> <i>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.</i><p>Sounds like the most interesting part would be learning what approaches the LLM <i>did</i> 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.
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).<p>> The seed is almost certainly Vitushkin's old rational "counterexample."<p>From <a href="https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0" rel="nofollow">https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0</a><p>This is not quite lore laundering, but it seems to be close.
I hate that Anthropic seemingly tries to make Claude act as if it was conscious or had feelings<p>> It's a strange feeling to admire the cleverness of something I did and can't remember doing.
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".
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:<p><a href="https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed56" rel="nofollow">https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...</a>
Also note the timestamp: he started working on this thread a few hours after the tweet.
> 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.<p>I mean, <i>clearly</i>, right?<p>You and me both, pal.
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 <i>then</i> we read "OK, I've activated Pro."
> <i>I'm bad at math ... he includes the GPT5 prompts for his conversation, which are easier to follow</i><p>You were kidding, right?
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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].<p>[1] <a href="https://news.ycombinator.com/item?id=44575026">https://news.ycombinator.com/item?id=44575026</a>
People are upvoting this because Tao is a celebrity.
I'm upvoting this because it's interesting. I don't have to fully understand something to find it interesting.
He’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.
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.
He’s a hacker news celebrity but not a global celebrity
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.”,<p>“Yes, you are exactly right.”<p>“You have gotten to the core issue.”<p>And non stop praise. Seems like sycophancy is still an issue lol.
After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?
> Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant.<p>I wouldn't have guessed this is true. I'm wondering what the proof looks like!
I’m fairly confident that the blog post is trying to say something like this:<p>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<p>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.<p>I do find the way it’s stated in the article to be confusing.
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation:<p>Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.<p>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.<p>What was conjectured is that this local invertibability property everywhere would mean global invertibility.<p>Turns out to not be the case.<p>For a simple case, the falsified conjecture is trivially true in 1D.<p>Specifically consider f(x) = x^2<p>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.<p>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.<p>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.<p>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).<p>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.
Thanks, I was curious for the motivation behind the conjecture, but it wasn’t mentioned in the Wikipedia article.
Sounds like 3 more iterations on K3/Fable, and we'll able to ask it to break <i>computational hardness assumptions</i> like DLP?
it doesn't overturn much. For example, here is a post from 2004<p><a href="https://www.math.columbia.edu/~woit/wordpress/?p=105" rel="nofollow">https://www.math.columbia.edu/~woit/wordpress/?p=105</a><p>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.<p>Anyway, in that post it says<p>> 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)*<p>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.
For the Jacobian determinant to be <i>constant</i> 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.<p>So Alpoge and Fable found an example of a function that was believed to be too strange to exist.
Not much.<p>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.
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 <i>not</i> constant.
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.
What’s a chance the counterexample was in the training?
I'd wager close to 0%
Extremely high. Or at least several partial solutions that can be smooshed together.<p>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.
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.
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.
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?
I'm not sure if you're taking the piss or genuinely don't know who he is
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.
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.<p>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.
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.<p>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.<p>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.
Goats though, they get it. Bang your head against a monitor until things start working.
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.
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Some people downvoting you, but I think it is a valuable illustration of IQ gap.<p>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.
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.<p>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.
Related:<p><i>Claude Fable produced a counterexample to the Jacobian Conjecture</i><p><a href="https://news.ycombinator.com/item?id=48973869">https://news.ycombinator.com/item?id=48973869</a><p><i>Human mathematicians are being outcounterexampled</i><p><a href="https://news.ycombinator.com/item?id=48983382">https://news.ycombinator.com/item?id=48983382</a>
Can we audit the CoT and work the AI did to generate such a remarkable cancellation?
I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.<p>I think a reasonable assumption is that there is an interaction between an LLM, a <a href="https://en.wikipedia.org/wiki/Computer_algebra_system" rel="nofollow">https://en.wikipedia.org/wiki/Computer_algebra_system</a> tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
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.
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.<p>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.
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.<p><a href="https://alpo.ge/" rel="nofollow">https://alpo.ge/</a>
Not just an assumption, I saw the LLM saying it used sympy.
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.<p>We're definitely still in the computer chess phase.
> a human prompting with deep math expertise<p>The original tweet implied that the whole thing was done while the author was watching the World Cup final.<p>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.
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.<p>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.<p>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.<p>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.<p>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.
<a href="https://news.ycombinator.com/item?id=48986943">https://news.ycombinator.com/item?id=48986943</a>
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I am sure he did the minimum effort needed to communicate what he wanted to communicate.<p>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?
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!
Terrence Tao's blog is better than most textbooks and everyone has been using it for the past 20 ish years lol
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.