An OpenAI model has disproved a central conjecture in discrete geometry (opens in new tab)

I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year.

I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.) But stuff like this is exactly the type of innovation LLMs are great at, and that doesn't discount the need for humans to also be good at "recombinant" innovation. We still seem to be able to do a lot that they cannot in terms of synthesizing new ideas.

I like to think of it as:

Imagine every bit of human knowledge as a discrete point within some large high dimensional space of knowledge. You can draw a big convex hull around every single point of human knowledge in a space. A LLM, being trained within this convex hull, can interpolate between any set of existing discrete points in this hull to arrive at a point which is new, but still inside of the hull. Then there are points completely outside of the hull; whether or not LLMs can reach these is IMO up for debate.

Reaching new points inside of the hull is still really useful! Many new discoveries and proofs are these new points inside of the hull; arguable _most_ useful new discoveries and proofs are these. They're things that we may not have found before, but you can arrive at by using what we already have as starting points. Many math proofs and Nobel Prize winning discoveries are these types of points. Many haven't been found yet simply because nobody has put the time or effort towards finding them; LLMs can potentially speed this up a lot.

Then there are the points completely outside of hull, which cannot be reached by extrapolation/interpolation from existing points and require genuine novel leaps. I think some candidate examples for these types of points are like, making the leap from Newtonian physics to general relativity. Demis Hassabis had a whole point about training an AI with a physics knowledge cutoff date before 1915, then showing it the orbit of Mercury and seeing if it can independently arrive at general relativity as an evaluation of whether or not something is AGI. I have my doubts that existing LLMs can make this type of leap. It’s also true that most _humans_ can’t make these leaps either; we call Einstein a genius because he alone made the leap to general relativity. But at least while most humans can’t make this type of leap, we have existence proofs that every once in a while one can; this remains to be seen with AI.

> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.

Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibniz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science.

For example, many belive that to sovle the Riemann Hypothesis, we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.

As others have pointed out, both can be true:

* LLMs do just interpolate their training data, BUT-

* That can still yield useful "discoveries" in certain fields, absent the discovery of new mechanics that exist outside said training data

In the case of mathematics, LLMs are essentially just brute-forcing the glorified calculators they run on with pseudo-random data regurgitated along probabilities; in that regard, mathematics is a perfect field for them to be wielded against in solving problems!

As for organic chemistry, or biology, or any of the numerous fields where brand new discoveries continue happening and where mathematics alone does not guarantee predicted results (again, because we do not know what we do not know), LLMs are far less useful for new discoveries so much as eliminating potential combinations of existing data or surfacing overlooked ones for study. These aren't "new" discoveries so much as data humans missed for one reason or another - quack scientists, buried papers, or just sheer data volume overwhelming a limited populace of expertise.

For further evidence that math alone (and thus LLMs) don't produce guaranteed results for an experiment, go talk to physicists. They've been mathematically proving stuff for decades that they cannot demonstrably and repeatedly prove physically, and it's a real problem for continued advancement of the field.

You have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand the full vastness of what would objectively be "math" and that it is possible that our perceived math is terribly wrong and a subset of a greater math. Whether that greater math has the same seemingly closed system properties is not something that can be known.

Recombining existing material is exactly right, and in this case LLMs were uniquely positioned to make the connection quicker than any group of humans.

The proof relies on extremely deep algebraic number theory machinery applied to a combinatorial geometry problem.

Two humans expert enough in either of those totally separate domains would have to spend a LONG time teaching each other what they know before they would be able to come together on this solution.

I feel this is the case whenever I "problem solve". I'm not really being creative, I'm pruning a graph of a conceptual space that already exists. The more possibilities I see, the easier it is to run more towards an optimal route between the nodes, but I didn't "create" those nodes or edges, they are just causal inevitabilities.

You can build a census of all gen-2, degree-2 formal products of polynomial like terms. If you insist on instituting your own rewrite rules and identity tables, it is straightforward — maybe an 15 minutes of compute time — to perform a complete census of all of the algebraic structures that naturally emerge. Every even vaguely studied algebra that fits in the space is covered by the census (you've got to pick a broad enough set of rewrite- and identity- operations). There's even a couple of "unstudied" objects (just 2 of the billion or so objects); for instance:

    (uv)(vu) = (uu)(vv)

Shows up as a primitive structure, quite often.

If you switch to degree-3 or generator-3 then the coverage is, essentially, empty: mathematics has analyzed only a few of the hundreds (thousands? it's hard to enumerate) naturally occurring algebraic structures in that census.

It’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

I'm just hoping we're almost past this phase of needing to assess LLM capabilities against an arbitrary one dimensional yard stick labeled 'Not Human' on end and "Beyond Human' on the other.

It's irrelevant and pointless. Irrelevant not just in the sense that when Deep Blue finally beat Kasparov, it didn't change anything but in the sense some animals and machines have always been 'better' on some dimensions than humans. And it's pointless because there's never been just one yardstick and even if there was it's not one dimensional or even linear. Everyone has their own yardstick and the end points on each change over time.

