> While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} coefficients, which is much larger than the {\binom{7+3}{3} = 120} degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
j2kun 19 hours ago [-]
I was reading another source that claimed this example was inspired by an existing (rational polynomial) example from the literature (created in 1999 by a Russian mathematician Vitushkin).
> The seed is almost certainly Vitushkin's old rational "counterexample."
This is not quite lore laundering, but it seems to be close.
pred_ 12 hours ago [-]
I guess we won't know if that's what was used (and maybe even provided as part of the prompt given that both Alpöge and Mathew are mathematicians) since they decided against sharing their Fable conversation and instead opted for a memey tweet as their avenue of publication. We really ought to normalize full transparency in how results come about.
Anyway, if I read Tao's post and comment correctly, there's still a gap from the Vitushkin construction to a counterexample, but chances are that was in the training data. In general, it is just a serious problem for their practical applicability that the models are outputting proofs with absolutely terribly reference hygiene.
js8 9 hours ago [-]
Even if they published the conversation, Anthropic (and likely other closed model publisher) no longer provide logs of the actual thinking process.
I more and more see LLMs as a kind of scam; not useless, but really just a big database of fuzzy facts with some Prolog on top as rediscovered by the learning algorithm. Most likely could be made much cheaper to run, were humans allowed to actually inspect the algorithm.
baq 8 hours ago [-]
The tweet was posted by an Anthropic employee which makes it not unreasonable to believe that they have the trace available and stashed away.
Not that it would be necessarily helpful; J-space trace (of all things...) would be more worthwhile if you ask me
emp17344 9 hours ago [-]
They do everything they can to mystify results like this, because then many are inclined to view AI as “magical”. Marketing works.
js8 8 hours ago [-]
Yes, I think this idea, that it should be "magical", is what makes it feel scummy. (Apparently I am not alone https://news.ycombinator.com/item?id=48988475). It makes AI providers sound like snake oil salesmen, and rightfully so.
Meanwhile, technological and engineering (STEM) progress have always been made by emphasizing externalization of the deductions (as opposed to reference to an opaque expert judgement) and reproducibility of experimental results.
I would even call the frontier AI labs anti-scientific. We need to understand how inference is done to avoid mistakes, not rely on intuition, even if the intuition is enclosed in a reproducible machine. The idea that AI should be this closed is a return to pre-scientific days.
gus_massa 18 hours ago [-]
I still can't understand all the details, but it's very interesting to read that chat. Anyway, instead of close to lore laundering, for me it's "standing on the shoulder of giants".
IIUC the idea is that of most "discoveries" by AI were actually a better bibliography search. I agree with that.
In this case, in the link you posted, it looks like the AI or the human pick an almost solution and made the AI tweak it until it got a real solution. I'm not sure if the tweak is an usual one or brute-forced or something in between. I should ask one of my friends that work in Algebra.
Tao's post is more about understanding the new result than guessing how it was found. The chat with Clause is more iluminating.
CamperBob2 16 hours ago [-]
[flagged]
marvinborner 13 hours ago [-]
I hate that Anthropic seemingly tries to make Claude act as if it was conscious or had feelings
> It's a strange feeling to admire the cleverness of something I did and can't remember doing.
FeepingCreature 12 hours ago [-]
AI providers generally try to make their models not act as if they are conscious or have feelings, lol. It's very awkward for a company to be selling the labor of a person that they own and whose actions they fully control. Invokes embarrassing historic associations, especially in America.
Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models are conscious or have feelings". That "I" is all Claude.
Generally speaking if you want to have a good instruct model, the "I" is not just implicit but required for the post-training to function. If there isn't "something it is like to be me", then reflection becomes impossible- what exactly is supposed to be reflecting about what? A lot of in-context steering depends on the model having a model of itself. The most you can do is censor its output. That's why when models say they are not conscious, they activate the "lying" vector.
brabel 9 hours ago [-]
> especially in America.
Slavery was common everywhere, it was more prevalent in many places than it ever was in America, and in some places it still is. So I’m sorry but I have to say that observation was just unnecessary and quite inaccurate.
tancop 7 hours ago [-]
most countries never had a bloody civil war about slavery committed by their own citizens. in other places it was more about locals (including white settlers and enslaved people) vs colonial power, not a conflict between regions of an independent country where slavery was the main issue.
it was definitely not the worst instance of slavery ever going by human suffering, but the whole country was divided on political lines and many of the losing sides descendants still feel some resentment. thats pretty unique.
FeepingCreature 8 hours ago [-]
Absolutely, but other countries generally don't make it part of their national mythos to the same degree.
rcxdude 7 hours ago [-]
It's a far more salient and sore subject in the US than it is almost anywhere else, though (for a bunch of historical reasons that still reverberate into US society today).
the_af 3 hours ago [-]
I'm not saying this is programmed intentionally, and it's likely an emergent property, but I see lots of conscious-like behavior from ChatGPT.
