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When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of th

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them? Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? How do we tell truth from fiction?

tumdum_ 2mo ago
Latest presentation of Terence Tao on what current advancements in AI mean for math discusses (among other things) those issues: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...
- glitchc 2mo ago
  Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
  - boothby 2mo ago
    No. More like "I am mathematician and I have digested this sloppy writing and understand this to be an application of Foo theory to the Bar problem with a Baz twist. The prior publication omits crucial references to..."
    - glitchc 2mo ago
      The writing's already pretty good, even Terrence Tao calls it flawless, if you read the slides. That's not where the LLM requires help.
serial_dev 2mo ago
I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)". Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?
- svachalek 2mo ago
  Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.
  - sarchertech 2mo ago
    To continue the analogy, it’s like writing answers that are so convoluted that no one can understand them.
  - HarHarVeryFunny 2mo ago
    Yes, although a counterpoint to that is that if these weren't long-standing published problems that have stumped humans, would anyone care? What is important here, the problem/proof, or the fact that they are famous problems that have stumped humans (and therefore make good bragging material). I'm not sure that a problem being famous, or hard for a human, is the same as it being important. Take Fermat's last theorem as an example - was a proof of it ever considered to be, then or now, consequential, or was it the Paris Hilton of problems - famous for being famous? A problem whose legend grew because it it so simple to state, with the challenge that there might be a simple proof, and that turned out to be so difficult to solve that it therefore became prestigious to do so?
  - deleted 2mo ago
    [deleted]
  - chrisjj 2mo ago
    > this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. Except OpenAI isn't showing any upvotes.
the_sleaze_ 2mo ago
I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?
- saalweachter 2mo ago
  So a good fraction of famous old math puzzles with no practical use are famous because they are in some way similar to problems people actually care about. If you can solve the toy problem that is in some way simpler, maybe you can use the same methods to solve the "real" problem. One open question is whether these machine solutions to these problems will act as springboards to future research, either when given to human mathematicians, or when used to train future machine models.
- Levitz 2mo ago
  It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.
  - stymaar 2mo ago
    True. Though ~99% is still useless in the end.
    - boothby 2mo ago
      We are not at the end, how could you possibly be confident in this claim?
    - wongarsu 2mo ago
      As they say: 80% of marketing is useless, you just don't know in advance which 80%. Same with math
- LPisGood 2mo ago
  One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.
qsort 2mo ago
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Huh? Mathematics is an extremely wide subject, it's perfectly normal even for two professional mathematicians not to understand each other's work. Have you considered that maybe they just don't work in that area? Are you implying OpenAI's paper (which was, by the way, edited by humans and provided Lean certificates for most of the proofs) is actually gibberish? That's flat-earth levels of conspiracy.
WarmWash 2mo ago
Mathematics is an unusually dense (if not the most dense...by a few large steps) field. So lots of areas of mathematics are extremely deep and narrow without any real shortcuts, even for seasoned mathematicians.
- gglitch 2mo ago
  Awesome observation. What would you consider denser?
  - Iolaum 2mo ago
    QWEN-3.6-27B P.S. Couldn't resist :p
    - HarHarVeryFunny 2mo ago
      Unlike 3.6 35B. Is it dense or MoE? It's a good model Sir!
  - Jensson 2mo ago
    Physics possibly.
    - WarmWash 2mo ago
      Physics has bounds of concern, where as mathematics would encompass all physics, as well as all possible alternative physics, in all possible forms. Maybe not unbounded, but close to it.
hgoel 2mo ago
This seems like an extreme exaggeration from a few people claiming to not understand a very recent result. This cycle of a new result being discovered, and reaearchers needing some time to truly digest and disassemble it, is normal.
a_conservative 2mo ago
I have the same questions about human mathematicians! I can't tell if xkcd #435 is still true, or if math is just as mushy as everything else seems to be. When a math proof can only be understood by a handful of people, what does that mean about that proof? I think the LLMs are pushing a problem that existed already and pushing it further. [0] https://xkcd.com/435/ https://xkcd.com/435/
- eru 2mo ago
  That's why OpenAI also published machine-checkable proofs. The process is very, very faintly similar to running a typechecker over your software sources.
