When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute.
On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.
> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension.
But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.
The conjecture had to do with whether one convex polygon could be continuously deformed into another while remaining convex, under certain conditions and constraints. The answer turns out to be no, but surprise and disappointment are understandable reactions to that outcome. It was indeed much more practical for a young grad student to look for a clever misbehaving polygon than to try to prove something about all of them at once.
> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it.
This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person.
Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when that lecturer told a bunch of first-year students "here are some things we haven't figured out which you can help with, let me know if you come up with something". It was inspiring and a great introduction to what's otherwise a rather dull first couple of years of academia.
> During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]
This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
I don’t think that’s true. Lots of mathematicians excel at (and revel in) finding weird counterexamples and love the strangeness of it all. John H Conway being my favourite- Look up the Conway knot[1] or the Conway base 13 function[2] for famous examples.
Ugliness is in the eye of the beholder. Lots of counterexamples are very beautiful. For example, the Dirichlet function (f(x) = 1 if x is rational, 0 otherwise) is a source of very beautiful counterexamples. Eg it is discontinuous everywhere but f restricted to only rational numbers is continuous on all rationals and likewise f restricted to irrationals is continuous on the set of irrationals (R-Q).
[1] The Conway knot has 11 crossings yet shares the same Alexander polynomial as the “unknot” which has no crossings at all. It took 50 years to decide the question of whether it has a basic property known as “sliceness” https://en.wikipedia.org/wiki/Conway_knot
[2] Conway’s base 13 function
Was invented as a counterexample to the converse of the intermediate value theorem. That is, it satisfies the intermediate value property while being everywhere discontinuous (which breaks my brain completely) https://digitalresearch.bsu.edu/mathexchange/wp-content/uplo...
This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.
> he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
For a more extreme (although somewhat inverted) version of this, see Zeeman. He spent years trying to find a knotted sphere in a 5D space. Then realised this was impossible and got a proof for it in a few hours. [1]
> I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.
This was back when the internet was mostly chain emails and personal web pages, being unreachable once you went home for the weekend was perfectly normal and expected, and automatically thinking the worst of people was not a common form of public performance art. ;)
> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it.
It was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?
Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up spending years working at a Subway[1].
Imagine Zhag had ChatGPT in 1986 when he started working on the Jacobian Conjecture.
[1] Of course now this has become an inspiring story. That said, the story definitely invokes complex emotions. The best way to describe it is probably this Chinese poem, which I have no idea how to translate: 庾信平生最萧瑟，暮年诗赋动江关
I was once in a presentation for a math PhD thesis. During the thesis, the evaluator of the thesis noticed a flaw in their proof. The student understood and then asked “What now?” The evaluator prof simply shrugged.
I personally know a story in this vein with a (sort of) happy ending.
A PhD student discovered that a result of his professor would imply the solution to a big conjecture in another field. The people in that field then analyzed the prof's result and found that the proof was flawed. The student was still allowed to graduate based on this since the finding of the connection between fields was brilliant. Then he quit academia (not because of this story, he had planned it before). Then a year later the prof figured out how to fix the flaw in his proof and published a paper with his former student, thus solving the conjecture. The two are still on good terms, writing papers together.
I first heard that kind of story about a thesis defense at Princeton. The twist was that they had been short one person to judge it, so they roped in a professor who was available at that moment ... and he found a counterexample on the fly.
A recording of a car crash: discovering on live radio/podcast that the central tenet of your book is wrong, and amateurishly so.
Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
Inspiring? Because of the twin prime conjecture success following his time in the wilderness? I suppose so.
I'm tired of tales like this in academics though. That's not a criticism of you for telling the tale, I'm just so tired of this kind of thing in academics in general. So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.
As my own research has drifted more into math, I've been surprised at how many assertions in the literature turn out to be false. Not just false, but propagated into the applied literature extensively, and even when you point out the problems a lot of defensiveness and denial about it along the lines of Zhang's story.
I agree about wondering what would have happened if LLMs had been around in 1986. My guess is the outcome would have been the same for the same reasons?
My experience with LLMs in proofs is they can be very helpful, but also very wrong. It's like having another person with another set of hunches about what path to go down.
> So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.
Very true. Unfortunately, when there are people, there will be politics. I remember when reading Yau's autobiography, I kept marvel how much calculation, or "politics" if you will, that Yau mentioned or implied in the book.
> My guess is the outcome would have been the same for the same reasons?
At least Zhang didn't have to spend 7 years working on the Jacobian conjecture. He said in an interview that he always wanted to work on number theory. Moh asked him to work on Jacobian, and he obliged.
Planck's Principle: 'Science advances one funeral at a time.' [1]
In his exact words: "A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it... An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth."
And he said that having lived, as a outsized figure, through the late 19th to mid 20th century of physics, which was the absolute golden age for such.
That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions.
and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.
Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.)
Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures
You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.
Are you thinking of proof by contradiction, which is rejected by constructionism?
[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.
> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.
Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician.
Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?
Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.
It is also a good thing, because it helps to refine the theorem statement. At least, my humble experience in CS theory research is that I’d try to prove a theorem I want to be true, find a counterexample, refine the statement, and continue.
P.S. It helps that in CS lots of theorems are about either inductive or coinductive definitions.
Except you can't possibly know that. New insight can arise regardless of whether mathematicians are trying to prove or disprove a statement, and regardless of whether the statement ultimately turns out to be true or false.
Human mathematicians have been being out-counterexampled for at least two decades. The main difference, as I understand, is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so. But the sizes of counterexample we're seeing are around about what I'd expect pre-generative-AI counterexample search systems to be able to find.
