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An OpenAI model has disproved a central conjecture in discrete geometry
https://x.com/wtgowers/status/2057175727271800912 https://x.com/wtgowers/status/2057175727271800912, https://xcancel.com/wtgowers/status/2057175727271800912 https://xcancel.com/wtgowers/status/2057175727271800912
- aussieguy1234 5mo agoSo we've got the proof, what are the practical applications of this?
- hajile 5mo agoIt proves the limit doesn't exist, but doesn't seem to show how to find one of these more optimized limits. Given that nobody has ever brute-forced a counter-example, it may be that calculating these solutions is P=NP hard which would mean sticking with the status quo unless we find a good algorithm to calculate it on quantum computers and can actually get quantum computers large enough to work on the problem. It's interesting as a math problem and test of AI, but not much else IMO.
- num42 5mo agoI am not surprised! The birth of computer science was rooted in the desire to automate mathematical discovery and proof writing.
- fromMars 5mo agoSeems rather depressing to me but maybe I am a Luddite.
- JacobAsmuth 5mo agoExactly. I would rather we let these discoveries stay hidden for a while longer such that human ingenuity may untangle them from the coils of reality. A machine? A mechanical man? Deins to produce something as pure as mathematics without the divine fervor of the ineffable spirit of Man?! It's just not what God wants.
- h4h4itsfunny 5mo ago[dead]
- pickleRick243 5mo agoHuman ingenuity is untangling perhaps the deepest question of all- what is the essence of Reason and the intellect that so privileges man? I don't know if it's what God wants, but it's certainly getting close to some existentially fundamental questions. While many seem to be anxious or pessimistic about the future of intellectual/artistic pursuits (understandable although I disagree), I do find the utter lack of curiosity or interest in the incredible machinery that is causing all the fuss to be striking.
- doright 5mo agoOne has to be in a secure enough position in life to be able to manifest such curiosity to begin with. On the contrary many people feel threatened.
- unmole 5mo agoI can't tell if this is satire.
- csallen 5mo agoI had the exact same thought. It depends entirely on what voice you read it in, I suppose.
- pickleRick243 5mo agoHaha, now I do think it is more likely satire. I don't think HN has many people who would post like this sincerely.
- aurareturn 5mo agoOne thing seems for certain is that OpenAI models hold a distinct lead in academics over Anthropic and Google models. For those in academics, is OpenAI the vendor of choice?
- FloorEgg 5mo agoGemini 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.
- aurareturn 5mo agoYes, I meant academic research.
- cute_boi 5mo agoGemini is like someone with short-term memory loss; after the first response, it forgets everything. That being said, I have checked multiple model and gemini can sometime give accurate answer.
- FloorEgg 5mo agoGemini is a series with a lot of individual models. What you are describing doesn't match my experience at all with Gemini 3 or 3.1, especially the pro version.
- snaking0776 5mo agoAgreed I usually use Gemini for explaining concepts and ChatGPT for getting things done on research projects.
- astrange 5mo agoGemini the chatbot has a very strange personality that intensely overindexes on your user profile and absolutely loves insane mixed metaphors. Its explanations are quite good but they're also hard to understand because it keeps trying to relate everything back to programming metaphors or what it thinks it knows about the streets in the neighborhood I live in.
- dist-epoch 5mo ago[flagged]
- ekjhgkejhgk 5mo agoYour understanding of this technology is out of date, and getting out of date faster as time goes by.
- throwaw12 5mo agoThanks for giving me a hope that there is a still place for human knowledge workers.
- embedding-shape 5mo ago> It's not a new result, LLMs can't produce new results Who else disproved this longstanding conjecture before the model did so, since obviously it must have been in the training data since before?
- empath75 5mo agoImportant note: this was not done with a special mathematics harness or specialized workflow.
- dwroberts 5mo agoHow/why should we know this, it does not explain the process in the text?
- horhay 5mo agoThis part of the announcement holds no value besides maybe taking a shot at the Deepmind Co-Mathematician paper. Nearly every mathematical success they've achieved around the GPT 5.2 generation has been done with general (and even public) models. Their last bountied problem solved was done with 5.4 Pro, also a general model.
- Jeff_Brown 5mo agoCan anyone find (or draw) a picture of the construction?
- pradn 5mo agoThey have a "before" picture but not an "after"!
- deleted 5mo ago[deleted]
- ninjha 5mo agoThey only proved that one exists; computing the actual construction is non-obvious (the naive way to construct it is computationally infeasible).
- deleted 5mo ago[deleted]
- gibspaulding 5mo agoThis 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.
- paulddraper 5mo agoYeah, unfortunately, they just proved there existed a better solution, they didn't construct it. (Though in some ways that's actually more impressive.)
- krackers 5mo agoSee https://news.ycombinator.com/item?id=48233034 https://news.ycombinator.com/item?id=48233034
- alansaber 5mo agoAI isn't going to supercharge science but I wouldn't be as dismissive as other posters here.
- OldGreenYodaGPT 5mo agoIsn’t that a joke? It already has supercharged science
- datsci_est_2015 5mo agoWhere are the second order effects of this supercharging of science? Or has it not been enough time for those to propagate?
- ks2048 5mo agoSince "supercharged science" is as ill-defined as AGI, ASI, etc., people will be able to debate it endlessly for no reason.
- vatsachak 5mo agoI absolutely believe that AI will supercharge science. I do not believe it will replace humans.
- seydor 5mo agoreplace, no. obsolete, yes
- dvfjsdhgfv 5mo agolol (That's the first time I used that expression on HN.)
- unsupp0rted 5mo agoI absolutely believe that AI will supercharge science and replace humans. Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together
- vatsachak 5mo agoAs I have stated before, AI will win a fields medal before it can manage a McDonald's A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation. LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.
