HN

Need help?
<- Back
Sharing AI progress in mathematics

Sharing AI progress in mathematics

OfficialTurkey

Comments (454)

  • jboggan
    I was a graph theory junkie long ago and even moved to Budapest for awhile to study among the greats. While I was there I started working on Barnette's Conjecture which came to occupy my thoughts over the next 24 years of my life, on and off as I worked in many different fields. Last summer I even thought for a few days that I had actually solved it.But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.
  • zone411
    A quick check shows that this list claims to fully solve 90 of the top 500 open problems in math (https://proofatlas.ai/open-problems/).The highest ranked would be:| 22 | Hilbert’s tenth problem over ℚ || 29 | Unique Games || 31 | Anderson-model extended states || 37 | Spacetime Penrose inequality || 48 | Nonexistence of Landau–Siegel zeros || 52 | Baum–Connes || 78 | Abundance || 80 | Hadwiger || 87 | Bose–Einstein condensation || 92 | Two-dimensional entanglement area law |
  • xanderlewis
    As Kevin Buzzard recently said:> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.
  • theoa
    What's missing for me for each result are the following:* Explain the result to me as if I'm a 10-year-old. * Create the infographic for this result. * Make a Khan Academy-style video to teach me this result.
  • prideout
    This includes a proof of Barnette's Conjecture, which is one of the graph theory conjectures that I tried attacking with SOTA models a few months ago. I like it because it is easy to understand with a basic knowledge of graph theory. I spent quite a bit of time on it and failed. Their proof looks approachable at first glance.https://github.com/openai/math/blob/main/preprints/Paired-st...
  • NotOscarWilde
    As a TCS/scheduling person, this one is definitely of lesser importance than UGC, but it has been an open problem since the book of Garey and Johnson in 1979:A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
  • rcr-anti
    In Stellaris you can play as a civilization of robots who keep their biological creator race alive as "bio trophies". The bio trophies don't do anything meaningful besides by existing satisfy the need their ancestors placed in the robots to take care of them. Starting to wonder if that's the best we can hope for, if these things will be, if they aren't already, better than us at anything that matters.
  • schleck8
    Levent Alpöge (Anthropic mathematician) comment on the significance:> Sure, mathematical history features a lot of incredible developments, like the invention of proof, zero, or the computer, and on the great problems our progress has been over timelines measured in decades or centuries. Obviously this technology didn’t appear today, but blurring our eyes a bit to combine the past ten years, with today a measurement of those developments, there is nothing comparable.
  • enoether
    Unique Games Conjecture [0] is a seminal conjecture in Complexity Theory, and is an underlying assumption for many, many inapproximability results. A valid proof is a big deal![0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu...
  • bcatanzaro
    “I think at the heart of this issue is that humans have two competing natures: a tendency to compete and a capacity to appreciate beauty,” said Kai Shaikh, a graduate student in mathematics at the University of Toronto. “To me this seems to be a case of the former attempting to strangle the latter.” [1]Beauty can be appreciated even when it is vast, even when it is beyond one's comprehension. I don't think this release should be primarily viewed as an outcome of competition. Instead it is revealing truths about the universe that were always there and always beautiful, even if we hadn't seen them yet. I believe there are infinitely more such beautiful truths currently hidden and waiting for us to discover.[1] https://www.nytimes.com/2026/10/06/science/openai-math-probl...
  • gizmodo59
    This is significant progress and released without all the drama. Some very important progress in Reinmann, Hodge and unique games theorem. Point the repo to your agent and ask for the significance! In a way this is probably 50-100 years of math progress by humans
  • sebmellen
    It’s fascinating to read through the reasoning traces: https://github.com/openai/math/tree/main/reasoning_tracesLook at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...
  • againstapples
    As an AI "doomer" can I ask the non-doomer people here how you interpret the significance of results like these, and what kind of progress you expect to see in the next 1-5 years?Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?
  • foota
    From their github: "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." That's pretty crazy.
  • kingstnap
    Some of these are interesting ngl.109. Integer multiplication below n log nSurprising that this is possible.158. The Euclidean plane cannot be colored with five colors.Only 6 and 7 remain!376. Universal computation in forced Navier–Stokes flows.Morning coffee proven turing complete
  • binlog
    So they set up a whole "Advisory Group on Mathematics and Artificial Intelligence", filled it with renowned academics, promised to listen to the group on "review and communication of emerging results"...and then a week later went nah, we are just going to dump it all on github. OpenAI truly is a special company - in the best and worst way.
