it deterministically robust (only probabilistically).
And it's not like this is something where we're loaned some top math genius for a limited amount of time and we have to make the most of it. Rather, this is a new high water mark. The accessibility of the results is no longer scarce. The scarcity has shifted, and that's where the focus of the math ecosystem should shift as well. And it doesn't help for a frontier community to saturate and take over messaging pipelines that were typically managed by the math ecosystem. It's not about "stay in your lane" but rather "we need coherence and be careful not to break the system."
Just two cents from someone who could screw up basic cashier math on any given day.
This is a transitive period. In a few years, verification and exchange between model instances will happen faster than humans can follow. Human input will be an ethical question, and not a productivity one, because it will be the bottleneck in any science.
> genuinely contributing to mathematics
What's the difference between the two? Proofs are no longer the goalpost?
OAI obviously have a fiscal incentive here, but to presume a year from now we won't see improvements and more succinct work on the results coming from models?
If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....
Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.
End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.
OP didn’t suggest that.
The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.
Those few thousand mathematicians are now getting a taste of their own medicine. After all, it was people with extraordinary mathematical talent who developed machine learning and large language models, leaving hundreds of millions of people who earn their living through speaking, writing, or teaching worried about their future job prospects.
Still, I believe almost everyone will be fine. Perhaps AI will also prove good at coming up with new conjectures, and some mathematicians may shift towards applied mathematics or other sciences.
Im not sure how that will work, but im convinced the current paradigm of just pushing agents into codebases for not much reason other than you can is going to make building software incredibly boring and push creative people away from the field and stagnate progress.
My prediction is software gets boring and building hardware projects will be the new frontier for creative engineers looking to push computing further. Which is probably a good thing.
This really expresses the heartburn you see across all fields, not exclusive to careerism. I certainly have friends in decomp and fan translation spaces that have been demotivated by the current rash of efforts happening there.
The rush to be "first" has always been over-celebrated, but it would be nice to believe there's a way to get beyond that thinking.
This I don't understand, seems like an obvious thing to automate, especially for byte-matching?
If I understand Tao correctly, he's saying that's going to have to be the focus going forward. I just default to thinking the models are going to be much better than us at that, too.
I wonder if this is true. The code produced by these models are not really getting any more elegant over time. On contrary, the models seem to be getting worse, often proposing really baroque architectures. You can use RL to optimize for correctness, optimizing the vibe seems much more difficult.
Which should improve collaboration, Research and Clarity.
I would really appreciate if we come up with protocols for using ai in STEM field's it might be award at first but we could regulate properly using this method.
Which is a simple measurable goal requiring little bureaucracy. The mythical "all you need is a pen and paper and a lifetime of dedication"
> "Math 2.0" will need to ... value mathematical progress more holistically
which is directionally the opposite
> community building ... AI can contribute positively
what is this belief based on? Any other communities can illustrate?
Sounds like Terence Tao would have said the same about Ramanujan who basically just "solved" problems without much explanation / reasoning / communication other than it just arrived from god.
In the case of Ramanujan, others took on the responsibility of socializing and community building knowing that he wouldn't do it himself. Why can't the same approach happen here?
There will be people who want to just "solve" math problems now that they have a new tool that lets them express themselves this way. Maybe the don't want to participate in the broader math community, etc. Why discourage them, or add friction / a barrier to them participating in their own way? Why not take on the burden of socializing, making sense of, and community building yourself?
There may be valid reasons here I'm missing, but to me this seems a bit like wanting others to approach a field in a particular way even though the field can support many ways.
1. Some present a unified line that the whole point of their craft is the human experience, and that automation is the antithesis of that. Marathon runners don't care that a car can get there faster, poets don't care that Poem Bot 2000 can write poems too. I think this is smart if you can credibly take this position. The difficulty is mostly convincing the buy side, which requires being very outspoken about your views.
2. Some appear to be undecided, with one faction taking the pro-human stance and another rushing to accelerate things with AI. A good example of this is mathematics, and I really wonder where they end up in the long haul. They have a very good claim on #1, because mathematics is pretty close to an art form and is robustly insulated from the pressures of the marketplace. But they can also choose option #3, below.
3. Some crafts prioritize results above all else, practitioners either rushing to extract as much money as possible before it all collapses, or believing that they can out-prompt everyone else forever and that their prompting skills are indispensable to their employers in the long haul. That's software engineering. I think this is going to be interesting to watch.