- soupspaces 5mo agoLee Sedol, Move 37 https://www.reddit.com/r/singularity/comments/1l0z5yk/the_moment_everything_changed_humans_reacting_to/ 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-contradiction-is-a-far-finer-gambit-than https://www.goodreads.com/quotes/902543-it-proof-by-contradi...
- vatsachak 5mo agoMy claim is that we haven't even witnessed the move 37 of math yet. I am claiming that math AI is going to get even better
- Lerc 5mo agoI 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.
- edbaskerville 5mo agoI just visited a McDonald's for the first time in a while. The self-order kiosk UI is quite bad. I think this is evidence in favor of the idea that an incompetent AI will soon be incompetently running a McDonald's.
- deleted 5mo ago[deleted]
- OldGreenYodaGPT 5mo ago[dead]
- yusufozkan 5mo ago"The proof came from a general-purpose reasoning model, not a system built specifically to solve math problems or this problem in particular, and represents an important milestone for the math and AI communities."
- Kwantuum 5mo agoI trust openAI's marketing team 100%
- krackers 5mo agoIt seems plausible given that people have been using off the shelf 5.5 xhigh to decent success with some erdos problems. There is likely still some scaffolding around it though (like parallel sampling or separate verifier step) since it's not clear if you can just "one shot" problems like this.
- seydor 5mo agoall reasoning is .. well problem reasoning. restricting black-box AIs to specific human-defined domains because we believe that's better is such a human-ist thing to do.
- horhay 5mo agoThe accomplishment is cool. But all Erdos problems and other complicated mathematical problems they solved were accomplished with general-purpose models too. In fact for some of those problems, including bountied ones, they were public models. So I don't get saying this
- endymi0n 5mo agoTo paraphrase Gwynne Shotwell: “Not too bad for just a large Markov chain, eh?”
- rhubarbtree 5mo agoErdos, or the model?
- phkahler 5mo agoI would have thought a triangular grid works better than a grid of squares. You get ~3n links vs ~2n for the square grid. Curious what the AI came up with.
- comboy 5mo agoYes, not providing visualization of the solution seems criminal.
- kmeisthax 5mo agoKnowing OpenAI, the solution's probably being withheld as a trade secret, lest it fall victim to distillation attacks (i.e. exactly the same shit they did to the open Internet).
- red_admiral 5mo agoUnless it's a non-constructive proof.
- kilotaras 5mo agoBoth 3n and 2n are linear, the broken conjecture is that you can't do better than linear.
- bustermellotron 5mo agoThe grid of squares actually gets > Cn for any C. (More in fact… C can grow like n^a/loglog(n).) The AI proved > n^{1 + b} for some small b > 0, which a human (Will Sawin) has now proved can be about b = 0.014. The grid can be rescaled so the edges are not necessarily length 1, but other pairs will have length 1; that is necessary to get more than 2n unit distances.
- dadrian 5mo agoWhile the result is impressive, this blog post is extremely disappointing. - It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive) - It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it. - It's description of the new proof just cites some terms of art with no effort made to actually explain the result. If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.
- Al-Khwarizmi 5mo agoIndeed, 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.
- deleted 5mo ago[deleted]
- changoplatanero 5mo agoapparently 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
- solomatov 5mo agoHow central is it in the discrete geometry? Could anyone with the knowledge in the field reply?
- energy123 5mo agoThere's pages of comments from like 8 mathematicians in the attached pdf
- sigmar 5mo agoThe blog post links a pdf that OpenAI put together of nine mathematicians that commented on the proof. Each is quite brief and written in accessible terms (or more accessible terms, at least). https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
- horhay 5mo agoThere may be years of investigation as to how far you can generalize these methods. As to how central it is, it's a longstanding problem that Erdos loves to cite for that branch of math. The thing is is that it seems a lot of the effort through the years (which is unquantifiable in scale as to how much time was spent and how many people focused their entire worklives on it if any) has gone for trying to look for the proof, and the search for the disproof seems minimal.
- Lockal 5mo agoNo separate Wikipedia page -> just another Erdős problem. There is no universally agreed-upon "central" conjecture (like "P vs. NP" in CS), but here are some pillars: 1) https://en.wikipedia.org/wiki/Happy_ending_problem https://en.wikipedia.org/wiki/Happy_ending_problem 2) https://en.wikipedia.org/wiki/Hadwiger_conjecture_(combinatorial_geometry) https://en.wikipedia.org/wiki/Hadwiger_conjecture_(combinato... 3) https://en.wikipedia.org/wiki/Hirsch_conjecture https://en.wikipedia.org/wiki/Hirsch_conjecture
- reactordev 5mo agoI dunno, I'm skeptical without proof. I've had the MAX+ plan for a while and I'm sorry, the quality between GPT vs Claude is night and day difference. Claude understands. GPT stumbles over every request I give it.
- nathan_compton 5mo agoWeird thing to say about a report which literally has the attached mathematical proof.
- reactordev 5mo agoExcept its not a proof. It's an existential proof of what? Projecting points and loosing density? Nah, it's wrong. At least with Edros you could solve f(x) or not solve it (inf). You can not with this. All they did was balance a really fancy quadratic equation. The projection from C^f to R² doesn't demonstrate geometric injectivity, so nⱼ = |X| isn't established, and the bound collapses.
- nathan_compton 5mo agoYou got downvoted, but I'm a bit curious - can you elaborate on these critiques?
- deleted 5mo ago[deleted]
- 0x5FC3 5mo agoIs there a reason why we only hear of Erdos problems being solved? I would imagine there are a myriad of other unsolved problems in math, but every single ChatGPT "breakthrough in math" I come across on r/singularity and r/accelerate are Erdos problems.
- tonfa 5mo agoAfaik this is because there is a community and database around them.