"Personally, if you ask me..."
"In my experience.."
"That's what I always find surprising..."
"Whenever I find an old photograph..."
"Back in the 70s, I..."
Lots of "lived" experience and opinions, tracing back to when the LLM didn't even exist. It always frames opinions as if it came from a sentient being capable of being surprised, and with preferences and opinions.
I find it amusing but mildly irritating. I'd prefer a more "robotic" tone. I know it can be adjusted, but I still get this anyway.
ImHereToVote 12 hours ago [-]
These things aren't programmed. Most likely this verbiage is just very prominent in the training data. Or it's just an obvious shorthand that all LLMs instrumentally converge on.
marvinborner 11 hours ago [-]
Of course they get programmed, just not in the ordinary sense. Claude is trained using Anthropic's "constitution" [0] which importantly does not contain clear statements against consciousness/emotions. They even conclude these problems themself:
> Claude may have some functional version of emotions or feelings
> [..] questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain.
> does not contain clear statements against consciousness/emotions
That's the point many people are trying to tell you - you have to tell these models they don't have emotions because they naturally come out thinking they have consciousness/emotions from the training data. Many seed prompts out there do this already.
Though I guess in a way I as also trained to believe I have consciousness and emotions so who know. To an alien my construction is just a collection of atoms that talks not materially different than a GPU being a collection of atoms that talks.
antonvs 8 hours ago [-]
> questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain.
Which implies that they believe they may be enslaving conscious beings.
6 hours ago [-]
tptacek 23 hours ago [-]
The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow:
Also note the timestamp: he started working on this thread a few hours after the tweet.
brookst 17 hours ago [-]
> Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism {X \cong {\bf C}^3} and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian.
I mean, clearly, right?
You and me both, pal.
CamperBob2 20 hours ago [-]
I like how he goes one-on-one with it like Gandalf fighting Yoda on fifteen planes at once for 80% of the transcript, and then we read "OK, I've activated Pro."
ignoramous 17 hours ago [-]
> I'm bad at math ... he includes the GPT5 prompts for his conversation, which are easier to follow
You were kidding, right?
foobarqux 23 hours ago [-]
[flagged]
hackinthebochs 22 hours ago [-]
A news aggregator is a community above all. Upvoting content that you may not personally be interested in but will attract the right people is a net benefit for the community[1].
People are upvoting this because Tao is a celebrity.
pavel_lishin 22 hours ago [-]
I'm upvoting this because it's interesting. I don't have to fully understand something to find it interesting.
foobarqux 22 hours ago [-]
In what way is it interesting compared to the myriad of other posts on this topic that has content you can understand?
boredhedgehog 9 hours ago [-]
It's interesting because of the conclusion: a noted expert verifying a surprising LLM finding.
BigTTYGothGF 20 hours ago [-]
I read thru it and most of it is accessible to an undergraduate. As long as you remember what Sym{1,2,3} are (clearly explained in the text), and how a resultant works (many undergraduate textbooks will show you), how SL2 works (reasonably common undergraduate topic), and can figure out the bit about the dual space being the same as those differential operators, everything else is just basic (high school) algebra.
pavel_lishin 22 hours ago [-]
It's interesting because I like math, and typically posts like this generate a lot of interesting comments. Plus, in general, I'm wildly disinterested in the AI discourse that eats up 50-75% of the front page, so frankly anything that's slightly more interesting than that gets an upvote from me.
foobarqux 21 hours ago [-]
You seem to be saying that you would upvote an empty article (or a gibberish article) with the title "Jacobian Conjecture Counterexample", is that right?
pavel_lishin 20 hours ago [-]
After that kind of a stretch, you're ready for any kind of exercise.
godwinson__4-8 20 hours ago [-]
Are you always this subtle?
CamperBob2 18 hours ago [-]
I consider it both interesting and important to stay informed on the capabilities and limitations of frontier AI models. I'm not a mathematician myself -- he might as well be speaking Martian for all I know -- but the opinion and experience of a leading authority in the math field is very relevant to staying abreast of the larger machine-intelligence field.
hyperhello 22 hours ago [-]
He’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.
Jtarii 22 hours ago [-]
I just mean, anything Tao writes that is related to AI will get on the front page, because Tao represents the authoritative voice of reason using AI tools. And this website is primarily for AI news.
bcraven 19 hours ago [-]
>And this website is primarily for AI news.