It's not easy to find a counterexample to the Jacobian conjecture, by any means – by which I mean to say that naïve brute-force search will take too long – but the scope of existing searches listed on Wikipedia[0] suggest that many tricks are already known, and that people just hadn't looked, systematically, for a counterexample in three variables before. Wikipedia writes:
> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.[17][18]
where reference 17 is from 1983, and reference 18 is a preprint with no given date. Knowing very little about this problem, my impulse is to side with the unnamed faculty member cited in the article:
> [who] said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem,
For context, the auto-generated counterexample is in three variables, has degree 7, and was discovered in 2026.
> about what I'd expect pre-generative-AI counterexample search systems to be able to find.
The difference today (and the reason why everyone is excited about it) is that the same system that does advanced math can write poetry, play an above average game of chess, code frontend/backend stuff and do cybersec. These are not "expert systems", nor are they trained for each task individually. That's the catch.
> just indicated that humans had not spent enough time thinking about the problem
Heh, this is a weak excuse. We've seen variations on this theme every time something cool gets solved by the models.
> The conjecture was first stated for two variables by Ludwig Kraus in 1884 [...] an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
> The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. It was notorious for the large number of published and unpublished false proofs that turned out to contain subtle errors.
> A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy. In fact during the workshop I learnt from Harvard PhD student Bryan Wang that Harvard were already giving free Fable access to all PhD students, post-docs and faculty at Harvard.
Yeah, given how much it accelerates grad students to produce meaningful output more quickly, why wouldn’t you make an investment of $2400/student/year. Seems like pennies overall.
Many graduate students view themselves as ethical beings, not machines that "produce meaningful output," and everybody here knows (useful) LLMs are indefensibly evil because of stolen training data and enormous environmental impact.
I suppose it will fall to AI as well to compose the mathematical equivalent of The Ballad of John Henry. Who will be the human champion, the last great hero who can deliver proofs "from the book" that a machine cannot outperform?
This is an unhealthy view of mathematics. It's mathematics as envisioned by football fans.
The most valuable things in mathematics are not beautiful proofs. We need more useful definitions. Actually coming up with useful definitions (and building good conjectures out of them - not even theorems, conjectures) is something LLMs have not yet tried to conquer.
It's probably not quite that dramatic yet, though it seems possible it will get there, maybe even soon.
There's no structural reason to expect acceleration any more or less than an asymptotic behavior (if even that, acceleration is probably the bigger ask). Different problems yield to a new solvent, maybe that's also more, but it could go either way and we definitionally don't know yet because we don't understand the convexity of AI capability, we cannot directly access it interiority, we don't know if it's sandbagging (other than that it does sometimes, it can). It's an emergent phenomemon that might actively resist measurement. Or it might be as predictable as a clock in a few years.
No one knows, or if they do, they aren't talking. The loud people don't know anything.
I wish I had LLM-built Lean formalisations in university, so much of the math in the slides had errors, and some professors are very bad and ungracious admitting it, while simultaneously rejecting requests for clarifications by saying "the proof is in the slides".
Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.
> Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.
I would object to the first part: Of course there is a nontrivial learning curve, but then I'd argue that non-slop Lean/Agda/Rocq/... formalizations are amazing for understanding proofs. A good formalization presents the outline and the key arguments in nicely structured form, and then, unlike pen-and-paper proofs, also allow you to get the details on every single step, exactly to your desired level of depth.
The proofs in Martín Escardó's TypeTopology Agda repository come immediately to my mind as an example. [An interactive Agda tutorial is here: lets-play-agda.quasicoherent.io]
In contrast, LLM-generated formalizations can currently be extremely messy. They certify truth and can also contain interesting arguments, but substantial work is required to bring them into a shape that contributes to the actual goal of improving our understanding of the mathematical landscape.
mathematicians have been using computers for well over half a century, but this was after "bounding" the problem first and then running through the cases with a computer. Now AI is doing the first part. However, mathematicians are still needed at crafting prompts, and knowing where to look, still. The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.
For now. I wonder if we will ever get to the point where the computer starts doing mathematics that we just can't understand. Surely there must be some limit to what we can understand (like how a gorilla will never understand prime numbers, there are probably limits to our intelligence as well).
> The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.
Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.
I don't think there was a 'prompt' for it, rather a long and dedicated work of a professional mathematician which involved LLM in some capacity. I'm sure the search step wasn't an ad-hoc script running in a Claude Code session (as somebody would naively assume), it was an optimized numerical code running in Anthropic's compute cluster. Note that details are not published yet and `__alpoge__` is officially working at Anthropic.
How do mathematicians view counter examples? Is it like an unexpected result in the physical sciences: annoying in the moment but potentially stupendously important as it reveals some inaccuracy in the current models? Or is it more like a bug report in coding… probably just, another little annoying detail?
Counterexamples are clarifying. For everyone condition in a proof, it's really handy to have a maximally simple, memorable counterexample that makes it fail because it violates that condition. Mathematicians tend to walk around with a bestiary of counterexamples in their heads. It also makes it really easy to recover a theorem because you try to sketch out the statement, and the spiky, memorable counterexamples jump out of your memory and you add conditions to constrain the domain away from them.
There are pedagogical books (CF. 'Counterexamples in Topology', 'Counterexamples in Analysis') that teach the nuances of subjects through counterexamples. They're popular as it's sometimes easier to learn details from a pathology or degenerate example than from just learning what is intended.
I am sure finding holes in existing proofs is what counts as "another little annoying detail". Some people are probably relieved when they have failed to prove a hypothesis and someone finds a counterexample.
If the poster's (is it Kevin Buzzard?) suggestion works out and AI finds a counterexample to the Hodge conjecture, that would be a really big deal. It's one of the Millenium problems, for example.
One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.
I think he was being provocative and maybe a bit tongue-in-cheek when he said that. A candidate object alone doesn't resolve the Hodge Conjecture. Any apparent counterexample would have to prove that no algebraic cycle exists, no invariant subspace exists, or that every element of an infinite ideal is nilpotent. Much harder, but not impossible.