Don't assume I'm handing "the win" to the AI supremacists either. LLMs can be very useful tools and will continue to dramatically improve but they'll never surpass humans on ALL the dimensions that some humans think are crucial. The supremacists are doomed to eternal frustration because there won't ever be a definitive list of quantifiable metrics, a metaphorical line in the sand, that an AI just has to jump over to finally be universally accepted as superior to humans in all ways that matter. That will never happen because what 'matters' is subjective.

There was a project long long ago where every piece of knowledge known was cross pollinated with every other piece of knowledge, creating a new and unique piece of knowledge, and it was intended to use that machine to invalidate the patent process - obviously everything had therefore been invented.

But that's not how new frontiers are conquered - there's a great deal of existing knowledge that is leveraged upon to get us into a position where we think we can succeed, yes, but there's also the recognition that there is knowledge we don't yet have that needs to be acquired in order for us to truly succeed.

THAT is where we (as humans) have excelled - we've taken natural processes, discovered their attributes and properties, and then understood how they can be applied to other domains.

Take fire, for example, it was in nature for billions of years before we as a species understood that it needed air, fuel, and heat in order for it to exist at all, and we then leveraged that knowledge into controlling fire - creating, growing, reducing, destroying it.

LLMs have ZERO ability (at this moment) to interact with, and discover on their own, those facts, nor does it appear to know how to leverage them.

edit: I am going to go further

We have only in the last couple of hundred years realised how to see things that are smaller than what our eye's can naturally see - we've used "glass" to see bacteria, and spores, and we've realised that we can use electrons to see even smaller

We're also realising that MUCH smaller things exist - atoms, and things that compose atoms, and things that compose things that compose atoms

That much is derived from previous knowledge

What isn't, and it's what LLMs cannot create - is tools by which we can detect or see these incredible small things

I'm not sure how feasible this is, but I love the thought experiment of limiting a training set to a certain time period, then seeing how much hinting it takes for the model to discover things we already know.

E.g. training on physics knowledge prior to 1915, then attempting to get from classical mechanics to general relativity.

This is a good point, and there’s some deep philosophical questions there about the extent to which mathematics is invented or discovered. I personally hedge: it’s a bit of both.

That said. I think it’s worth saying that “LLMs just interpolate their training data” is usually framed as a rhetorical statement motivated by emotion and the speaker’s hostility to LLMs. What they usually mean is some stronger version, which is “LLMs are just stochastically spouting stuff from their training data without having any internal model of concepts or meaning or logic.” I think that idea was already refuted by LLMs getting quite good at mathematics about a year ago (Gold on the IMO), combined with the mechanistic interpretatabilty research that was actually able to point to small sections of the network that model higher concepts, counting, etc. LLMs actually proving and disproving novel mathematical results is just the final nail in the coffin. At this point I’m not even sure how to engage with people who still deny all this. The debate has moved on and it’s not even interesting anymore.

So yes, I agree with you, and I’m even happy to say that what I say and do in life myself is in some broad sense and interpolation of the sum of my experiences and my genetic legacy. What else would it be? Creativity is maybe just fortunate remixing of existing ideas and experiences and skills with a bit of randomness and good luck thrown in (“Great artists steal”, and all that.) But that’s not usually what people mean when they say similar-sounding things about LLMs.

If anything, this is more illustration of how llms are not useful to us...

They will do their own thing, don't need us. In fact, we will be in the way...

We can choose to study them and their output, but they don't make us better mathematicians...

To every proof, there is a corresponding program. This makes proofs expressible in a language made up of finite grammatical rules and terminal symbols. Knowledge accessible by proof is thus always a form of interpolating data whether made up by an AI model or a human mathematician. The people dismissing AI because of claims that it can only interpolate data don't have a good understanding of what it means to know something. Now of course not everything can be known via proof but for the sorts of things that we want to know via a computer this is a fine compromise.

"LLMs just interpolate their training data"

Cracks me up.

What exactly do we think that human brains do?

There is a creational aspect in math - definitions and rules are created.

It is the old discovery vs invention in mathematics.

Or of you prefer philosophy: Parmenides (nothing changes) vs Heraclitus (you cannot bath twice in the same river aka everything changes all the time).

Postmodernism also claimed that everything has been done already. IMO these 2 are points of view that one can adopt, not truths based on fact. So the distinction is a matter of taste or perspective, not of truth, IMO.

We know that LLMS "just interpolate" their training data. Maybe there's a mystery about what "just interpolate" means when the data set gets enormous. But we know what LLMs do.

Pretty much everything that appears novel in life is derivative of other works or concepts.

You can watch a rock roll down a hill and derive the concept for the wheel.

Seems pretty self evident to me

> held that mathematical truths don’t report new facts about the world

I'm not as familiar with the early work, but later Wittgenstein held this belief too.

Side note: don't underestimate how much literal, physical time and energy "unfold" implies. Proofs occur on physical substrates.

This is the second reference to Wittgenstein I’ve seen today in totally different contexts. Reminded me how much I vibe with his Tractatus.