  • dekhn
    I'm a software engineering/biology/ML guy who loves when clever math ideas get turned into real solutions (https://en.wikipedia.org/wiki/Compressed_sensing). I am curious if any of the results have immediate applications in any kind of engineering or science.It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
  • davegoldblatt
  • lynndotpy
    Two weeks ago, I heard rumblings that UGC, RL=L, and a matrix multiplication lower bound (even lower than the one here, but so unbelievably lower I think it was a typo), and a few others were about to be proven by someone at OpenAI. I find myself saying "big if true" a lot lately.Even those on their own were enough to make your head spin. But seeing about 100x that? Geeze.
  • pavitheran
    From the GitHub description: “On average, each result used 3 hours of ChatGPT Pro thinking compute”
  • karahime
    Extremely unfortunate that gate keeping got to the point where they felt the need to ask for permission to share math.
  • open592
    Let's hypothetically say I'm a PHD student who is half way through my studies and I have a halfway written version of one of these "preprints" - what do I do?Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?
  • ks2048
    I think they should put human names on the papers as someone who has reviewed the result, even if just a preliminary review. (I’m assuming they didn’t just pipe their model output directly to the internet and these had some amount of review?)
  • ravenical
  • binlog
    So happy this is shared on GitHub rather than some gatekeeping paid journal. Truly a new age for science.
  • sigbottle
    Unique games conjecture and matmul <= 2.25. What the hell.
  • 7373737373
    It may be useful to publish a formalization of ALL known mathematics at this point. Like every book ever printed, every paper on arXiv etc.How many Gigabytes would that be, compressed? Wikipedia once fit on a DVDThis might also allow for some interesting meta-mathematics
  • ed
  • TheMrZZ
    These results are wild. Several individual findings are crazy good and use mostly unexplored methods (the improvement over Riemann for example)... I'm pretty sure some of these results would have been Fields-worthy.But having so many of them at once? Damn. We really live in the future.
  • AmazingEveryDay
    I think Alan Turing would be quite intrigued by these developments, and maybe wondering what took so long.
  • trostaft
    Not much computational mathematics here, but I do see some of interest in the MCMC community. In particular, 139, 93, and 101 are very interesting. Also (I'm not too familiar), I remember attending some talks on the Crouzeix conjecture 325 attacks, should sharpen some rates for Krylov methods. Obviously need to read deeper, some of these don't have corresponding Lean formalizations.Cool!
  • williamhm
    And this is the result of a discussion between users and the platform; it's great that they listened.
  • karannb
    I think such "process-oriented" fields shouldn't be subjected to mass automation (or at least not till humans have sufficiently leveled up our game and understanding). What I mean is progress in math comes from having gained a deeper understanding of the problem for subseq
  • edward_d
    This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model?
  • rinconrex
    The math equivalent of AI code reviews piling up. I wonder where the incentives will align and the equilibrium turns out.
  • patcon
    I'm a little concerned, but I'm also glad they're publishing quick before the department of war starts making national security claims of related to using the knowledge for cryptography and/or weapons
  • bashtoni
    Is AI going to put Mathematicians out of a job, or is it going to create many new jobs reviewing proofs it creates?I'm not sure it's clear right now.
  • avd201
    Wow, FFT faster than O(nlog(n))? I wonder if that will open the floodgates for further improvement or not. I don't understand anything about most of the fields these results touch, but I can say that this in particular is very surprising.
  • closetheloopdev
    Hopefully the techniques and results here will be in the training dataset for the next models, so that each new release will give us more interesting techniques and results!It seems that OpenAI has a proof machine that keeps multiplying fruitful proofs!
  • dualvariable
    How many of these results are incorrect?I doubt the answer to this is "none".And how many of them are just exploiting some loophole that will need to be closed in the problem definition?
  • karannb
    I think such "process-oriented" fields shouldn't be subjected to mass automation (at least not till we as humans have sufficiently leveled up our game and understanding).What I mean is progress in math comes from having gained a deeper understanding of the problem for subsequent attack of more problems and IMPORTANTLY applications! Right now the first one is trivially satisfied (given oai maintains some memory across models) but the second one is not! It's generating proofs faster than anyone can validate and so only the model can use these. Consequently if it just keeps doing more theory it's... not very helpful or at least not optimally helpful. This is just bragging rights for now.More "application-oriented" research would be awesome, where it tries to achieve some desirable effect and then produces relevant theory and experiments around it. Fields like CS, Physics, Chemistry, etc. This would also benefit a wider section of the population rather than the 10 people who understand most of these proofs.