Indeed I was being slightly tongue-in-cheek -- but if you ask geometers whether they believe the Hodge conjecture then you certainly don't always get an unqualified "yes"! This is in contrast to e.g. asking number theorists whether they believe Birch--Swinnerton-Dyer, where they are almost always very confident.
A lot of this math is beyond my comprehension, but it often seems to talk of proofs of theorems. What I want to know is if we continue on this accelerated AI mathematics trajectory, will we eventually be discovering new forms of math that will in turn have some applications down the line in engineering or biomedicine etc? I guess what I’m asking is are we on the cusp of a huge breakthrough for humanity, or largely just proving what was already known?
It’s possible. Compressed sensing is one example of what I suppose you could call a new form of math with applications in biomedicine. It can be used to significantly shorten the time required to obtain a MRI scan, which can improve the patient experience and enable more patients to access MRIs.
Engineering and biomedicine, probably in the long term (if at all). But accelerated development of new mathematical methods has a possibility of proving to be relevant for fundamental physics research.
Occasionally large improvements in our models of the universe have been associated with the development of mathematical tools that allow those models to be expressed and/or tested.
You can personally work out how apply this one in engineering today!
If you have a spare hour or two, I'd encourage you to have a go at learning (however you learn best - I like just asking smarter people or robots stupid questions) what the maths means and why it's important.
And then once you feel like you have a vague grip on the principles, think about a problem in a domain you know a lot about. Try to see if the maths - and how it's changed our perception - could be used as a tool to solve that problem, or if the solution is analogous to a solution you could try in your own domain of expertise.
LLMs are good at speeding up, I think, the journey an idea has to take between "theoretical academic stuff for academics" and "a usable idea for regular people", because they increasingly allow you to ask an infinite number of stupid questions and give you (hopefully) reasonably good responses.
I've had loads of fun doing this today - specifically seeing if the idea this counter (from what I understand: a many-to-one conversion that kind of does and kind of does not preserve meaning) can tell me anything about the relationship between language and meaning.
I'm sure everything I've done today while mucking around has been the equivalent of a monkey with a typewriter (and Codex), but I think the huge breakthrough(s) you ask whether we're on the cusp of are relatively dependent on how many monkeys are throwing typewriters at problems they know a little bit about, after learning a bit about new ideas like this one. Historically, that's a really good way for broad cultural innovation to happen - distributed information applied across multiple domains by experts in them.
Most of maths is remarkably abstract and impractical, but sometimes real-world scenarios turn out to be related to some obscure branch of math. Internet security is based on elliptic curves in finite fields, and why would you ever study that if it wasn't powering internet security? Well some people did study it before, for no good reason, and that's how we knew about it.
> A member of the faculty (who I won’t name) said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem, implying that a 60-year-old question of Grothendieck was not actually that interesting to work on. I didn’t tell him that at some point earlier in my career I had spent a week working hard on the problem. In my mind my colleague is just going through the five stages of grief; right now they seem to be in the denial phase.
It seems to me also that the very vocal anti-LLM crowd are in the denial phase of grief.
When I was still much more skeptical of AI, I asked ChatGPT to write me a proof of the Goldbach Conjecture. Of course it didn't, but it gave me a several-screens-long research program for how one might get there, with a few alternate paths and what pieces are still missing from each one. Maybe it cribbed that all from some grad student's blog, but it was still pretty impressive.
A week or two ago I asked Claude Code to write a comprehensive testing plan for a new Postgres feature I wrote, UPDATE/DELETE FOR PORTION OF. I was a bit anxious about how many bugs were discovered as soon as it was merged this spring. Claude found some untested areas, then it wrote more tests for them. I'm not talking about LOC covered, but feature combinations. (I've been meaning to submit this as a followup patch. . . .) Fortunately it didn't find any more bugs.
I sense that testing plan has affinity with the findings in the OP, even though it is far more humble than research mathematics. Even better would be if we knew good ways to express invariants about Postgres's behavior, and then we could ask LLMs to violate them. I'm sure there are good ways already, and the "we" who is not knowing is not "all humans" but "the Postgres team" or just "me".
As a counterexample to the article (heh): even though Claude didn't uncover any new bugs, a human did, just a few days ago. Alas!
Isn't this exactly what we should expect/hope for?
At least initially, chess computers were better than humans not because they were more creative or inspired, but because they thought harder/deeper. That's exactly the sort of "find counterexamples to this if they exist" work that we're seeing here.
Eventually computer chess got to the point where humans look at some moves and say, "That's an amazing, inspired move. That's not a 'bot' move at all, I can learn from this." We're just not there yet with AI/math in general.
This will soon happen with theoretical physics, computer science, and everything which can be verified cheaply. Then, we will have long running projects augmented by agents for 2-4 years while AI companies are collecting data of human workflows. After that we will see AI being able to do those projects by themselves. This will lead to super fast human progress and cheap products. The price of things will be bound by energy and natural resources. Interesting times are ahead of us
I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math.
Maybe in the future the work of Mathematicians will be like the ones of SWEs with AI, check thousands of lines of AI generated proof and find the subtle errors
>I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math.
That point had come some time ago. Nowadays the literature is both enormous and littered with false proofs and an unknown, but nonzero, number of false published results.
Yes it compresses the proofs but still "Sol had generated 1.2 million lines of Lean code in the three weeks that it had worked on the project". I mean how do you even verify that?
This comment gives a sliver of hope for us currently outside academia and in low income academic institutions: enthusiastic laypeople with access (read: funds) to the SOTA models can collaborate with destitute researchers to produce research the academic alone could not.
Plus, as a non-expert, you will naturally want to understand more about what you are proving together with the expert. LLMs can help there, too, by carving a path from elementary mathematics to the research problem more efficiently and in a more targeted manner than a generic exposition or survey.