- 0x5FC3 5mo agoInteresting. OpenAI could also be trying to solve other problems, but Erdos problems maybe falling first?
- CSMastermind 5mo agoNo, Erdos problems were accepted as sort of a benchmark. There's a bunch of reasons they're favorable for this task: 1. They have a wide range of difficulties. 2. They were curated (Erdos didn't know at first glance how to solve them). 3. Humans already took the time to organize, formally state, add metadata to them. 4. There's a lot of them. If you go around looking for a mathematics benchmark it's hard to do better than that.
- ric2b 5mo agoI'm curious how the relative difficulty between the problems can be assessed when no one knows how to solve any of them.
- throw-the-towel 5mo agoThey're just famous because Erdos was a great mathematician, kinda like the Hilbert problems a century earlier.
- empath75 5mo agoIt'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.
- bradleykingz 5mo agook. so what are the implications of for math
- jaimex2 5mo agoabsolutely none
- Fraterkes 5mo agoI guess if this stuff is going to make my employment more precarious, it’d be nice if it also makes some scientific breakthroughs. We’ll see
- ausbah 5mo agoshame we won’t see any of these medical breakthroughs when we all lose our jobs and thus our healthcare
- karmasimida 5mo agoThere is a world that AI makes medical breakthroughs, but there is 0 guarantee it is going to be affordable
- cubefox 5mo agoBreakthroughs in pure mathematics aren't scientific though. They say us nothing about the world, and they are not useful.
- zozbot234 5mo agoThe summarized chain of thought for this task (linked in the blogpost) is 125 pages. That's an insane scale of reasoning, quite akin to what Anthropic has been teasing with Mythos.
- estetlinus 5mo agoToday I generated the equivalent of two LOTR books just to fix three missing rows in my SQL models (and open a PR), so +1
- wayeq 5mo agoor put differently, you melted x cubic meters of polar ice
- FuckButtons 5mo agoBased on some napkin math, that would be about ~100 watt hours of electricity on an H100 cluster, or, roughly the same amount of energy needed to boil a kettle for a cup of tea.
- nhinck2 5mo agoThat's an exceptionally fast output you have there... Mind showing your working out?
- FuckButtons 5mo agoLotr ~1.4k pages 1 page ~1k tok -> ~700k tok 1tok ~0.5J -> 350kJ ~100Wh ~1 cup of tea
- Petersipoi 5mo agoDoing anything melts x cubic meters of polar ice.
- devttyeu 5mo ago
- m-hodges 5mo agoTo the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
- dvt 5mo ago> 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.
- Tenobrus 5mo agowhat basis do you have for assuming an LLM is fundamentally incapable of doing this?
- dvt 5mo agoBecause by definition LLMs are permutation machines, not creativity machines. (My premise, which you may disagree with, is that creativity/imagination/artistry is not merely permutation.)
- KoolKat23 5mo agoIt pretty much is, otherwise it is randomness or entropy.
- nh23423fefe 5mo agogod of the gaps
- taimurshasan 5mo agoI wonder how much this cost vs a Math Professor or a team of Math Professors.
- forgot_old_user 5mo agoit will only get cheaper in the long run
- dvfjsdhgfv 5mo agofor a sufficiently long definition of long
- aspenmartin 5mo agoNo for a very short definition of long, look at data on: how fast do prices decrease for a constant level of performance
- aspenmartin 5mo ago40x cheaper per year if trends continue
- Karrot_Kream 5mo agoSadly math professors aren't very expensive. Academics are paid terrible wages. Until recently, tenure was the carrot at the end of a grueling education. But now that tenure positions are getting rarer (well, tenure positions aren't growing vs the number of aspiring academics is), they continue to be cheap highly educated labor.
- seydor 5mo agocan the AI please tell us what to do now that all knowledge work will become unemployment?
- throw-the-towel 5mo agoSee the longstanding debate on whether new math is "invented" or "discovered". Most mathematicians I knew thought it's discovered.
- soupspaces 5mo agoRegardless 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?)
- atmosx 5mo ago...long standing indeed. It can be traced back to Plato's works.
- skybrian 5mo agoAny design already exists as a possibility, so it could be said to be both invented and discovered, depending on how you look at it.
- cubefox 5mo agoAll inventions are discoveries, though not all discoveries are inventions.
- FrustratedMonky 5mo agoDepending on your point of view? I see what you did there. Who knew Obi-one was just smoking and pontificating on Wittgenstein.
- lubujackson 5mo agoFor anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time. Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements. Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. 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.
- cubefox 5mo agoThere is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-economy https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...
- zem 5mo agowow, that was indeed a brilliant essay. i particularly liked the framing that "solving a big conjecture was a cryptographic proof that you had come up with a genuine conceptual innovation".
- svieira 5mo ago> The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics. I just wanted to highlight this very correct human-centric thought about the purpose of intellection.
- toraway 5mo agoThank you for sharing, that was one of the most insightful long form pieces I've read in a long time! And the writing was enjoyable to read even as a math layperson. I was going to say you should submit it but I saw you did a few days ago but it only got a few votes... If Dang sees this IMO it would be extremely deserving of the second chance pool as I wouldn't be surprised to see easily jump to the front page with a different roll of the dice.
- pizzao 5mo agoCan someone explain to me what is their "prompting-scaffolding" to make it work ?