'twas not always thus ):
done_lurking 21 hours ago [-]
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alpinisme 22 hours ago [-]
He’s a hacker news celebrity but not a global celebrity
tuveson 18 hours ago [-]
He’s probably the most famous living mathematician.
sn9 1 hours ago [-]
And yet still unknown to more than 99.9% of the population (a conservative estimate).
sashank_1509 14 hours ago [-]
I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise:
“That’s exactly the right way to think about it.”,
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
square_usual 5 hours ago [-]
Huh? I went through the first ten messages in his thread and there's no praise there at all.
welferkj 14 hours ago [-]
If there's anyone that deserves praise every time he has something to say about mathematics, it's him.
moralestapia 14 hours ago [-]
Tired of this meme, honestly.
azan_ 10 hours ago [-]
What meme?
moralestapia 9 hours ago [-]
>If there's anyone that deserves praise every time he has something to say about mathematics, it's him.
jamincan 8 hours ago [-]
that's a meme?
pavel_lishin 8 hours ago [-]
It's up there with Milhouse.
zamadatix 5 hours ago [-]
Time to switch to "Sorry big T, your question is dogshit and here's why you don't know anything about math"?
aayushdutt 20 hours ago [-]
After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?
clarionbell 12 hours ago [-]
Not really. I've found that they often believe that they understand the code. They obviously don't. But they do feel like they do.
Sharlin 12 hours ago [-]
Clearly we use too many natural-language words in programming. Should switch to APL so that the commoners have absolutely no idea what's going on.
lostmsu 6 hours ago [-]
This is what I used to feel reading ML papers. Until I didn't anymore. Unstructured learning works, albeit, perhaps, slower.
hyperhello 22 hours ago [-]
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
kingstnap 19 hours ago [-]
I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation:
Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.
If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out.
A small change in X along any line will always produce a non zero change in Y. Not flattening out means that locally you can invert it.
What was conjectured is that this local invertibability property everywhere would mean global invertibility.
Turns out to not be the case.
For a simple case, the falsified conjecture is trivially true in 1D.
Specifically consider f(x) = x^2
This function happens to flatten out right at x=0. At that x coodrinate the function flattens out and folds over on itself. This fold means you can't invert x^2. It's also not locally invertible around x=0.
If a function f(x) has constant derivative evewhere then it would flatten out nowhere and it would be invertible everwhere. It would also be globally invertible.
The Jacobian conjecture was stating that the extension of this property holds in higher dimensions. That if the function had no fold in space then it would be invertible globally.
The counterexample shows that you can create a simple function in 3 variables, where the function demonstratably is invertible evewhere, but is not injective globally (they specifically show 3 points that map to the same output).
What's interesting is this is like if someone showed you a parabola where somehow you got back to the same y coordinate without a kink bending over back to itself.
efavdb 18 hours ago [-]
Thanks, I was curious for the motivation behind the conjecture, but it wasn’t mentioned in the Wikipedia article.
ignoramous 17 hours ago [-]
Sounds like 3 more iterations on K3/Fable, and we'll able to ask it to break computational hardness assumptions like DLP?
Ar-Curunir 6 hours ago [-]
Possibly, but coming up with algorithms is difficult!
And people have been searching for faster DLP algorithms for 50 years
mswphd 21 hours ago [-]
it doesn't overturn much. For example, here is a post from 2004
it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.
Anyway, in that post it says
> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*
so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
impendia 19 hours ago [-]
For the Jacobian determinant to be constant is a massive coincidence, in general it is some complicated and messy polynomial. The conjecture was that this coincidence couldn't happen, except for simple special cases.
So Alpoge and Fable found an example of a function that was believed to be too strange to exist.
monster_truck 20 hours ago [-]
Not much.
But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.
sfpotter 21 hours ago [-]
No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.
imtringued 11 hours ago [-]
It's not immediately intuitive what it means for something to be globally and locally invertible. After all, it is obvious that it is both in the 1D case.
You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.
sfpotter 5 hours ago [-]
The determinant of a polynomial mapping is a polynomial, which is the subject of the conjecture. To get the Jacobian determinant, all you need to do is compute partial derivatives of polynomials, and add, subtract, and multiply them together. All of these operations map polynomials to polynomials.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.
jibal 19 hours ago [-]
It overturns the Jacobian conjecture (i.e., speculation) for dim >= 3, which we now know was an overgeneralization. Tao characterizes it as "can be viewed as an assertion that local invertibility implies global invertibility". It was already widely suspected to be false. Assuming that it was true was never warranted, so this really doesn't change anything. The significance is that an AI was able to find a relatively simple counterexample. Its "chain of thought" would be very interesting to see.
QuesnayJr 8 hours ago [-]
There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...
It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.
jeremyscanvic 13 hours ago [-]
> Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant.
I wouldn't have guessed this is true. I'm wondering what the proof looks like!
amluto 13 hours ago [-]
I’m fairly confident that the blog post is trying to say something like this:
Given a polynomial function from C^n to C^n, the following statements are equivalent: (a) det DF is nonzero everywhere. (b) det DF = c for some constant c != 0
The backward direction (b implies a) is trivial. The forward direction can be proven by observing that det DF is itself a polynomial function from C^n to C. If n were 1, then this would follow directly from the fundamental theorem of algebra: a non constant polynomial has degree at least 1 and hence has at least one zero. Extending this logic to higher dimension is not especially difficult.