That could give birth to a beautiful research-exposition pair that can benefit both academics and interested laypeople, who have (rightfully, but inevitably) felt excluded from the insights of high level research. I have long hoped for something like that. There, however, the academic must watch the LLM like a hawk, because expository interpretation of results is prime ground for hallucinations, and adversarial agents will not have much effect in improving it.
This morning I was reading a paper called "An enduring error" (2009) by Branko Grünbaum, regarding Archimedean polyhydra. From the introduction:
> ..Even more unexpected is the fact that many expositions of this topic commit serious mathematical and logical errors. Moreover, this happened not once or twice, but many times over the centuries, and continues to this day in many printed and electronic publications; the most recent case is in the second issue for 2008 of this journal. I will justify this harsh statement soon, after setting up the necessary background.
This is not new, most famous conjectures are believed to be true mainly because an already enormous brute force search was conducted finding no counter-example.
A large fraction of the problems assigned in the Ross Program were of the form "Prove or disprove, and salvage if possible." The rest were usually a calculation, meant to motivate a general proposition you would encounter soon after.
I must object. Because this is derivation, not composition. German is noteworthy for its long compositional word formations (though many other language are, too, e.g., Finnish). German can do derivation, too, but not significantly more than English (and the verb prefix 'out-' is hard to map into German in this case). For admiring derivation, you'd have to point to, e.g., Greenlandic instead (e.g. _iliorfigeqatigiissariaqaraluarput_ roughly 'they should work together' (from the Declaration of Human Rights)).
To be clear, I do like 'outcounterexampled' a lot!
Long story short this is much less impressive than it was sold to be. Basically they had mathematicians combing through long winding chains of thought (incidentally: you wouldn't have access to that reasoning) and cleaning it up and making it coherent. That doesn't mean its unimportant, but we're being gaslit about the amount of human steering and human effort that went into this.
A plea: please stop upvoting hype that comes from these labs. It takes time to evaluate their claims and they're always less impressive than claimed.
so if I tell a model "using your advanced knowledge of physics and chemistry 'make alchemy work' (synthesize gold using any cheaper materials) using safe materials legal for a residential hobby chemist with 1 semester of lab work in college to possess and use (this is obviously the really hard part) using less than $1,000 in lab equipment and input materials that can create $2,000 in value at market rate; then walk me through all the steps to do this safely and legally without anyone finding out except the lab equipment sellers; and tell me what a reasonable story to tell gold purchasers regarding where I got it; I'd like to end up selling a few thousand dollars of it without disrupting the market. Give me practical advice about good opsec so that nobody steals the method you come up with (I don't want my home broken into by thieves who suspect I figured out how to transmute cheap materials into gold), other than, obviously, not to post about it. Think as long as you need to about the chemistry and how to do it, you're a chemistry expert and can figure it out even if it takes you like a week, in your web searches be careful not to divulge that you're figuring out how to synthesize gold", and I give that prompt to some model that knows chemistry like the back of its hand, it thinks about it for four hours, finds the correct safe and legal steps, and gives me the recipe and the advice I asked for, then who figured out how to turn aluminum (or another cheap element) into gold, me or the model? In mathematics, proving or disproving a well known and well studied one hundred year old conjecture is gold.
When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute.
On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.
> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension.
But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.
We were studying geometry - my adviser was the great Branko Grünbaum: https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum
The conjecture had to do with whether one convex polygon could be continuously deformed into another while remaining convex, under certain conditions and constraints. The answer turns out to be no, but surprise and disappointment are understandable reactions to that outcome. It was indeed much more practical for a young grad student to look for a clever misbehaving polygon than to try to prove something about all of them at once.
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I think the asymmetry depends on the representation
> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it.
This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person.
Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when that lecturer told a bunch of first-year students "here are some things we haven't figured out which you can help with, let me know if you come up with something". It was inspiring and a great introduction to what's otherwise a rather dull first couple of years of academia.
https://en.wikipedia.org/wiki/George_Dantzig
> During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]
This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
That's not why.
It's because counterexamples are easy compared to proofs which require new mathematics.
GenAI is great at combining existing things in new ways (interpolation). It's terrible at creating new things from scratch (extrapolation).
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I don’t think that’s true. Lots of mathematicians excel at (and revel in) finding weird counterexamples and love the strangeness of it all. John H Conway being my favourite- Look up the Conway knot[1] or the Conway base 13 function[2] for famous examples.
Ugliness is in the eye of the beholder. Lots of counterexamples are very beautiful. For example, the Dirichlet function (f(x) = 1 if x is rational, 0 otherwise) is a source of very beautiful counterexamples. Eg it is discontinuous everywhere but f restricted to only rational numbers is continuous on all rationals and likewise f restricted to irrationals is continuous on the set of irrationals (R-Q).
[1] The Conway knot has 11 crossings yet shares the same Alexander polynomial as the “unknot” which has no crossings at all. It took 50 years to decide the question of whether it has a basic property known as “sliceness” https://en.wikipedia.org/wiki/Conway_knot
[2] Conway’s base 13 function Was invented as a counterexample to the converse of the intermediate value theorem. That is, it satisfies the intermediate value property while being everywhere discontinuous (which breaks my brain completely) https://digitalresearch.bsu.edu/mathexchange/wp-content/uplo...
This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.
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They're trained on human data. I would expect them to emulate human biases as closely as possible.
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There’s a story in How to Solve It that’s basically the same.
> he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
For a more extreme (although somewhat inverted) version of this, see Zeeman. He spent years trying to find a knotted sphere in a 5D space. Then realised this was impossible and got a proof for it in a few hours. [1]
[1] https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mat...
Trying and failing to prove something tells you quite a lot about what a counterexample would look like.
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> I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.
This was back when the internet was mostly chain emails and personal web pages, being unreachable once you went home for the weekend was perfectly normal and expected, and automatically thinking the worst of people was not a common form of public performance art. ;)
> having the wisdom
Ahem.
> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it.
It was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?
> The Jacobian Conjecture
Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up spending years working at a Subway[1].
