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A digestion of the Jacobian conjecture counterexample
- dumpstate 2mo agoWhat’s a chance the counterexample was in the training?
- zeusdclxvi 2mo agoI'd wager close to 0%
- TeriyakiBomb 2mo agoExtremely 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.
- pertymcpert 2mo agoI think you’re just speculating.
- 3form 2mo agoThe 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 2mo ago[dead]
- BurdensomeCount 2mo agoClose 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.
- imtringued 2mo agoThe 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 2mo agoAnthropic could audit the model to find the answer. It’s telling that they won’t do this.
- gus_massa 2mo agoFrom a comment by j2kun https://news.ycombinator.com/item?id=49000833 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.
- dumpstate 2mo agoRight. Nothing obvious in public. Where I'm heading is that all labs pay for solutions to software engineering task and never publicize those, I assume they do the same with maths?
- gus_massa 2mo ago>>> What’s a chance the counterexample was in the training? My guess is it was not in the training. Only the 2D "almost" countraexample was in the training, but it was not clear how to fix it. It's not clear how much steering Levent Alpöge did to get the result. He said he did during the final match of the World Cup, but he is from Turkey and living in USA and the game sadly was quite one sided, so I guess he does not care too much. So my guess is that he had a long chat with Fable. I'm guessing too much, but if I can guess one more time the problem was probably too difficult to get solved by Levent Alpöge alone and by Fable alone, and it's a genuine Centaur solution.
- greenavocado 2mo ago[flagged]
- fhdkweig 2mo agoHonest 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?
- hyperhello 2mo agoIt 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 2mo agoWhen 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.
- greenavocado 2mo agoFable, 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.
- raphlinus 2mo agoActually 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 2mo agoI 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.
- tptacek 2mo agoThe 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: https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed56 https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
- foobarqux 2mo ago[flagged]
- Jtarii 2mo agoPeople are upvoting this because Tao is a celebrity.
- hyperhello 2mo agoHe’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.
- ChrisArchitect 2mo agoRelated: Claude Fable produced a counterexample to the Jacobian Conjecture https://news.ycombinator.com/item?id=48973869 https://news.ycombinator.com/item?id=48973869 Human mathematicians are being outcounterexampled https://news.ycombinator.com/item?id=48983382 https://news.ycombinator.com/item?id=48983382
- hyperhello 2mo agoOkay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
- sfpotter 2mo agoNo. 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 2mo agoIt'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 2mo agoThe 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.
- mswphd 2mo agoit doesn't overturn much. For example, here is a post from 2004 https://www.math.columbia.edu/~woit/wordpress/?p=105 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.
- vanderZwan 2mo ago> 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 2mo agoI 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." From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0 https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0 This is not quite lore laundering, but it seems to be close.
- gus_massa 2mo agoI 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".
- CamperBob2 2mo ago[flagged]
- j2kun 2mo agoFWIW, you may enjoy https://www.argmin.net/p/lore-laundering-machines https://www.argmin.net/p/lore-laundering-machines
- brcmthrowaway 2mo ago[dead]
- zzzeek 2mo agoreading 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 2mo agoSome 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 2mo agoI 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.
- TMWNN 2mo agoI recently saw a photo of an alligator lying on top of a pool floatie shaped like an alligator. That alligator could be trained in every way possible for ten years, and never ever recognize the mutual resemblance. Like you, I wonder if we will soon cross the boundary where we are the alligator. (I suspect that we have been the alligator all along, outside the context of AI.)
- drivebyhooting 2mo agoCan we audit the CoT and work the AI did to generate such a remarkable cancellation?
- castedo 2mo agoI 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 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.
- p-e-w 2mo 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.
- monster_truck 2mo agoI 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.
- jmward01 2mo agoFinding 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 2mo 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!
- aayushdutt 2mo agoAfter reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?
- clarionbell 2mo agoNot 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 2mo agoClearly 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 2mo agoThis is what I used to feel reading ML papers. Until I didn't anymore. Unstructured learning works, albeit, perhaps, slower.
- richard_chase 2mo ago[flagged]
- tclancy 2mo agoWhy 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.
- lubujackson 2mo agoI 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?
- vatsachak 2mo agoTerrence Tao's blog is better than most textbooks and everyone has been using it for the past 20 ish years lol
- zaptheimpaler 2mo agoYou'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!
- sashank_1509 2mo agoI 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.
- welferkj 2mo agoIf there's anyone that deserves praise every time he has something to say about mathematics, it's him.
- moralestapia 2mo agoTired of this meme, honestly.
- azan_ 2mo agoWhat meme?
- moralestapia 2mo ago>If there's anyone that deserves praise every time he has something to say about mathematics, it's him.
- jamincan 2mo agothat's a meme?
- pavel_lishin 2mo agoIt's up there with Milhouse.
- zamadatix 2mo agoTime to switch to "Sorry big T, your question is dogshit and here's why you don't know anything about math"?
- jeremyscanvic 2mo 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 2mo agoI’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 2mo agoAh right this makes a lot of a sense - thanks!