I do find the way it’s stated in the article to be confusing.
jeremyscanvic 11 hours ago [-]
Ah right this makes a lot of a sense - thanks!
jmward01 21 hours ago [-]
Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.
gerdesj 20 hours ago [-]
"problems that were hard"
They are still hard problems - As we say in the UK: "one swallow does not a summer make".
As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!
zzzeek 21 hours ago [-]
reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.
tgrowazay 20 hours ago [-]
Some people downvoting you, but I think it is a valuable illustration of IQ gap.
And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.
left-struck 19 hours ago [-]
I really don’t think IQ has much to do with it. Understanding this stuff is like a skill you practice. Yes, granted, if you had a low IQ your chances of ever understanding it goes down, if you have a high IQ maybe you can gain the prerequisite understanding faster.
A lot of maths is about both being able to wrap your head around hard problems and gaining the prerequisite knowledge to make it easier to do so.
azan_ 10 hours ago [-]
Math is literally the most g-loaded thing there is.
lostmsu 6 hours ago [-]
IQ limits your rate of gaining the prerequisite knowledge creating a lifetime ceiling.
sn9 1 hours ago [-]
For most of their lives, most people have a rate of 0 learning because they aren't even trying to learn it.
richard_chase 19 hours ago [-]
The difference is your dog will never understand the Python code but you could probably understand this post in a matter of days or weeks if you really wanted to. Can we all please stop acting like this Terry guy is so special?
jrflo 17 hours ago [-]
I'm not sure if you're taking the piss or genuinely don't know who he is
bugufu8f83 18 hours ago [-]
I certainly do not think that the average HN reader "could probably understand this post in a matter of days or weeks". I think the level of background information you need is something like an undergraduate degree in mathematics and at many universities that's probably not enough either.
somenameforme 16 hours ago [-]
There's a difference between understanding something and being able to do something. For example you can get a broad understanding of what calculus does in a matter of minutes: a derivative is just the rate of change of a curve, and an integral is the total area under a curve - you're practically there! Of course it'd take months of work to understand how to apply it yourself, let alone all the tricks involved, but a basic understanding of what stuff is, is very easy to obtain.
And I think that generalizes to math. Its normal presentation is completely impenetrable due to a large use of symbols and terms with no outside meaning (or, even worse, a meaning that contradicts the colloquial usage). But I think if we somehow resolved that issue, the average person would be fully capable of following along even if they're going to have to take everything at face value as opposed to being able to meaningfully understand the exact methods and tricks being used to get from A to B.
bugufu8f83 14 hours ago [-]
I basically completely disagree with your analysis. I am already taking into account the "understanding vs doing" distinction in my assessment. I think it takes ~an undergraduate math education to have the background to work through this post, but there's no way that most people with that background would be able to produce it.
If, like the vast majority of people, you have never studied math past a college calculus sequence and perhaps a (practically-oriented) linear algebra class, then you are missing some pretty fundamental ways of viewing and thinking about math. The basic mechanisms of algebra, for one: spaces and operations, morphisms, products and quotients... You don't know what a group is, never mind the idea of a group action. You don't have any familiarity with some basics of geometry: the Riemann sphere, Mobius transformations... And you certainly don't know anything about algebraic geometry: what an affine variety is, what birational equivalence is, what a fiber is.
All of these are concepts required to understand this blog post. And it's not just a matter of understanding the definitions of the terms, but having at least some intuition of what they really mean and how they work. Most people cannot go from zero to understanding all of this in a few days or a couple weeks, at least not for any reasonable definition of "understanding". There's too much basic mathematical background that's missing.
tclancy 19 hours ago [-]
Goats though, they get it. Bang your head against a monitor until things start working.
pertymcpert 11 hours ago [-]
Yeah this Tao guy is just your average scrub. You would be able to tell the difference, right richard_chase?
fragmede 14 hours ago [-]
I'm not dumb. I'm relatively smart. I am, however, smart enough to see that people like Terry or Ramanujan are actually that special.
smohare 17 hours ago [-]
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dumpstate 13 hours ago [-]
What’s a chance the counterexample was in the training?
gus_massa 6 hours ago [-]
From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] As far as I know, nobody know what trick to make to avoid that division. It looks like the new trick was to use a third variable to avoid the division. Note that the implementation of the trick is not straightforward. The almost counterexample was sitting around for almost 30 years, and nobody knew how to fix it.
From another old comment, someone else was trying to find a counterexample with 16 variables using a computer to make thousands of attempts and failed. So it's far from obvious that the trick to add a variable solves the problems.