I think someone should be talking to Godel.

this is an excellent point, new ground isn't necessarily novel, it's a rearrangement of existing pieces

Post hoc ergo propter hoc

I think you are conflating composition and prediction. LLMs don't compose higher abstractions from the "axioms, symbols and rules", they simply predict the next token, like a really large spinning wheel.

> A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation.

However, this was not verified in Lean. This was purely plain language in and out. I think, in many ways, this is a quite exciting demonstration of exactly the opposite of the point you're making. Verification comes in when you want to offload checking proofs to computers as well. As it stands, this proof was hand-verified by a group of mathematicians in the field.

> manage a McDonald's

Dystopia vibes from the fictional "Manna" management system [0] used at a hamburger franchise, which involved a lot of "reverse centaur" automation.

> At any given moment Manna had a list of things that it needed to do. There were orders coming in from the cash registers, so Manna directed employees to prepare those meals. There were also toilets to be scrubbed on a regular basis, floors to mop, tables to wipe, sidewalks to sweep, buns to defrost, inventory to rotate, windows to wash and so on. Manna kept track of the hundreds of tasks that needed to get done, and assigned each task to an employee one at a time. [...]

> At the end of the shift Manna always said the same thing. “You are done for today. Thank you for your help.” Then you took off your headset and put it back on the rack to recharge. The first few minutes off the headset were always disorienting — there had been this voice in your head telling you exactly what to do in minute detail for six or eight hours. You had to turn your brain back on to get out of the restaurant.

[0] https://en.wikipedia.org/wiki/Manna_(novel)

I disagree. It will be able to perform work deserving if a fields medal before it is capable of running a McDonalds. I think it will be running a McDonalds well before either of those things happen, and a fields medal long after both have happened.

The proof is not written in Lean, though. It’s written in English and requires validation by human experts to confirm that it’s not gibberish.

Stockfish did not teach itself to play chess. You are probably thinking of Leela Chess Zero - an open re-implementation of AlphaZero - both were given nothing but the rules of chess and a board and played millions of games against themselves until they were the strongest engine available at the time.

Stockfish's neural net evaluation model was trained on millions of its positions with its own original algorithmic evaluation function (entirely developed by humans) and search tree. The result was a much smaller model than Leela's that requires little computation (not even a GPU), paired with its already extremely efficient search/pruning algorithms that made it stronger than Leela in competitive play. Leela's evaluation function is much stronger (at one ply it has an ELO of around 2300, Stockfish is probably closer to 1800), but it requires vastly more resources and those are always bounded in a match.

Humans haven't learned as much new information about chess from Stockfish as we have from Leela.

Lee Sedol, Move 37 https://www.reddit.com/r/singularity/comments/1l0z5yk/the_mo... Edit: I wasn't necessarily disagreeing. But on second thought the chessboard in this math analogy is being built, not just played in. This Hardy quote comes to mind https://www.goodreads.com/quotes/902543-it-proof-by-contradi...

The issue with this prediction is the gulf between problem-solving using known tools, versus creating new concepts for problems where existing tools aren't enough.

All AI proofs so far, including this one, are using existing tools in new ways, rather than inventing new tools. This is not surprising if you know how these models are trained. These existing tools are in distribution. New tools are not.

Problems worth of a Fields Medal likely require new tools to be invented. Thus it is not clear whether progress within the confines of the current paradigm is enough.

We could get this weird spiky situation where the AI is insanely superhuman at all problem solving, but completely incapable of coming up with a single new tool. It discovers everything there is to discover, subject to existing axioms and concepts.

Timothy Gowers gives some commentary on this in the attached PDF.

> A difficult part was constructing a chess board on which to play math

We have that chess board for quite a while now, over 40 years. And no, there is nothing special about Lean here, it is just herd mentality. Also, we don't know how much training with Lean helped this particular model.

I think your analogy is good but I don't believe modern LLMs use Lean or any lean-like structure in their proofs. At least recent open source ones like DeepSeek can do advanced math without it (maybe the most cutting edge ones are doing it I can't say).

AI is already too old for that.

Managing a McDonalds is a question of integration and modalities at this point. I don't think anyone still doubts that these models lack the reasoning capability or world knowledge needed for the job. So it's less of a fundamental technical problem and more of a process engineering issue.

> we'll see more specialized math AI resembling StockFish soon

Heuristically weighted directed graphs? Wow amazing I'm sure nobody has done that before.

Hey ChatGPT, if a person spills hot McCoffee on themselves who is at fault?

I dunno. Is AI less than forty years old?

the only thing keeping the mcdonalds from happening will be political, likewise the same with fields medal

We're automating art and science so that we can flip burgers. This future sucks.

our local AI models are already capable of running McDonalds.

Nonsense. Have you been watching the figure live stream? Or the Unitree video from yesterday with real time novel action generation? We’re less than a year away. If you can cook a burger, assemble a sandwich and clean up surfaces you’re all of the way there.

