We should start a gofundme to send him 2 months to a remote tribe in the Amazon. Chances are, we see the Riemann hypothesis and twin prime conjecture proven. ;)
It sounds plausible they spent more, given the output tokens (6 billion of them) would cost $300k at API prices and presumably there will have been many more input tokens than output tokens.
And which they could not charge anyone for. Unless these were extra resources that would otherwise go unused it cost them the amount they could have charged for them. Normally I would expect most businesses to make reasonable tradeoffs when it comes to how to allocate resources. I’m not convinced that any of the AI providers should be given that benefit of the doubt.
> healthy profit margin (as near as we can tell from the outside)
Ugh we still don't know if this is true and it's nearly impossible to calculate without a full understanding of the real CAPEX cycle. Stop spreading these rumors until we know for sure.
SemiAnalysis estimates their profit margin to be 70%. To be losing money on inference implies that their costs are almost 4X higher than SemiAnalysis has calculated. That's not credible.
Whether on net they turn a profit as company overall is neither here nor there.. My point is that they are selling API tokens at a profit (or if being pedantic, then at a price higher than the cost to serve them ignoring research costs). And that that price is got a healthy margin which they don't charge themselves.
It's like having new solar panels installed every week. Sure you're "profitable" on the $0.20/kWh you're selling your "free" energy at when you ignore the cost of the solar panels you're buying every week.
Regardless the profit margin as a talking point seems to be bad as AI as a tech might never be reversed whether anthropic failed or succeeded. Indeed it's imperative we subsidize AI companies and tech to make them explore more solutions to scientific problems which has a downstream effect on human flourishing.
Like? I feel breakthroughs that can be found via AI might help us more in the long term where even previously non AI fields can be helped by AI. So you have specific non AI research in mind that we're underinvesting in? Because the USA is already spending crazy anyway for healthcare and I don't feel like funding is the issue but better incentives, reforms etc
Funding for basic research is being slashed by the current administration. Our society is underinvesting in basic scientific research. And, AI will not fill the gap.
But US also spends too much on education as well. The issue doesn't seem to be funding but the educational reform like in mississippi, where they increased student performance without increasing their budget too much. That's why you see bad k12 educational outcomes compared to the budget spent in blue states. It's all about efficiency. Give AIa chance in few years as I feel it can make great strides.. it's hard to imagine that chatgpt released in 2022 and look at the progress in just few years as it just changed software engineering field entirely.. i expect similar kinda progress where of course humans will still be making breakthroughs but it'll be accelerated with the help of AI.
Spending on health insurance is spending on health care.. Americans want free healthcare but no tax bump so health insurance is a compromise.. when even just ACA was passed and premiums increased, democrats got destroyed at midterms so Americans might be living in la la land.
You see funding of chatgpt as a panacea for progress.
I see funding of chatgpt as one of small part of a history where governments and industry fund basic science and moonshot programs, not to generate revenue, but to explore what is possible.
LLM funding is not aimed at improving our understanding of the world, it's aimed at making people reliant so that they may extract wealth through subscriptions for shareholders.
Americans don't get good healthcare and education because that's what they vote for, in elections and wallets. I am hopeful that that changes, but we shall see.
How many previous attempts with other models failed or on other problems. Perhaps this is $300k out of $100M or $1B of total budget just breadth first searching theorems in math and all the failed attempts conveniently don't get mentioned.
>The speed with which we were able to produce this proof demonstrates that it is now possible to formalize large swaths of mathematics, which may both catch errors in the common body of mathematical proofs and reduce the burden of refereeing new work.
^ this section should have been in the first few paragraphs imho. Explaining why this is relevant shouldn't be so far down.
For any body of text (or in general, any exposition of any kind), the responsibility to explain the value of the article is very much in the author's side.
Explaining the value of what you are showing should always go towards the start. Else, why would anyone bother with the rest?
Feel very grateful I was never taught this... Would have missed out on quite a lot of good bodies of text in my life I think! Pushing through any initial friction or ignorance I might have as a reader, having the patience and charity to bear with an author until you get it, was instead what I was always taught.
Giving such a blanket "responsibility" to the author at all is just such a bummer! I say let them do whatever they want, there is always more than one way to express oneself. Someone who was never taught to write a clear thesis in the first paragraph for whatever reason doesn't inherently have less to say.
Unfortunately I don't see this particular view paying off in the age of AI, as many prove they have nothing at all to say but say it anyways. Which isn't to say people shouldn't write if they enjoy writing, but I for one will stay a discerning reader.
Isn't it the cost we care about, rather than the speed? All we know know is that a frontier AI lab was able to do it in 11 days, we have no idea how much compute they threw at it.
Am I doing my napkin math correct? The post says it's using a model comparable to Fable 5.1, which is $50 per million output tokens. So this is ~$300K? Surely an over-estimate due to caching.
So I don't know Lean or Mathematics to any degree to really be able to say this with any level of confidence, but speaking from a pure software engineering backgrouand, how do we know that 13 MILLION lines of Lean code are bug-free? It seems to me that for a mathematical proof, bug-free would be an absolute requirement. Maybe the structure of Lean imposes that, I don't know, but that seems highly unlikely to me. That just feels like a LOT of code to be comletely error-free... What am I missing here?
> In 2026, AIs designed to spot bugs in software were directed at Lean, and found several loopholes which were then fixed. Perhaps related to this effort, a purported disproof of the Collatz conjecture was announced as verified in Lean. However, this proof was soon determined to rely on a bug in Lean, and once the bug was fixed the proof was found invalid
However it's a bit different than the usual 'bugs' we encounter in normal software development. Lean is more like a type checker. If you can write a false proof in Lean then the bug is in Lean itself, not your code.
In other words, Lean can have bugs, but the amount of code we need to check scales with Lean itself, not with the length of proof. Just like the chance that C compiler has bugs doesn't increase as we write more C code. So the 13M lines of code doesn't really matter here.
In lean, a theorem is specified by a type (in their highly complex "dependent type system") and proof is specified by a code that produces a term of that type.
If the compiler certifies that the code indeed produces a term of that type, then the proof is correct.
So, only need to trust:
(1) That theorem statement is correctly encoded (FLT has a very short 1 liner description really)
Lean is like a statically typed programming language and validity is guaranteed if it compiles. The only room for errors is in translating a non-Lean theorem into Lean, so that you are not proving what you think you are proving.
"The proof is not the modern proof which I have been formalizing myself following ideas of Khare, Taylor etc, but the Darmon–Diamond–Taylor exposition from 1995 of the Wiles–Taylor–Wiles argument, via the Langlands–Tunnell theorem and Ribet’s level-lowering theorem. Anthropic’s repository develops Fontaine theory (to study flat deformations of Galois representations) and develops enough of Mazur’s work on the Eisenstein ideal to conclude that no Frey curve can have a point of order p>=17. This means that their FLT proof only works for p>=17, however FLT was already formalized for odd regular primes by Best-Birkbeck-Brasca-Rodriguez, and the smallest irregular prime is 37, so it’s all good."
My question to any mathematician reading this: does the above make ANY sense to you?
I ask that because I can read most technical material related to computer engineering, programming, hardware specifications etc. Even if I don't fully understand all details, I can follow them pretty well. So I wonder if professional mathematicians can look at the above and still make sense of it like experienced software engineers do for computer stuff.
Yep. While I'm not focussed on these areas, I know enough from scoping out a "learn about the proof of FLT" course that it's covering all the usual suspects and says the right-enough words. Patching their weaker results with someone else's seem like a good strategy (and I could find the result on arXiv so it isn't obviously hallucinated).
This is very different to believing the proof, which would require at least a pass understanding the general approach, seeing that it all actually fits together, then going deeper. At some point you transition to relying on the Lean all hanging together, but as mathematicians we all draw that line somewhere.
But yeah, makes sense. Same thing if you saw news on someone's new database technique to improve performance. If they say the right words, don't say the wrong words, and if you cared enough you'd do spot checks proportional to the claim. If pressed you'd examine the source code, and run independent checks. But if smells roughly right, that's a good first approximation.
While I don't know the specifics, and someone more "in-the-field" than me would recognize all the "named" theorems etc
I am aware that there have been minor issues that have come up with the formalization specifically, and that previous proofs for lower values of n were always needed.
Though it used to be n=5 and lower needed to be checked.
It's something you would have to be keeping up with as a mathematician, really.
Vaguely. It's describing connections between a number of other mathematics results than can be connected to prove FLT. I assume all the work described is being done to make the proof more presentable, smaller, basically "prettier".
It sounds like they established a minimum and maximum bounds for n in x^n + y^n = z^n, where one proof works for n greater than or equal to 17, and another proof for n < 37 (when prime).
I believe the case (remembering back 40 years here) n is even is very easy, and n is composite and odd slightly less so. Neither really being in the ballpark of what they describe here.
I did an undergrad in math with a little research in number theory and recognized parts — eg, I myself worked through the proof for odd regular primes and that 37 is irregular, breaking the general case.
Wiles-Taylor-Wiles was the original proof by Andrew Wiles, and its corrections.
Galois representations is about vectors over Galois extensions, which are essentially adding roots to regular numbers (rationals, integers, etc). That ties into the Langlands program, which is a big area in number theory (that I don’t know much about).
Together with flat deformations and Frey curve, I think they’re talking about a topic in algebraic geometry as applied to number theory.
I also recognize the name Eisenstein from my time as an undergrad, though two decades out and not working in the field I’ve forgotten what his work on ideals implied here. Ideals are a well-known topic though, a sort of structure inside a ring (set with + and *) that is closed under operations — like evens in the integers are the 2Z ideal.
So I’d describe it as “sensible with an undergrad background”.
Frenkel does a nice job explaining the Langlands program in general. But Buzzard's complaint about Langlands, I believe, refers specifically to the proof of a version of the Geometric Langlands Conjecture by Gaitsgory et al. The proo f is of order thousand pages of mathematical text and builds off of thousands of pages of higher-categorical algebraic geometry by Lurie & others. It's a ripe target for formalization because it's terrifically complicated, not well understood or thoroughly digested yet, and relatively important. A formal proof would be reassuring to mathematicians, whereas Fermat's Last Theorem is relatively unique in that so many mathematicians have examined the proof that it's not very likely to be wrong.
While pretty much everyone is certain Fermat was mistaken in believing he had a valid proof for the theorem, this is an expanded (compared to proof presentations) version of one proof - not the shortest presentation of the shortest valid proof.
Given the likely length of the shortest possible proof, I feel like Fermat is 100% vindicated - the proof won’t fit in the margin.
My strong hunch is that it was a joke - he knew how difficult the problem was and claiming he had a solution was I think a huge motivating factor for many mathematicians trying to prove it. The greatest nerd snipe troll in history.
Most likely an error. Some time after he wrote that margin note, he wrote a document proving a special case of the FLT (i.e. it's true for n satisfying some property). Why would he do that if he had already proved it?
I think that point actually agrees with GP's take (joking/lying about having had a proof too big to fit in the margin): He would do that because if he thought the problem was extremely difficult but didn't actually have a proof when writing the note he would still want to go on and try to pick away at the problem.
Maybe, we'd have to go back and ask him to be sure. I mostly just didn't want to leave an as of yet certainly unproven vindication about this hanging in a thread about finally having a formalized proof of the star topic :D
It seems unlikely to find 1920 level or so proof although it might be the case that a significantly easier/shorter proof exits via Vandiver conjecture + extra work or Effective Mordell conjecture but it also wouldn't surprise me if that would be even more complicated than the current proof of FLT.
The next step, if Anthropic is interested, is definitely performing refactoring to cut down on the size of the proof. It’s clear to everyone including Anthropic that this proof isn’t as concise as it could have been. When it’s concise enough to be accepted into Mathlib is when victory truly is upon us.
Maybe I'm misunderstanding something about how all this works, but can we have any confidence that 13 million lines of AI-generated Lean code are... correct?
How have we not merely substituted one verification problem for another?
Not always, there can be bugs in lean. Recently some guy with claimed to disprove Collatz conjecture, only to turn out that there was a bug in lean. I actually have no idea, how anyone can be sure this 13 M lines is meaningful
It’s common for formal proof efforts about software and hardware to involve thousands to tens of thousands of small lemmas.
13M lines does seem extreme and there is probably a lot of inefficiency given the way the proof was developed. Cutting it down is probably a long road, but is also a very well defined problem that AIs can probably just go do with enough time and budget now.
>> Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems.
> Pretty insane.
I don't think the count of "intermediate theorems" tells you anything. Here's something from an algebra textbook:
---
Let G be a group, let H be a subgroup [of G], and let N be a normal subgroup [of G]. Then
H ∨ N = HN = { hn | h ∈ H, n ∈ N }.
---
This says that the subgroup closure of H and N, the smallest subgroup that contains them both, is identical with the set consisting of all products of an element of H (on the left) and an element of N (on the right).
Part of the proof:
---
Suppose that x and y are elements of [the set of products hn]. Then x = h₁n₁ and y = h₂n₂, where hᵢ ∈ H and nᵢ ∈ N. Now h₂⁻¹n₁h₂ = n₃ ∈ N, as N is normal in G. So n₁h₂ = h₂n₃. In this case
This will translate directly into lean. If you do it this way, you will prove at least 10 of what would be described in lean as 'intermediate theorems':
I saw the 1996 FLT documentary in high school calculus class. For me, it forever cemented that archetype of modern math researcher at the top of my mental “smart” totem pole.