  - parineum 2mo ago
    They're publishing machine checkable proofs because that's the only way they, themselves, can check them. They don't understand the math either.
    - inigyou 2mo ago
      For 3.5 days we had a machine checkable proof the Collatz conjecture was false. There turned out to be a bug in the machine checker.
      - eru 2mo ago
        Yes, however any LLM agent you asked about that proof could tell you that. (And even most humans who know a little bit about the bug.) You are right that Lean isn't great in this respect, and people are working on proof formalisations that are less prone to bugs.
wat10000 2mo ago
This has been an ongoing debate since the computerized proof of the four color theorem fifty years ago.
eru 2mo ago
> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.
- lstodd 2mo ago
  I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.
  - goodmythical 2mo ago
    Being familiar with the basics of fission power generation does not put the design of a competitive plant within your wheelhouse.
    - eru 2mo ago
      Nor understanding and judging a proposed design, yes.
  - neutrinobro 2mo ago
    Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?
    - lstodd 2mo ago
      I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught to us back then.
HarHarVeryFunny 2mo ago
I get the impression that the value of unproven conjectures is more in the new math and techniques that may be discovered - by humans - trying to prove/disprove them, rather than much utility in any eventual result. Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not. These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new. However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go. Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.
- lacunary 2mo ago
  > are using known math to solve them rather than inventing anything new. Isn't a new proof new math? If not, what qualifies as new math?
  - davidivadavid 2mo ago
    My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy. An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?
nsxwolf 2mo ago
Maybe the "beautiful, elegant" math is really just accidentally that way, just the tiny cross section our dumb human brains can understand. The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.
- stymaar 2mo ago
  > The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless. It is, provably: per Curry–Howard correspondence, any program you write is a proof of a theorem, and it is indeed mathematically meaningless.
  - inigyou 2mo ago
    Generating a value of type "Either (Int, String) Bool" is proving that there's at least one integer and at least one string, or there's at least one valid boolean value. Except in Haskell, where it could also be an infinite loop.
goodmythical 2mo ago
The field has been dealing with this for a long time. Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0]. The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B. Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new. This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research. [0] https://www.newscientist.com/article/1997488-wikipedia-size-maths-proof-too-big-for-humans-to-check/ https://www.newscientist.com/article/1997488-wikipedia-size-...
yzydserd 2mo ago
They claimed them as advances, not breakthroughs.
chasd00 2mo ago
i doubt anyone really wants, or cares, to understand what the AI comes up with. If it's not related to your work or your name isn't associated with the discovery then I would think time is better spent on things that are.
nater5000 2mo ago
>If a math problem falls in the forest but nobody is around to understand it does it make a sound? This is not new nor unique. There is plenty of research, especially in math, which can really only be understood by a few people in the entire world. It is not uncommon for a proof to be presented by a mathematician which, initially, is only understood to that mathematician, and it can take a long time for even another mathematician who is an expert in the same field to be able to confidentially say they understood it. >I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. This is meaningless in a vacuum. If you give a novel proof in some niche subfield of topology to a competent mathematics researcher who focuses in number theory, they'd say the same thing regardless of if a human or machine wrote the proof. A bunch of "career mathematicians" on Twitter proclaiming this doesn't mean anything other than these people aren't currently equipped to understand the contents of the proofs. That's fine and normal, but the idea that anybody with a PhD in Math should be able to pick up one of these proofs and give it a skim and be able to say, "ahh, yes, quite clever, it seems so obvious in retrospect," is absurd. That's not how this kind of research works. >Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? Nobody is blindly accepting these proofs as valid. ChatGPT isn't spitting out a wall of text and proclaiming that they've solved a previously unsolved math problem while everyone is saying, "well if an LLM says it, it must be true!" lol These proofs are being checked by automated systems (which have been in-use well before