- yusufozkan 5mo ago"This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves." - Noam Brown (OpenAI reasoning researcher) on X
- famouswaffles 5mo agoAnother entry in a growing list of the last couple months (interestingly mostly Open AI): 1. Erdos 1196, GPT-5.4 Pro - https://www.scientificamerican.com/article/amateur-armed-with-chatgpt-vibe-maths-a-60-year-old-problem/ https://www.scientificamerican.com/article/amateur-armed-wit... There are a couple of other Erdos wins, but this was the most impressive, prior to the thread in question. And it's completely unsupervised. Solution - https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba9c https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba... 2. Single-minus gluon tree amplitudes are nonzero , GPT-5.2 https://openai.com/index/new-result-theoretical-physics/ https://openai.com/index/new-result-theoretical-physics/ 3. Frontier Math Open Problem, GPT-5.4 Pro and others - https://epoch.ai/frontiermath/open-problems/ramsey-hypergraphs https://epoch.ai/frontiermath/open-problems/ramsey-hypergrap... 4. GPT-5.5 Pro - https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/ https://gowers.wordpress.com/2026/05/08/a-recent-experience-... 5. Claude's Cycles, Claude Opus 4.6 - https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cyc...
- catigula 5mo agoEvery time I interact even with OpenAI's pro model, I am forced to come to the conclusion that anything outside the domain of specific technical problems is almost completely hopeless outside of a simple enhanced search and summary engine. For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not. Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this. Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.
- dvfjsdhgfv 5mo agoYeah, I remember it was one of my biggest disappointments with LLMs.
- brcmthrowaway 5mo agoEnd times are approaching
- throwaway2027 5mo agoNot to dismiss the AI but the important part is that you still need someone able to recognize these solutions in the first place. A lot of things were just hidden in plain sight before AI but no one noticed or didn't have the framework either in maths or any other field they're specialized in to recognize those feats.
- xiaod 5mo ago[flagged]
- dwroberts 5mo agoWould be interesting to know what kind of preparatory work actually went into this - how long did it take to construct an input that produced a real result, and how much input did they get from actual mathematicians to guide refining it
- lacewing 5mo agoWhy? 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.
- JacobAsmuth 5mo agoOr it means that this was a brand new model, they tried it and were instantly rewarded with a hit that was so interesting that several mathematicians pushed to publish the results.
- lacewing 5mo agoAnthropic and OpenAI don't do PR this way. This is not a side project for a publicly-traded BigCo. The bulk of their valuation hinges on being first to AGI / best at AGI.
- rohitsriram 5mo ago[dead]
- ShadowPulse4709 5mo ago[flagged]
- arsan87 5mo agoneato. can we do any thing with this new found knowledge or is this mathematical sports? can we please put these ground breaking AIs to work on actual problems humans have?
- clarle 5mo agoPeople thought neural networks were just an interesting thought exercise a few decades ago and not for practical ML problems, and look what happened since then.
- voooduuuuu 5mo agoAsk an LLM to invent a new word and post it here. You will see that it simply combines words already in the training data.
- dpoloncsak 5mo ago[dead]
- dmos62 5mo ago[dead]
- deleted 5mo ago[deleted]
- konart 5mo agoSo LLM is german?
- SparkyMcUnicorn 5mo ago[dead]
- satvikpendem 5mo agoFunny that the replies are dead. It's true that generally we shouldn't have AI output on HN but this case is an exception as we are explicitly asking for it, so it's interesting that people still flag the replies.
- CamperBob2 5mo agoAnd this is really not OK. I've been a victim of the same filter. Dang/Tomhow, are you reading this? Would it make sense to modify your slop filter to avoid auto-flagging/killing replies that credit the LLM explicitly? Otherwise valid discussions will continue to get hosed.
- dmos62 5mo agoAre you saying comments are getting shadow banned?
- ks2048 5mo agoTimothy Gowers' tweet about this: "If you are a mathematician, then you may want to make sure you are sitting down before reading futher.". woah.
- missyougowers 5mo agoUnfortunately 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 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.
- aroman 5mo agoAre you saying this result is uninteresting and therefore AI slop or puffery? Obviously OpenAI has a motivation to "market" the accomplishment as much as possible, but surely you agree it IS a remarkable achievement?
- missyougowers 5mo agoI'll let the mathematicians in the field determine the level of "interest" in this result, but saying "you may want to make sure you are sitting down" is pure puffery. > has a motivation to "market" the accomplishment as much as possible I am so sick of HN promoting unethical behaviour as virtuous due to it's financialization worship at the foot of "valuations". > but surely you agree it IS a remarkable achievement? If you could define the bounds of "remarkable" I could answer this question.
- horhay 5mo agoIt's remarkable, its not out of the bounds of the pattern of success that AI has had with math recently to the point that people should sound alarm bells. A lot of the weight this holds is the fact that it's an old problem and that its difficulty hinges on the lack of investigation the disproof side of hypothesis. The model basically took a contrarian path and found tools and methods that support that a disproof is viable. So the (unquantified amount of) mathematicians out there were all dedicating their resources on the notion that this can be proved. Some with hindsight would say that if they a had team of experts who are driven to the goal of disproof that this would have been achievable by humans, and one of the mathematicians of the paper state as much,this still has value in terms of reliability measurement, and possibly human-aided endeavors when the methods scrounged by the model can be used in other solutions.
- Kye 5mo agoIs this something that can be made explainable to someone without any of the relevant background, or is this one of those things where all that background is needed to understand it? Because I have no idea what's going on here, but would like to.
- mooreat 5mo agoI think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by grounding it in a larger theory while with the counterexample the model has to just perform a more advanced form of search to find the correct construction. Obviously this search is impressive not naive and requires many steps along the way to prove connections to the counterexample, but instead of developing new deep mathematics the model is still just connecting existing ideas. Not to discount this monumental achievement. I think we're really getting somewhere! To me, and this is just vibes based, I think the models aren't far from being able to theory craft in such a way that they could prove more complicated conjectures that require developing new mathematics. I think that's just a matter of having them able to work on longer and longer time horizons.
- davebren 5mo ago> 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.