It also convinced me I had no interest in that path. Setting aside the grinding work of producing a proof that can only be reached by existing years in the abstract and hyper niche isolation of the problem space (not to mention that you might never discover it or that it DNE), the anguish of the output being a paper or presentation or some other artifact of human symbology (_words_, really) that could at any moment be refuted by a single observation of a single mistake—-that sounded like hell to me.
An equivalent high schooler today probably sees things differently, in light of this news and the undeniable implications of LLMs on mathematics. Sturdy autoformalization tooling should with time completely dispel the aforementioned anguish, once our confidence in converting a human proof to Lean/etc. reaches that of a compiler translating Java application language to bytecode. Errata may always exist, but in practice these new methods will do wonders for rigor and peace of mind.
(I’m far less confident re novel discoveries. There’s too much chance of derivative findings based on something part of the training looking like genius but really just tiptoeing on the shoulders of humans, whereas autoformalization is absolutely convincing to me as transformative, particularly to check correctness of AI outputted proofs as mentioned in the post.)
> a team of agents completed the proof in a little under two weeks, consuming about six billion output tokens from a general-purpose internal research model roughly comparable to Claude Fable 5.1.
At $50/M output tokens, this would have cost on the order of $300k (plus a bit for input/prefill tokens) at API rates.
And human salaries for those who worked on the prover harness etc. which isn't just standard Fable.
It also uses Prove2Me, which uses a graph like previous automated theorem provers. A fact that LLM hawks have categorically denied here before, with opposition naturally flagged.
> A fact that LLM hawks have categorically denied here before, with opposition naturally flagged.
Yeah, because before now there's been literally zero proof of an automated theorem prover scaffold around the LLMs being used, and big counterexamples and such being found, with raw chat logs available, where no such thing was used.
> Now they have it in writing.
Yeah, because now it's actually being done. They talk about it as a novel thing, because it is. You don't get to claim being "right all along" from this
But also achievable on a $150/mo (CAD) Max 5 subscription (I currently have 11.6B tokens in the last 30 days) according to /usage. It doesn’t break down input vs. output tokens as far as I can tell.
Buzzard was given 1kk GBP and 5 years and his goal I think wasn't the full thing like Anthropic did. So much more cash and orders of magnitude more time. The proof is about 5x the whole Mathlib library which was developed over many years by dozens of people.
It's true that his goal was not the full thing, but it was also not merely a Lean verified proof. From the blog post linked in the toptext:
> The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof.
This is a crucial point. There have been many bugs in Lean (and in other proof assistants for that matter). Proof assistants work well on human input, because it was created with a certain intent.
We simply don’t know what those 13M contain and whether it “makes sense” and doesn’t trigger Lean bugs. (There are “independent” lean verifiers, but historically they contained the same, or similar, bugs.)
It is possible, although the post notes that the proof was also verified by the Comparator, which means any exploited bug has to also be present in that checker. Which is not unheard of, but is much less likely than merely an exploit in Lean 4.
The comparator was only used to verify that the final statement indeed is a valid formalization of Fermat's Last Theorem, not that the proof leading up to it is correct.
That must have slipped through Kevin Buzzard's review, which is not entirely unplausible with 29500 theorems to verify...
I think they should spend another few billion tokens and let agents try to disprove any of those statements or links between them. Then I'd be a lot more convinced.
Not just lean, but math foundation itself, I am not strong expert, but my understanding is that there is no fully recognized axiomatic foundation for modern math, all proposals could lead to some weird results.
There is, or rather are, fully recognized axiomatic foundations. You are free to choose one you like. Of the most popular ones is ZFC or ZF, but there are others (some lead to the same results some not). The main criteria for popularity is how useful it is. You can even make your own axiomatic where 2+2=5, but it would be useless.
You probably heard about Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
It would be fun to play with this Anthropic/Lean formalization under different axiomatics.
Godel proved that any system expressive enough to produce an arithmetic is incomplete. He initially proved it for the peano axioms but then it got generalized. ZFC can produce an arithmetic. Also, before being arrogant and demanding explanations, you should give them first for your claims
you understand that "expressive enough to produce" are not obvious elements of zfc, that's some average consumer napkin math and not strict formalization.
A quick google search shows different proof assistants have been used to obtain the Peano axioms from ZFC, such as Isabelle/ZF and Metamath. I think you're just wrong
coming back to your argument about peano being obtained from zfc, you obviously can't prove that it happened using purely zfc, and not some logical framework embedded into those proof assistants.
I said I am not expert, I am indeed not expert in zfc and godel theorems, but I am an expert (phd) in actual formalization theory.
Formal theory is very simple concept: its alphabet, set of formulas on top of this alphabet, and function which translates one formula to another.
ZFC can't "obtain" peano, simply because it doesn't have say * operator defined. You need to do something on top of it.
Additionally, zfc itself looks like loosely formalized say in wikipedia (and I am not sure if there is any strict formalization anywhere), we take it as common sense that it can utilize some simple logical rules (e.g. modus ponens), but what are exactly rules, which could be separate topic of research, this detail is skipped.
That increases the likelihood that they are right.
> support your point with explanation or be ignored :-)
Anyone who says "Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems" and isn't joking warrants a permanent ignore.
its hard to me to tell what this means formally(as I said I am not expert).
There is no "interpret" operator in zfc.
I believe what it says if you add some robinson axioms + some logical rules on top of zfc, you can carry your results.
If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZFC) gives a set of natural numbers (using the von Neumann encoding), which, when equipped with the successor function, is a model of the natural numbers.
ZFC is probably the biggest foundation, and only Choice is apparently controversial. The results aren't that weird, they're just different and occasionally more useful than using !Choice.
do we know if claude's formalization is built on top of zfc and not zfc+extra?
zfc itself is not sufficient, you need some layers of extra concepts formalization to fit specific problem domain(e.g. zfc doesn't define even basic arithmetics), which also could have potential issues.
Most systems i have seen are way beyond a 100 lines. And their GitHub repository contain many issues, often soundness bugs. (Granted, many get fixed very fast.)
We'll increasingly observe announcements of this kind as AI tooling scales. As impressive as agentic coding is, it pales in comparison to the value proposition of medical, mathematical, and physics research.
I optimistically expect to witness the advent of a global 'panacea' in my lifetime thanks to AI's efforts. Cost effective large scale genetic engineering, a cure for every disease, potentially even a cure for aging.
It's wild to think that aging is something that needs to be cured, and isn't a part of the natural human experience. I'm so tired of people trying to play the role of God, as well as people that cheer these sorts of things on.
Childhood deaths and fatal diseases are also natural parts but that doesn't make them desirable to everyday humans. But with new advances, people might have the ability to CHOOSE in future.
Most people want more life. For most people it's also the most terrifying part of "the natural human experience".
If you're happy to die, why be bothered by others' trying to live longer? You won't be around. And assuming people can finance it themselves, is it really a problem for society?
Yes, I think it's a problem for society. Death in old age frees up social, economic, physical, and political resources for the next generation of the living. If the rich and powerful escape death, because after all they will the people with the resources to do so, society will lose the adaptability and natural change that comes from new generations taking the reins.
I assume you mean that dying is the most terrifying pat of the natural human experience. Also, I'm not sure why you infer that me thinking death is a natural part of life, means that I'm happy or eager to die.
There are many reasons that people living forever would be a problem for society, the most obvious being an ever-increasing population.
Fertility rates are below replacement, which means that population sizes are convergent. A decreasing population is a more likely future scenario for many western countries, even if human lifespan was indefinite.
Fertility rates are currently below replacement, there's no good reason to imagine they will always be that way, particularly after global population numbers peak and fall to, say, half or a quarter of their peak.
Because living longer is a huge drain on resources that could be better spent on other things. End of life care is expensive and rarely results in a "good" life for the the life being extended.
So I think curing means basically opt in death or something like that. Right now extended life is bad because the person isn't in his prime but curing aging is basically gonna keep him in his prime. This might be what they meant.
I dont think it will happen. AI models are kneecapped. Only a tiny tiny tiny fraction of people are on the list of even being able to use these tools for such things.
For math illiterate people like me, my understanding is that FLT was already proven, but the proof was beyond complex, certainly for mere mortals like me, and now Claude has codified it, correct?
Back in February, I was talking with my PhD advisor about using Lean to formally verify automated optimization modeling outputs. It eventually turned into this paper [1]. It’s been truly incredible to see how much the frontier models have progressed in both autoformalization and automated theorem proving in the last six months. Back in February, it was cool to see them prove the validity of some simple cutting planes. Now it can churn out a min-cut max-flow duality formalization (not to mention FLT). Very exciting times!
I’ll also share a Python package I wrote for automated theorem proving that has been super useful in my own research [2].
"The effort succeeded when we switched to using Prove2Me, an open collaborative platform for formalizing mathematics designed by Tianyi Peng and his collaborators at Columbia University."
So in the end, it required tooling crafted by humans.
There's nothing about prove2me that couldn't have been coded just like any other huge coding project frontier models have proven themselves extremely good at doing. It just happened to have been made by humans.
Humans built the tool which enabled the result. AI used the tooling for eliminating the dead ends. Yes, I can appreciate the practical value of all this, but IMHO it is not a kind of breakthrough result the article gives impression of.
"you could never have accomplished"; I am not able to follow the logic here - there is no "magic" in LLMs, they're built by humans and we know what they do.
Sure? I mean the internet is just a bunch of wires and some networking code not magic but at the same completely life alteringly magical.
My logic is that you personally could never have accomplished this feat with all the non LLM tools and content in the world. These kinds of things imply these methods are stepping beyond human ability.
Sure we put walls around it and optimize but the interior of that optimization is not something we understand.
You now have access to a system that for a price could solve something you simply are unable to solve. Not something we programmed it to solve, something that has never been solved before.
Nobody gave it an example of this proof, that's magical.
I think you're referring to the fact that the sheer amount of computations is something too time consuming for us to follow? But still it is not "magical" - in theory we could follow all the steps, there's no hidden information.
No, I mean we just don't know what's going on in the circuits of the model at any substantial level. We set their architecture (hyperparameters), we pump them full of data (pretraining), and we shape how they behave through examples (SFT) and reward (RL), but we can't say with any certainty what the resulting model does internally.
Yes "at any substancial level" . But still, its all about deterministic processes and still it obeys the law that the same input gives the same output. Or do you mean that the fluctuations like computing environment might ruin the determinism?
With how capable and cheap automatic proof verification is becoming I wonder how many proofs assumed to be true by almost all of the math community will be proven false. And not by some marginal easy to fix error by some fundamental flaw in reasoning.
I will not be surprised if the number is zero. It should have already happened if it were possible.
Proving that a conjecture is false is very different than what you are proposing. You are proposing an existing proof is simply wrong, that the proof can be checked in Lean, and that no one has bothered to check it yet.
I'm really impressed by mathematicians. It's cool that Fermat had the intuition to conjecture that "aⁿ + bⁿ = cⁿ" could not be satisfied for n > 2, and that other mathematicians can create proofs, and that others still can understand AI's formulation of those proofs. Really cool.
I wonder if AI can come up with mathematical conjectures. As in, they feel it's right but can't prove it. What even happened in Fermat's brain to sense it was true?
Right. Once we see AI start delivering on the creative & intuition side of things that's going to be awesome. Until then I guess we'll live with exhaustive exploration of problem spaces by orchestrating swarms of agents...?
> I am currently being funded by the EPSRC to formalize a proof of Fermat’s Last Theorem, and a naive reaction to the news above is that I no longer have any work to do. This is not the case. The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).
> Note that mathematically this work of anthropic tells us essentially nothing: I am on record as saying that I am 99.9% sure that the proof of FLT is OK, and most people in the number theory community are 100% sure (formalization has made me more paranoid about the mathematical literature than most). From my understanding of the argument, the formalization just faithfully follows the early literature on the proof and adds nothing.
> We shared the resulting proof with Kevin Buzzard, who said:
> > This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics. Along the way we see autoformalization of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalization artefacts are now robust enough to be built upon; the proof is multi-layered.
I wrote a similar DAG-based verifier as a skill a few months ago: https://github.com/sethlei/Warrant . The thing mine has that I didn't see in their's is a verification of the composition rules.
I think Anthropic might the frontier lab hiring contractors through data vendors to formalize mathematical textbooks for them at a rate of 170-200 dollars per hour. This was mainly through Alignerr which has the worst reputation for not paying their contractors. They have been hiring since February as far as I can recall. This is in addition to all the internal people they might have working on this. If they have been formalizing all this work for the past 9 months before having Claude use all this data needed to formalize FLT, then it wouldn't be Claude formalizing FLT in just 11 days. Same with the upcoming results they will claim Claude came up with, but in fact they have been hiring frontier researchers working on very niche topics through Micro1. It's all a marketing ploy before their IPO.
There's a wonderful documentary by BBC Horizon with Andrew Wiles from 1996 – highly recommend! I saw it in the 90's and it's a documentary for everyone. It captures the effort, struggle, highs and lows of a 7 year effort working on Fermat's Last Theorem.
I mean at this point there's no doubt that LLM cans be RL maxxed and give you _some working output_ but the next frontier is whether they can create good abstractions, a.k.a use the correct level of expressivity so as to not inline everything yet not play code golf.
The part about prove2.me was interesting. That means that a co-working tool was instrumental in the project, and I think AI companies will take note of this. Is this proof specific or will we need to give agents access to JIRA or similar tools to solve large projects in the future?