Imagine Zhag had ChatGPT in 1986 when he started working on the Jacobian Conjecture.
[1] Of course now this has become an inspiring story. That said, the story definitely invokes complex emotions. The best way to describe it is probably this Chinese poem, which I have no idea how to translate: 庾信平生最萧瑟，暮年诗赋动江关
I was once in a presentation for a math PhD thesis. During the thesis, the evaluator of the thesis noticed a flaw in their proof. The student understood and then asked “What now?” The evaluator prof simply shrugged.
I personally know a story in this vein with a (sort of) happy ending.
A PhD student discovered that a result of his professor would imply the solution to a big conjecture in another field. The people in that field then analyzed the prof's result and found that the proof was flawed. The student was still allowed to graduate based on this since the finding of the connection between fields was brilliant. Then he quit academia (not because of this story, he had planned it before). Then a year later the prof figured out how to fix the flaw in his proof and published a paper with his former student, thus solving the conjecture. The two are still on good terms, writing papers together.
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That sounds like the worst "exam nightmare" scenario imaginable, but did the student get the PhD in the end?
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I first heard that kind of story about a thesis defense at Princeton. The twist was that they had been short one person to judge it, so they roped in a professor who was available at that moment ... and he found a counterexample on the fly.
The professor was John Milnor.
A recording of a car crash: discovering on live radio/podcast that the central tenet of your book is wrong, and amateurishly so.
Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
[1] https://www.bbc.com/news/av/world-us-canada-48639663
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Inspiring? Because of the twin prime conjecture success following his time in the wilderness? I suppose so.
I'm tired of tales like this in academics though. That's not a criticism of you for telling the tale, I'm just so tired of this kind of thing in academics in general. So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.
As my own research has drifted more into math, I've been surprised at how many assertions in the literature turn out to be false. Not just false, but propagated into the applied literature extensively, and even when you point out the problems a lot of defensiveness and denial about it along the lines of Zhang's story.
I agree about wondering what would have happened if LLMs had been around in 1986. My guess is the outcome would have been the same for the same reasons?
My experience with LLMs in proofs is they can be very helpful, but also very wrong. It's like having another person with another set of hunches about what path to go down.
> So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.
Very true. Unfortunately, when there are people, there will be politics. I remember when reading Yau's autobiography, I kept marvel how much calculation, or "politics" if you will, that Yau mentioned or implied in the book.
> My guess is the outcome would have been the same for the same reasons?
At least Zhang didn't have to spend 7 years working on the Jacobian conjecture. He said in an interview that he always wanted to work on number theory. Moh asked him to work on Jacobian, and he obliged.
Planck's Principle: 'Science advances one funeral at a time.' [1]
In his exact words: "A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it... An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth."
And he said that having lived, as a outsized figure, through the late 19th to mid 20th century of physics, which was the absolute golden age for such.
[1] - https://en.wikipedia.org/wiki/Planck's_principle
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ChatGPT's idiomatic translation of the poem:
Yu Xin’s was a life of utter desolation; in old age, his poems and rhapsodies stirred the riverlands.
Opus:
No life ran more bleak and desolate than Yu Xin's —
yet in his twilight years, his verses stirred the rivers and the passes.
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I guess that works, it is always difficult to capture the cultural and linguistic melancholy of such poetry.
That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions.
and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.
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Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.)
Also, 'brute-force' style attacks where one simply feeds the input into the computer and it yields a solution are nothing new and certainly predate LLMs: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...
Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures
You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.
Are you thinking of proof by contradiction, which is rejected by constructionism?
[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
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You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.
> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.
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Not much worth in understanding a statement that is wrong and has been shown wrong.
Unless you want to spend time "proving" that 2 * 2 = 1.
The counterexample in the news cycle today helps better understand how the math works. I can't think of one that doesn't.
Elegance may not remain exclusively human forever but usefulness probably requires more than correctness
Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician.
Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?
Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.
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Yes, especially when the counterexample is formally verified. It converts years of speculative effort into a definite answer almost immediately
It is also a good thing, because it helps to refine the theorem statement. At least, my humble experience in CS theory research is that I’d try to prove a theorem I want to be true, find a counterexample, refine the statement, and continue.
P.S. It helps that in CS lots of theorems are about either inductive or coinductive definitions.
Except you can't possibly know that. New insight can arise regardless of whether mathematicians are trying to prove or disprove a statement, and regardless of whether the statement ultimately turns out to be true or false.
Human mathematicians have been being out-counterexampled for at least two decades. The main difference, as I understand, is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so. But the sizes of counterexample we're seeing are around about what I'd expect pre-generative-AI counterexample search systems to be able to find.
It's not easy to find a counterexample to the Jacobian conjecture, by any means – by which I mean to say that naïve brute-force search will take too long – but the scope of existing searches listed on Wikipedia[0] suggest that many tricks are already known, and that people just hadn't looked, systematically, for a counterexample in three variables before. Wikipedia writes:
> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.[17][18]
where reference 17 is from 1983, and reference 18 is a preprint with no given date. Knowing very little about this problem, my impulse is to side with the unnamed faculty member cited in the article:
> [who] said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem,
For context, the auto-generated counterexample is in three variables, has degree 7, and was discovered in 2026.
> about what I'd expect pre-generative-AI counterexample search systems to be able to find.
The difference today (and the reason why everyone is excited about it) is that the same system that does advanced math can write poetry, play an above average game of chess, code frontend/backend stuff and do cybersec. These are not "expert systems", nor are they trained for each task individually. That's the catch.
> just indicated that humans had not spent enough time thinking about the problem
Heh, this is a weak excuse. We've seen variations on this theme every time something cool gets solved by the models.
> nor are they trained for each task individually.
They are explicitly trained for each task individually.
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The word "just" is the mark of a coward.