This is the main thing that I keep harping about that human knowledge is too vast today for a person or even a group of people and llm will change that many discoveries that require serendipity in the past will be more likely than ever

Terence Tao gave a recent talk about this issue (lack of attribution). He called it the decoupling of implicit and explicit goals. AI is only good at solving the explicit goals for now, and humans don't have the bandwidth or the institutions to know how to integrate AI into the field.

https://youtu.be/Uc2zt198U_U?si=OkwO3xT8-zhSABwh

I am also using these models to accelerate scientific discovery. Yes, they are making all the difference at the frontier. At least, they feel they are. The messy thing is that we still need to communicate with each other and that's not getting dramatically faster or better. As you note the models need to be built so they do more work to participate in our communication economy. Or we will do so much, alone, to get nowhere fast because so much of our behavior is still bound up in old (good, tested, but clunky) ways of building shared knowledge.

> I dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.

This is a very important point, especially when the output is from a non-deterministic random walk with some unknown probability distribution.

Yes, I share your optimism overall, although I think it is raising a question of what the future role of the researcher is (much like the current debate on developer roles).

I attended a conference on AI for maths and open science a few weeks ago, and was struck by just how many examples of AI-supported solutions there already are. Virtually every speaker had an example of either their own use of (often the frontier) AI models in solving a problem that was previously too hard (for various definitions of hard).

I wrote up a few notes [1], and most of the speaker videos are available via the conference website [2].

[1] https://scholarlyfutures.substack.com/p/ai-and-the-practical...

[2] https://www.newton.ac.uk/event/ooew11/

> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.

AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of theorems proven, it’s how proving theorems affects the working operational models of mathematicians. Only a rare few new theorems in mathematics nowadays have direct real world applicability.

If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.

I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a proof or explore some problem) and sharing the resulting text with others, which of course anybody else could have found if they simply also knew the right point of the textbook.

To be blunt, this seems incredibly uninteresting to me. I enjoy learning mathematics, sure, but I just don't find much inherent meaning in reading a textbook or a paper. The meaning comes from the taking those ideas and applying them to my own problems, be it a direct proof of a conjecture or coming up with the right framework or tools for those conjectures. But, of course, in this future, those proofs and frameworks are already in the textbook. So what's the point? If someone cared about these answers in the first place, they probably could have found the right prompt to extract it from this phantom textbook anyways.

You could argue for there being work still like marginal improvements and applying the returned proof to other scenarios as happened in this case, but as above, what is really there to do if this is already in the phantom textbook somewhere and you just need to prompt better? The mathematicians in this case added to the exposition of the proof, but why wouldn't the phantom textbook already have good enough exposition in the first place?

I think my complete dismissal of the value of things like extending the proofs from an LLM or improving exposition is too strong -- there is value in both of them, and likely will always be -- but it would still represent a sharp change in what a mathematician does that I don't think I am excited for. I also don't think this phantom textbook is contained even in the weights of whatever internal model was used here just yet (especially since as some of the mathematicians in the article pointed out, a disproof here did not need to build any new grand theories), but it really does seem to me it eventually will be, and I can't help but find the crawl towards that point somewhat discouraging.

Can you describe what the reaction to these results has been like in your department? Obviously many people are excited, but what else? How do grad students feel about this? Are any professors getting worried about becoming obsolete?

Maybe I'm misunderstanding how these models work, but isn't it more the responsibility of the harness and its prompts rather than the model itself to make sure that a result is generated with explicit sources?

> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.

And by opening the door to LLM-generated results, you'll see greater and greater amounts without any hope of ever navigating this field again without machine help.

It's a little like a software project which more and more gets extended by a AI agents with less and less review by human software engineers and in the end the complexity and spaghetti design are so incomprehensible by humans that the maintenance requires an AI agent. The risk is that math as a whole (the field itself) will experience that effect.

I am curious if LLMs are better at some kinds of problems than others. IIRC this and another big recent one were cases of the LLM producing a counterexample to a conjecture.

Nice response to read

> But, for scientists, I find that these tools address the problem of the > exploding complexity barrier in the frontier.

They do the opposite by locking the results the produce within the slop presentation that needs more AI to comprehend.

> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.

Always, always always, the problem with research and development is leadership, not insufficient supportive technology. It is a political problem, there is absolutely, positively no shortage of technologies to support research. Your optimism is totally misplaced. The NSF funding cuts have negatively impacted math more than AI has benefitted it. And guess who supports the administration that cut NSF funding? The people who ousted the PhDs from OpenAI.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless.

Along with all the rest of what humans find meaningful and fulfilling.

What you describe here has always been true in all sciences, but also in medicine. But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers.

So the crossdomain pollination that used to exist in scientists is not only not encouraged. It's also actively punished by society.

Many breakthroughs come from taking an idea from one field and applying it somewhere else. But, almost every serious field is now so deep/complex/huge that humans rarely get the time, or even have enough practically useable memory, to understand and correlate multiple unrelated areas properly.

And this is where machines, such as these reasoning LLMs, can help. Because they can remember patterns across many domains and try absolutely bonker weird connections and ideas.