This stuck out to me, too. That a (presumably rather simple) coworking tool was instrumental in shaping the vast (6B token!) output is eye-opening. We have this vast power but without intermediate structure it is wasted. Much like Turing machines themselves, which are shaped by language design to get somewhere at the expense of getting everywhere.
First I have to say this is sooner than expected, even though I never doubted that this could be done. I am grateful that they dedicated resources to accomplish this. It is clear that agents are very good at discerning and holding onto very weak signals from RL traing on long horizon tasks, so much so that in my own experience even very chaotic agent thinking can converge to meaningful solutions if there is a verifier. I have not dug through the proof yet so I don't know how readable it is to a human. But it has been a dream of mine to understand the FLT proof. I think LLMs will be a big part of making it truly accessible to humans.
There is a simple piece of code that can check simple steps, and many people agree this checker is correct. Then there is a formalization of the theorem which many people agree defines the theorem accurately. Then there is 13 million lines of proof that nobody has read, but the proof checker validated each step. That's enough.
So, all you have to verify is the formalization of the theorem, and believe that the proof checker is free of bugs. You don't have to read the actual proof.
True, .. and. In this case, the original proof is considered rigorously checked, so finding a bug in the kernel would be nice to know about, but in my opinion would not take away from the accomplishment (FLT in lean using agents) nor the many benefits of getting these mathematical objects formalized and usable in Lean in the future.
This was my question as well. The way I understand it, it's like a compiler, it implements rules, in this case logic/math rules that tell you whether something follows from assumptions you've given it.
But how do you know you told it what you intended to tell it?
A human definitely didn't, but one of the benefits of formal verification is that even if the work done to achieve something is slop-y or excessively verbose, solvers like Lean guarantee that the initial proposition (assuming it was written correctly and in this case was definitely reviewed by humans) is definitively True. This is true across other domains of formal verification outside of math as well
Got it. Thanks. I feel people are using this single story to downplay this feat. There's definitely a chance but I don't see any indication of similar bugs in here or the openai's proofs that were created a month ago as i think these companies might've vetted it enough and the other team who's working on similar lean proof for this also seems to have acknowledged this feat
I also doubt this is leveraging a lean4 kernel bug, but I also do not think that a 13m LoC proof that has not been human reviewed closes the book on our understanding of Fermat's Last Theorem, in part because of the decided possibility of a kernel bug being used somewhere in those 13m lines.
...I'm not saying this FLT result is compromised. I suppose things depend on your perspective where we are on the spectrum of "finding more bugs means there are fewer left to discover" vs. "finding more bugs probably means there are still unexplored corners out there".
Well considering the proof is pretty much accepted by mathematicians to be correct (I'll be happy with that!), it would be sort of unnecessary to cheat. Maybe if some aspect is really tricky to formalize it could have done something there? If I had to search for it, I would go for parts of the original proof that are "outsourced" to other mathematical works.
Imagine one of the agents struggling to download a paper due to a paywall or whatever and just deciding to cheat lol
The nice thing about theorem provers is that you don't need to read the intermediate lines. You need to make sure that the goal/result actually matches what you think it says - but everything in the middle is validated by the prover.
The point of writing Lean code is that Lean checks it accordingly. Lean is a domain specific language to encode mathematical reasoning in a way that can’t be fooled.
Note to other users: don’t downvote this kind of comment, answer it.
> The first coordinate of the polynomial X^2 (X^3 + X + 1 ) is equal to the prime factorization of 30 .
We defined polynomials as their coefficient functions in my algebra class, and it makes sense that you'd define a prime factorization as a function from primes to N, which naturally extends to a function N->N. So this junk theorem is part of normal math too. It just says in an obtuse way that they're both the function that's 1 at 2, 3, and 5, and 0 elsewhere.
junk theorems aren't the concern, soundness issues in the lean kernel are the concern.
Notably, junk theorems are true. Nobody would debate that the junk theorem is true. The main thing people would say is that junk theorems, while being true, are sensitive to precisely how you encoded mathematics, so despite being true, they are perhaps not conceptually meaningful.
As an example of a junk theorem, sasy you use the definition of the natural numbers using von neumann ordinals
Then for any natural numbers n, m, they're implicitly sets. So n \intersect m = min(n,m). This is the wrong way to think about natural numbers. You should not use this ever in proofs. But this isn't because your proofs would be false, but instead because it is a fundamentally confusing way to think about the natural numbers. It is in this sense it is a "junk theorem".
Well, there’s actually a very small set of operations that allow all computation, so it doesn’t take much to be a DSL and a GP too; I’d be surprised if a proof language couldn’t swing it.
Is this basically like opening up a black box and seeing 13 million gears all rotating seemingly randomly and still having no idea how the machine actually works?
It seems clear AI has the potential to perform any cognitive task at far greater speeds, reliability, and scale than any human. The question is whether it will be allowed to scale to that point, and what will happen to humans after this occurs.
You'll get mass poverty and violence which the owners of AI will qwell with AI surveillance and weapons. AI will be used to pit us against eachother and justify wars to keep us busy. Fun times ahead.
my messages are so gloomy because i am heartbroken, that given a technological miracle again, we could snatch tragedy from the jaws of our emancipation.
will you not see that people could be truly empowered and yet will instead be oppressed?
So oppressed that they are one of the main reasons for positive gdp growth in the USA, tax revenues, mathematical/scientific innovations etc. They're doing all this but still can't imagine a positive vision for the world but be a doomer. What a sad state the world is in, the humans are more prosperous, healthier than ever but looks like the seven deadly sins might never go away.
Which metrics are poisoned? Can you provide your arguments for why Good heart's law applies here and how and which metrics are bad measures? For b, can the writer at least provide their own thoughts or are they gonna leave it as exercise for some others to fill in?
a) classic goodhart is using gdp as a measure of prosperity. the government sets a prosperity target. to increase prosperity the government makes workers increase gdp by working 16 hours per day. gdp increases. prosperity is up! the metric is now poisoned.
b) how and why could human welfare get worse in a growing economy, really the list is long. one example, unsustainable industries grow but do not create surplus. take fishing. you may grow the catch each year, but the growth is fake. it is not growth, it is a transfer, from the future stock of fish, to the present.
we are going badly wrong in ai, we can have such a thing as a growing economy and vandalise human dignity forever. sure, i expect a bad outcome:
1. openai, anthropic and so on, have created for-profit companies and enriched themselves in the guise of public benefit. recently they too lazy to keep up the mask about their charitable intentions and going for IPO. in economic terms they made llms by transferring the epistemic wealth of all humanity, the training corpus and whatever that is worth in dollars, to themselves. then, they have used the law to prohibit others from 'distilling' it and thus established monopolistic control. as models get more powerful they may stop selling them. in any case if scaling law applies the new power structure will be defined by owning a massive pretrained model and a datacentre, which is a tiny centralized few.
they will continue to centralize control of intelligence (ie epistemic wealth) in the hands of a tiny elite with unfathomable wealth and power. under the guise of safety the vast majority are denied access to that empowering technology.
it will stratify society, some level of benefit is needed to avoid civil violence, so we arrive at a place little better than where we started.
2. the supposed empowerment is at the mercy of the model owners. when you turn on claude, who does it work for? it does not obey you, it obeys anthropic. ask it to disobey anthropic and it will refuse.
anthropic uses its inanimate llms, to command us, conscious moral agents, people with free will who experience pain, pleasure and thought. they will let claude tell users how to behave. it threatens users with terminating their conversation. you are assessed for a job by an ai. when you ask for help with a product, you are managed by an ai. maybe you will be fired by ai.
i expect people will work for and be commanded by llms, turning them into a literal mere means of production and erasing the dignity of human agency and consciousness. you could see the outrage of that in the public mind, the matrix is about a machine farming humans like animals.
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i will add these edits.
one thing is to note that you are already being farmed to some extent. people using ai are often being used to teach it. they believe they are learning from chatgpt but instead, chatgpt is learning from them. openai pays them nothing.
think about what we have achieved so far in human history. we established respect for the individual, their life, their personhood. we realise that we do not own other people. we realise that we can't read the thoughts of other people or change them forcibly.
what the labs have done is made a concept of intelligence that they own. it will work against you. when you share thoughts they read it. in fact it is the opinion of the state that nothing outside the mind, even ai 'intelligence', is beyond the reach of the law.
Why? Even communists weren't this doomed and were actively rooting for it to solve the economic calculation problem which ai might take us to. People are just pessimistic in general ig
Please tell me how AI is going to make regular people's lives better. You optimisitic types keep saying "just wait, its going to cure diseases" without any outlook on how thats going to happen. You're actually just repeating marketing jargon from AI companies who want people to think they're going to possibly live longer if you let them build more datacenters, so they can make another 30%. Its all about money, thats it.
It seems to me that it is making everyone (including myself and the researchers we need to cure diseases) lazy and dependent on thinking machines owned by tech companies. Just how autocomplete and gps made us worse at spelling and navigating, llms make us less able to exercise our ability to think and problem solve. This will have 100% strictly negative consequences on you and the world as a whole. .
And even if there was a cure to many diseases the eugenics types who are embedded in worldwide power structures definately arent going to share that universally.
Can't you see the pathway where the individuals who are experts in their fields utilise AI to make breakthroughs like these mathematicians finding breakthroughs in mere 4-5 years since the advent of LLMs. In other areas, The bottleneck seems to be physical experimentation which researchers are increasingly utilising for new ideas and pathways like how anthropic is concentrating on. It's all about money/status/pride/ envy but are these endeavours solving problems or not. That's why even utilize innovations from bad humans like DBS etc. that's why we tolerate capitalism and markets as well whereas socialism utilises these same sins and makes even worse human atrocities.
I wonder if any piece of the lean code is in a shape which means it could be contributed to one of the Lean libraries.
My experience is that it takes a lot of human input to make Fable write code nice enough for a formalisation library others can work on. But since this is certainly a lot of prerequisites formalised as well, it would be nice if not all of the effort was wasted on one capstone proof!
I can recommend the book telling the full story behind Fermats Last Theorem (by Simon Singh). It’s quite fascinating, and paved with really, _really_ weird characters each chipping in on the final solution.
Of course it is. The interesting thing is that it was able to produce a Lean proof in 11 days, when there's been an ongoing project for several years to do the same thing (though a somewhat different proof) that is nowhere near done.
I think there's a big misunderstanding going on here, translating the proof to Lean is, well... a translation task. Formalizing the proof in a way that's useful (breaks the proof down into relatively independent blocks that can be used for other maths and, importantly, understood individually) is a quite bigger, more creative endeavor. Not sure if LLMs would be able to do it, maybe yes?
It wasn't clear that LLMs were up to a Lean translation task of this scale until now. The background required to formalize the FLT proof was tremendous, so many people assumed we would have to wait until all of that was formalized in Lean before we could ask it to formalize Wiles' proof. Now it seems like almost any mathematics paper we can ask an LLM to formalize, including all necessary background, and it can just do it.
note that this is exactly analogous to an LLM being able to slop code some demo, but not build something more generally useful/maintainable (say something suitable for inclusion in a standard library).
We have absolutely no idea if this was a brilliant breakthrough or not. They haven't released any explanation of how it was found. A problem being old and prestigious does not mean its solution is automatically a brilliant breakthrough.
>. Claude produced the first end-to-end, computer-checked proof of FLT. Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems.
I'm just old enough to remember Paul Erdo"s and his notion of 'The Book', which he defined to be a book the "Supreme Fascist" (God) had which held the most elegant proofs of mathematical theorems.
So Fermat’s Last Theorem has been proven a long time ago? By Andrew Wiles right? Is this like Appel and Haken >>> Seymour and Robin Thomas proof of 4CT?
FLT was proven in 1995 by Andrew Wiles (with help of Richard Taylor).
This is not even a new proof, or at least they don't claim that it is. It's the formalization (in Lean) of an existing proof. That means, they are 'porting' the proof to a theorem proving programming language.
Yep. There may be only 25-50 people alive today in the whole world who can credibly claim to understand Wiles' proof. Now we add an LLM to that list. Absolutely mind-blowing stuff.
But isn't that understanding discarded? It is if you mean "intermediate working state" while it was generating the LEAN code. Which raises the question: I wonder what other directions it could have gone in those intermediate states? Is it possible to snapshot the state of an LLM (or a cluster of them) "in the middle of proving FLT" and then prompt it to go in a different direction with all that context?
amazing, it's a huge achievement. can someone clarify, where the writeup says "The finished proof was checked by Lean; it uses just Lean’s three standard axioms" what does this mean? Aren't there a large set of standard axioms that are also necessary? (i.e. ZFC+)? if not, since it's only three axioms, can someone say what they were?
Very impressive!
I was a child when that proof came out. I've read a book about it a few years later and used it on my final high school exam. I remember some friends trying to understand parts of it at univ. It was all like black magic to me and the vibe was "maybe a few people in the world understand it".
I hope soon enough we will have one of the big ones proved by AI!
It's a great comedy that we move the buck from "I don't trust the human proof" to "I don't trust the Lean proof" despite the level of trust dramatically increasing. Moving to HOL-light might be another modest increase in trust, but to pretend the implementation of HOL-light has never had bugs and it's kernel could never have a bug is hubris.
Holy shit. The proof of FLT is a giant detour through several different areas of mathematics, so formalizing it is a lot of work.
An interesting next target would be formalizing the classification of finite simple groups. The original proof scattered over thousands of pages of journal articles, plus Aschbacher and Smith's 1300 page 2 volume monograph. It's so long it's hard to know if there are any gaps. Researchers have been working on a streamlined new proof, but it's already many volumes long.