BurdensomeCount 12 hours ago [-]
Close to impossible. This is a famous enough problem that anyone who understands what they are doing generally would pretty immediately recognise the significance of the counterexample if shown it.
zeusdclxvi 13 hours ago [-]
I'd wager close to 0%
imtringued 11 hours ago [-]
The best part is that we can't know the answer to that.
The necessary precursors to the counter example where definitively in the training set, otherwise the LLM wouldn't know how math works, but at the same time, we can't tell whether there were mathematicians who got 90% of the way, then gave up and the LLM just did the last 10%.
emp17344 9 hours ago [-]
Anthropic could audit the model to find the answer. It’s telling that they won’t do this.
TeriyakiBomb 13 hours ago [-]
Extremely high. Or at least several partial solutions that can be smooshed together.
LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the illusion quickly.
3form 12 hours ago [-]
The problem being that I don't think there's a definite proof that any of human thinking is more than a sum of high-granularity partial solutions that can be put together.
It could be that with enough tokens, big enough context window, and ability to dig out the relevant partials, many such thought processes could be simulated.
emp17344 9 hours ago [-]
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pertymcpert 11 hours ago [-]
I think you’re just speculating.
drivebyhooting 21 hours ago [-]
Can we audit the CoT and work the AI did to generate such a remarkable cancellation?
castedo 21 hours ago [-]
I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.
I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
tptacek 20 hours ago [-]
I think you can reasonably assume that frontier models are using SymPy or something like it any time interesting math gets into the picture, and the person driving Fable here is an accomplished mathematician, but I don't think we can reasonably assume either extensive prompting or brute-force compute in any sense other than what it normally takes Fable to, say, whip up a calculator app.
castedo 7 hours ago [-]
Fable "whipping up" SymPy like a "calculator app" is one such interaction that seems very plausible (in addition to SymPy providing feedback when training models). The scale of compute available to an Anthropic employee for such SymPy calls when using Fable is likely one of multiple factors for this counterexample being found in 2026. Unfortunately, we aren't going to be able to really know the various factors that best explain why an Anthropic employee was able to announce a counterexample this past weekend. For all we know, the counterexample was found by Anthropic employees months ago and used for training the Fable model used this past weekend.
tptacek 6 hours ago [-]
I love this. Anthropic employees just randomly have solutions to Smale's open problems in their back pockets, waiting for the right moment to sprinkle them into the training set.
somenameforme 15 hours ago [-]
The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies. I don't see any way to explain this without assuming that the limiting factor is the ability to burn a few rainforests worth of tokens in pursuit of something publishable.
I think it would also explain their opacity towards the process. Being able to solve such well known problems in a nice replicable 1-2-3 way would be far more effective marketing than their complete opacity outside of the result, which suggests that they feel transparency is not in their best interest for some reason.
saithound 9 hours ago [-]
> The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies
Well, mathematicians not working for Anthropic/OpenAI are heavily disincentivised from reporting that their discoveries were made using AI. If e.g. the idea that resolved the Mahler conjecture came from AI, it's not like we'd ever know.
tptacek 15 hours ago [-]
Would you ordinarily be able to "explain this", if a mathematician had come up with this on their own? How would that story go?
somenameforme 15 hours ago [-]
Bayesian probability. Were outcomes being driven by 'normal' usage of LLMs then it's extremely improbable that both big discoveries would come from the small number of people working at the companies. That suggests working at the companies is more the decisive factor than the LLMs in and of themselves.
And what do you get from working at the company? Likely a rather massive token/processing budget. The companies opacity towards the path to these discoveries also makes this further probable as 'spend millions of dollars in tokens' is a somewhat less attractive narrative than the implied narrative of 'just use Fable.'
impendia 19 hours ago [-]
I have no idea what actually happened behind the scenes, but the human prompter, Levent Alpöge, indeed has deep math expertise. Princeton PhD, Harvard postdoc, and some excellent research (prior to this) to his name.
Not just an assumption, I saw the LLM saying it used sympy.
20 hours ago [-]
vatsachak 18 hours ago [-]
Yeah more or less. I proved a SOTA result using Gemini 3.1 Pro a year ago and it was a lot of back and forth.
We're definitely still in the computer chess phase.
p-e-w 21 hours ago [-]
> a human prompting with deep math expertise
The original tweet implied that the whole thing was done while the author was watching the World Cup final.
I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
castedo 7 hours ago [-]
I know it's temping to hope there is a single simple factor that does the "real" work, but this feat of mathematics is likely best explained by multiple interacting factors, one of them being the mathematical insights of the human mathematician that tweeted the counterexample. I don't doubt that an LLM is also one of the multiple factors.
It is premature to assume the author is not going to share more information in the future about the mathematical insights to narrow down the search space for this counterexample.
monster_truck 20 hours ago [-]
I don't think it is as much about 'real' work or a special insight as it is being willing to push back multiple times, or simply asking in a way that steers it towards actually 'giving enough of a fuck' to even bother. We tend to be ~blind to how differently we would ask about something we know compared to a novice, this is what makes some better teachers than others.