We, the humans still have to verify the work (at least as of now). But, the "maybe this tool, or idea, or trick, from that completely unrelated field applies here" reasoning/experimentation could become much easier.

I have always said this and will say it again: reasoning is just experimentation with a feedback loop and continuous refinement.

I’ve always been skeptical about the role of LLMs in mathematics, but this is the first time I’ve seen this argument, and I actually find it very compelling. Maybe LLMs will help us develop more horizontal understanding of the field.

I think you're on point, and you've explained it very well.

As we're becoming hyper specialised, they become an invaluable tool to merge the horizon in, so to speak.

One of the challenges I had in graduate mathematics was just trying to keep all the concepts in my brain. It doesn’t help that you end up with things like homomorphism and homeomorphism tangling one’s brain thanks to their superficial similarities. Heck, just keeping track of basic theorems and definitions is a challenge.

Yep. The thing is people (maybe because of our limited scope) just focus on the depth and not the breadth. Because this is a general purpose model - it also has PhD+ knowledge in Physics, Biology, History, etc.

I think we still don't really comprehend how much can be achieved by a single "mind" that has internalized so much knowledge from so many areas.

It’s as if the body of human knowledge is our I’ve mind. It used to be expensive to access that, but no more.

Cool thing is now when someone contributes something to the hive mind, it can instantly be applied to any other problem people are working on.

That's the whole point of LLM, connecting all the missing dots no single human could possible keep in working knowledge, even just for a subfield of mathematics alone. The era of polymaths is over for a reason, so we build a new one to tackle that. If LLMs can build on top of that once all remaining ones are found or if this stalls is yet to be proven, but humans stalled out there too.

There are so many research papers; just finding a solution to, say, a bio problem in a deep math paper would be a gold mine of opportunity. Very exciting times!

Check out Ashby's Law of Requisite Variety

To me, AI feels like the morbidity of Star Trek teleportation, where it's actually copying the person at to the other end and zapping the original one out of existence. The original human never benefits from the fast transportation.

Similarly, we're creating tools to improve knowledge, but we're progressively zapping the human out of the equation. Knowledge is created for something, but it's unclear if very soon humans will be able to understand it, or really benefit from it, except billionaires, etc.

It's too bad that we're not improving humans nearly as fast as we're replacing ourselves.

like the research team that rediscovered calculus for treating diabetics

As a mathematician, new, conceptual math is when I'll become interested in reading LLM output.

I appreciate very much the work done so far, but this sort of asymptotic/quantitative result didn't interest me much even when it was done by humans.

(This is not snobbery, just a personal preference.)

No it is not Leibniz/Euler/Galois. More like writing good papers that contribute to the broader understanding of a theory. I think if one evaluated a mathematicians research output and it consisted of mostly the kinds of problems AI has solved so far, it would give the impression that this person is somehow very good at picking accessible problems to target, but has not made a larger impact on the field.

What were you imagining when OpenAI said that their models would have "PhD-Level Intelligence"? Were you imagining that there were specific tasks they could do that were on par with what a human with a PhD could do? Because by that definition, many computer tools have "PhD-Level Intelligence". By that definition, Wolfram Alpha has "PhD-Level Intelligence".

What I assumed they were saying is that their LLMs would be as intelligent as a human with a PhD across all, or at least most, knowledge tasks, and they clearly are not.

My only complaint is the claims always start spreading 6-12 months before the delivery. A little patience goes a long way in what's possible with AI and we all just have to wait and see what parts actually grow this next cycle or not. Guessing at it based on trend lines only leads to people getting excited when it matches their particular guess and ignoring it when it doesn't.

PhDs used to mean publishing a novel mathematical result: when has that changed?

Not denying that these advances are impressive, but it is important to consider that this is a cherry-picked result. This doesn’t mean that AI can now be expected to do problems of similar or lower difficulty, but that it happened to work well on one problem. What you won’t see is how many others they had to try to get this result.

It would be more impressive if it wasn’t behind the closed doors of a rich company. For all we know, they could’ve paid some mathematicians to work on the problem and pretend that their results are from ChatGPT.

Yet it still codes like a junior developer that memorized all of stack overflow.

What's laughable is an OpenAI employee invented the term "PHD level intelligence" and you think that " PHD Level intelligence" is a real term that describes a real thing and you are repeating it here.

I still laugh.

And the goalposts will keep getting moved all the way to the singularity. And then those people will/would say "Oops. I was wrong."

large language models do not have pigeon-level intelligence. They can't even feed themselves.

Erdős problems form a substantial fraction of all mathematical problems that have been explicitly stated but not solved; are sufficiently famous that people care about them; and are sufficiently uninteresting that people have not spent that much effort trying to solve them.

Solving problems people have already stated is a niche activity in mathematical research. More often, people study something they find interesting, try to frame it in a way that can be solved with the tools they have, and then try to come up with a solution. And in the ideal case, both the framing and the solution will be interesting on their own.

Erdos problems are easier to state, thus they make a great benchmark for the first year of AI mathematics.