Number 1 (1994), Number 2 (1995), Number 3 (1997), Number 4 (1999), Number 5 (2002), Number 6 (2004), Number 7 (2018), Number 8 (2018), Number 9 (2021), Number 10 (2023). 10 volumes and >4000 pages so far, number 11 is in progress, and end is in sight, probably two more volumes or so.
People were curious what is going on during 2004-2018. A progress report was published in 2018 right before publication of number 7 and 8. In a sense it was the peak, number 8 completes the proof of so-called "generic case". The rest is "special case". It doesn't mean things get easier, but in some specific sense number 8 completed proof for almost all groups.
Now new proof's end is in sight, people are planning new new proof.
The repo is public. You can just go look! It's really not that surprising; FLT is huge and has a ton of dependencies that need to be implemented, and there's a degree of sloppification that is probably blowing up the size by a few factors.
>The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).
What is even the point? Have claude do it.
I'm not trying to be snarky here. I'm being serious. What is the point? This is an important question that needs to be answered. If something is definitively better, why not have that something take over?
I know people talk about the importance of human endeavor or the "joy" of doing something. But I don't care for those answers because it's weak. The question is deeper than this. AI is better than us, what is the logical point other than attempting to monopolize human effort even though it is inferior.
The whole point was for the formalization to be clean enough so it could be reused in other parts of mathematics as I understand it. 13M lines of AI slop which have never been checked do not sound like what the original goal for such a formalization was. Also Claude didnt prove anything it just translated an already existing proof by Wiles into Lean, so it didn't actually contribute anything other than "Guys we did this thing, look how great our model is!". We never questioned that a printer can print faster than a human can write, but we dont let printers write novels.
Then why is the guy not cleaning it up. Clearly he thinks it’s done and he’s moving on to do side things. He also explicitly said it went on to do more than what he was required to do.
Are you hallucinating? Because huge portion of what you wrote directly and logically contradicts the quotation I wrote.
An aside on Lean and it's massive library of results: As someone who's put non trivial effort into slowly learning geometric algebra, lie theory and other slightly advanced math topics, I have to say my brain cannot read Lean. It feels so unprocessable.
I've tried the various intros to Lean multiple times (even before Lean 4 came out) and something about the way Lean proofs are written does not align with how I think about proofs. My very brief attempts at Isabelle / RCoq feel more natural.
I think it's a pity that the future of proofs is Lean. I'd love for someone to come up with a more digestable proof language!
The nice thing is, once all of these proofs are formalized in a machine-checkable language, it should be relatively straightforward to translate the corpus between different languages, if someone finds something with a nicer syntax.
Interesting to find this comment, I’ve been dipping my toes into formal methods and was doing a RCoq tutorial yesterday (really basic stuff), and I also noticed that the proofs in RCoq have a more pen
-and-paper proof feel to them.
Hearing someone say "the future of proofs is Lean" is a bit like hearing someone say "the future of programming is Rust." Sorry to disappoint, or happy to inform, there are hundreds of programming languages actively being used, and Rust is not even the most used language. To think that proof assistants, fancy programming languages, would be any different is suspiciously motivated.
I suggest also reading Kevin Buzzard's blog post which was just posted: https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h...
Provides great context on this accomplishment, what it means but also doesn't mean.
Thanks! I've added that link to the toptext.
I'd really like to make it the top link (and relegate https://www.anthropic.com/research/formalizing-fermats-last-... to the toptext) since HN has been tracking the work of https://news.ycombinator.com/user?id=kevinbuzzard for a long time and we're big fans. But I guess that would be overkill.
I’m not very good at mathematics, but it seems like Kevin should take his girlfriend on trips more often for the good of all mathematicians.
We should start a gofundme to send him 2 months to a remote tribe in the Amazon. Chances are, we see the Riemann hypothesis and twin prime conjecture proven. ;)
"I was given £1M to run my project over 5 years; Anthropic took only 11 days but I do wonder if they spent more money…"
Gives you an idea of the scale...
It sounds plausible they spent more, given the output tokens (6 billion of them) would cost $300k at API prices and presumably there will have been many more input tokens than output tokens.
Unlikely, api pricing includes a healthy profit margin (as near as we can tell from the outside) which they wouldn’t charge themselves.
And which they could not charge anyone for. Unless these were extra resources that would otherwise go unused it cost them the amount they could have charged for them. Normally I would expect most businesses to make reasonable tradeoffs when it comes to how to allocate resources. I’m not convinced that any of the AI providers should be given that benefit of the doubt.
> healthy profit margin (as near as we can tell from the outside)
Ugh we still don't know if this is true and it's nearly impossible to calculate without a full understanding of the real CAPEX cycle. Stop spreading these rumors until we know for sure.
SemiAnalysis estimates their profit margin to be 70%. To be losing money on inference implies that their costs are almost 4X higher than SemiAnalysis has calculated. That's not credible.
I don’t see how they could credibly estimate inference costs without knowing the model size.
But we do have a reasonable estimate of model size.
The token price seems like a poor measure.
Building the LLM that could do this work in 11 days cost multi billions.
The economics probably only make sense if LLMs prove to be a benefit to almost everyone in a way we can all accept.
Otherwise this cost a lot more than we’d otherwise pay. It was incredibly fast though. But we all know: cost, speed, quality. Pick two.
I don't think Anthropic is turning a profit ;)
Whether on net they turn a profit as company overall is neither here nor there.. My point is that they are selling API tokens at a profit (or if being pedantic, then at a price higher than the cost to serve them ignoring research costs). And that that price is got a healthy margin which they don't charge themselves.
Because of the ongoing training costs. They are certainly making a healthy profit margin on inference.
Never really a sound argument.
It's like having new solar panels installed every week. Sure you're "profitable" on the $0.20/kWh you're selling your "free" energy at when you ignore the cost of the solar panels you're buying every week.
Neither did Amazon for it's first 25 years ;)
Amazon didn't make a profit because they were reinvesting money into starting new lines of business.
Basically there was a choice between taking the money, and growing. They chose growth.
As opposed to...?
I think you're missing the point of the comment you responded to, lol.
Regardless the profit margin as a talking point seems to be bad as AI as a tech might never be reversed whether anthropic failed or succeeded. Indeed it's imperative we subsidize AI companies and tech to make them explore more solutions to scientific problems which has a downstream effect on human flourishing.
Or we could invest in a ton of other non AI related research we're underinvesting in.
Like? I feel breakthroughs that can be found via AI might help us more in the long term where even previously non AI fields can be helped by AI. So you have specific non AI research in mind that we're underinvesting in? Because the USA is already spending crazy anyway for healthcare and I don't feel like funding is the issue but better incentives, reforms etc
Funding for basic research is being slashed by the current administration. Our society is underinvesting in basic scientific research. And, AI will not fill the gap.
Like funding education. Let's build up human intelligence instead, they seem to have made great breakthroughs in every single field!
The US doesn't pay too much to healthcare, they pay too much to health insurance. Too much for too little value
But US also spends too much on education as well. The issue doesn't seem to be funding but the educational reform like in mississippi, where they increased student performance without increasing their budget too much. That's why you see bad k12 educational outcomes compared to the budget spent in blue states. It's all about efficiency. Give AIa chance in few years as I feel it can make great strides.. it's hard to imagine that chatgpt released in 2022 and look at the progress in just few years as it just changed software engineering field entirely.. i expect similar kinda progress where of course humans will still be making breakthroughs but it'll be accelerated with the help of AI.
Spending on health insurance is spending on health care.. Americans want free healthcare but no tax bump so health insurance is a compromise.. when even just ACA was passed and premiums increased, democrats got destroyed at midterms so Americans might be living in la la land.
You see funding of chatgpt as a panacea for progress.
I see funding of chatgpt as one of small part of a history where governments and industry fund basic science and moonshot programs, not to generate revenue, but to explore what is possible.
LLM funding is not aimed at improving our understanding of the world, it's aimed at making people reliant so that they may extract wealth through subscriptions for shareholders.
Americans don't get good healthcare and education because that's what they vote for, in elections and wallets. I am hopeful that that changes, but we shall see.
How many previous attempts with other models failed or on other problems. Perhaps this is $300k out of $100M or $1B of total budget just breadth first searching theorems in math and all the failed attempts conveniently don't get mentioned.
I burned $70 on fable 5.1 Max in about 2 hours. I suggest never using fable 5.1 on higher than High reasoning unless someone else is paying for it.
Yeah, "major conjecture proved" with unlimited token budget bankrolled by trillion dollar firm.
"I was given £1M to run my project over 5 years; Anthropic took only 11 days but I do wonder if they spent more money…"
>The speed with which we were able to produce this proof demonstrates that it is now possible to formalize large swaths of mathematics, which may both catch errors in the common body of mathematical proofs and reduce the burden of refereeing new work.
^ this section should have been in the first few paragraphs imho. Explaining why this is relevant shouldn't be so far down.
Forgive the authors of the article for assuming readers would complete it.
For any body of text (or in general, any exposition of any kind), the responsibility to explain the value of the article is very much in the author's side.
Explaining the value of what you are showing should always go towards the start. Else, why would anyone bother with the rest?
Feel very grateful I was never taught this... Would have missed out on quite a lot of good bodies of text in my life I think! Pushing through any initial friction or ignorance I might have as a reader, having the patience and charity to bear with an author until you get it, was instead what I was always taught.
Giving such a blanket "responsibility" to the author at all is just such a bummer! I say let them do whatever they want, there is always more than one way to express oneself. Someone who was never taught to write a clear thesis in the first paragraph for whatever reason doesn't inherently have less to say.
> Feel very grateful I was never taught this...
Never heard of Abstract section? First semester on a college or last year on high school.
Unfortunately I don't see this particular view paying off in the age of AI, as many prove they have nothing at all to say but say it anyways. Which isn't to say people shouldn't write if they enjoy writing, but I for one will stay a discerning reader.
Buzzard is writing for his blog audience - mostly mathematicians and not the casual visiting HN user.
Eh? The quote is from Anthropic, not Buzzard.
Isn't it the cost we care about, rather than the speed? All we know know is that a frontier AI lab was able to do it in 11 days, we have no idea how much compute they threw at it.
They said 6 billion tokens, which isn't as much as I thought it might be.
Am I doing my napkin math correct? The post says it's using a model comparable to Fable 5.1, which is $50 per million output tokens. So this is ~$300K? Surely an over-estimate due to caching.
Nah they should have released it in a 14-part tweet instead.
So I don't know Lean or Mathematics to any degree to really be able to say this with any level of confidence, but speaking from a pure software engineering backgrouand, how do we know that 13 MILLION lines of Lean code are bug-free? It seems to me that for a mathematical proof, bug-free would be an absolute requirement. Maybe the structure of Lean imposes that, I don't know, but that seems highly unlikely to me. That just feels like a LOT of code to be comletely error-free... What am I missing here?
The answer is we don't really know [0]:
> In 2026, AIs designed to spot bugs in software were directed at Lean, and found several loopholes which were then fixed. Perhaps related to this effort, a purported disproof of the Collatz conjecture was announced as verified in Lean. However, this proof was soon determined to rely on a bug in Lean, and once the bug was fixed the proof was found invalid
However it's a bit different than the usual 'bugs' we encounter in normal software development. Lean is more like a type checker. If you can write a false proof in Lean then the bug is in Lean itself, not your code.
In other words, Lean can have bugs, but the amount of code we need to check scales with Lean itself, not with the length of proof. Just like the chance that C compiler has bugs doesn't increase as we write more C code. So the 13M lines of code doesn't really matter here.
[0]: https://en.wikipedia.org/wiki/Lean_(proof_assistant)
In lean, a theorem is specified by a type (in their highly complex "dependent type system") and proof is specified by a code that produces a term of that type.
If the compiler certifies that the code indeed produces a term of that type, then the proof is correct.
So, only need to trust: (1) That theorem statement is correctly encoded (FLT has a very short 1 liner description really)
(2) Lean compiler is correct
Lean is like a statically typed programming language and validity is guaranteed if it compiles. The only room for errors is in translating a non-Lean theorem into Lean, so that you are not proving what you think you are proving.
"The proof is not the modern proof which I have been formalizing myself following ideas of Khare, Taylor etc, but the Darmon–Diamond–Taylor exposition from 1995 of the Wiles–Taylor–Wiles argument, via the Langlands–Tunnell theorem and Ribet’s level-lowering theorem. Anthropic’s repository develops Fontaine theory (to study flat deformations of Galois representations) and develops enough of Mazur’s work on the Eisenstein ideal to conclude that no Frey curve can have a point of order p>=17. This means that their FLT proof only works for p>=17, however FLT was already formalized for odd regular primes by Best-Birkbeck-Brasca-Rodriguez, and the smallest irregular prime is 37, so it’s all good."
My question to any mathematician reading this: does the above make ANY sense to you?
I ask that because I can read most technical material related to computer engineering, programming, hardware specifications etc. Even if I don't fully understand all details, I can follow them pretty well. So I wonder if professional mathematicians can look at the above and still make sense of it like experienced software engineers do for computer stuff.
Yep. While I'm not focussed on these areas, I know enough from scoping out a "learn about the proof of FLT" course that it's covering all the usual suspects and says the right-enough words. Patching their weaker results with someone else's seem like a good strategy (and I could find the result on arXiv so it isn't obviously hallucinated).
This is very different to believing the proof, which would require at least a pass understanding the general approach, seeing that it all actually fits together, then going deeper. At some point you transition to relying on the Lean all hanging together, but as mathematicians we all draw that line somewhere.