LLMs have existed) as well as being checked over by actual experts who are actually capable of (and motivated to) verifying these proofs. But that work still isn't done. There's enough evidence that these companies are confident in saying these proofs are correct, but there's going to be a lot of ongoing work from people to continue to verify and, more importantly, understand these proofs. It's literally some of these people's full-time jobs to do this. >How do we tell truth from fiction? When was the last time you verified even a classical, relatively simple mathematical assertion? How often are you just relying on a larger system of experts to ensure that we're not just blindly accepting fiction as truth? That's not to try to stick it to you personally, but it's just highlight that there's an entire system in-place here that you're not aware of and that you don't have an understanding of that is working just fine including in this context. Real mathematicians aren't going to lazily start letting OpenAI assert whatever they want about their products solving these kinds of problems without heavy scrutiny.
butokai 2mo ago
Your observation is perfectly on point, I think the season of companies announcing breakthroughs might be over soon (unless they somehow manage a major achievement, P vs NP or similar). At the same time, mathematicians will be left with superintelligent machines solving the actual math for them, much like software engineers nowadays. This was unexpected, and unexpected at this scale up to a couple of months ago.
glitchc 2mo ago
My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.
- kaashif 2mo ago
  Why does being a compressed corpus of knowledge mean it can't discover things? What if it has knowledge of techniques to discover proofs? It does and that's what's happening.
- insanitybit 2mo ago
  That axiom is clearly false. AI can interact with external systems, which means it is not just compressed knowledge - it has the ability to access new information.
- anthk 2mo ago
  Pattern matching and replacing can be Turing Complete, from sed to rewriting languages. So it's just a matter of combinatorics. https://wiki.xxiivv.com/site/rewriting.html https://wiki.xxiivv.com/site/rewriting.html
somenameforme 2mo ago
I find some interesting parallels in chess, which often has lots of analogs with math to begin with. But chess went from a pure human endeavor to one where supercomputers aided by world class players finally managed to eek out a slightly suspicious win against a world champion (approximately where we are now in math) and to now a days - where your phone could easily crush the world's strongest player, who is also probably the strongest player of all time. The way the chess world adapted this was initially to try to understand the machine. After all chess, like math, is complete information - so you can easily see the computers 'thoughts' in terms of the exact moves its saying are best in a variation and how it might respond to any other idea. But it quickly became clear that this wasn't working so well. Players would regularly get positions that the computer says 'and black wins' and then proceed to lose it convincingly, simply because the positions were so extremely weird and difficult to play that even if it might be technically winning, it's the sort of position where you're walking a fine line with lots of complex moves to find. Humans aren't computers and even the best of us can't play like one in weird positions. Now a days they're taken more in balance. The computer's evaluation of a position is probably about as good as you can get, but playability matters much more in practical terms. Knowing the eval of a position doesn't really matter if you don't understand the position. Knowing the answer can help with understanding (for instance computers have radically reshaped and improved human understanding of space in chess as we noticed computers obsessing over it) but I think the days of 'oh the computer says it's winning, so I should be able to take it from here' are near to gone.
dieselgate 2mo ago
In college the punchline for all the engineer, physicist and mathematician jokes were something like "The mathematician says: Yes, there is a solution." In all serious "I don't understand any of this it's way over my head."
torginus 2mo ago
There's no honor in being stupid, yet I must admit I am stupid, as there's even less honor in being stupid and pretending otherwise. A lot of people were super hyped about OpenAI's 10 discoveries, but I still don't understand what they mean, and even if I did, what are the implications. Like, what are non-sofic groups, and what follows from the conclusion that they exist? I mean in the sense that quantum mechanics might make my head spin, but it's because of that that we have stuff like semiconductors, which have been one of the most significant discoveries. The Fourier transform is one of the reasons we have fast telecommunications and radars. What practical things are possible or might be possible due to these results?
- kadoban 2mo ago
  > What practical things are possible or might be possible due to these results? Not much, they're all fairly minor problems that have been solved, of ~entirely niche academic interest. It's so far more just that AI _can_ solve novel math problems, ones that humans didn't accidentally train the answer in and it just spit it back out.