Afaik this is because there is a community and database around them.

It's a large set of problems that are both interesting and difficult, but not seen as foundational enough or important enough that they have already had sustained attention on them by mathematicians for decades or centuries, and so they might actually be solvable by an LLM.

As others have written, Erdős was a lifelong curator of mathematical problems, from high-school level problems to the types that will land you a Fields medal. Like the Collatz conjecture.

Most new math problems appear in other papers, doctoral dissertations, etc. Usually you'll find them in the "future work" / "future research" section.

So obviously in order to present and formalize these problems, you either need the author(s) to do it, or some reader. At this level of math, there are many extremely niche fields, where the papers might only be read by a small amount of people.

In short, it is a visibility problem.

But, I figure, there's some potential use in AI models to extract and present these problems, which would make them available to a larger audience.

That is exactly what Erdős did. His life revolved around math, and seeking mathematical questions.

They're just famous because Erdos was a great mathematician, kinda like the Hilbert problems a century earlier.

It's not just Erdos problems - https://news.ycombinator.com/item?id=48213189

I was promised a cure for cancer, but all I got was this disproof of an Erdos problem.

Erdos problems are well-posed for AI — elementary statements, exact counterexample targets, extensively catalogued. selection bias: these are exactly the problems AI can actually search

The models can't actually so good work on practical problems so openai tasks them on stuff nobody cares about

OpenAI specifically targeted Academia a lot and gave out a lot of free/unlimited usage to top academics and universities/researchers.

They also offer grants you can apply for as a researcher. I'm sure other labs may have this too but I believe OpenAI was first to this.

Hasn't AlphaFold been used to make real discoveries for a few years now?

I think the mathematicians on X are all using GPT 5.5 Pro

From my limited testing, Gemini can dig out hard to find information given you detail your prompt enough.

Given that Google is the "web indexing company", finding hard to find things is natural for their models, and this is the only way I need these models for.

If I can't find it for a week digging the internet, I give it a colossal prompt, and it digs out what I'm looking for.

OpenAI models seem to have been trained on a lot of auto-generated theorem proving data; GPT 5.5 is really good at writing Lean.

Gemini seems better trained for learning and I think Google has made a more deliberate effort to optimize for pedagoical best practices. (E.g. tutoring, formative feedback, cognitive load optimization)

As far as academic research is concerned (e.g. this threads topic), I can't say.

A simpler explanation is that more people are using ChatGPT

Yes, this and every internet forum will still be doing this two years hence. Your life will be better if you take to heart this famous passage from Nietzsche:

I do not want to wage war against what is ugly. I do not want to accuse; I do not even want to accuse those who accuse. Looking away shall be my only negation.

People are afraid for their livelihood. What do you expect?

I find it understandable, it is common to evaluate human intelligence vs AI as a zero-sum competition, because that is how employers typically understand it and LM providers market it. AI proving itself moves the needle in an uncomfortable direction for all of us without very robust job security.

> I wonder if we’ll still be doing this two years hence.

It is going to take some time for people to recognize that AI has a very different set of competencies that compliments human intelligence rather well. It is unlikely to eclipse human intelligence at scale, and the companies betting on that will fall behind. That is when the conversation will start to shift.

I think part of it is that one side throws rocks and so it never even matters was is in the article. It becomes a battle if the article is good or the article is shit.

Yes, I'm tired too. I want you have real discussions about these things. But the problem is everyone believes their reality is real and anyone's reality that disagrees is fake. It just escalates. I take long breaks from HN because I realize I just come to the forums and end up being angry. Why do we do this to ourselves? The reality is that at a core level we usually want the same things.

What I don’t understand is why people dismiss this kind of progress with false claims. Especially when discussing programming, people start to act irrational using arguments from back in 2022.

I think that you can easily address your concerns about this new technology (since we all are concerned about the future) but at the same time acknowledge how revolutionary it is.

We won’t be doing it in 2 years. By then my side will have won!

Lets just be real its because a lot of programmer's ego is built on intelligence/being a coding wizard and this threatens that ego

If suddenly anyone can code we're not that special anymore.

This is like saying a sculpture always existed, the sculptor just had to remove the superfluous material.

Or like a musical octave has only 12 semitones, so all music is just a selection from a finite set that already existed.

Sure the insane computation we're throwing at this changes our perspective, but still there is an important distinction.

Any design already exists as a possibility, so it could be said to be both invented and discovered, depending on how you look at it.

Regardless of which, both Newton and Leibniz imprint in their findings a 'voice' and understanding different from each other and that of an LLM (for now?)

One can argue that mathematical facts are discovered, but the tools that allow us to find, express them and prove them, are mostly invented. This goes up to the axioms, that we can deliberately choose and craft.

Math is an abstraction of reality, it had to be invented, so more inventions or discoveries could be made within it.

...long standing indeed. It can be traced back to Plato's works.

> Was the question asked by a mathematician?