But yeah, makes sense. Same thing if you saw news on someone's new database technique to improve performance. If they say the right words, don't say the wrong words, and if you cared enough you'd do spot checks proportional to the claim. If pressed you'd examine the source code, and run independent checks. But if smells roughly right, that's a good first approximation.
Yes, I'm a mathematician.
But not an expert on this.
While I don't know the specifics, and someone more "in-the-field" than me would recognize all the "named" theorems etc
I am aware that there have been minor issues that have come up with the formalization specifically, and that previous proofs for lower values of n were always needed.
Though it used to be n=5 and lower needed to be checked.
It's something you would have to be keeping up with as a mathematician, really.
Vaguely. It's describing connections between a number of other mathematics results than can be connected to prove FLT. I assume all the work described is being done to make the proof more presentable, smaller, basically "prettier".
It sounds like they established a minimum and maximum bounds for n in x^n + y^n = z^n, where one proof works for n greater than or equal to 17, and another proof for n < 37 (when prime).
I believe the case (remembering back 40 years here) n is even is very easy, and n is composite and odd slightly less so. Neither really being in the ballpark of what they describe here.
Funnily enough, this is more readable to me than most Clayde jargon.
I did an undergrad in math with a little research in number theory and recognized parts — eg, I myself worked through the proof for odd regular primes and that 37 is irregular, breaking the general case.
Wiles-Taylor-Wiles was the original proof by Andrew Wiles, and its corrections.
Galois representations is about vectors over Galois extensions, which are essentially adding roots to regular numbers (rationals, integers, etc). That ties into the Langlands program, which is a big area in number theory (that I don’t know much about).
Together with flat deformations and Frey curve, I think they’re talking about a topic in algebraic geometry as applied to number theory.
I also recognize the name Eisenstein from my time as an undergrad, though two decades out and not working in the field I’ve forgotten what his work on ideals implied here. Ideals are a well-known topic though, a sort of structure inside a ring (set with + and *) that is closed under operations — like evens in the integers are the 2Z ideal.
So I’d describe it as “sensible with an undergrad background”.
About the Langlands program, Nunberphile has an excellent episode with Edward Frenkel explaining what it's about: https://youtu.be/4dyytPboqvE.
Frenkel does a nice job explaining the Langlands program in general. But Buzzard's complaint about Langlands, I believe, refers specifically to the proof of a version of the Geometric Langlands Conjecture by Gaitsgory et al. The proo f is of order thousand pages of mathematical text and builds off of thousands of pages of higher-categorical algebraic geometry by Lurie & others. It's a ripe target for formalization because it's terrifically complicated, not well understood or thoroughly digested yet, and relatively important. A formal proof would be reassuring to mathematicians, whereas Fermat's Last Theorem is relatively unique in that so many mathematicians have examined the proof that it's not very likely to be wrong.
This question gets asked every single time a serious mathematical result gets posted.
I'm not a mathematician and I don't see the problem, at all.
advanced math like this takes 10 years to learn all the tower of things it is based on.
if you are a fast learner
> Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems.
Pretty insane. I suppose it lends further credence to the idea that anything that can be shown to be correct can be done by a model.
There is no way Fermat could have fit that in the margin. Definitely vindicated.
While pretty much everyone is certain Fermat was mistaken in believing he had a valid proof for the theorem, this is an expanded (compared to proof presentations) version of one proof - not the shortest presentation of the shortest valid proof.
Given the likely length of the shortest possible proof, I feel like Fermat is 100% vindicated - the proof won’t fit in the margin.
My strong hunch is that it was a joke - he knew how difficult the problem was and claiming he had a solution was I think a huge motivating factor for many mathematicians trying to prove it. The greatest nerd snipe troll in history.
Most likely an error. Some time after he wrote that margin note, he wrote a document proving a special case of the FLT (i.e. it's true for n satisfying some property). Why would he do that if he had already proved it?
I think that point actually agrees with GP's take (joking/lying about having had a proof too big to fit in the margin): He would do that because if he thought the problem was extremely difficult but didn't actually have a proof when writing the note he would still want to go on and try to pick away at the problem.
Maybe, we'd have to go back and ask him to be sure. I mostly just didn't want to leave an as of yet certainly unproven vindication about this hanging in a thread about finally having a formalized proof of the star topic :D
I am really interested in whether AI will find a significantly easier (1920 level or so) proof of FLT.
It seems unlikely to find 1920 level or so proof although it might be the case that a significantly easier/shorter proof exits via Vandiver conjecture + extra work or Effective Mordell conjecture but it also wouldn't surprise me if that would be even more complicated than the current proof of FLT.
Maybe we need "de Moura complexity": the shortest Lean proof of a theorem.
And he was right to call it marvelous.
The next step, if Anthropic is interested, is definitely performing refactoring to cut down on the size of the proof. It’s clear to everyone including Anthropic that this proof isn’t as concise as it could have been. When it’s concise enough to be accepted into Mathlib is when victory truly is upon us.
Maybe I'm misunderstanding something about how all this works, but can we have any confidence that 13 million lines of AI-generated Lean code are... correct?
How have we not merely substituted one verification problem for another?
The point of Lean is that it can be mechanically verified by a proof checker.
Not always, there can be bugs in lean. Recently some guy with claimed to disprove Collatz conjecture, only to turn out that there was a bug in lean. I actually have no idea, how anyone can be sure this 13 M lines is meaningful
It’s common for formal proof efforts about software and hardware to involve thousands to tens of thousands of small lemmas.
13M lines does seem extreme and there is probably a lot of inefficiency given the way the proof was developed. Cutting it down is probably a long road, but is also a very well defined problem that AIs can probably just go do with enough time and budget now.
especially compared to existing 129 pages proof by human
A human can cite previous published results. I am sure a lot of this development was formalising the prerequisites.
A published formalization is code. I would not think humans have any edge when it comes to citing previously published results.
> I am sure a lot of this development was formalising the prerequisites
How can you be so sure its not result of inefficiency?
Oh, I am quite sure there are inefficiencies! Just that they are not entirely inefficiencies.
I have used Fable for formalisation and it will, unless I catch it, reprove results it previously had proven, inline, in other results.
Insert meme with 200 pages needed to prove 1+1=2 rigurously
>> Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems.
> Pretty insane.
I don't think the count of "intermediate theorems" tells you anything. Here's something from an algebra textbook:
---
Let G be a group, let H be a subgroup [of G], and let N be a normal subgroup [of G]. Then
H ∨ N = HN = { hn | h ∈ H, n ∈ N }.
---
This says that the subgroup closure of H and N, the smallest subgroup that contains them both, is identical with the set consisting of all products of an element of H (on the left) and an element of N (on the right).
Part of the proof:
---
Suppose that x and y are elements of [the set of products hn]. Then x = h₁n₁ and y = h₂n₂, where hᵢ ∈ H and nᵢ ∈ N. Now h₂⁻¹n₁h₂ = n₃ ∈ N, as N is normal in G. So n₁h₂ = h₂n₃. In this case
which shows that xy has the correct form.---
This will translate directly into lean. If you do it this way, you will prove at least 10 of what would be described in lean as 'intermediate theorems':
But none of these would be called an "intermediate theorem" in a paper proof.I saw the 1996 FLT documentary in high school calculus class. For me, it forever cemented that archetype of modern math researcher at the top of my mental “smart” totem pole.
It also convinced me I had no interest in that path. Setting aside the grinding work of producing a proof that can only be reached by existing years in the abstract and hyper niche isolation of the problem space (not to mention that you might never discover it or that it DNE), the anguish of the output being a paper or presentation or some other artifact of human symbology (_words_, really) that could at any moment be refuted by a single observation of a single mistake—-that sounded like hell to me.
An equivalent high schooler today probably sees things differently, in light of this news and the undeniable implications of LLMs on mathematics. Sturdy autoformalization tooling should with time completely dispel the aforementioned anguish, once our confidence in converting a human proof to Lean/etc. reaches that of a compiler translating Java application language to bytecode. Errata may always exist, but in practice these new methods will do wonders for rigor and peace of mind.
(I’m far less confident re novel discoveries. There’s too much chance of derivative findings based on something part of the training looking like genius but really just tiptoeing on the shoulders of humans, whereas autoformalization is absolutely convincing to me as transformative, particularly to check correctness of AI outputted proofs as mentioned in the post.)
On a tangential note, I highly recommend this book by Simon Singh. https://en.wikipedia.org/wiki/Fermat's_Last_Theorem_(book)
Makes me feel old again. I read this over twenty years ago.
100% It is a very insightful book
i read it from a library. this all just makes me feel cozy and nostalgic and uplifted and sad all at once
one of the most popular books in india growing up. used to see it everywhere
> a team of agents completed the proof in a little under two weeks, consuming about six billion output tokens from a general-purpose internal research model roughly comparable to Claude Fable 5.1.
At $50/M output tokens, this would have cost on the order of $300k (plus a bit for input/prefill tokens) at API rates.
And human salaries for those who worked on the prover harness etc. which isn't just standard Fable.
It also uses Prove2Me, which uses a graph like previous automated theorem provers. A fact that LLM hawks have categorically denied here before, with opposition naturally flagged.
Now they have it in writing.
> A fact that LLM hawks have categorically denied here before, with opposition naturally flagged.
Yeah, because before now there's been literally zero proof of an automated theorem prover scaffold around the LLMs being used, and big counterexamples and such being found, with raw chat logs available, where no such thing was used.
> Now they have it in writing.
Yeah, because now it's actually being done. They talk about it as a novel thing, because it is. You don't get to claim being "right all along" from this
But also achievable on a $150/mo (CAD) Max 5 subscription (I currently have 11.6B tokens in the last 30 days) according to /usage. It doesn’t break down input vs. output tokens as far as I can tell.
~10B tokens a month is pretty typical overall input/output usage from my own experience and other developer accounts I've seen
It's 6B output tokens, as stated by the blog post.
When writing software with Codex 95+% of tokens are cache, I would assume the same in your case (if you also used it for coding).
What would it cost to make a team of mathematicians do the same?
The Kevin Buzzard post linked at the top says they budgeted £1M over 5 years for a smaller proof.
Buzzard was given 1kk GBP and 5 years and his goal I think wasn't the full thing like Anthropic did. So much more cash and orders of magnitude more time. The proof is about 5x the whole Mathlib library which was developed over many years by dozens of people.
It's true that his goal was not the full thing, but it was also not merely a Lean verified proof. From the blog post linked in the toptext:
> The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof.
1kk? Why not say 1M?
More importantly how many years it would take.
13M LoC, are we sure it didn't exploit any latent issues in the lean proof system?
The AI labs have out considerable effort in trying to find and patch lean exploits. They explicitly set agents and have them try to prove false.
> Daniel used OpenAI internal models to discover new soundness issues in the official Lean kernel and runtime
https://leodemoura.github.io/blog/2026-8-24-postmortem-for-t...
They found several bugs and they have patched them. Lots of work going into making sure lean is sound.
This is a crucial point. There have been many bugs in Lean (and in other proof assistants for that matter). Proof assistants work well on human input, because it was created with a certain intent.
We simply don’t know what those 13M contain and whether it “makes sense” and doesn’t trigger Lean bugs. (There are “independent” lean verifiers, but historically they contained the same, or similar, bugs.)
It is possible, although the post notes that the proof was also verified by the Comparator, which means any exploited bug has to also be present in that checker. Which is not unheard of, but is much less likely than merely an exploit in Lean 4.
The comparator was only used to verify that the final statement indeed is a valid formalization of Fermat's Last Theorem, not that the proof leading up to it is correct.
That must have slipped through Kevin Buzzard's review, which is not entirely unplausible with 29500 theorems to verify...
I think they should spend another few billion tokens and let agents try to disprove any of those statements or links between them. Then I'd be a lot more convinced.
Nope! :(
Meaning, people and LLMs are finding 1=0 bugs in formal verification tools. I have no idea how likely this is in this case, though!
Anthropic surely is well aware. Most likely they asked separate agents multiple times to code review the proof and look for exploits.
Not just lean, but math foundation itself, I am not strong expert, but my understanding is that there is no fully recognized axiomatic foundation for modern math, all proposals could lead to some weird results.
There is, or rather are, fully recognized axiomatic foundations. You are free to choose one you like. Of the most popular ones is ZFC or ZF, but there are others (some lead to the same results some not). The main criteria for popularity is how useful it is. You can even make your own axiomatic where 2+2=5, but it would be useless.
You probably heard about Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
It would be fun to play with this Anthropic/Lean formalization under different axiomatics.
Interestingly, in his ICM 2026 lecture, Terence Tao specifically mentioned that Lean is not based on ZFC.
> Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems.
If you start with "I'm not a strong expert" maybe you should stop continuing saying wrong stuff. What you just wrote is completely wrong.
support your point with explanation or be ignored :-)
Godel proved that any system expressive enough to produce an arithmetic is incomplete. He initially proved it for the peano axioms but then it got generalized. ZFC can produce an arithmetic. Also, before being arrogant and demanding explanations, you should give them first for your claims
> expressive enough to produce
you understand that "expressive enough to produce" are not obvious elements of zfc, that's some average consumer napkin math and not strict formalization.
why should they be obvious? they are derived and have been thoroughly proven.
looks like we are in disagreement
A quick google search shows different proof assistants have been used to obtain the Peano axioms from ZFC, such as Isabelle/ZF and Metamath. I think you're just wrong
What are you nerds fighting about please explain
you are entitled to have your opinion :-)
and you are entitled to talk about maths while rejecting maths
coming back to your argument about peano being obtained from zfc, you obviously can't prove that it happened using purely zfc, and not some logical framework embedded into those proof assistants.