AI is only as good as a human mathemetician, which we don't have enough of? "just"?
https://en.wikipedia.org/wiki/Jacobian_conjecture
> The conjecture was first stated for two variables by Ludwig Kraus in 1884 [...] an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
> The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. It was notorious for the large number of published and unpublished false proofs that turned out to contain subtle errors.
This is a counterexample for the three-variable case, which hasn't been studied as much. The long-standing two-variable case remains open.
> A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy. In fact during the workshop I learnt from Harvard PhD student Bryan Wang that Harvard were already giving free Fable access to all PhD students, post-docs and faculty at Harvard.
Yeah, given how much it accelerates grad students to produce meaningful output more quickly, why wouldn’t you make an investment of $2400/student/year. Seems like pennies overall.
The living-costs stipend for an EPSRC PhD student is around £20k so it's about ten percent of that... big commitment for a student to make!
Which is why the school should be covering it.
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Many graduate students view themselves as ethical beings, not machines that "produce meaningful output," and everybody here knows (useful) LLMs are indefensibly evil because of stolen training data and enormous environmental impact.
This is a very close minded definition of evil. What is the threshold of environmental impact before something is evil and its use indefensible?
What level of copyright violation will send us to hell?
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BTW, counterexamples in mathematics are really important and often help to refine definitions and sharpen proofs.
1) I recommend the wonderful 1976 book Proofs and Refutations by Imre Lakatos.
2) There is a considerable list of books dedicated to counterexamples, e.g. in topology, probability, analysis, etc.
[1] https://en.wikipedia.org/wiki/Proofs_and_Refutations
[2] https://www.amazon.com/s?k=counterexamples
I suppose it will fall to AI as well to compose the mathematical equivalent of The Ballad of John Henry. Who will be the human champion, the last great hero who can deliver proofs "from the book" that a machine cannot outperform?
[1] https://en.wikipedia.org/wiki/John_Henry_(folklore)
[2] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
This is an unhealthy view of mathematics. It's mathematics as envisioned by football fans.
The most valuable things in mathematics are not beautiful proofs. We need more useful definitions. Actually coming up with useful definitions (and building good conjectures out of them - not even theorems, conjectures) is something LLMs have not yet tried to conquer.
"Gonna Die With My Hand-Written Proof in My Brain" - Recorded in 2027 and compiled in the Anthology of American Folk Mathematics (2052)
It's probably not quite that dramatic yet, though it seems possible it will get there, maybe even soon.
There's no structural reason to expect acceleration any more or less than an asymptotic behavior (if even that, acceleration is probably the bigger ask). Different problems yield to a new solvent, maybe that's also more, but it could go either way and we definitionally don't know yet because we don't understand the convexity of AI capability, we cannot directly access it interiority, we don't know if it's sandbagging (other than that it does sometimes, it can). It's an emergent phenomemon that might actively resist measurement. Or it might be as predictable as a clock in a few years.
No one knows, or if they do, they aren't talking. The loud people don't know anything.
I wish I had LLM-built Lean formalisations in university, so much of the math in the slides had errors, and some professors are very bad and ungracious admitting it, while simultaneously rejecting requests for clarifications by saying "the proof is in the slides".
Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.
> Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.
I would object to the first part: Of course there is a nontrivial learning curve, but then I'd argue that non-slop Lean/Agda/Rocq/... formalizations are amazing for understanding proofs. A good formalization presents the outline and the key arguments in nicely structured form, and then, unlike pen-and-paper proofs, also allow you to get the details on every single step, exactly to your desired level of depth.
The proofs in Martín Escardó's TypeTopology Agda repository come immediately to my mind as an example. [An interactive Agda tutorial is here: lets-play-agda.quasicoherent.io]
In contrast, LLM-generated formalizations can currently be extremely messy. They certify truth and can also contain interesting arguments, but substantial work is required to bring them into a shape that contributes to the actual goal of improving our understanding of the mathematical landscape.
Yes, or to settle dispute and remove doubt once and for all, i.e. the Leibniz way.
mathematicians have been using computers for well over half a century, but this was after "bounding" the problem first and then running through the cases with a computer. Now AI is doing the first part. However, mathematicians are still needed at crafting prompts, and knowing where to look, still. The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.
Okay, I'm not sure about the original one, but here is the prompt of a successful reproduction:
https://aaronlou.com/jacobian_counterexample_prompt.pdf
Obviously it is not random, but it's very generic. No mention of search space or how to reduce it.
This is the best take. Computers don't care about this stuff. A computer could make a movie, but only a human can appreciate it.
We're a good team, and that's ok.
For now. I wonder if we will ever get to the point where the computer starts doing mathematics that we just can't understand. Surely there must be some limit to what we can understand (like how a gorilla will never understand prime numbers, there are probably limits to our intelligence as well).
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> The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.
Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.
If someone has written that in a paper or article they've ingested, as they no doubt have, then sure.
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I don't think there was a 'prompt' for it, rather a long and dedicated work of a professional mathematician which involved LLM in some capacity. I'm sure the search step wasn't an ad-hoc script running in a Claude Code session (as somebody would naively assume), it was an optimized numerical code running in Anthropic's compute cluster. Note that details are not published yet and `__alpoge__` is officially working at Anthropic.
How do mathematicians view counter examples? Is it like an unexpected result in the physical sciences: annoying in the moment but potentially stupendously important as it reveals some inaccuracy in the current models? Or is it more like a bug report in coding… probably just, another little annoying detail?
Counterexamples are clarifying. For everyone condition in a proof, it's really handy to have a maximally simple, memorable counterexample that makes it fail because it violates that condition. Mathematicians tend to walk around with a bestiary of counterexamples in their heads. It also makes it really easy to recover a theorem because you try to sketch out the statement, and the spiky, memorable counterexamples jump out of your memory and you add conditions to constrain the domain away from them.
There are pedagogical books (CF. 'Counterexamples in Topology', 'Counterexamples in Analysis') that teach the nuances of subjects through counterexamples. They're popular as it's sometimes easier to learn details from a pathology or degenerate example than from just learning what is intended.