  - mrkandel 2mo ago
    Sofic groups are a pretty big things in group theory. Its the big project of one of one of our researchers. I have no idea what it's about because im not in algebra but the algebraists say its a big thing.
randomizedalgs 2mo ago
Even the experts on the problem being solved find the writeups nearly impossible to read. Example: https://nitter.poast.org/henryquantum/status/2083623695436623915 https://nitter.poast.org/henryquantum/status/208362369543662... Seems like a disservice to the community that openai put so little effort into producing good writeups...
- boothby 2mo ago
  I think this is where Terry Tao hit the nail on the head. OpenAI has made claims, but the mathematics community does not accept garbage. It's less that they've done a disservice to the community; more that they've done a disservice to their own reputation.
- mlpoknbji 2mo ago
  I agree. Prompting 5.6 Sol to "Review these writeups and include appropriate citations and credit where appropriate" probably would have been a step in the right direction. Then again, this would require them to actually care about the mathematics.
twotwotwo 2mo ago
The field of mathematics is smart about this and knows the difference between a pile of Lean code and understanding, and mathematicians try to get from the unintuitive explanations to something that makes more sense, e.g. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the... (where incidentally Tao used a chatbot to help take apart the problem, but with a lot of interaction and work from his side). Making things understandable is mathematics, and more generally a kind of intelligence, and is crucial to continued progress. You couldn't use algebraic geometry to disprove a conjecture if people hadn't organized (what could have been just) a pile of random observations into something called algebraic geometry. Historically LLMs have done best where it's possible to train using an objectively verifiable reward function. Computer programs are pretty good on this front and so are Lean proofs. (Of course, they don't only do things you can RLVR heavily, but those have progressed fastest.) Not sure where 'making mathematical knowledge more understandable' falls on that spectrum. Understandability isn't only important for advanced math. Keeping computer programs from becoming a mess is a challenge in high-level organization too, and the chat with the user is an explanation task. If you look online at what people say about large LLM-built codebases (SlopCodeBench is a neat effort to make make it concrete, but common wisdom seems to mostly agree on the general problem) and chatbot prose, I don't think everyone considers those solved problems! It's hard to tell how thoroughly the labs grasp and care about this at an organization-wide level. I'm sure at least some maybe-results exist inside labs but haven't been published because the humans couldn't verify them and didn't want to be embarrassed with a false result. (Maybe also why counterexamples are a lot of the first results published: often simple to verify, even if hard to obtain.) A good sign would be if results in a few months come out more like what mathematicians consider well-written papers explaining results in a more intuitive way, fewer shocking announcements of bare counterexamples in tweets. It's probably a slow climb to get there.
HoldOnAMinute 2mo ago
I, for one, welcome our giant supercomputer overlords.
bluefirebrand 2mo ago
> Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands An unsettling number of people do seem to be more than ready for this. In fact I'd say they seem almost gleeful about it. Thinking is hard and they don't seem to like doing it!
layer8 2mo ago
> How can we possibly make use of these breakthroughs if we don’t understand them? I mean, arguably the development of LLMs matches that description.
lo_zamoyski 2mo ago
> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them? Theoretical science isn't about usefulness per se, but knowledge and understanding for its own sake, so a better way to say this is to point out that you only benefit in such cases if you understand things yourself. If understanding is the goal - which it is in the case of true theory - then the only way to attain that goal is to actually attain understanding. What good is it if an LLM produces a valid proof, but no one grasps it? This isn't like digging a ditch where it doesn't matter who does it or how he does it as long as you have a ditch. Here, the ditch is knowledge, as it were.
casey2 2mo ago
It's more like FOSS project using and you ask a developer who works at a different company how to build this project and he says "I don't know automake, perl and there are lots of macros" What are the odds that this random FOSS project has solved the problem of building software and nobody noticed. Close to 0. Don't confuse accidental complexity for transcendental depth.
scotty79 2mo ago
> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them? Isn't that what people from more practical sciences said about math anyways? All math is eventually applied math.