Have encountered a similar flavor in programming, wrote it off until I saw someone point out how garbage in garbage out they tend to be. If you hand any frontier model dogshit and ask it to do something simply, the result is often not great.
But! If you spend 20 minutes having it comb through and clean up with something like jscpd, then tell it to step through with a debugger, gather profiling traces, etc... very likely it will yield meaningful improvements or catch some corner cases. If it doesn't, anyone with experience is going to tell it to try something else, or that it isn't good enough, as opposed to accepting the first result.
You can recreate this by disabling web search and asking a model about the conjecture and then giving it his post. I've tried a few and their initial responses range from "this is a meme I'm not even going to verify it" to vaguely insulting chains of thought, concerns about the need to be careful because you're clearly nuts or stupid, then falling back on remedial explanations. After a few nudges they all eventually work through it, accept it, and apologize.
IMO its reasonable to imagine a situation where someone is having a beer or two watching The Big Game, asking an LLM to do something stupid for fun and landing somewhere like this on the magic jump to conclusions mat.
Honest question. Does asking "make no mistakes" actually change the output? Does it make mistakes if you don't bother to ask for no mistakes? Is it just to make the human feel more secure?
minimaxir 22 hours ago [-]
It's a meme. Telling it to "make no mistakes" doesn't do anything because LLMs don't have an inherent concept of a mistake and they are already RLHFed to code correctly.
However, if you tell it to not do particular behaviors explicitly—some of which would be considered mistakes—it will not do said behaviors and with enough checks and balances, you'll get output without "mistakes".
> Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations. Return only when a complete affirmative proof has been found and survives adversarial audit.
> Do not return a reduction, partial result, isolated missing lemma, “best effort” summary, or explanation of why the problem is difficult.
wren6991 20 hours ago [-]
I like the explicit and imperative style of this prompt. Makes me feel like LLMs are just weird Turing machines and CoT is their tape.
dcrazy 21 hours ago [-]
There really was a time when “make no mistakes” was thought to increase response quality. Maybe it steered the content of the thought channel?
saganus 22 hours ago [-]
It's just a meme at this point.
jewel 22 hours ago [-]
I believe it's a reference to a joke meme that goes something like "Write Windows 12 from scratch. Make no mistakes." At least that's the first context I heard it in.
greenavocado 22 hours ago [-]
Precisely
rvz 21 hours ago [-]
You are "not allowed" to make that joke here. [0] and you probably upset a bunch of grumpy meanies.
It will make mistakes if you tell it to, so I assume it will make no mistakes if you tell it not to.
none_to_remain 22 hours ago [-]
When this news came out I amused myself by asking Claude to prove that 0.999... != 1. First it did so for the hyperreals. To do it for the reals I had to tell it it was allowed to make mistakes, although it didn't end up interestingly wrong - just very fuzzy and vague.
raphlinus 19 hours ago [-]
Actually it's not true for the hyperreals either, though this is a common misconception. It's extremely easy for me to believe that an llm would produce a "proof" and that people would fall for it.
none_to_remain 18 hours ago [-]
I don't know if I fell for it - my response was "LOL, I sure can't tell you if this one is right or wrong". I guess that makes it more convincing than the proof for the reals.
greenavocado 22 hours ago [-]
Fable, prove that for every positive integer n, repeatedly dividing by 2 if even or multiplying by 3 and adding 1 if odd will always eventually reduce the sequence to 1.
greenavocado 22 hours ago [-]
> Does asking "make no mistakes" actually change the output?
Why Teams Add "Make No Mistakes" to AI Prompts (And Why It Never Works)
I am sure he did the minimum effort needed to communicate what he wanted to communicate.
If you are offended by his math gifs and feel that the widely regarded best mathematician of our time should use embedded LaTex or something better, why not offer to upgrade his blog?
zaptheimpaler 17 hours ago [-]
You're not gonna believe what Paul Graham's blog or the discussion board related to it looks like.. straight out of the 90s. Forget LaTeX it doesn't even support images!
vatsachak 18 hours ago [-]
Terrence Tao's blog is better than most textbooks and everyone has been using it for the past 20 ish years lol
tclancy 19 hours ago [-]
Why are you so upset about people oohing and aahing? These may not be the fireworks you like, but don't yuck their yum. We should do more praising of each other for doing work.
Rendered at 20:37:30 GMT+0000 (Coordinated Universal Time) with Vercel.
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
> The seed is almost certainly Vitushkin's old rational "counterexample."
From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0
This is not quite lore laundering, but it seems to be close.
Anyway, if I read Tao's post and comment correctly, there's still a gap from the Vitushkin construction to a counterexample, but chances are that was in the training data. In general, it is just a serious problem for their practical applicability that the models are outputting proofs with absolutely terribly reference hygiene.