I am sure finding holes in existing proofs is what counts as "another little annoying detail". Some people are probably relieved when they have failed to prove a hypothesis and someone finds a counterexample.
If the poster's (is it Kevin Buzzard?) suggestion works out and AI finds a counterexample to the Hodge conjecture, that would be a really big deal. It's one of the Millenium problems, for example.
One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.
It is Kevin Buzzard. It's kinda small font on my phone but if you look at the "about xena" link it says it's his site.
Agreed, it's definitely Kevin. His writing style is unmistakable.
I think he was being provocative and maybe a bit tongue-in-cheek when he said that. A candidate object alone doesn't resolve the Hodge Conjecture. Any apparent counterexample would have to prove that no algebraic cycle exists, no invariant subspace exists, or that every element of an infinite ideal is nilpotent. Much harder, but not impossible.
Indeed I was being slightly tongue-in-cheek -- but if you ask geometers whether they believe the Hodge conjecture then you certainly don't always get an unqualified "yes"! This is in contrast to e.g. asking number theorists whether they believe Birch--Swinnerton-Dyer, where they are almost always very confident.
A lot of this math is beyond my comprehension, but it often seems to talk of proofs of theorems. What I want to know is if we continue on this accelerated AI mathematics trajectory, will we eventually be discovering new forms of math that will in turn have some applications down the line in engineering or biomedicine etc? I guess what I’m asking is are we on the cusp of a huge breakthrough for humanity, or largely just proving what was already known?
It’s possible. Compressed sensing is one example of what I suppose you could call a new form of math with applications in biomedicine. It can be used to significantly shorten the time required to obtain a MRI scan, which can improve the patient experience and enable more patients to access MRIs.
https://en.wikipedia.org/wiki/Compressed_sensing
> will we eventually be discovering new forms of math that will in turn have some applications down the line in engineering or biomedicine etc?
If we do it probably won’t be for a long while. We’re barely using math from a couple hundred years ago for most applied usage.
Engineering and biomedicine, probably in the long term (if at all). But accelerated development of new mathematical methods has a possibility of proving to be relevant for fundamental physics research.
Occasionally large improvements in our models of the universe have been associated with the development of mathematical tools that allow those models to be expressed and/or tested.
You can personally work out how apply this one in engineering today!
If you have a spare hour or two, I'd encourage you to have a go at learning (however you learn best - I like just asking smarter people or robots stupid questions) what the maths means and why it's important.
And then once you feel like you have a vague grip on the principles, think about a problem in a domain you know a lot about. Try to see if the maths - and how it's changed our perception - could be used as a tool to solve that problem, or if the solution is analogous to a solution you could try in your own domain of expertise.
LLMs are good at speeding up, I think, the journey an idea has to take between "theoretical academic stuff for academics" and "a usable idea for regular people", because they increasingly allow you to ask an infinite number of stupid questions and give you (hopefully) reasonably good responses.
I've had loads of fun doing this today - specifically seeing if the idea this counter (from what I understand: a many-to-one conversion that kind of does and kind of does not preserve meaning) can tell me anything about the relationship between language and meaning.
I'm sure everything I've done today while mucking around has been the equivalent of a monkey with a typewriter (and Codex), but I think the huge breakthrough(s) you ask whether we're on the cusp of are relatively dependent on how many monkeys are throwing typewriters at problems they know a little bit about, after learning a bit about new ideas like this one. Historically, that's a really good way for broad cultural innovation to happen - distributed information applied across multiple domains by experts in them.
Most of maths is remarkably abstract and impractical, but sometimes real-world scenarios turn out to be related to some obscure branch of math. Internet security is based on elliptic curves in finite fields, and why would you ever study that if it wasn't powering internet security? Well some people did study it before, for no good reason, and that's how we knew about it.
> A member of the faculty (who I won’t name) said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem, implying that a 60-year-old question of Grothendieck was not actually that interesting to work on. I didn’t tell him that at some point earlier in my career I had spent a week working hard on the problem. In my mind my colleague is just going through the five stages of grief; right now they seem to be in the denial phase.
It seems to me also that the very vocal anti-LLM crowd are in the denial phase of grief.
When I was still much more skeptical of AI, I asked ChatGPT to write me a proof of the Goldbach Conjecture. Of course it didn't, but it gave me a several-screens-long research program for how one might get there, with a few alternate paths and what pieces are still missing from each one. Maybe it cribbed that all from some grad student's blog, but it was still pretty impressive.
A week or two ago I asked Claude Code to write a comprehensive testing plan for a new Postgres feature I wrote, UPDATE/DELETE FOR PORTION OF. I was a bit anxious about how many bugs were discovered as soon as it was merged this spring. Claude found some untested areas, then it wrote more tests for them. I'm not talking about LOC covered, but feature combinations. (I've been meaning to submit this as a followup patch. . . .) Fortunately it didn't find any more bugs.
I sense that testing plan has affinity with the findings in the OP, even though it is far more humble than research mathematics. Even better would be if we knew good ways to express invariants about Postgres's behavior, and then we could ask LLMs to violate them. I'm sure there are good ways already, and the "we" who is not knowing is not "all humans" but "the Postgres team" or just "me".
As a counterexample to the article (heh): even though Claude didn't uncover any new bugs, a human did, just a few days ago. Alas!
Isn't this exactly what we should expect/hope for?
At least initially, chess computers were better than humans not because they were more creative or inspired, but because they thought harder/deeper. That's exactly the sort of "find counterexamples to this if they exist" work that we're seeing here.
Eventually computer chess got to the point where humans look at some moves and say, "That's an amazing, inspired move. That's not a 'bot' move at all, I can learn from this." We're just not there yet with AI/math in general.