  • Xcelerate
    > 241. Rigidity of the Turing degrees. Every order automorphism of the Turing degrees is the identity.Wow. This is just crazy.
  • lf88
    In some ways, this feels more like an ominous warning about the times to come than something to celebrate.
  • rifty
    As AI works steadily through the already discovered unsolved problems in mathematics, who is currently discovering new ones?
  • chickenjoseph
    This achievement feels like a reasonable candidate for the moment where LLMs are officially "more intelligent" than any person. How can we justify moving the goalposts yet again? How can people have grown so numb to seeing advancements that they don't read this as significant? I feel like I'm standing at the foot of the exponential. I am not excited for the future, and I don't see how humans retain meaningful control over the future if we continue on this trajectory.I have been lurking for quite some time. I made an account to post this, but I honestly don't know what to say. I would like to get off this wild ride.
  • xydac
    i wonder what it means for maths researchers, and how it aligns with how they approach math problems.
  • NegativeLatency
    Why should I care?
  • blurbleblurble
    This just looks entirely obscene from a PR perspective. It's like the cable companies networks owning the networks. At least form some partnerships to obscure the total narcissism party.
  • dyauspitr
    This is like that meme where death goes door-to-door. Currently, he has visited the software development and mathematics doors. I wonder what’s next.
  • curtis-jm
  • anon
    undefined
  • yewenjie
    A lot of these seem to be proving conjectures rather than finding counterexamples, a lot of people used that to claim that these models are not really smart/creative etc.That copium didn't last for what, three months?
  • connor11528
    will this make the math for building data centers work?
  • nnoman7808
    Roblox
  • TeeWEE
    This feels like a huge AI slop dump. Who validated these results. Why are they not mentioned. Human understanding is key here.In my experience AI (frontier models) sometimes does weird stuff that needs human review. Not that’s incorrect but sometimes overly complex language or weird use of language.
  • rafterydj
    I don't know, this does not feel like the message hit OpenAI where it needed to hit, if this is their primary response.
  • i_idiot
    If only AI can better humans in meditation...
  • lokl
    Do applied math next.
  • matapassiones
    Valency has the papers up on Valency Hub
  • jrflo
    Glad to seem them changing their tact with the whole NS debacle. Hopefully we can all focus on the results now rather than the surrounding drama.
  • anon
    undefined
  • kevinwang
    wow
  • pugfugly
    holy fucking shit
  • anon
    undefined
  • Catloafdev
    This is a pretty hilarious thing to read juxtaposed with AGMAI's requests.Basically "Here you go, have fun with this, fuck all your demands, by the way we're gonna be releasing the model stay tuned!"
  • aaraujo002
    The Advisory Group states in its recommendations [1]:"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?[1] https://agmai.org/general-sep29/
  • k2xl
    Can someone knowledgeable about the subject outline the most significant portions of the results?
  • hi__dang
    Mathematics is solved.
  • tootie
    Seemingly none are vetted and reviewed yet
  • applicative
    I wonder if the Lean compiler can change it's license so that a for-profit corporation can only use it if it pays, say, a few hundred billion dollars. This is the correct path.
  • nautilus12
    Have any real mathematicians working on these problems reviewed any of these and determined if they are just gobbledegook or not?The ones with lean proofs could still be formulated incorrectly
  • baggy_trough
    Stochastic parrot truthers in shambles.
  • mathisfun123
    With so many results in so many different areas no way they even remotely spot checked well enough.Prediction: one of these is wrong and this (publicity stunt) will backfire.Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
  • dpweb
    [dead]
  • philipwhiuk
    [dead]
  • philipfweiss
    [flagged]
  • redox99
    The stochastic parrots have predicted the next token once again.
  • applicative
    Why didn't they just give mathematicians access, so they could at least understand and write up the results in publishable form?
  • digitaltrees
    Gross. After the accusations training on mathematicians conversations and unique methods, to dump this volume of unsubstantiated papers is at best tone deaf. Are they hoping humans review and validate this? The amount of free that they are coat tailing is outrageous.
  • sandworm101
    So the million monkeys at a million typewriters have churned out 700 shakespeares, but they need me for spellcheck?
  • mi_lk
    Curious if Sébastien Bubeck still work at OpenAI? He came out quite dirty after Navier-Stokes drama
  • dgacmu
    I find the claimed matrix multiply result (w<= 2.25) shocking. I hope it holds up.
  • senderista
    Good to see they're engaging with the mathematical community, even if they had to be publicly shamed into doing so.