I'd end up with the buckets
a) Fully human
b) Hybrid human-AI
c) Fully AI
It’s more likely that instead of spending a 100K/year direct grant on two PhD students, PIs will hire 1 and have the student spend 50K on AI.
AI only take us as far as our imagination thinks to ask it. This can be exhilarating when new models drop every month and we can continually reach a new threshold, basically for free. But it is only a one time gain and ultimately short-sighted. Where I find continuous value is using LLMs to help my understanding, full stop.
I use LLMs all day long as a SWE and I have tried many approaches, but the most satisfying and consistent approach is to lean heavily into understanding a problem space and a solution space. Yes, it whips up architecture and code, but I spend most of my time peppering it with questions about the design and how it handles certain situations, what about this edge case and that security concern and this future product need. I have it write a report breaking down the feature and how it integrates with existing code and if the report is too confusing I have it simplify either the report or the code until it makes sense to me, sometimes scaling back the work to a more manageable state. I do all of this before I look at any of the code it writes.
The difference from this approach is that I am not suffering reading through 3000 lines of AI slop but I am reviewing a PR that I fully understand. I can eyeball it quickly for anything that doesn't fit my mental model and dig deeper or quickly revise it. Only after I am happy with the bones do I consider the meat and skin of the code.
What I find most concerning is how frontier AI companies all seem to have this Math 1.0 perspective that they only want to type "solve Riemann" into the chat box and have the magic to happen. It is the same problem Google ran into, where a simple, no thinking solution serves most of the people best and most profitably, so you fully ignore or remove everything else (boolean operators, exact phrase search, verticals, filters, infinite pages of results, "nothing found" if there isn't, etc.) But that choice leads to the situation Google is in now, scrambling to stay relevant. In a different world, Google would have continuously augmented their search capabilities and eventually built a smooth, guidable AI interface.
But no, we must only have an input box and a Go button.
Everything looks like a nail when you build hammers, sell hammers, have infinite hammers to play with however you like and your company mission is to build a hammer starship to explore the hammerverse, whether or not that is even possible.
Everything else is secondary (or the last of our priorities) and would be better automated?
This is a hard pill to swallow
The job of professional mathematician might be the first to be completely eliminated by LLMs, save for those who can make money from a patron. I am hoping they are able to figure something out to save their profession, as other professions could use it as a blueprint as AI comes for them next.
Strong disagree.
Do you work in a math adjacent field? I do and I find having a mathematician around invaluable.
It's like a non-software person writing software. Yes, using a LLM will get you to a solution that works. But just talking with a software engineer will make the quality of that solution enormously better.
I find the same with math - I can get something to work using an LLM, but if I speak to a mathematician they'll say some magic words to try and I put that in the LLM and it is "oh yes this is a much better solution".
This is very different work to generating proofs though. Its things like "I'm trying to get my confidence intervals to properly deal with census like sampling but at small sample sizes" (yes, I know stats not pure math but still..)
You may say that currently people pay for math salaries even though they dont understand the math or the economic outcome. But there are now mathematitians trying to convince not to fund.
In this new reality, you will get "mathematitians" trying to convince that only AI maths matter. And on the other side someone speaking about "understanding", "taste", "community". And the people deciding to fund dont have the skills to differentiate. So the will fund the AI boosters with a higher probability.
Thats how the job of professional mathematitian dissapears. By being replaced by something that on the surface looks similar, but its just an ugly copy.
On the other hand, it puts a premium on resources. AI is not cheap for mathematicians. Folks are fancy universities in rich countries with forward thinking ministries of science will have an advantage over the rest.
What is clearly in immediate crisis is the traditional model of doctoral education. Most of the problems that were "given" to ordinary doctoral students are solvable (quickly) even by something like Claude pro. Mathematicians need to adopt training models more like what is done in experimental and laboratory sciences - collaborative and structured.
Where Tao is wrong is in regards to exposition. AI already writes better lecture notes, problems, and exercises for mid level undergrad math classes than do most of my colleagues. It's exposition is generally well structured and clear and it can adjust level on request quite well. It writes research better than most professional mathematicians too.
There will be no gap in understanding. Now there is because the models are discovering things at the edge of what they can do and so suck at explaining it. There's nothing particularly special about a newly solved problem in terms of learning it.
If we accept AI can explain all of existing math nicely, why shouldn't it be able to explain new proofs?