tumdum_ 2mo ago

Latest presentation of Terence Tao on what current advancements in AI mean for math discusses (among other things) those issues: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...

glitchc 2mo ago
Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
- boothby 2mo ago
  No. More like "I am mathematician and I have digested this sloppy writing and understand this to be an application of Foo theory to the Bar problem with a Baz twist. The prior publication omits crucial references to..."
  - glitchc 2mo ago
    The writing's already pretty good, even Terrence Tao calls it flawless, if you read the slides. That's not where the LLM requires help.

serial_dev 2mo ago

I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)". Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?

svachalek 2mo ago
Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.
- sarchertech 2mo ago
  To continue the analogy, it’s like writing answers that are so convoluted that no one can understand them.
- HarHarVeryFunny 2mo ago
  Yes, although a counterpoint to that is that if these weren't long-standing published problems that have stumped humans, would anyone care? What is important here, the problem/proof, or the fact that they are famous problems that have stumped humans (and therefore make good bragging material). I'm not sure that a problem being famous, or hard for a human, is the same as it being important. Take Fermat's last theorem as an example - was a proof of it ever considered to be, then or now, consequential, or was it the Paris Hilton of problems - famous for being famous? A problem whose legend grew because it it so simple to state, with the challenge that there might be a simple proof, and that turned out to be so difficult to solve that it therefore became prestigious to do so?
- deleted 2mo ago
  [deleted]
- chrisjj 2mo ago
  > this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. Except OpenAI isn't showing any upvotes.

the_sleaze_ 2mo ago

I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?

saalweachter 2mo ago
So a good fraction of famous old math puzzles with no practical use are famous because they are in some way similar to problems people actually care about. If you can solve the toy problem that is in some way simpler, maybe you can use the same methods to solve the "real" problem. One open question is whether these machine solutions to these problems will act as springboards to future research, either when given to human mathematicians, or when used to train future machine models.
Levitz 2mo ago
It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.
- stymaar 2mo ago
  True. Though ~99% is still useless in the end.
  - boothby 2mo ago
    We are not at the end, how could you possibly be confident in this claim?
  - wongarsu 2mo ago
    As they say: 80% of marketing is useless, you just don't know in advance which 80%. Same with math
LPisGood 2mo ago
One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.

qsort 2mo ago

> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Huh? Mathematics is an extremely wide subject, it's perfectly normal even for two professional mathematicians not to understand each other's work. Have you considered that maybe they just don't work in that area? Are you implying OpenAI's paper (which was, by the way, edited by humans and provided Lean certificates for most of the proofs) is actually gibberish? That's flat-earth levels of conspiracy.

WarmWash 2mo ago

Mathematics is an unusually dense (if not the most dense...by a few large steps) field. So lots of areas of mathematics are extremely deep and narrow without any real shortcuts, even for seasoned mathematicians.

gglitch 2mo ago
Awesome observation. What would you consider denser?
- Iolaum 2mo ago
  QWEN-3.6-27B P.S. Couldn't resist :p
  - HarHarVeryFunny 2mo ago
    Unlike 3.6 35B. Is it dense or MoE? It's a good model Sir!
- Jensson 2mo ago
  Physics possibly.
  - WarmWash 2mo ago
    Physics has bounds of concern, where as mathematics would encompass all physics, as well as all possible alternative physics, in all possible forms. Maybe not unbounded, but close to it.

hgoel 2mo ago

This seems like an extreme exaggeration from a few people claiming to not understand a very recent result. This cycle of a new result being discovered, and reaearchers needing some time to truly digest and disassemble it, is normal.