- mooreat 5mo agoI currently operate under the assumption that humans are at most as powerful as Turing Machines. And from what I understand these models internally are modeling increasingly harder and larger DFAs, so they're at least as powerful as regular languages. Assuming humans are more powerful than regular languages I could maybe agree that these methods may not eventually yield entirely human like intelligence, but just better and better approximations. The vibe I get though is that we aren't more powerful than regular languages, cause human beings feel computationally bounded. So I could see given enough "human signal" these things could learn to imitate us precisely.
- buddhahastha 5mo ago[flagged]
- CGMthrowaway 5mo agoHow do you even get an LLM to try to solve one of these problems? When I ask it just comes back with the name of the problem and saying "it can't be done"
- KalMann 5mo agoMaybe you need to phrase it better. Like with a more specific direction of thinking.
- lovecg 5mo agoBy making it think for 100+ pages https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf 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.
- atleastoptimal 5mo agoTo all AI skeptics: What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?
- enoint 5mo agoThat’s one possibility. If it fails to convince a critical mass that it’s a net improvement in their lives, then the impediment to continual improvement will be sabotage.
- xandrius 5mo agoYou should really look up a video about what GPTs fundamentally are.
- Rover222 5mo agoYou should also really look up a video about what neural synapses really are.
- rzmmm 5mo agoMaybe after decades. 2022 models were microscopic compared to latest models.
- KalMann 5mo agoI think there's been natural but steady progress with since 2024 with the release of the o1 model, which showed impressive reasoning capabilities. But I think it's wrong to look at the magnitude of the accomplishments and assume that will be field independent. We don't know the range of problems reasoning techniques are useful for. What we see here is refinement of capabilities that have been noticeable for years.
- davebren 5mo ago> What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? No matter how much compute time it's given to combine training samples with each other and run through a validation engine it will still be missing some chunk of the "long tail". To make progress in the long tail it would need to have understanding, and not just a mimicry of understanding. Unless that happens they will always be dependent on the humans that they are mimicking in order to improve.
- somewhereoutth 5mo agoThe real test would be if an LLM makes an important conjecture.
- cpard 5mo agoThe proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break through the increasing super specialization that humans in science have to go through today. Which in one hand is important on the other hand does limit the person in terms of the tooling and inspiration it has access to.
- nashadelic 5mo agoThere 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!
- doubledamio 5mo agoI’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.
- cpard 5mo agoIt's up to us I think. We can use LLMs to generate web pages in candy crash style and end up dumper by outsourcing thinking to the machines or we can use it to expand our cognitive capabilities. What makes me more of an optimist in this case is that people who today decide to go into these sciences are mostly people who are driven by intellectual activity so I feel they are the right ones to figure this out, probably more so than us the engineers.
- Ar-Curunir 5mo agoUnfortunately, LLMs might lead to the demise of the primary institution that allows for people that are in it for the love of intellectual activity to do that activity, namely research universities. Certainly the people proposing the tech are quite opposed to the modern university.
- trostaft 5mo agoSpeaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have been experimenting with LLMs in our research process. I've had pretty great success, though fairly rarely do they solve my entire research question outright like this. Usually, I end up with a back and forth process of refinements and questions on my end until eventually the idea comes apparent. Not unlike my traditional research refinement process, just better. Of course, I don't have access to the model they're using =) . Nevertheless, one thing that struck me in this writeup, was the lack of attribution in the quoted final response from the model. In a field like math, where most research is posted publicly and is available, attribution of prior results is both social credit and how we find/build abstractions and concentrate attention. The human-edited paper naturally contains this. 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. One more note: the comments on articles like these on HN and otherwise are usually pretty negative / downcast. There's great reason for that, what with how these companies market themselves and how proponents of the technology conduct themselves on social media. Moreover, I personally cannot feel anything other than disgust seeing these models displace talented creatives whose work they're trained on (often to the detriment of quality). 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. 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.
- julianozen 5mo agoNice response to read
- School-Cotton 5mo agoWhy 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.
- recitedropper 5mo agoThis is impressive, no question. Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". 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. It isn't likely that someone solved this and then just put it in the training data, although I honestly wouldn't put that past OpenAI. More interesting though is the extent to which they've generated training data that may have touched on most or all of the "original" tenets found in this proof. We can't know, of course. But until these things are built in a non-clandestine manner, this question will always remain.
- Rover222 5mo agoSeems like a very tin-foil-hat-take to me
- recitedropper 5mo agoI'm not letting the government read my brainwaves. In all seriousness though: My suggestion is that those shepherding the frontier of AI start acting with more transparency, and stop acting in ways that encourage conspiratorial thinking. Especially if the technology is as powerful as they market it as.
- Rover222 5mo agoHard to argue with that
- mrdependable 5mo agoHow is that a "tin-foil-hat" take? It's not a secret, and in fact widely reported, that these companies are spending billions on creating training data.
- dmix 5mo ago
- deleted 5mo ago[deleted]
- _heimdall 5mo agoAs this becomes more common it makes me wonder where the LLM ends and the harness begins. The underlying model may still effectively be a stochastic parrot, but used properly that can do impressive things and the various harnesses have been getting better and better at automating the use of said parrot.
- raincole 5mo agoI like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)
- zeofig 5mo agoI still laugh.
- johnfn 5mo agoHave you updated your priors after this announcement? If not, why not?
- xyzsparetimexyz 5mo agoPrior whats?
- Jblx2 5mo agohttps://en.wikipedia.org/wiki/Prior_probability https://en.wikipedia.org/wiki/Prior_probability
- se4u 5mo agoWhen a qualifying noun is absent , then priors means prior beliefs.
- xyzsparetimexyz 5mo agoPrior to what? Why not just say beliefs?