As per the report, the prompt used to solve the problem is AI-written and the solution was initially graded by an AI grading pipeline. They don't say this explicitly, but it seems like OpenAI has an automatic pipeline where they prompt models for solutions to famous math problems (which wouldn't be unexpected given how flashy a solution to a famous math problem looks)

> Was the paper right from a get-go or was there someone who pointed out mistakes?

Also as per the report, the output of the model isn't really a "paper"; it's a very terse 2 page solution which is apparently correct. The paper was later written based on this solution to make it more presentable.

> How much attempts were made before solution was found?

Given that this appears to be from an automated pipeline, I would say that it had many attempts. But either way, the blogpost says that with enough test-time compute, the model finds this same solution 50% of the time.

[1] https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...

>I am cautious about AI "discoveries" after Mythos paper.

Can you be more specific? I'm still under the impression that Mythos was a huge deal:

https://xcancel.com/hlntnr/status/2052479493801975987

> I will eat my words if an AI oneshotted that one without any external help

like having a colleague peer review your paper, or bouncing ideas off a mentor before you write them down?

I agree there's a lot of AI marketing BS at the moment, but revising approaches based on feedback is a good thing.

I'm also wondering about the process. What was the prompt, what they fed into the model, what it was trained on, etc. The article reads like a marketing post.

Nevertheless new maths is exciting and might lead to what I find slightly more interesting - new physics.

No need to be cautious.

If a for-profit (because... you know, OpenAI isn't at all what it initially was) huge corporation (again, not a cute startup trying to help humanity) publish anything it's a piece of marketing. Every single word a corporation say is marketing.

So... that's also that, a piece of marketing to sell more of whatever their potential client can buy. It's not a piece of research. It's an ad. That's it.

> the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.

Isn't this just anthropocentrism? Why is understanding only valid if a human does it? Why is knowledge only for humans? If another species resolved the contradictions between gravity and quantum mechanics, does that not have meaning unless they explain it to us and we understand it?

There is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...

>>But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.

Future of code is pretty much a bunch of guys shepherding a bunch of agents to get them to your goal.

I don't see how math might not go that way as well.

It is not only unsurprising ; it was always expected. There is no difference between programs and proofs. They are the same thing

for now the LLMs will build off human understanding, eventually we will be left behind

This only a proof that a field with more connections is possible, not what it looks like.

I’m very out of my depth, but the structure of the proof seems to follow a pattern similar to a proof by contradiction. Where you’d say for example “assume for the sake of contradiction that the previously known limit is the highest possible” then prove that if that statement is true you get some impossible result.

They only proved that one exists; computing the actual construction is non-obvious (the naive way to construct it is computationally infeasible).

They have a "before" picture but not an "after"!

See https://news.ycombinator.com/item?id=48233034

Yeah, unfortunately, they just proved there existed a better solution, they didn't construct it.

(Though in some ways that's actually more impressive.)

Searching for a proof and disproof are sometimes not so different. In most cases, you nibble the borders to simplify the problem.

For example, to prove something is impossible let's say you first prove that there are only 5 families, and 4 of them are impossible. So now 80% of the problem is solved! :) If you are looking for counterexamples, the search is reduced 80% too. In both cases it may be useful

In counterexamples you can make guess and leaps and if it works it's fine. This is not possible for a proof.

On the other hand, once you have found a counterexample it's usual to hide the dead ends you discarded.

One of the mathematicians in the video describes the process as:

> the AI has been able to explore all these possibilities much more comprehensibly, and doing that it found a path, it found a way to the solution.

Finding a counterexample of a mathematical conjecture strikes me as not that different from finding a vulnerability in a complex codebase.

That's proof by contradiction: https://en.wikipedia.org/wiki/Reductio_ad_absurdum

> I think that's just a matter of having them able to work on longer and longer time horizons.

No this will never do the kind of math that humans did when coming up with complex numbers, or hell just regular numbers ex nihilo. No matter how long it's given to combine things in its training data.

That's here for anyone wondering - https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...

Today I generated the equivalent of two LOTR books just to fix three missing rows in my SQL models (and open a PR), so +1

I note that (though summarized), this is ~100k tokens. Anyone who routinely works with Codex (or any agentic harness really) can tell you how trivial it is to eat up 100k tokens doing complex work. I've personally had plenty of codex 5.5 xhigh sessions where just the pure chain of thought token count in a single turn exceeds 200k (and I assume doesn't go further only due to compaction meta-guidance; the harness will push the model to stay under 256k per turn/thinking block) .

I think the more interesting question is how many tokens were spent all told; the most interesting graph in the article imo is the success rate by log test-time compute: how many tokens are being spent on the right of the graph to hit a winning CoT/solution like this >50% of the time?

AFAIK, extremely long CoT is fairly routine for those math-finetuned private models. Also, is Mythos unusually verbose compared to Opus?

Exactly. Maybe OpenAI paid mathematicians to keep this discovery quiet, then added their proof to the training data, then manipulated a second team into prompting for this question such that the model could regurgitate the solution. This would plausibly explain why the model seems so capable at doing things like refuting fundamental theorems of mathematics while in things like competitive programming, biology, and physics it's merely only in the top 99.9%.