a_conservative 2mo ago

I have the same questions about human mathematicians! I can't tell if xkcd #435 is still true, or if math is just as mushy as everything else seems to be. When a math proof can only be understood by a handful of people, what does that mean about that proof? I think the LLMs are pushing a problem that existed already and pushing it further. [0] https://xkcd.com/435/ https://xkcd.com/435/

eru 2mo ago
That's why OpenAI also published machine-checkable proofs. The process is very, very faintly similar to running a typechecker over your software sources.
- parineum 2mo ago
  They're publishing machine checkable proofs because that's the only way they, themselves, can check them. They don't understand the math either.
  - inigyou 2mo ago
    For 3.5 days we had a machine checkable proof the Collatz conjecture was false. There turned out to be a bug in the machine checker.
    - eru 2mo ago
      Yes, however any LLM agent you asked about that proof could tell you that. (And even most humans who know a little bit about the bug.) You are right that Lean isn't great in this respect, and people are working on proof formalisations that are less prone to bugs.

wat10000 2mo ago

This has been an ongoing debate since the computerized proof of the four color theorem fifty years ago.

eru 2mo ago

> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.

lstodd 2mo ago
I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.
- goodmythical 2mo ago
  Being familiar with the basics of fission power generation does not put the design of a competitive plant within your wheelhouse.
  - eru 2mo ago
    Nor understanding and judging a proposed design, yes.
- neutrinobro 2mo ago
  Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?
  - lstodd 2mo ago
    I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught to us back then.

HarHarVeryFunny 2mo ago

I get the impression that the value of unproven conjectures is more in the new math and techniques that may be discovered - by humans - trying to prove/disprove them, rather than much utility in any eventual result. Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not. These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new. However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go. Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.

lacunary 2mo ago
> are using known math to solve them rather than inventing anything new. Isn't a new proof new math? If not, what qualifies as new math?
- davidivadavid 2mo ago
  My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy. An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?

nsxwolf 2mo ago

Maybe the "beautiful, elegant" math is really just accidentally that way, just the tiny cross section our dumb human brains can understand. The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.

stymaar 2mo ago
> The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless. It is, provably: per Curry–Howard correspondence, any program you write is a proof of a theorem, and it is indeed mathematically meaningless.
- inigyou 2mo ago
  Generating a value of type "Either (Int, String) Bool" is proving that there's at least one integer and at least one string, or there's at least one valid boolean value. Except in Haskell, where it could also be an infinite loop.

goodmythical 2mo ago

The field has been dealing with this for a long time. Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0]. The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B. Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new. This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research. [0] https://www.newscientist.com/article/1997488-wikipedia-size-maths-proof-too-big-for-humans-to-check/ https://www.newscientist.com/article/1997488-wikipedia-size-...

yzydserd 2mo ago

They claimed them as advances, not breakthroughs.

chasd00 2mo ago

i doubt anyone really wants, or cares, to understand what the AI comes up with. If it's not related to your work or your name isn't associated with the discovery then I would think time is better spent on things that are.

nater5000 2mo ago

>If a math problem falls in the forest but nobody is around to understand it does it make a sound? This is not new nor unique. There is plenty of research, especially in math, which can really only be understood by a few people in the entire world. It is not uncommon for a proof to be presented by a mathematician which, initially, is only understood to that mathematician, and it can take a long time for even another mathematician who is an expert in the same field to be able to confidentially say they understood it. >I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. This is meaningless in a vacuum. If you give a novel proof in some niche subfield of topology to a competent mathematics researcher who focuses in number theory, they'd say the same thing regardless of if a human or machine wrote the proof. A bunch of "career mathematicians" on Twitter proclaiming this doesn't mean anything other than these people aren't currently equipped to understand the contents of the proofs. That's fine and normal, but the idea that anybody with a PhD in Math should be able to pick up one of these proofs and give it a skim and be able to say, "ahh, yes, quite clever, it seems so obvious in retrospect," is absurd. That's not how this kind of research works. >Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? Nobody is blindly accepting these proofs as valid. ChatGPT isn't spitting out a wall of text and proclaiming that they've solved a previously unsolved math problem while everyone is saying, "well if an LLM says it, it must be true!" lol These proofs are being checked by automated systems (which have been in-use well before LLMs have existed) as well as being checked over by actual experts who are actually capable of (and motivated to) verifying these proofs. But that work still isn't done. There's enough evidence that these companies are confident in saying these proofs are correct, but there's going to be a lot of ongoing work from people to continue to verify and, more importantly, understand these proofs. It's literally some of these people's full-time jobs to do this. >How do we tell truth from fiction? When was the last time you verified even a classical, relatively simple mathematical assertion? How often are you just relying on a larger system of experts to ensure that we're not just blindly accepting fiction as truth? That's not to try to stick it to you personally, but it's just highlight that there's an entire system in-place here that you're not aware of and that you don't have an understanding of that is working just fine including in this context. Real mathematicians aren't going to lazily start letting OpenAI assert whatever they want about their products solving these kinds of problems without heavy scrutiny.