- niek_pas 5mo ago"Update your priors" is a common expression in English: https://en.wiktionary.org/wiki/update_one%27s_priors#English https://en.wiktionary.org/wiki/update_one%27s_priors#English
- analognoise 5mo agoBack when “term rewriting” was “AI”, multiple math tools were released that took known math facts and did tricks like uncovering new integrals - apply the pattern in some depth in a tree, see what pops out. What was discovered were numerous mistakes in the published literature on the subject. “New math! AI!” No, just mechanical application of rules, human mistakes. There were things that were theorized, but couldn’t be exhaustively checked until computers were bigger. Once again, a tool is applied, it has the AI label - its progress! But it isn’t something new. It’s just an LLM. There’s a consistent under appreciation of AI (and math, honestly), but watching soulless AI mongers declare that their toy has created the new is something of a new low; uninspired, failed creatives, without rhyme or context; this is a bigger version of declaring that your spell checker has created new words. The result is more impressive than what was done with tables of integrals and SAINT in 1961, sure. Apparently if you add a “temperature” knob to a text predictor, otherwise sane individuals piss themselves and call it new. Then again I thought NFTs, crypto, and the Metaverse were stupid, so what do I know.
- auggierose 5mo agoWhich model did this? Is it available to the public?
- agentultra 5mo agoI’m curious about the “autonomous” claim. Usually these systems require a human to guide and verify steps, clarify problems, etc. are they claiming that the reinforcement model wasn’t given any inputs, tools, guidance, or training data from humans?
- deleted 5mo ago[deleted]
- alsetmusic 5mo ago> AI is about to start taking a very serious role in the creative parts of research, and most importantly AI research itself. While this progress is not unexpected, it reinforces the urgency we feel about understanding this next phase of AI development, the challenges of aligning very intelligent systems, and the future of human-AI collaboration. I find this hyperbolic, but ya gotta juice up the upcoming IPO. I hate that they took an interesting announcement and reminded me why I hate tech and our society at the end.
- dwa3592 5mo agoFew questions that the blog did not answer, if anyone knows that'll be great: - Does anyone know if this was a 1 minute of inference or 1 month? - How many times did the model say it was done disproving before it was found out that the model was wrong/hallucinating? - One of the graphs say - the model produced the right answer almost half the times at the peak compute??? did i understand that right? what does peak compute mean here?
- mrcwinn 5mo agoThe back and forth in this discussion reveals to me we are sorting through a kind of philosophical debate about intelligence. That alone tells me LLMs are doing something novel.
- SubiculumCode 5mo agoI wonder whether there will be progress in string theory from these kinds of applications of AI.
- iLoveOncall 5mo agoAbsolutely no proof that any LLM actually found the result, and just a mention of an "internal model". Served to you by one of the biggest liars in the world. Why would anyone believe this to be true even for a split second?
- varenc 5mo agoThis has been an unsolved open problem for 80 years. What you're suggesting is that someone connected to Open AI solved this very hard math problem, but then rather than taking credit for it, falsely attributed it to AI? The point of having an AI solve an unsolved problem, is to make it very clear that the insight must have come from the AI and wasn't in the training data. Sure, it's possible OpenAI had access to some math professors that solved it and then let an AI model take the credit... but seems unlikely. That human would be turning down a potential Fields Medal for this discovery. The abridged chain-of-thought from the model also serves as some evidence of LLM origin: https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d... (could be fake, though I'm unsure what proof of LLM origin couldn't be faked)
- Antibabelic 5mo ago> That human would be turning down a potential Fields Medal for this discovery. While interesting, this result is not Fields Medal material.
- geraneum 5mo ago> but seems unlikely. That human would be turning down a potential Fields Medal for this discovery. I also don’t like the tin foil hatty theories and don’t know what OpenAI actually did, but an NDA does wonders! Just pointing out that this line of operations is not really unlikely.
- iLoveOncall 5mo ago> What you're suggesting is that someone connected to Open AI solved this very hard math problem, but then rather than taking credit for it, falsely attributed it to AI? I'm suggesting that OpenAI invested a lot of resources and money in having someone (or a group of people) disprove this conjecture, so they could claim their LLM disproved it. Yes. I'm not sure why you're surprised by this, given that everything that Altman has said in the past has turned out to be a lie. The fact that they gave an EDITED (even rewritten, from the PDF itself) chain of thought is just further proof. Why not give the raw one alongside? No reason at all, except if it doesn't exist.
- spacebacon 5mo ago[flagged]
- zuzululu 5mo agoThis topic and discussion is out of my league what is the implication here ? LLMs aren't a dead end ?
- adt 5mo agohttps://lifearchitect.ai/asi/ https://lifearchitect.ai/asi/
- 3422817 5mo agoNice. By the year 2100 200 Erdos problems will have been solved by AI. Let's build more data centers.
- neuroelectron 5mo agoI wonder if it has anything to do with the fact that AI is a grid of grid-calculating grids. It seems like it would be especially well suited to finding solutions about grids. That is until you consider the fact that even 1 trillion billion grids is still not anywhere close to an infinite grid. So, probably slop.
- sinuhe69 5mo agoHow did they jump from finding counter-examples (disproof) to a proof?
- zone411 5mo agoI actually tried using GPT-5.5 Pro on this problem recently. It thought it was making progress on one path, but it made so many mistakes that it didn't feel worth it pushing further. It'll be interesting to check whether it's the same route. I got partial results (proved in Lean) that improve on the best-known results for four Erdős problems with GPT-5.5 Pro
- overgard 5mo agoI think it's worth being skeptical of this.. there's a way too common pattern of "AI Lab Shows AI Doing Something Only Humans Can Do" only for a bunch of important caveats and limitations to be discovered after the initial hype. And of course, the correction never seems to be as viral as the hype. I'll believe it when a mathematician actually reads the 100+ pages of reasoning.