This type of discourse is just inane and more reflective of the author's sensibilities than anything it claims.

Congrats to the OpenAI team for one of the most significant breakthrough discoveries in AI history.

Seems like a very tin-foil-hat-take to me

> The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain original ideas.

Really? Any references to read more?

Any answer to this question must also consider the current cost/token and its downward trajectory as algo and hardware advances drive down costs.

Good q. You can see that tantalizing graph where compute is displayed on a log axis and pass@1 goes up to like 50/60%. I’d love to know just how much compute is encoded on that axis.

I guess you can get some estimate from the excerpted CoT, but that CoT might be backed by quite a lot of parallel compute.

I would like to know this as well, including how long was it working on this problem?

Why?

It's clearly not yet a tool that can deliver new math at a scale. I say this because otherwise, the headline would be that they proved / disproved a hundred conjectures, not one. This is what happened with Mythos. You want to be the AI company that "solved" math, just like Anthropic got the headlines for "solving" (or breaking?) security.

The fact they're announcing a single success story almost certainly means that they've thrown a lot of money at a lot of problems, had experts fine-tuning the prompts and verifying the results, and it came back with a single "hit". But that doesn't make the result less important. We now have a new "solver" for math that can solve at least some hard problems that weren't getting solved before.

Whether that spells the end of math as we know... I don't think so, but math is a bit weird. It's almost entirely non-commercial: it's practiced chiefly in the academia, subsidized from taxes or private endowments, and almost never meant to solve problems of obvious practical importance - so in that sense, it's closer to philosophy than, say, software engineering. No philosopher is seriously worried about LLMs taking philosopher jobs even though they a chatbot can write an essay, but mathematicians painted themselves into a different corner, I think.

Says in the papers. "...which was first mathematically generated in one shot by an internal model at OpenAI, and then expositionally refined through human interactions with Codex."

Doesn't really matter the prep-work, what they say is it's a one-shot result, achieved by AI. The blog doesn't claim it was done by a currently public Model.

shame we won’t see any of these medical breakthroughs when we all lose our jobs and thus our healthcare

Breakthroughs in pure mathematics aren't scientific though. They say us nothing about the world, and they are not useful.

By making it think for 100+ pages https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d... Regular ChatGPT users don’t have a way to do that, this is something they do internally only.

edit: apparently that’s only the _condensed summary_ of the chain of thought.

Maybe you need to phrase it better. Like with a more specific direction of thinking.

apparently the proof is not constructive in the sense of not giving an easy to compute recipe for generating a set of points that you can plot on a 2d plane

Indeed, it's a pity. While many advanced math problems are highly abstract or convoluted to explain to a layman audience, this one in particular is about points in a 2D plane and distances. A drawing would have been nice.

One of the authors said in a reddit comment (and I hope I am summarizing accurately) that it's impossible to show a diagram as the smallest instance of the technique gives like 10^1000000 points.

Erdos, or the model?

The last two are straightforward. The proof relies on a result called the Golod-Shafarevich theorem that gives a criterion for a group to be infinite. Golod and Shafarevich proved this a long time ago (1964). Moreover, if you look at how Golod and Shafarevich used this criterion, it's the same way it's used in the proof: They apply it to some Galois groups that appear in number theory, prove these are infinite in certain cases, and deduce that there exists an infinite tower of number fields with some surprising properties.

Much more recently (2021), Hajir, Maire, and Ramakrishna figured out how to apply the Golod-Shafarevich theorem to a slightly different Galois group to produce an infinite tower of number fields with some even more surprising properties. This is used in the new proof. It requires very slightly modifying the construction of Hajir, Maire, and Ramakrishna to produce the fields needed in this proof, but the explanation of how to do this takes only a paragraph in the human-written summary. (The explanation is more laborious in the original AI writeup).

The relation to Ellenberg-Venkatesh is more indirect. This is where "in retrospect" comes in because this work was not cited in the original AI proof. This has to do with the next step of the proof, after you construct the number field, you need to find many elements of this field with the same norm to produce many vectors of the same length. To do this, the proof uses a pigeonhole argument which uses small split primes of the field (constructed via Hajir, Maire, and Ramakrishna's argument) to construct many ideals. By the pigeonhole principle, you can guarantee two ideals lie in the same class. When two ideals lie in the same class, you get an element of the field. You can rig things so these elements all have the same norm. Ellenberg and Venkatesh had an argument which also used the pigeonhole prnciple to guarantee two ideals lie in the same class to produce elements of the field. They were working on a different problem so their argument was slightly different, but similar.

Unfortunately Gowers has taken Tao's lead on this one.

Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs

It is disheartening to see him jump into this GenAI puffery.

I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work with these labs.

My guess is Gowers wants in on that action, or his University does.

Either way, it makes me sad. If its self motivated... even sadder.

That will require AGI.

Yeah, I remember it was one of my biggest disappointments with LLMs.

For those of us who care about the answers to these questions, rather than who gets credit for doing it, we will welcome any faster means of solving these problems.