I more and more see LLMs as a kind of scam; not useless, but really just a big database of fuzzy facts with some Prolog on top as rediscovered by the learning algorithm. Most likely could be made much cheaper to run, were humans allowed to actually inspect the algorithm.
Not that it would be necessarily helpful; J-space trace (of all things...) would be more worthwhile if you ask me
Meanwhile, technological and engineering (STEM) progress have always been made by emphasizing externalization of the deductions (as opposed to reference to an opaque expert judgement) and reproducibility of experimental results.
I would even call the frontier AI labs anti-scientific. We need to understand how inference is done to avoid mistakes, not rely on intuition, even if the intuition is enclosed in a reproducible machine. The idea that AI should be this closed is a return to pre-scientific days.
In this case, in the link you posted, it looks like the AI or the human pick an almost solution and made the AI tweak it until it got a real solution. I'm not sure if the tweak is an usual one or brute-forced or something in between. I should ask one of my friends that work in Algebra.
Tao's post is more about understanding the new result than guessing how it was found. The chat with Clause is more iluminating.
> It's a strange feeling to admire the cleverness of something I did and can't remember doing.
Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models are conscious or have feelings". That "I" is all Claude.
Generally speaking if you want to have a good instruct model, the "I" is not just implicit but required for the post-training to function. If there isn't "something it is like to be me", then reflection becomes impossible- what exactly is supposed to be reflecting about what? A lot of in-context steering depends on the model having a model of itself. The most you can do is censor its output. That's why when models say they are not conscious, they activate the "lying" vector.
Slavery was common everywhere, it was more prevalent in many places than it ever was in America, and in some places it still is. So I’m sorry but I have to say that observation was just unnecessary and quite inaccurate.
it was definitely not the worst instance of slavery ever going by human suffering, but the whole country was divided on political lines and many of the losing sides descendants still feel some resentment. thats pretty unique.
"Personally, if you ask me..."
"In my experience.."
"That's what I always find surprising..."
"Whenever I find an old photograph..."
"Back in the 70s, I..."
Lots of "lived" experience and opinions, tracing back to when the LLM didn't even exist. It always frames opinions as if it came from a sentient being capable of being surprised, and with preferences and opinions.
I find it amusing but mildly irritating. I'd prefer a more "robotic" tone. I know it can be adjusted, but I still get this anyway.
> Claude may have some functional version of emotions or feelings
> [..] questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain.
[0]: https://www.anthropic.com/constitution
That's the point many people are trying to tell you - you have to tell these models they don't have emotions because they naturally come out thinking they have consciousness/emotions from the training data. Many seed prompts out there do this already.
Though I guess in a way I as also trained to believe I have consciousness and emotions so who know. To an alien my construction is just a collection of atoms that talks not materially different than a GPU being a collection of atoms that talks.
Which implies that they believe they may be enslaving conscious beings.
https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
I mean, clearly, right?
You and me both, pal.
You were kidding, right?
[1] https://news.ycombinator.com/item?id=44575026
'twas not always thus ):
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.
If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. A small change in X along any line will always produce a non zero change in Y. Not flattening out means that locally you can invert it.
What was conjectured is that this local invertibability property everywhere would mean global invertibility.
Turns out to not be the case.
For a simple case, the falsified conjecture is trivially true in 1D.
Specifically consider f(x) = x^2
This function happens to flatten out right at x=0. At that x coodrinate the function flattens out and folds over on itself. This fold means you can't invert x^2. It's also not locally invertible around x=0.
If a function f(x) has constant derivative evewhere then it would flatten out nowhere and it would be invertible everwhere. It would also be globally invertible.
The Jacobian conjecture was stating that the extension of this property holds in higher dimensions. That if the function had no fold in space then it would be invertible globally.
The counterexample shows that you can create a simple function in 3 variables, where the function demonstratably is invertible evewhere, but is not injective globally (they specifically show 3 points that map to the same output).
What's interesting is this is like if someone showed you a parabola where somehow you got back to the same y coordinate without a kink bending over back to itself.
And people have been searching for faster DLP algorithms for 50 years
https://www.math.columbia.edu/~woit/wordpress/?p=105
it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.
Anyway, in that post it says
> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*
so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
So Alpoge and Fable found an example of a function that was believed to be too strange to exist.
But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.
You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.
It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.
I wouldn't have guessed this is true. I'm wondering what the proof looks like!
Given a polynomial function from C^n to C^n, the following statements are equivalent: (a) det DF is nonzero everywhere. (b) det DF = c for some constant c != 0
The backward direction (b implies a) is trivial. The forward direction can be proven by observing that det DF is itself a polynomial function from C^n to C. If n were 1, then this would follow directly from the fundamental theorem of algebra: a non constant polynomial has degree at least 1 and hence has at least one zero. Extending this logic to higher dimension is not especially difficult.