- throwaway613746 5mo ago[dead]
- ferris-booler 5mo agoWhat strikes me in this case (and I haven't seen in other comments) is that it's a _disproof_ of a conjecture put forth by Erdős and supported (at least according to OpenAI) by other professional mathematicians. Erdős, one of the greats, thought that the limit was O(n^{1 + o(1)}), which GPT disproved. We can argue about recombination/interpolation of training data in LLMs, but even if this was an interpolation, the result was contrarian rather than a confirmation. Any system that can identify an error in Erdős's thinking seems very useful to me (though perhaps he did not spend much time thinking about or checking this particular conjecture).
- __0x01 5mo agoFrom the companion paper: > The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. Can someone please elaborate on this?
- awdfeswavcra 5mo agoThe 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.
- libraryofbabel 5mo agoThis HN thread depressed me. I’m still thinking about why. Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers’ remarks. This is really interesting stuff! Yet the comments are just a battleground of people rehearsing the same tired arguments about LLMs from 2023, refutations of those arguments, angry counters, etc. Does it make anyone else sad that the battle lines seem to have been drawn 3 years ago and we just seem to have the same fights over and over? I wonder if we’ll still be doing this two years hence.
- jryan49 5mo agoPeople are afraid for their livelihood. What do you expect?
- conception 5mo agoLivelihoods and lives.
- JacobAsmuth 5mo agoExactly. And when one's life is threatened, what are we to do if not fight? Fight! Fight! Fight!
- heuristicsearch 5mo ago[flagged]
- libraryofbabel 5mo agoWell yes, but there is a choice being made here and I would love to believe we can do better. The rational response to being afraid about your livelihood isn’t to spend time filling every HN thread on LLMs with embittered negativity. Not to mention all the flat denials that LLMs can do mathematics and write decent code, which is almost a self-contradictory position if you are worried they are going to replace you. There are a lot of big issues at stake here and just because a person is interested in what AI can do and curious to discuss it does not make them uncritically positive about it’s effects on society, the economy, and the world. Yet that is often the assumption and it leads to battle lines being drawn, on every AI discussion, over and over again. It means the serious discussion gets swamped and that makes me sad.
- dev1ycan 5mo agoWouldn't surprise me if they're just paying math geniuses to do math research and attribute it to AI models.
- armanj 5mo agouseless fact: there is no mention to "gpt" in this article. the ai is referred to as "An internal OpenAI model".
- precision1k 5mo agoI see mixed emotions here. I understand both. On one hand it's exciting and fascinating. On the other it's concerning. One concern I haven't seen mentioned is the possibility that, as these models become larger and more powerful, their capability to solve frontier math problems will also grow. Does there become a point where humans are no longer the driver of innovation and research in this world, and instead are relegated to become stewards of the AI models whose purpose is to push the boundaries of mathematics, theoretical physics and other academic disciplines?
- noslenwerdna 5mo agoFor 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.
- gpderetta 5mo agowhat if at some point we will no longer be able to understand the answers?
- greenknight 5mo agoI believe that the answer is yes it will happen... the question is when will that point will occur. Right now, we are in a transition period... Models are improving, but they are not capable just yet to take over. Where do you see it being in a years time? or 2? or 5?
- epicsagas 5mo ago[flagged]
- zmmmmm 5mo agoAs a side observation, it is striking but also not surprising in retrospect that the big successes in AI are coming from domains where things are fundamentally verifiable. Both software and math are either fully verifiable or low-cost verifiable (breaking a test is not the same cost as building a bridge and watching it fall down to see if it worked). Other domains are extracting value but I feel like there's an order of magnitude difference. It raises the question, what other domains fit into these categories where the AI itself has pretty much free reign to verify its own results?
- oscord 5mo agoCan it model a sustainable economy model, with human happiness and fulfilment indexes and planet preservation focus? Current capitalism and the red thing are so tired!
- ninjagoo 5mo agoMany folks are upset about the supplanting of human effort by ai. Umanwizard voiced this valid concern below [1], but his comment got downvoted, unfairly, IMHO, instead of just being addressed. So putting out at least my response as its own top-level comment for visibility. > the closer the expertise you spent your whole life building is to being worthless. Perhaps it is time for life to be considered intrinsically valuable, instead of being "worthy" only based on output or capability. Disability, animal and environmental advocates have been fighting for this for a long time. Not too long ago women and minorities were in the same boat. Even now, there are many advocating and fighting for a return to the dark old days. > Along with all the rest of what humans find meaningful and fulfilling. Some humans. Many are content to enjoy simply existing, and the beauty of life and the universe around us. Just like many non-scientists today enjoy and benefit from the work of scientists, tomorrow too many will enjoy learning from, and applying the coming advancements and leaps in many fields. And those of a scientist or other research-type mindset? No doubt they will contribute meaningfully by studying the frontier, noting what remains unanswered, and then advancing the frontier, just like researchers do today; just because scientists in the past solved many questions doesn't mean that there aren't any questions to answer today. IMHO, AI means that the frontier expands faster, not that it is obliterated. Even AI cannot overcome the laws and limitations of physics/universe: even Dyson spheres only capture the energy of one star, thus setting a limit on the amount of compute, and thereby a limit on intelligence. And we are a loooong way from a Dyson sphere. [1] https://news.ycombinator.com/item?id=48215122 https://news.ycombinator.com/item?id=48215122
- momo26 5mo agoI'm curious that giving an counter-example is kind of easy to disprove. But can the model really prove something correctly and rigorously? Cuz now it seems like all the knowledge is based on the existed thing, and none of them can prove a myth.