butokai 2mo ago

Your observation is perfectly on point, I think the season of companies announcing breakthroughs might be over soon (unless they somehow manage a major achievement, P vs NP or similar). At the same time, mathematicians will be left with superintelligent machines solving the actual math for them, much like software engineers nowadays. This was unexpected, and unexpected at this scale up to a couple of months ago.

glitchc 2mo ago

My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.

kaashif 2mo ago
Why does being a compressed corpus of knowledge mean it can't discover things? What if it has knowledge of techniques to discover proofs? It does and that's what's happening.
insanitybit 2mo ago
That axiom is clearly false. AI can interact with external systems, which means it is not just compressed knowledge - it has the ability to access new information.
anthk 2mo ago
Pattern matching and replacing can be Turing Complete, from sed to rewriting languages. So it's just a matter of combinatorics. https://wiki.xxiivv.com/site/rewriting.html https://wiki.xxiivv.com/site/rewriting.html

somenameforme 2mo ago

I find some interesting parallels in chess, which often has lots of analogs with math to begin with. But chess went from a pure human endeavor to one where supercomputers aided by world class players finally managed to eek out a slightly suspicious win against a world champion (approximately where we are now in math) and to now a days - where your phone could easily crush the world's strongest player, who is also probably the strongest player of all time. The way the chess world adapted this was initially to try to understand the machine. After all chess, like math, is complete information - so you can easily see the computers 'thoughts' in terms of the exact moves its saying are best in a variation and how it might respond to any other idea. But it quickly became clear that this wasn't working so well. Players would regularly get positions that the computer says 'and black wins' and then proceed to lose it convincingly, simply because the positions were so extremely weird and difficult to play that even if it might be technically winning, it's the sort of position where you're walking a fine line with lots of complex moves to find. Humans aren't computers and even the best of us can't play like one in weird positions. Now a days they're taken more in balance. The computer's evaluation of a position is probably about as good as you can get, but playability matters much more in practical terms. Knowing the eval of a position doesn't really matter if you don't understand the position. Knowing the answer can help with understanding (for instance computers have radically reshaped and improved human understanding of space in chess as we noticed computers obsessing over it) but I think the days of 'oh the computer says it's winning, so I should be able to take it from here' are near to gone.

dieselgate 2mo ago

In college the punchline for all the engineer, physicist and mathematician jokes were something like "The mathematician says: Yes, there is a solution." In all serious "I don't understand any of this it's way over my head."

torginus 2mo ago

There's no honor in being stupid, yet I must admit I am stupid, as there's even less honor in being stupid and pretending otherwise. A lot of people were super hyped about OpenAI's 10 discoveries, but I still don't understand what they mean, and even if I did, what are the implications. Like, what are non-sofic groups, and what follows from the conclusion that they exist? I mean in the sense that quantum mechanics might make my head spin, but it's because of that that we have stuff like semiconductors, which have been one of the most significant discoveries. The Fourier transform is one of the reasons we have fast telecommunications and radars. What practical things are possible or might be possible due to these results?

kadoban 2mo ago
> What practical things are possible or might be possible due to these results? Not much, they're all fairly minor problems that have been solved, of ~entirely niche academic interest. It's so far more just that AI _can_ solve novel math problems, ones that humans didn't accidentally train the answer in and it just spit it back out.
- mrkandel 2mo ago
  Sofic groups are a pretty big things in group theory. Its the big project of one of one of our researchers. I have no idea what it's about because im not in algebra but the algebraists say its a big thing.