So much in AI is dependent on which of these two outcomes occur.
Take a look at this interview from two days ago: https://m.youtube.com/watch?v=oQypVVv1u1o
The interviewee is worried about the future of math research. He is not strictly worried about being replaced, instead he is worried that he will no longer be able to launder math-as-a-hobby through math-as-something-useful as is the case today. He lays out very clearly that grant proposals claim to have useful outcomes while the proposers know those claims are nonsense.
Business as usual in math, and frankly in all the other sciences, is to do research that furthers the researchers careers or personal interests and pretend that it’s somehow useful. This would be absolutely fine if it were privately funded, but it’s not, this is public money.
In every other endeavour, lying in order to get money is considered fraud.
We have collectively wasted a huge amount of taxpayer money and human time, entire careers, on things not likely to ever matter to anyone.
I look forward to science becoming automated so that we can have real progress instead of the current broken system.
Edit: If you'd like a better medicine based one, look to radiology, where AI is an omnipresent tool but claims that radiologists are no longer needed, based on an ignorant view that a radiologist's job is "classify images according to what diseases they indicate" have only contributed to a crippling worldwide shortage of radiologists.
Up until now the prize in pure (as opposed to applied) mathematics was the _understanding_ and the machine can't do that for you. What does it mean if we get "super powered alien maths" but humans can't do it? It's like inter univeral teichmuller theory but imagine if Mochizuki was right and it came with a lean proof?
In my view, over the last century, math has turned into an intellectual analogue of extreme bodybuilding competitions. A navel gazing runaway optimization in making useless stuff just to demonstrate cleverness. That's fine, why not. But society has no obligation to fund that, just as it doesn't fund other extreme hobbies. Ideally if we ever get something like UBI, math can be still their hobby.
Maybe AI will take over some roles of doctors, but that's independent of curing diseases. Think about diseases that have a cure - do people with those diseases not go see a doctor?
There is a world where we get to the edge of AI capabilities, and we build on top of that. As humans have always done with every new technology.
There is another more pessimistic view where LLMs just replace every human capability, and our economic overlords dont need us for anything and we just eat the small pieces of bread that are left.
This comes down to the fact of:
is human existence/intelligence just the simbolic representations we make in our brain? Or are they just a tool?
I tend to think of Godels incompleteness theorem as a proof that on the limit LLMs are useless. The real question for me is at what point approaching this limit becomes an issue, and if it has any practical consequences.
We don't build on top of that. No need for us to. AI does. That's sort of the whole point of this endeavor is it not? Humans need not apply.
In my experience, every new model release allows me to go further, although every time i see every time the limitations, and i identify where i add value. And this value gets bigger every time.
But on the other hand, every model release reduces the amount of people that are able to value this "added value", because it requires more skills.
So we have the paradox that the added value i can bring on top gets bigger and bigger, but the perception of the economic value for the majority of the population gets smaller.
I believe in the future, we're going to see a similar shift in "programmer" - instead of a human programming the computer, you'll give the ai an idea and it will spit out a program.
And just like how automating the act of computation revolutionized what we could compute, automating the act of writing code will change the act of programming - hopefully, as you described, allowing us to do things that simply were not practical in the past.
People wrote many books about software engineering, all from valuable experience from buildng expensive software systems. But in the age of AI, is there still anything learnable from generated code?
Personally I always ask AI to summarize its findings and lessons in a .md file. And I always learn something from it.
But could AI utilize some new patterns and paradigms I wasn't aware of? Very likely. Because we only learn from our personal grave mistakes, a summary from others gets neglected and forgotten
The S fell short in actual reality for the most part, as it was merely a hiring requirement. A hiring requirement that didn't even make sense, because the skillset of academic CS only marginally overlaps with the skillset one wants to hire for.
Material engineering at least for the most part has actual real-world applications where one can push humanity further. CS (as practiced, not necessarily the idea of real CS but the CS we got due to it being used as a hiring filter) for the most part is just self-referential spinning with mostly unclear results.
There is real impressive work being done in that field, of course, but I'd argue that the majority of it over the last decade or so at least was just performative nonsense.
Maybe by again allocating new resources to other fields, what hides under the label CS can become more pure actual CS again. I think that would also be a much less miserable experience for everyone involved.
Sort of. An elegant proof is useful beyond what it shows. It hints at new mathematics, and can prompt discovery in applied fields. I don’t think I’ve heard of elegant code leading to discovery on its own.