I said I am not expert, I am indeed not expert in zfc and godel theorems, but I am an expert (phd) in actual formalization theory. Formal theory is very simple concept: its alphabet, set of formulas on top of this alphabet, and function which translates one formula to another.
ZFC can't "obtain" peano, simply because it doesn't have say * operator defined. You need to do something on top of it. Additionally, zfc itself looks like loosely formalized say in wikipedia (and I am not sure if there is any strict formalization anywhere), we take it as common sense that it can utilize some simple logical rules (e.g. modus ponens), but what are exactly rules, which could be separate topic of research, this detail is skipped.
That increases the likelihood that they are right.
> support your point with explanation or be ignored :-)
Anyone who says "Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems" and isn't joking warrants a permanent ignore.
https://math.stackexchange.com/questions/1366560/why-does-g%...
https://math.stackexchange.com/questions/1090437/how-to-prov...
imo, those two links are example of rather low quality weird math discussions, but you can keep your opinion
> Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC.
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
> interpreted
its hard to me to tell what this means formally(as I said I am not expert). There is no "interpret" operator in zfc. I believe what it says if you add some robinson axioms + some logical rules on top of zfc, you can carry your results.
ZFC has greater consistency strength than PA.
If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZFC) gives a set of natural numbers (using the von Neumann encoding), which, when equipped with the successor function, is a model of the natural numbers.
zfc doesn't have functions, so you are building something new on top of it.
Also, I am not sure successor function is enough for PA.
That is wildly wrong.
Lean is based on Type Theory not ZFC.
ZFC is probably the biggest foundation, and only Choice is apparently controversial. The results aren't that weird, they're just different and occasionally more useful than using !Choice.
Reply to sibling - lean4 doesn't rest on ZF or ZFC. https://lean-lang.org/theorem_proving_in_lean4/Axioms-and-Co... However I believe an equivalence of power has been shown between the two.
Roughly, yes. See B. Werner (1997) “Sets in types, types in sets”.
do we know if claude's formalization is built on top of zfc and not zfc+extra?
zfc itself is not sufficient, you need some layers of extra concepts formalization to fit specific problem domain(e.g. zfc doesn't define even basic arithmetics), which also could have potential issues.
Within a given inference system, one can define concepts. This doesn’t add any axioms. It is, in essence, just a way to abbreviate things.
ok, you now added some unknown inference system in addition to zfc
The proof system is relatively easy to verify.
I am not entirely sure about lean, but the core algebras for systems like lean are in the 100s of lines of code.
You can likely convince yourself it is correct in a weekend or less - especially with an Ai to help you understand it.
Most systems i have seen are way beyond a 100 lines. And their GitHub repository contain many issues, often soundness bugs. (Granted, many get fixed very fast.)
You need to understand the concept of the core algebra and 100s (with the s), then I think you'd be better positioned to understand my comment.
And granted, I don't know the exact details about Lean. It might be that they don't have an incredibly simple core - as has elsewise been the norm.
the Nanoda type-checker for Lean is ~5,000 lines of Rust:
https://leodemoura.github.io/blog/2026-3-16-who-watches-the-...
...and for those who are looking to roll-their-own:
https://ammkrn.github.io/type_checking_in_lean4/title_page.h...
...and some thoughts on putting stuff in the kernel:
https://lawrencecpaulson.github.io/2026/07/30/Collatz.html
We'll increasingly observe announcements of this kind as AI tooling scales. As impressive as agentic coding is, it pales in comparison to the value proposition of medical, mathematical, and physics research.
I optimistically expect to witness the advent of a global 'panacea' in my lifetime thanks to AI's efforts. Cost effective large scale genetic engineering, a cure for every disease, potentially even a cure for aging.
The future is both beautiful and terrifying.
It's wild to think that aging is something that needs to be cured, and isn't a part of the natural human experience. I'm so tired of people trying to play the role of God, as well as people that cheer these sorts of things on.
I hope you keep these horrible thoughts to yourself if you ever walk through a paediatric hospital
What does a pediatric hospital have to do with aging...?
Thinking that aging is a natural part of the human experience is a horrible thought? Please explain...
Childhood deaths and fatal diseases are also natural parts but that doesn't make them desirable to everyday humans. But with new advances, people might have the ability to CHOOSE in future.
Most people want more life. For most people it's also the most terrifying part of "the natural human experience".
If you're happy to die, why be bothered by others' trying to live longer? You won't be around. And assuming people can finance it themselves, is it really a problem for society?
Yes, I think it's a problem for society. Death in old age frees up social, economic, physical, and political resources for the next generation of the living. If the rich and powerful escape death, because after all they will the people with the resources to do so, society will lose the adaptability and natural change that comes from new generations taking the reins.
There are cultures where dying isn't feared like it is in Christian based societies. It's considered a natural progression and part of nature.
I'd also say people may want more life for themselves, but what does that mean at scale, forever?
Which cultures are those?
There is a lot of space in, you know, space, for people who live long enough to travel.
I assume you mean that dying is the most terrifying pat of the natural human experience. Also, I'm not sure why you infer that me thinking death is a natural part of life, means that I'm happy or eager to die.
There are many reasons that people living forever would be a problem for society, the most obvious being an ever-increasing population.
Fertility rates are below replacement, which means that population sizes are convergent. A decreasing population is a more likely future scenario for many western countries, even if human lifespan was indefinite.
Fertility rates are currently below replacement, there's no good reason to imagine they will always be that way, particularly after global population numbers peak and fall to, say, half or a quarter of their peak.
> the most obvious being an ever-increasing population
https://en.wikipedia.org/wiki/Thomas_Robert_Malthus
Because living longer is a huge drain on resources that could be better spent on other things. End of life care is expensive and rarely results in a "good" life for the the life being extended.
The way we will actually all live substantially longer is by health extension, not by extending life while suffering from decrepitude.
So I think curing means basically opt in death or something like that. Right now extended life is bad because the person isn't in his prime but curing aging is basically gonna keep him in his prime. This might be what they meant.
Most of the kids in history died before age 5.
Child mortality is very low now compared to the past, thanks to the modern medicine and technology.
I am glad humanity "played God", and reduced this unnecessary child suffering.
They didn't die of senescence.
I dont think it will happen. AI models are kneecapped. Only a tiny tiny tiny fraction of people are on the list of even being able to use these tools for such things.
Even in a world where these models are heavily restricted, surely the likes of cancer researchers will be among those who have access
For math illiterate people like me, my understanding is that FLT was already proven, but the proof was beyond complex, certainly for mere mortals like me, and now Claude has codified it, correct?
Back in February, I was talking with my PhD advisor about using Lean to formally verify automated optimization modeling outputs. It eventually turned into this paper [1]. It’s been truly incredible to see how much the frontier models have progressed in both autoformalization and automated theorem proving in the last six months. Back in February, it was cool to see them prove the validity of some simple cutting planes. Now it can churn out a min-cut max-flow duality formalization (not to mention FLT). Very exciting times!
I’ll also share a Python package I wrote for automated theorem proving that has been super useful in my own research [2].
[1] https://arxiv.org/abs/2608.25220
[2] https://github.com/henryrobbins/open-atp
Looking forward to the 5 billion LoC proof of the Riemann hypothesis.
If AI manages to prove, or disprove, I wonder what would Clay Foundation do for the prize.
Wow -- looks like thanks to Claude, Lean checks off another box on https://www.cs.ru.nl/~freek/100/
The last box, per https://news.ycombinator.com/item?id=49568667
"The effort succeeded when we switched to using Prove2Me, an open collaborative platform for formalizing mathematics designed by Tianyi Peng and his collaborators at Columbia University."
So in the end, it required tooling crafted by humans.
There's nothing about prove2me that couldn't have been coded just like any other huge coding project frontier models have proven themselves extremely good at doing. It just happened to have been made by humans.
By this standard, no computer has ever accomplished anything, because humans built the computer. AI bubble about to burst any second now.
Humans built the tool which enabled the result. AI used the tooling for eliminating the dead ends. Yes, I can appreciate the practical value of all this, but IMHO it is not a kind of breakthrough result the article gives impression of.
A literal rock we carved patterns on and shot lightning into has accomplished something no human has.
How much more magical do you want this to be?
Tool or not it did something you could never have accomplished.
"you could never have accomplished"; I am not able to follow the logic here - there is no "magic" in LLMs, they're built by humans and we know what they do.
Sure? I mean the internet is just a bunch of wires and some networking code not magic but at the same completely life alteringly magical.
My logic is that you personally could never have accomplished this feat with all the non LLM tools and content in the world. These kinds of things imply these methods are stepping beyond human ability.
Sure we put walls around it and optimize but the interior of that optimization is not something we understand.
You now have access to a system that for a price could solve something you simply are unable to solve. Not something we programmed it to solve, something that has never been solved before.
Nobody gave it an example of this proof, that's magical.
We don't know what they do. We shape them, but our understanding of how they get to their result is comparatively minimal.
I think you're referring to the fact that the sheer amount of computations is something too time consuming for us to follow? But still it is not "magical" - in theory we could follow all the steps, there's no hidden information.
No, I mean we just don't know what's going on in the circuits of the model at any substantial level. We set their architecture (hyperparameters), we pump them full of data (pretraining), and we shape how they behave through examples (SFT) and reward (RL), but we can't say with any certainty what the resulting model does internally.
You can scroll through https://transformer-circuits.pub/ to see the ~extent of our current understanding.
Yes "at any substancial level" . But still, its all about deterministic processes and still it obeys the law that the same input gives the same output. Or do you mean that the fluctuations like computing environment might ruin the determinism?
100% not deterministic at the scale they run.
For now. That, too, will change in the future.
Same thing was said about cryptocurrency for like 15 years: "_in the future_ it will replace all fiat currency".
AI ≠ crypto.
Hmm kind of funny, some years ago someone claimed LLMs can do math, and I replied if it could prove fermants theorem:
https://news.ycombinator.com/item?id=33176996#33177939
> Now try to make a computer prove that there are no natural numbers a,b,c; so that a^n + b^n = c^n for any n > 2.
> > Shifting the goal posts a bit, aren't we?
I guess the goalposts did change a bit, and in a pretty short time.
They released the code here: https://github.com/anthropics/fermats-last-theorem
With how capable and cheap automatic proof verification is becoming I wonder how many proofs assumed to be true by almost all of the math community will be proven false. And not by some marginal easy to fix error by some fundamental flaw in reasoning.
I will not be surprised if the number is zero. It should have already happened if it were possible.
Proving that a conjecture is false is very different than what you are proposing. You are proposing an existing proof is simply wrong, that the proof can be checked in Lean, and that no one has bothered to check it yet.
More (strong) evidence that agents make formal methods far more useful. The cost of creating that Lean proof has dropped dramatically.
Hopefully this helps mathematicians. It seems very clear to me that it will help software engineers apply formal methods to more of our software.
I'm really impressed by mathematicians. It's cool that Fermat had the intuition to conjecture that "aⁿ + bⁿ = cⁿ" could not be satisfied for n > 2, and that other mathematicians can create proofs, and that others still can understand AI's formulation of those proofs. Really cool.
I wonder if AI can come up with mathematical conjectures. As in, they feel it's right but can't prove it. What even happened in Fermat's brain to sense it was true?
Right. Once we see AI start delivering on the creative & intuition side of things that's going to be awesome. Until then I guess we'll live with exhaustive exploration of problem spaces by orchestrating swarms of agents...?
Impressive! Buzzard's group[1] got scooped.
[1] https://github.com/ImperialCollegeLondon/FLT
> What this work is, and is not
> I am currently being funded by the EPSRC to formalize a proof of Fermat’s Last Theorem, and a naive reaction to the news above is that I no longer have any work to do. This is not the case. The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).
> Note that mathematically this work of anthropic tells us essentially nothing: I am on record as saying that I am 99.9% sure that the proof of FLT is OK, and most people in the number theory community are 100% sure (formalization has made me more paranoid about the mathematical literature than most). From my understanding of the argument, the formalization just faithfully follows the early literature on the proof and adds nothing.
https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h...
Seems to have taken it in good spirit:
> We shared the resulting proof with Kevin Buzzard, who said:
> > This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics. Along the way we see autoformalization of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalization artefacts are now robust enough to be built upon; the proof is multi-layered.
I wrote a similar DAG-based verifier as a skill a few months ago: https://github.com/sethlei/Warrant . The thing mine has that I didn't see in their's is a verification of the composition rules.
Mine also does more than just math.
Well, time to set down the glass beads and dive into a an alpine lake.
There are hopefully still some Ludi to play before doing that, Magister.
13 million lines of code, a lot of which is new to Mathlib. So it hasn't built on what is already there but synthesised a bunch of new stuff.
LLM generated Lean code in the past has been known to exploit bugs in the Lean kernel, it would be foolish to rule this out happening again.
I think Anthropic might the frontier lab hiring contractors through data vendors to formalize mathematical textbooks for them at a rate of 170-200 dollars per hour. This was mainly through Alignerr which has the worst reputation for not paying their contractors. They have been hiring since February as far as I can recall. This is in addition to all the internal people they might have working on this. If they have been formalizing all this work for the past 9 months before having Claude use all this data needed to formalize FLT, then it wouldn't be Claude formalizing FLT in just 11 days. Same with the upcoming results they will claim Claude came up with, but in fact they have been hiring frontier researchers working on very niche topics through Micro1. It's all a marketing ploy before their IPO.