randomizedalgs 2mo ago

Even the experts on the problem being solved find the writeups nearly impossible to read. Example: https://nitter.poast.org/henryquantum/status/2083623695436623915 https://nitter.poast.org/henryquantum/status/208362369543662... Seems like a disservice to the community that openai put so little effort into producing good writeups...

boothby 2mo ago
I think this is where Terry Tao hit the nail on the head. OpenAI has made claims, but the mathematics community does not accept garbage. It's less that they've done a disservice to the community; more that they've done a disservice to their own reputation.
mlpoknbji 2mo ago
I agree. Prompting 5.6 Sol to "Review these writeups and include appropriate citations and credit where appropriate" probably would have been a step in the right direction. Then again, this would require them to actually care about the mathematics.

twotwotwo 2mo ago

The field of mathematics is smart about this and knows the difference between a pile of Lean code and understanding, and mathematicians try to get from the unintuitive explanations to something that makes more sense, e.g. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the... (where incidentally Tao used a chatbot to help take apart the problem, but with a lot of interaction and work from his side). Making things understandable is mathematics, and more generally a kind of intelligence, and is crucial to continued progress. You couldn't use algebraic geometry to disprove a conjecture if people hadn't organized (what could have been just) a pile of random observations into something called algebraic geometry. Historically LLMs have done best where it's possible to train using an objectively verifiable reward function. Computer programs are pretty good on this front and so are Lean proofs. (Of course, they don't only do things you can RLVR heavily, but those have progressed fastest.) Not sure where 'making mathematical knowledge more understandable' falls on that spectrum. Understandability isn't only important for advanced math. Keeping computer programs from becoming a mess is a challenge in high-level organization too, and the chat with the user is an explanation task. If you look online at what people say about large LLM-built codebases (SlopCodeBench is a neat effort to make make it concrete, but common wisdom seems to mostly agree on the general problem) and chatbot prose, I don't think everyone considers those solved problems! It's hard to tell how thoroughly the labs grasp and care about this at an organization-wide level. I'm sure at least some maybe-results exist inside labs but haven't been published because the humans couldn't verify them and didn't want to be embarrassed with a false result. (Maybe also why counterexamples are a lot of the first results published: often simple to verify, even if hard to obtain.) A good sign would be if results in a few months come out more like what mathematicians consider well-written papers explaining results in a more intuitive way, fewer shocking announcements of bare counterexamples in tweets. It's probably a slow climb to get there.

HoldOnAMinute 2mo ago

I, for one, welcome our giant supercomputer overlords.

bluefirebrand 2mo ago

> Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands An unsettling number of people do seem to be more than ready for this. In fact I'd say they seem almost gleeful about it. Thinking is hard and they don't seem to like doing it!

layer8 2mo ago

> How can we possibly make use of these breakthroughs if we don’t understand them? I mean, arguably the development of LLMs matches that description.

lo_zamoyski 2mo ago

> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them? Theoretical science isn't about usefulness per se, but knowledge and understanding for its own sake, so a better way to say this is to point out that you only benefit in such cases if you understand things yourself. If understanding is the goal - which it is in the case of true theory - then the only way to attain that goal is to actually attain understanding. What good is it if an LLM produces a valid proof, but no one grasps it? This isn't like digging a ditch where it doesn't matter who does it or how he does it as long as you have a ditch. Here, the ditch is knowledge, as it were.

casey2 2mo ago

It's more like FOSS project using and you ask a developer who works at a different company how to build this project and he says "I don't know automake, perl and there are lots of macros" What are the odds that this random FOSS project has solved the problem of building software and nobody noticed. Close to 0. Don't confuse accidental complexity for transcendental depth.

scotty79 2mo ago

> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them? Isn't that what people from more practical sciences said about math anyways? All math is eventually applied math.