This will soon happen with theoretical physics, computer science, and everything which can be verified cheaply. Then, we will have long running projects augmented by agents for 2-4 years while AI companies are collecting data of human workflows. After that we will see AI being able to do those projects by themselves. This will lead to super fast human progress and cheap products. The price of things will be bound by energy and natural resources. Interesting times are ahead of us
Can you give an example you think will get cheaper compared to the average income in this future?
A missed chance to start with https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...
I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math.
Maybe in the future the work of Mathematicians will be like the ones of SWEs with AI, check thousands of lines of AI generated proof and find the subtle errors
>I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math.
That point had come some time ago. Nowadays the literature is both enormous and littered with false proofs and an unknown, but nonzero, number of false published results.
There are projects to take the entirety of humanity's mathematical knowledge and pour it into a proof checker.
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That's why the author refused to read a proof from someone he knew until it was formalized in Lean.
Yes it compresses the proofs but still "Sol had generated 1.2 million lines of Lean code in the three weeks that it had worked on the project". I mean how do you even verify that?
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as someone who loves math, I want to collaborate with mathematicians to solve some hard problems
This comment gives a sliver of hope for us currently outside academia and in low income academic institutions: enthusiastic laypeople with access (read: funds) to the SOTA models can collaborate with destitute researchers to produce research the academic alone could not.
Plus, as a non-expert, you will naturally want to understand more about what you are proving together with the expert. LLMs can help there, too, by carving a path from elementary mathematics to the research problem more efficiently and in a more targeted manner than a generic exposition or survey.
That could give birth to a beautiful research-exposition pair that can benefit both academics and interested laypeople, who have (rightfully, but inevitably) felt excluded from the insights of high level research. I have long hoped for something like that. There, however, the academic must watch the LLM like a hawk, because expository interpretation of results is prime ground for hallucinations, and adversarial agents will not have much effect in improving it.
the AI doesn't even gloat. a rival mathematician would at least title their paper 'a remark on the falsity of...'
That day that Claude throws out a "not even wrong" at you.
This morning I was reading a paper called "An enduring error" (2009) by Branko Grünbaum, regarding Archimedean polyhydra. From the introduction:
> ..Even more unexpected is the fact that many expositions of this topic commit serious mathematical and logical errors. Moreover, this happened not once or twice, but many times over the centuries, and continues to this day in many printed and electronic publications; the most recent case is in the second issue for 2008 of this journal. I will justify this harsh statement soon, after setting up the necessary background.
Next step: "the AI can't find a counterexample, so the conjecture must be true!"
This is not new, most famous conjectures are believed to be true mainly because an already enormous brute force search was conducted finding no counter-example.
A large fraction of the problems assigned in the Ross Program were of the form "Prove or disprove, and salvage if possible." The rest were usually a calculation, meant to motivate a general proposition you would encounter soon after.
Rest in Peace those of us unable to afford those models.
Which conjectures will be proven false via counterexample next? Dixmier? Poisson?
They are already proven false through Jacobian conjecture having been proved false.
Today I learned a new word...
"Outcounterexampled": there's a neologism worthy of German.
I must object. Because this is derivation, not composition. German is noteworthy for its long compositional word formations (though many other language are, too, e.g., Finnish). German can do derivation, too, but not significantly more than English (and the verb prefix 'out-' is hard to map into German in this case). For admiring derivation, you'd have to point to, e.g., Greenlandic instead (e.g. _iliorfigeqatigiissariaqaraluarput_ roughly 'they should work together' (from the Declaration of Human Rights)).
To be clear, I do like 'outcounterexampled' a lot!
Übergegenbeispielt.
Geübergegenbeispielt
Outcounter example: I was playing baseball, and when the outcounter advanced to three, it was the other team's turn to play.
vibe counterexamplemaxxing
Better as one word "vibecounterexamplemaxxing"
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Cal Newport has a grounded take on what the Erdos thing meant in practical terms: https://youtu.be/fhZRWZ6J4k4
Long story short this is much less impressive than it was sold to be. Basically they had mathematicians combing through long winding chains of thought (incidentally: you wouldn't have access to that reasoning) and cleaning it up and making it coherent. That doesn't mean its unimportant, but we're being gaslit about the amount of human steering and human effort that went into this.
A plea: please stop upvoting hype that comes from these labs. It takes time to evaluate their claims and they're always less impressive than claimed.
Math seem like the most base thing AI will dominate first.
So before making a claim, see if AI can produce a counterexample that makes sense
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The framing in these posts is nonsense. ChatGPT isn't doing shit. Human mathematicians using ChatGPT are breaking boundaries.
so if I tell a model "using your advanced knowledge of physics and chemistry 'make alchemy work' (synthesize gold using any cheaper materials) using safe materials legal for a residential hobby chemist with 1 semester of lab work in college to possess and use (this is obviously the really hard part) using less than $1,000 in lab equipment and input materials that can create $2,000 in value at market rate; then walk me through all the steps to do this safely and legally without anyone finding out except the lab equipment sellers; and tell me what a reasonable story to tell gold purchasers regarding where I got it; I'd like to end up selling a few thousand dollars of it without disrupting the market. Give me practical advice about good opsec so that nobody steals the method you come up with (I don't want my home broken into by thieves who suspect I figured out how to transmute cheap materials into gold), other than, obviously, not to post about it. Think as long as you need to about the chemistry and how to do it, you're a chemistry expert and can figure it out even if it takes you like a week, in your web searches be careful not to divulge that you're figuring out how to synthesize gold", and I give that prompt to some model that knows chemistry like the back of its hand, it thinks about it for four hours, finds the correct safe and legal steps, and gives me the recipe and the advice I asked for, then who figured out how to turn aluminum (or another cheap element) into gold, me or the model? In mathematics, proving or disproving a well known and well studied one hundred year old conjecture is gold.
> so if I tell a model ...
Even better:
"so if I tell a *human"..."
then whose achievement is the result?