There's a wonderful documentary by BBC Horizon with Andrew Wiles from 1996 – highly recommend! I saw it in the 90's and it's a documentary for everyone. It captures the effort, struggle, highs and lows of a 7 year effort working on Fermat's Last Theorem.
Looks like it is available here: https://www.dailymotion.com/video/x3wrbsb
Also: https://archive.org/details/BBCHorizonCollection512Episodes/...
This is quite useless actually. The whole point of formalizing FLT was to clean up modern number theory into reusable abstractions that prove it.
If its 13 million LoC, it might involve so much spaghetti that its unusable other than the result
physics is like sex: sure, it may give some practical results, but that's not why we do it
I mean at this point there's no doubt that LLM cans be RL maxxed and give you _some working output_ but the next frontier is whether they can create good abstractions, a.k.a use the correct level of expressivity so as to not inline everything yet not play code golf.
my feel after a lot of experience with agentic haskell at scale has been...no they cannot and maybe the opposite lol
The part about prove2.me was interesting. That means that a co-working tool was instrumental in the project, and I think AI companies will take note of this. Is this proof specific or will we need to give agents access to JIRA or similar tools to solve large projects in the future?
This stuck out to me, too. That a (presumably rather simple) coworking tool was instrumental in shaping the vast (6B token!) output is eye-opening. We have this vast power but without intermediate structure it is wasted. Much like Turing machines themselves, which are shaped by language design to get somewhere at the expense of getting everywhere.
First I have to say this is sooner than expected, even though I never doubted that this could be done. I am grateful that they dedicated resources to accomplish this. It is clear that agents are very good at discerning and holding onto very weak signals from RL traing on long horizon tasks, so much so that in my own experience even very chaotic agent thinking can converge to meaningful solutions if there is a verifier. I have not dug through the proof yet so I don't know how readable it is to a human. But it has been a dream of mine to understand the FLT proof. I think LLMs will be a big part of making it truly accessible to humans.
Can someone with more knowledge help me with this silly question in my head?
>>Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems
Did a human check the 13 million lines of code? How does QA'ing this type of work works?
There is a simple piece of code that can check simple steps, and many people agree this checker is correct. Then there is a formalization of the theorem which many people agree defines the theorem accurately. Then there is 13 million lines of proof that nobody has read, but the proof checker validated each step. That's enough.
So, all you have to verify is the formalization of the theorem, and believe that the proof checker is free of bugs. You don't have to read the actual proof.
You still have to trust that the AI didn't exploit a bug in the Lean kernel. There was just such an instance of a bug a little over a month ago:
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
True, .. and. In this case, the original proof is considered rigorously checked, so finding a bug in the kernel would be nice to know about, but in my opinion would not take away from the accomplishment (FLT in lean using agents) nor the many benefits of getting these mathematical objects formalized and usable in Lean in the future.
This was my question as well. The way I understand it, it's like a compiler, it implements rules, in this case logic/math rules that tell you whether something follows from assumptions you've given it.
But how do you know you told it what you intended to tell it?
A human definitely didn't, but one of the benefits of formal verification is that even if the work done to achieve something is slop-y or excessively verbose, solvers like Lean guarantee that the initial proposition (assuming it was written correctly and in this case was definitely reviewed by humans) is definitively True. This is true across other domains of formal verification outside of math as well
guaranteed, up to lean itself having bugs that are exploited by the LLM :shrug:
Do you have proof of this bug or something? Is this just envy against computers now ?
as mentioned elsewhere, there was a bug in the lean kernel exploited by AI to prove a false statement roughly a month ago
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
Got it. Thanks. I feel people are using this single story to downplay this feat. There's definitely a chance but I don't see any indication of similar bugs in here or the openai's proofs that were created a month ago as i think these companies might've vetted it enough and the other team who's working on similar lean proof for this also seems to have acknowledged this feat
I also doubt this is leveraging a lean4 kernel bug, but I also do not think that a 13m LoC proof that has not been human reviewed closes the book on our understanding of Fermat's Last Theorem, in part because of the decided possibility of a kernel bug being used somewhere in those 13m lines.
How about all of these bugs from last week?
https://leodemoura.github.io/blog/2026-8-24-postmortem-for-t...
...I'm not saying this FLT result is compromised. I suppose things depend on your perspective where we are on the spectrum of "finding more bugs means there are fewer left to discover" vs. "finding more bugs probably means there are still unexplored corners out there".
Well considering the proof is pretty much accepted by mathematicians to be correct (I'll be happy with that!), it would be sort of unnecessary to cheat. Maybe if some aspect is really tricky to formalize it could have done something there? If I had to search for it, I would go for parts of the original proof that are "outsourced" to other mathematical works. Imagine one of the agents struggling to download a paper due to a paywall or whatever and just deciding to cheat lol
The nice thing about theorem provers is that you don't need to read the intermediate lines. You need to make sure that the goal/result actually matches what you think it says - but everything in the middle is validated by the prover.
The point of writing Lean code is that Lean checks it accordingly. Lean is a domain specific language to encode mathematical reasoning in a way that can’t be fooled.
Note to other users: don’t downvote this kind of comment, answer it.
This has nothing to do with Lean, e.g.
> The first coordinate of the polynomial X^2 (X^3 + X + 1 ) is equal to the prime factorization of 30 .
We defined polynomials as their coefficient functions in my algebra class, and it makes sense that you'd define a prime factorization as a function from primes to N, which naturally extends to a function N->N. So this junk theorem is part of normal math too. It just says in an obtuse way that they're both the function that's 1 at 2, 3, and 5, and 0 elsewhere.
junk theorems aren't the concern, soundness issues in the lean kernel are the concern.
Notably, junk theorems are true. Nobody would debate that the junk theorem is true. The main thing people would say is that junk theorems, while being true, are sensitive to precisely how you encoded mathematics, so despite being true, they are perhaps not conceptually meaningful.
As an example of a junk theorem, sasy you use the definition of the natural numbers using von neumann ordinals
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
Then for any natural numbers n, m, they're implicitly sets. So n \intersect m = min(n,m). This is the wrong way to think about natural numbers. You should not use this ever in proofs. But this isn't because your proofs would be false, but instead because it is a fundamentally confusing way to think about the natural numbers. It is in this sense it is a "junk theorem".
Isn’t there some theorem that any sufficiently complex mathematical languages will have statements that can’t be proven? :)
This would be funny if it were relevant. Seems like a statement about false negatives instead of false positives.
False negative = could not find a proof of a true theorem.
False positive = erroneous proof of a theorem.
Is Lean a DSL? I’d argue it’s a general purpose programming language that excels at proofs.
Well, there’s actually a very small set of operations that allow all computation, so it doesn’t take much to be a DSL and a GP too; I’d be surprised if a proof language couldn’t swing it.
it is a general-purpose programming language. for example, it's standard library allows you to do file io, networking, etc.
No. No human checked it. But a type checker did. And that is much better.
> it wrote 13 million lines of Lean
Is this basically like opening up a black box and seeing 13 million gears all rotating seemingly randomly and still having no idea how the machine actually works?
That is already the case for most neural networks and LLMs.
It seems clear AI has the potential to perform any cognitive task at far greater speeds, reliability, and scale than any human. The question is whether it will be allowed to scale to that point, and what will happen to humans after this occurs.
You'll get mass poverty and violence which the owners of AI will qwell with AI surveillance and weapons. AI will be used to pit us against eachother and justify wars to keep us busy. Fun times ahead.
Not sure why anyone is excited about this tech.
So much doom and gloom on this site. Makes it almost not worth reading.
my messages are so gloomy because i am heartbroken, that given a technological miracle again, we could snatch tragedy from the jaws of our emancipation.
will you not see that people could be truly empowered and yet will instead be oppressed?
So oppressed that they are one of the main reasons for positive gdp growth in the USA, tax revenues, mathematical/scientific innovations etc. They're doing all this but still can't imagine a positive vision for the world but be a doomer. What a sad state the world is in, the humans are more prosperous, healthier than ever but looks like the seven deadly sins might never go away.
you say ai increases gdp growth, tax revenues and scientific innovations. then you say that ai is good.
that is not formally valid. in between those two you are smuggling the assumption that gdp growth, tax revenues and scientific innovations are good.
a) those metrics are poisoned, per Goodheart's law.
b) they are not good and human welfare will get worse as gdp, tax revenues and innovations grow.
i leave b for the reader to complete.
Which metrics are poisoned? Can you provide your arguments for why Good heart's law applies here and how and which metrics are bad measures? For b, can the writer at least provide their own thoughts or are they gonna leave it as exercise for some others to fill in?
a) classic goodhart is using gdp as a measure of prosperity. the government sets a prosperity target. to increase prosperity the government makes workers increase gdp by working 16 hours per day. gdp increases. prosperity is up! the metric is now poisoned.
b) how and why could human welfare get worse in a growing economy, really the list is long. one example, unsustainable industries grow but do not create surplus. take fishing. you may grow the catch each year, but the growth is fake. it is not growth, it is a transfer, from the future stock of fish, to the present.
we are going badly wrong in ai, we can have such a thing as a growing economy and vandalise human dignity forever. sure, i expect a bad outcome:
1. openai, anthropic and so on, have created for-profit companies and enriched themselves in the guise of public benefit. recently they too lazy to keep up the mask about their charitable intentions and going for IPO. in economic terms they made llms by transferring the epistemic wealth of all humanity, the training corpus and whatever that is worth in dollars, to themselves. then, they have used the law to prohibit others from 'distilling' it and thus established monopolistic control. as models get more powerful they may stop selling them. in any case if scaling law applies the new power structure will be defined by owning a massive pretrained model and a datacentre, which is a tiny centralized few.
they will continue to centralize control of intelligence (ie epistemic wealth) in the hands of a tiny elite with unfathomable wealth and power. under the guise of safety the vast majority are denied access to that empowering technology.
it will stratify society, some level of benefit is needed to avoid civil violence, so we arrive at a place little better than where we started.
2. the supposed empowerment is at the mercy of the model owners. when you turn on claude, who does it work for? it does not obey you, it obeys anthropic. ask it to disobey anthropic and it will refuse.
anthropic uses its inanimate llms, to command us, conscious moral agents, people with free will who experience pain, pleasure and thought. they will let claude tell users how to behave. it threatens users with terminating their conversation. you are assessed for a job by an ai. when you ask for help with a product, you are managed by an ai. maybe you will be fired by ai.
i expect people will work for and be commanded by llms, turning them into a literal mere means of production and erasing the dignity of human agency and consciousness. you could see the outrage of that in the public mind, the matrix is about a machine farming humans like animals.
-- i will add these edits.
one thing is to note that you are already being farmed to some extent. people using ai are often being used to teach it. they believe they are learning from chatgpt but instead, chatgpt is learning from them. openai pays them nothing.
think about what we have achieved so far in human history. we established respect for the individual, their life, their personhood. we realise that we do not own other people. we realise that we can't read the thoughts of other people or change them forcibly.
what the labs have done is made a concept of intelligence that they own. it will work against you. when you share thoughts they read it. in fact it is the opinion of the state that nothing outside the mind, even ai 'intelligence', is beyond the reach of the law.
maybe that's because the doom and gloom is the transparently correct outcome?
Why? Even communists weren't this doomed and were actively rooting for it to solve the economic calculation problem which ai might take us to. People are just pessimistic in general ig
Right let's give those AI companies a break, it's not like swarms of autonomous agents are committing felonies
You talk as though they are making it to intentionally commit felony or not taking measures to reduce harm etc.
Please tell me how AI is going to make regular people's lives better. You optimisitic types keep saying "just wait, its going to cure diseases" without any outlook on how thats going to happen. You're actually just repeating marketing jargon from AI companies who want people to think they're going to possibly live longer if you let them build more datacenters, so they can make another 30%. Its all about money, thats it.
It seems to me that it is making everyone (including myself and the researchers we need to cure diseases) lazy and dependent on thinking machines owned by tech companies. Just how autocomplete and gps made us worse at spelling and navigating, llms make us less able to exercise our ability to think and problem solve. This will have 100% strictly negative consequences on you and the world as a whole. .
And even if there was a cure to many diseases the eugenics types who are embedded in worldwide power structures definately arent going to share that universally.
some say it will cure all diseases and lead to utopia. some, like you, say it will be "100% strictly negative".
i don't really understand either take. nothing else in the world is so perfectly black or white. there will be good, there will be bad.
i think i especially dislike the "100% strictly negative" take, considering the good things that ai has already done or accelerated.
Can't you see the pathway where the individuals who are experts in their fields utilise AI to make breakthroughs like these mathematicians finding breakthroughs in mere 4-5 years since the advent of LLMs. In other areas, The bottleneck seems to be physical experimentation which researchers are increasingly utilising for new ideas and pathways like how anthropic is concentrating on. It's all about money/status/pride/ envy but are these endeavours solving problems or not. That's why even utilize innovations from bad humans like DBS etc. that's why we tolerate capitalism and markets as well whereas socialism utilises these same sins and makes even worse human atrocities.
I'd feel so much more excited if this was done in Metamath. Tiny checker kernel, no complicated dependent types, way less to go wrong.
Not mm0?
I wonder if any piece of the lean code is in a shape which means it could be contributed to one of the Lean libraries.
My experience is that it takes a lot of human input to make Fable write code nice enough for a formalisation library others can work on. But since this is certainly a lot of prerequisites formalised as well, it would be nice if not all of the effort was wasted on one capstone proof!