- cwmoore 5mo agoFrom the meandering and self-loving article: “ For decades, it was widely believed that this rate was essentially the best possible, and no construction could improve significantly over the square grid. In technical terms, Erdős conjectured an upper bound of n 1 + o ( 1 ) n 1+o(1) in which the additional o ( 1 ) o(1) indicates a term tending to 0 0 with n n. Our new result disproves this conjecture. More precisely, for infinitely many values of n n, the proof constructs configurations of n n points with at least n 1 + δ n 1+δ unit-distance pairs, for some fixed exponent δ > 0 δ>0. (The original AI proof does not give an explicit δ δ, but a forthcoming refinement due to Princeton mathematics professor Will Sawin has shown one can take δ = 0.014 δ=0.014.)”
- foota 5mo agoThey should feed it the classification of finite simple groups and get it to simplify it/turn it more constructive.
- yathartha 5mo ago[dead]
- Topology1 5mo agoAs someone starting grad school for pure mathematics, this has me both excited and nervous, but mainly the latter...
- globulus2023 5mo agoIn the article there is a nice clear diagram of the “square grid” arrangement that was previously thought to be optimal. Can anyone point me to a diagram of the newly found optimal arrangement?
- globulus2023 5mo agoIn the article there is a diagram of the “square grid” arrangement that achieves approximately 2n points separated by unit distance. Can anyone point me to a diagram of what the newly found solution looks like?
- deleted 5mo ago[deleted]
- purpleidea 5mo agoYou'd think a billion dollar company would be able to normalize the sound level on their video :/
- llagerlof 5mo agoThat will require AGI.
- NexraGear 5mo ago[flagged]
- ai_fry_ur_brain 5mo agoIm convinved they target these pure math problems because math is very occulted to the masses, and therefor can use math "discoveries" as a way to make an LLM seem more impressive than it is. Everything is a grift. What are the odds that if they ran the same prompt from scratch, with the same context and instructions that it would arrive at the same answer? Unlikely. I think its more likely that this is a 1:500000 chance and OpenAI can afford to brute force this result and justify the expense for marketing.
- ccvannorman 5mo agoI looked at all linked articles and could not find an example of the points (they show a square grid of points with n~=100 but no other ordering of points to show the more optimal layout(s)). Is there anywhere an image example of a superior layout for example with n>={100,1000,10000}..? I would love to see it. I am imagining it would look somewhat like a sloppy pizza.
- lg5689 5mo agoOne 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.
- krackers 5mo agoThat was for this particular technique, but the comment suggested that a similar construction can (but is unproven) to beat erdos and at least gives some flavor of what this looks like https://x.com/i/status/2057490144546927046 https://x.com/i/status/2057490144546927046
- lesostep 5mo agoI am cautious about AI "discoveries" after Mythos paper. What was the process of a writing a paper? Was the question asked by a mathematician? Was the paper right from a get-go or was there someone who pointed out mistakes? How much attempts were made before solution was found? I will eat my words if an AI oneshotted that one without any external help, but for know I am left wandering whether it's a new way to attribute discoveries to companies instead of people who put the work in
- crnkofe 5mo agoI'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.
- andy12_ 5mo ago> 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/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
- 0xDEAFBEAD 5mo ago>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 https://xcancel.com/hlntnr/status/2052479493801975987
- callamdelaney 5mo agoThe only relevant question is, how much did it cost?
- Quentak 5mo agoI'd like to know how many tokens in total went into solving this problem. Have they talked about this? It matters whether they got this result in 10 million tokens or 10 billion. Whether it's closer to 1 human working on this for 1 year or 1000 humans for 1 year. The news feels different when the probability of one AI run solving this is 1 in a thousand vs 1 a million. Approximately I'm asking about the amount of money it cost to solve it, which has to include the failed parallel runs.
- ApolloRising 5mo agoI would like to know this as well, including how long was it working on this problem?
- _diyar 5mo agoAny answer to this question must also consider the current cost/token and its downward trajectory as algo and hardware advances drive down costs.
- vessenes 5mo agoGood 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.
- DiogenesKynikos 5mo agoCalling all LLM skeptics. How did a "stochastic parrot" just disprove an Erdős conjecture that mathematicians couldn't figure out for decades?
- isolli 5mo agoQuestion: The conjecture was about an upper bound for the maximum number of pairs. It has been disproven. Was the Erdos problem the conjecture itself, or was it about the actual maximum number of pairs? (In which case it will probably never be solved.) The problem is defined in the narrow version here: https://www.erdosproblems.com/90 https://www.erdosproblems.com/90
- AlexToaniAI 5mo agoSo nowadays. AI may use different field and get lots of break through that migh human can't done! That's nuts!
- smailbkf 5mo ago[dead]
- kevinwang 5mo agoNitpicky/not important, but they say: Since loglog(n) tends to infinity with n, the additional term in the exponent tends to 0, meaning these constructions achieve growth only slightly faster than linear. Would anyone else describe the previous asymptotic behavior like that? I mean obviously loglogn to O(1) is a quantum leap, but wouldn't you describe loglogn as "grows so slowly it's almost constant", so the constructions achieve growth "almost n^{1+c}"? But I guess that might be overcorrecting too hard.
- kevinwang 5mo agoJust reread this comment and it's not easy to understand, but the edit window is passed. What I meant is that they describe loglogn the same way you could describe O(n) or O(n^2) -- it "tends to infinity with n", even though my mental model for loglogn is to treat it as barely more than constant. See: https://cs.stackexchange.com/questions/148197/who-said-first-in-practice-log-log-n-is-at-most-single-digit-number https://cs.stackexchange.com/questions/148197/who-said-first...
- mathsolved 5mo ago[dead]
- wuwei78 5mo ago[flagged]
- anonymousDan 5mo agoCome back to me when it has solved P=NP.
- PeterStuer 5mo agoThere's still parrots parroting 'stochastic parrot'.