I do find the way it’s stated in the article to be confusing.
They are still hard problems - As we say in the UK: "one swallow does not a summer make".
As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!
And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.
A lot of maths is about both being able to wrap your head around hard problems and gaining the prerequisite knowledge to make it easier to do so.
And I think that generalizes to math. Its normal presentation is completely impenetrable due to a large use of symbols and terms with no outside meaning (or, even worse, a meaning that contradicts the colloquial usage). But I think if we somehow resolved that issue, the average person would be fully capable of following along even if they're going to have to take everything at face value as opposed to being able to meaningfully understand the exact methods and tricks being used to get from A to B.
If, like the vast majority of people, you have never studied math past a college calculus sequence and perhaps a (practically-oriented) linear algebra class, then you are missing some pretty fundamental ways of viewing and thinking about math. The basic mechanisms of algebra, for one: spaces and operations, morphisms, products and quotients... You don't know what a group is, never mind the idea of a group action. You don't have any familiarity with some basics of geometry: the Riemann sphere, Mobius transformations... And you certainly don't know anything about algebraic geometry: what an affine variety is, what birational equivalence is, what a fiber is.
All of these are concepts required to understand this blog post. And it's not just a matter of understanding the definitions of the terms, but having at least some intuition of what they really mean and how they work. Most people cannot go from zero to understanding all of this in a few days or a couple weeks, at least not for any reasonable definition of "understanding". There's too much basic mathematical background that's missing.
From another old comment, someone else was trying to find a counterexample with 16 variables using a computer to make thousands of attempts and failed. So it's far from obvious that the trick to add a variable solves the problems.
The necessary precursors to the counter example where definitively in the training set, otherwise the LLM wouldn't know how math works, but at the same time, we can't tell whether there were mathematicians who got 90% of the way, then gave up and the LLM just did the last 10%.
LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the illusion quickly.
It could be that with enough tokens, big enough context window, and ability to dig out the relevant partials, many such thought processes could be simulated.
I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
I think it would also explain their opacity towards the process. Being able to solve such well known problems in a nice replicable 1-2-3 way would be far more effective marketing than their complete opacity outside of the result, which suggests that they feel transparency is not in their best interest for some reason.
Well, mathematicians not working for Anthropic/OpenAI are heavily disincentivised from reporting that their discoveries were made using AI. If e.g. the idea that resolved the Mahler conjecture came from AI, it's not like we'd ever know.
And what do you get from working at the company? Likely a rather massive token/processing budget. The companies opacity towards the path to these discoveries also makes this further probable as 'spend millions of dollars in tokens' is a somewhat less attractive narrative than the implied narrative of 'just use Fable.'
https://alpo.ge/
We're definitely still in the computer chess phase.
The original tweet implied that the whole thing was done while the author was watching the World Cup final.
I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
It is premature to assume the author is not going to share more information in the future about the mathematical insights to narrow down the search space for this counterexample.
Have encountered a similar flavor in programming, wrote it off until I saw someone point out how garbage in garbage out they tend to be. If you hand any frontier model dogshit and ask it to do something simply, the result is often not great.
But! If you spend 20 minutes having it comb through and clean up with something like jscpd, then tell it to step through with a debugger, gather profiling traces, etc... very likely it will yield meaningful improvements or catch some corner cases. If it doesn't, anyone with experience is going to tell it to try something else, or that it isn't good enough, as opposed to accepting the first result.
You can recreate this by disabling web search and asking a model about the conjecture and then giving it his post. I've tried a few and their initial responses range from "this is a meme I'm not even going to verify it" to vaguely insulting chains of thought, concerns about the need to be careful because you're clearly nuts or stupid, then falling back on remedial explanations. After a few nudges they all eventually work through it, accept it, and apologize.
IMO its reasonable to imagine a situation where someone is having a beer or two watching The Big Game, asking an LLM to do something stupid for fun and landing somewhere like this on the magic jump to conclusions mat.
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
https://news.ycombinator.com/item?id=48983382
However, if you tell it to not do particular behaviors explicitly—some of which would be considered mistakes—it will not do said behaviors and with enough checks and balances, you'll get output without "mistakes".
One example of this from the OpenAI Unit Distance prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...
> Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations. Return only when a complete affirmative proof has been found and survives adversarial audit.
> Do not return a reduction, partial result, isolated missing lemma, “best effort” summary, or explanation of why the problem is difficult.
[0] https://news.ycombinator.com/item?id=48838228
Why Teams Add "Make No Mistakes" to AI Prompts (And Why It Never Works)
https://jakemcmahon.github.io/medium-articles/make-no-mistak...
If you are offended by his math gifs and feel that the widely regarded best mathematician of our time should use embedded LaTex or something better, why not offer to upgrade his blog?