Usually it's the opposite. "That's in prod? And it works? It shouldn't work and I thought it was doing something else. Why does it work?"
I’ve not heard of it either, but code is an abstraction of math, so I don’t see why this couldn’t theoretically happen.
Anecdotally, I’ve started spending time advancing my math skills beyond the early college level I stopped at and I’ve frequently found I already know concepts of more advanced math - I just didn’t know what they were called or how to apply them to an equation on paper, but I’ve been using them for years and intrinsically grasped the underlying academics.
> things that I do and that my colleagues do, this kind of like curiosity-driven, you know, applied math, computational physics type research has always been justified by, I would argue, intentionally blurring the line between what I would call, you know, science as product versus science as process
> science as product is very kind of clear-cut. It’s, you know, things like, you know, cure cancer, solve nuclear fusion, generate, you know, clean energy.
> And then there’s science as process, which is kind of the curiosity-driven stuff about, you know, like, “I want to understand protein folding,” or, “I want to understand, you know, turbulence,” or, “I want to understand quantum gravity,” or something. And broadly speaking, we have tended to justify the latter by kind of laundering it through the former
And the examples he gives are actually the more defensible ones, he talks about a friend of his working on some abstract algebra under the false guise of cryptography research later.
And it’s not just him saying it, this is simply true. He should be lauded for admitting it publicly, this is the only way any progress is made. At least, it used to be. Now it’ll be AI instead.
People thought, back in the 17 century, that imaginary were useless (except as a trick for some calculations). Turns out the research into these numbers back then is amazingly useful today, 300 years later, in electronics and such.
Publicly funded maths research should continue, even if some taxpayers feel it's a waste of money.
However, with AI it actually may become so cheap that the scattergun random approach becomes more viable rather than less. It’s when human time and resources are scarce that you need to optimise. The hobbyist approach may therefore ironically continue, but without the hobbyists.
Note, I think the debate is mainly over what research should be funded, not whether any research should be funded.
Thats great! So in other words it should be perfectly fine for AI to solve all of the supposedly useless math problems so that it might possibly be useful later. No need to worry about the mathematicians hobbies here.
How much..?
Do you really think a society with zero human mathematicians or scientists will outperform one with both human and AI ones?
So as measured by utility, I absolutely believe we don’t need humans doing science into the future. I’m sure people will continue doing it, but not for utility, for enjoyment - as a hobby. Probably we’ll all end up as dedicated hobbyists.
Without human understanding you also might literally have no words for the thing you would otherwise want to ask for.
I think it'll be wildy useful but I also suspect human competence will still matter.
In fact, the taxpayer should be funding fundamental research because it's so hard to justify profit from it - but that funding would benefit all in the long future.
So leaving the applied research that have commercial value be funded by private, commercial interests, would make more sense.
of course not.
Hobbies can still be done even if AI gets it done more quickly. Just like today, where knives are made much faster/cheaper in presses and CNC machines, vs a blacksmith hammering. But still, there are hobby blacksmiths.
Thats one of the timeless human debates.
We are now living in the perfect combo of low morality and general human automation. So i expect the next few decades dominated by people who think (and have a "proof") that doing something without an expected economic gain is useless.
It’s fine that OpenAI posted their findings. It’s not fair to claim these problems have been solved. Not until someone can understand and verify the proof and then communicate the core, novel methodological element to someone else.
Your phrasing is illuminating that perhaps they aren't engaged in the creation, understanding, or integration of these proofs by humanity; they just have them. For them, this is a slidedeck they can pass to investors, creditors, the marketing department. Something they can add to the employee onboarding pamphlet.
What should they do? Hyperbolic maybe, but perhaps engage with humanity.
This isn't only bad for Math — it's bad for English too.
'Proof' is going to become the 2026 Most Misapplied Word of the Year.
The fact that they did not do so can only mean that either (1) their agents currently lack the capability to do it, or (2) OpenAI are completely indifferent and do not care in the slightest if the proofs are understood or not.
The concern is being raised without evidence, because the evidence points to the gap simply being frontier models have just started to be able to get a raw proof out. Why, given existing progress, should we expect them to be unable to distill insights from those proofs?
Certainly this even more likely doesn't matter at all for applications: if I can send a radio signal further because my AIs design it a certain way, that's an unambiguous result. Which is really the next step here: turn a proof into a "mechanical" application.