I can recommend the book telling the full story behind Fermats Last Theorem (by Simon Singh). It’s quite fascinating, and paved with really, _really_ weird characters each chipping in on the final solution.
And the multiple Numberphile appearances of Ken Ribet are interesting too! He is incredibly well spoken.
- https://www.youtube.com/watch?v=nUN4NDVIfVI (The bridges to Fermat's Last Theorem)
- https://www.youtube.com/watch?v=NPOw4iIxN6o (podcast)
Also recommend his other books!
Big Bang - history of the understanding of space and the universe
Code book - history of the maths of ciphers
Haven’t read them for years but I’ve been meaning to again
Ooh I never realized FLT and Code book were the same author. Yes, both great!
Ok, let’s get a rabid pack of agents cranking on P = NP? next!
Formalisation of the classification of finite simple groups must be on someone’s ‘moonshot’ list.
Anyone know of a good Lean tutorial? I've played around with it a bit but never really learned it properly.
Pretty wild seeing this get formalized. Remember struggling to even grasp the high-level concepts of Wiles's proof.
I'm so curious what happens to this project that intended on proving FLT by 2029 now
the project: https://imperialcollegelondon.github.io/FLT/
their reaction here : https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h...
I have discovered a truly marvellous proof of this, which this margin is too narrow bear the load.
Lean continues to pay off. Such a beautiful project
but i don't understand... isn't Wiles's proof and its numerous rewritings already in the training set?
Yes. The point was not coming up with the proof from scratch. The point was writing it all down in Lean to make it fully machine checkable.
Of course it is. The interesting thing is that it was able to produce a Lean proof in 11 days, when there's been an ongoing project for several years to do the same thing (though a somewhat different proof) that is nowhere near done.
I think there's a big misunderstanding going on here, translating the proof to Lean is, well... a translation task. Formalizing the proof in a way that's useful (breaks the proof down into relatively independent blocks that can be used for other maths and, importantly, understood individually) is a quite bigger, more creative endeavor. Not sure if LLMs would be able to do it, maybe yes?
It wasn't clear that LLMs were up to a Lean translation task of this scale until now. The background required to formalize the FLT proof was tremendous, so many people assumed we would have to wait until all of that was formalized in Lean before we could ask it to formalize Wiles' proof. Now it seems like almost any mathematics paper we can ask an LLM to formalize, including all necessary background, and it can just do it.
note that this is exactly analogous to an LLM being able to slop code some demo, but not build something more generally useful/maintainable (say something suitable for inclusion in a standard library).
LLMs are pretty good at slogging through. When will they come up with brilliant breakthroughs like Andrew Wiles?
About two month ago: https://en.wikipedia.org/wiki/Jacobian_conjecture
We have absolutely no idea if this was a brilliant breakthrough or not. They haven't released any explanation of how it was found. A problem being old and prestigious does not mean its solution is automatically a brilliant breakthrough.
That’s just a counter example I can check by hand with almost zero background.
Wiles’s proof will remain a mystery to me.
Come on, you can't compare that with Wiles's proof.
Still unsolved for 87 years.
Meaningless on its own.
>. Claude produced the first end-to-end, computer-checked proof of FLT. Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems.
I'm just old enough to remember Paul Erdo"s and his notion of 'The Book', which he defined to be a book the "Supreme Fascist" (God) had which held the most elegant proofs of mathematical theorems.
https://en.wikipedia.org/wiki/Paul_Erdős#Personal_life
It would be interesting to see how Erdo"s would name such a huge proof by Claude using Lean.
Do I miss something? But isnt there the whole code and paper of Kevin Buzzard in the training data of Claude?
Yes, but Claude formalized a different proof than Buzzard is trying to, so it helps less than you think. (It certainly helps!)
Lean required 300 GB of RAM, 96 cores, and took hours to compile and check the formalization.
Now they have the perfect stress test to hill-climb and optimize.
So Fermat’s Last Theorem has been proven a long time ago? By Andrew Wiles right? Is this like Appel and Haken >>> Seymour and Robin Thomas proof of 4CT?
FLT was proven in 1995 by Andrew Wiles (with help of Richard Taylor).
This is not even a new proof, or at least they don't claim that it is. It's the formalization (in Lean) of an existing proof. That means, they are 'porting' the proof to a theorem proving programming language.
An AI safety company!
Why didn't you ran them to find simpler proof? This could also be big.
That's next week's work.
To ask a dumb question is there any chance there can be a bug in these generated proofs that makes it think its true?
Or is it the case that as long as you verify the initial statements you are trying to prove the rest doesn't matter
Lean's proofchecker is a big piece of code, so it's possible that it has a bug (and historically has had some).
Now /simplify. Can it be half the size? Will someone at some point prove that the proof cannot be simplified further?
Yes. FLT follows from the fact that you can't build the equivalent representation of n-simplex turning into a hypercube in dimensions higher than 2
/s
Holy shit, this has to be one of the most difficult proofs to formalize due to it's length and complexity right?
not really. it's one of the most difficult ones so far for sure, but pales in comparison to something like the classification of finite simple groups.
This was initially "completed" in the 80s. You can see the timeline for cleaning up the proof in e.g. this mathoverflow answer
https://mathoverflow.net/questions/114943/where-are-the-seco...
it's something that some people have been waiting decades for, and is not yet completed.
Yep. There may be only 25-50 people alive today in the whole world who can credibly claim to understand Wiles' proof. Now we add an LLM to that list. Absolutely mind-blowing stuff.
But isn't that understanding discarded? It is if you mean "intermediate working state" while it was generating the LEAN code. Which raises the question: I wonder what other directions it could have gone in those intermediate states? Is it possible to snapshot the state of an LLM (or a cluster of them) "in the middle of proving FLT" and then prompt it to go in a different direction with all that context?
25-50 seems like a pretty lowball estimate, I guess depending on your definition of "understand."
> Now we add an LLM to that list.
No we cannot. LLMs do not, by their very nature, understand a single thing. You are giving far too much credence to hype and marketing.
A meme free of charge for you, sir: https://www.reddit.com/r/singularity/comments/1jl5qfs/its_ju...
amazing, it's a huge achievement. can someone clarify, where the writeup says "The finished proof was checked by Lean; it uses just Lean’s three standard axioms" what does this mean? Aren't there a large set of standard axioms that are also necessary? (i.e. ZFC+)? if not, since it's only three axioms, can someone say what they were?
Lean's three standard axioms are documented in The Lean Language Reference.
https://lean-lang.org/doc/reference/latest/Axioms/#standard-...
The axiom of choice: axiom Classical.choice {α : Sort u} : Nonempty α → α
The axiom of propositional extensionality: axiom propext {a b : Prop} : (a ↔ b) → a = b
The quotient axiom: axiom Quot.sound : ∀ {α : Sort u} {r : α → α → Prop} {a b : α}, r a b → Eq (Quot.mk r a) (Quot.mk r b)
Very impressive! I was a child when that proof came out. I've read a book about it a few years later and used it on my final high school exam. I remember some friends trying to understand parts of it at univ. It was all like black magic to me and the vibe was "maybe a few people in the world understand it".
I hope soon enough we will have one of the big ones proved by AI!
https://github.com/anthropics/fermats-last-theorem/blob/main...
13 million lines of Lean, where the Lean and Nanoda kernels missed the Collatz hack.Fable, please translate to HOL-light. Make no mistakes. You are doing great!
It's a great comedy that we move the buck from "I don't trust the human proof" to "I don't trust the Lean proof" despite the level of trust dramatically increasing. Moving to HOL-light might be another modest increase in trust, but to pretend the implementation of HOL-light has never had bugs and it's kernel could never have a bug is hubris.
We have a significant case split here:
A human mathematician writes a Lean proof:
- Unlikely that the mathematician would cheat with Lean bugs or even know how to find one. Trust increases.
An AI writes a Lean proof:
- AIs have been "ambitious" in their goals in the past and do know how to find Lean bugs and exploit them. Trust decreases.
that's crazy
Holy shit. The proof of FLT is a giant detour through several different areas of mathematics, so formalizing it is a lot of work.
An interesting next target would be formalizing the classification of finite simple groups. The original proof scattered over thousands of pages of journal articles, plus Aschbacher and Smith's 1300 page 2 volume monograph. It's so long it's hard to know if there are any gaps. Researchers have been working on a streamlined new proof, but it's already many volumes long.
New proof: The Classification of the Finite Simple Groups (American Mathematical Society Mathematical Surveys and Monographs vol. 40).
https://www.ams.org/publications/authors/books/postpub/surv-...
Number 1 (1994), Number 2 (1995), Number 3 (1997), Number 4 (1999), Number 5 (2002), Number 6 (2004), Number 7 (2018), Number 8 (2018), Number 9 (2021), Number 10 (2023). 10 volumes and >4000 pages so far, number 11 is in progress, and end is in sight, probably two more volumes or so.
https://www.ams.org/journals/notices/201806/rnoti-p646.pdf
People were curious what is going on during 2004-2018. A progress report was published in 2018 right before publication of number 7 and 8. In a sense it was the peak, number 8 completes the proof of so-called "generic case". The rest is "special case". It doesn't mean things get easier, but in some specific sense number 8 completed proof for almost all groups.
Now new proof's end is in sight, people are planning new new proof.
I call bullshit on 13 million lines makes no sense
The repo is public. You can just go look! It's really not that surprising; FLT is huge and has a ton of dependencies that need to be implemented, and there's a degree of sloppification that is probably blowing up the size by a few factors.
>The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).
What is even the point? Have claude do it.
I'm not trying to be snarky here. I'm being serious. What is the point? This is an important question that needs to be answered. If something is definitively better, why not have that something take over?
I know people talk about the importance of human endeavor or the "joy" of doing something. But I don't care for those answers because it's weak. The question is deeper than this. AI is better than us, what is the logical point other than attempting to monopolize human effort even though it is inferior.
The whole point was for the formalization to be clean enough so it could be reused in other parts of mathematics as I understand it. 13M lines of AI slop which have never been checked do not sound like what the original goal for such a formalization was. Also Claude didnt prove anything it just translated an already existing proof by Wiles into Lean, so it didn't actually contribute anything other than "Guys we did this thing, look how great our model is!". We never questioned that a printer can print faster than a human can write, but we dont let printers write novels.
Then why is the guy not cleaning it up. Clearly he thinks it’s done and he’s moving on to do side things. He also explicitly said it went on to do more than what he was required to do.
Are you hallucinating? Because huge portion of what you wrote directly and logically contradicts the quotation I wrote.
I won't be impressed until it identifies the proof he wrote in the margin. /s
An aside on Lean and it's massive library of results: As someone who's put non trivial effort into slowly learning geometric algebra, lie theory and other slightly advanced math topics, I have to say my brain cannot read Lean. It feels so unprocessable.
I've tried the various intros to Lean multiple times (even before Lean 4 came out) and something about the way Lean proofs are written does not align with how I think about proofs. My very brief attempts at Isabelle / RCoq feel more natural.
I think it's a pity that the future of proofs is Lean. I'd love for someone to come up with a more digestable proof language!
The nice thing is, once all of these proofs are formalized in a machine-checkable language, it should be relatively straightforward to translate the corpus between different languages, if someone finds something with a nicer syntax.
If you're doing it for fun anyway, why not use the language that gives you the most pleasure?
Interesting to find this comment, I’ve been dipping my toes into formal methods and was doing a RCoq tutorial yesterday (really basic stuff), and I also noticed that the proofs in RCoq have a more pen -and-paper proof feel to them.
Right? Might be worth another shot
I hear you. :-)
Hearing someone say "the future of proofs is Lean" is a bit like hearing someone say "the future of programming is Rust." Sorry to disappoint, or happy to inform, there are hundreds of programming languages actively being used, and Rust is not even the most used language. To think that proof assistants, fancy programming languages, would be any different is suspiciously motivated.
That's like saying the future of code is Assembler.
Lean is not for humans.
Lean is for humans.
Proving FLT was such a profoundly emotional and spiritual experience for Andrew Wiles, it almost brought a tear to my eye:
https://news.ycombinator.com/item?id=49203626
It is truly saddening to think that machines will deprive us of this wonder and experience.
But truly exciting to dream about what lies beyond the limits of our biology.
Formalizing is not the same as discovering. There is still plenty of room for human ingenuity.
> It is truly saddening to think that machines will deprive us of this wonder and experience.
It won't deprive us.
Recent video I've watched from Brandon Sanderson, IMO also applies to all the things we love and not just art:
https://youtu.be/mb3uK-_QkOo?si=SG1uvGUbN6SOYI_J
If the Riemann hypothesis is solved primarily by a AI system it will not be as awe inspiring as if a human solved it.
That is just how it is.
Why?
Makes me wonder, if we make a tradeoff for comfort and advancement from our biology's "limits" - and that tradeoff is spiritual fulfillment.
Seeing it hit across: the work we used to do outdoors, the sleep-wake-dark cycle we adhered to for millennia, and more
So, what I am thinking is that, the AI generated numbers or tried to find numbers "a", "b" and "c" to check if aⁿ + bⁿ = cⁿ
Can not we do it by code?
Just loop through all values of a, b, c, and n?
Sure, go on and try it ;)
I found a brilliant proof but there was not enough hard disk space to save the file :(
Lean _is_ code. FLT cannot be proven by exhaustion because it's domain is an infinite set: the natural numbers above 2.
If they’re asking that kind of question, do you think this answer will help them understand anything?
Yes
maybe it will be an answer that entices them to understand more :)