20 ms·

Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

• field10DOTcom 3mo ago
  [flagged]
• jason_s 3mo ago
  Are any of ChatGPT's statements erroneous? (how come Terence Tao gets this high quality output but I get hallucinations when I ask simpler math stuff?)
• vb-8448 3mo ago
  Maybe it's silly, but from someone who is ignorant on this topics, what are the consequences of this kind of "discoveries"? Is it something "revolutionary" or just another small brick that will pile up until something really "revolutionary" will happen?
  • rickypp 3mo ago
    "Hey Fable, please generate me the next 1000 undiscovered bitcoin hashes"
    • echelon 3mo ago
      You're joking, but perhaps LLMs will find a way to mathematically break the complexity of factorization. Maybe they'll find a solution where P=NP. That could really throw a wrench into the whole internet thing. It seems they need an expert human driver for now.
      • ConceptJunkie 3mo ago
        I'm sorry, I can't do that, but here is the design for a stable quantum computing platform that should allow you to generate those keys yourself...
        • NitpickLawyer 3mo ago
          Some materials are readily available on eMazon and aBay, so I've taken the liberty of ordering those for you. Your credit card bill will be a bit high this month, but it'll be worth it. There weren't any sellers for the advanced EUV lithography machines, so I've hacked into the only place on earth that makes them, changed their records and had them ship it to you. Expect to receive a "pinball machine" from Amsterdam, soon. I've instructed the roomba connected to the local network to start assembling stuff while we wait for the other materials. Oh, and you're gonna need a new toaster.
  • sdwr 3mo ago
    It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.
    • hyperhello 3mo ago
      To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.
      • Legend2440 3mo ago
        Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors." It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
        • hyperhello 3mo ago
          I shouldn’t, but: F1 = x^3y^3z + 3x^2y^4 + 3x^2y^2z + 7xy^3 + 3xyz + 4y^2 + z F2 = 3x^3y^2z + 9x^2y^3 + 6x^2yz + 12xy^2 + 3xz + y F3 = -x^3z - 3x^2y + 2x That’s the counterexample. Low integer coefficients, power 7 in three variables. If someone said it was there, couldn’t we all have written a pretty simple brute force solution for the search space, especially with the constraints that the symbolic determinant had to cancel to a constant?
          • traes 3mo ago
            Honestly just try it. You'll figure out the problem very quickly.
          • elisbce 3mo ago
            I don't think you even understand the problem. The determinant needs to be a non-zero constant AND you need to prove that particular map is not globally injective, meaning you have to find at least two points mapping to the same value. Of course it looks easy when someone shows you the counterexample.
  • arm32 3mo ago
    To me, this shows that extremely talented and qualified mathematicians (can) use frontier-level LLMs to automate their personal grind-y workloads that would otherwise (probably) take more time to accomplish with natural intelligence. By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
  • fragmede 3mo ago
    Practically, from this specific one? Nothing, it's very much a math thing. It's like art or music at this level. Are there consequences to a van Gogh?
  • hyperhello 3mo ago
    [dead]
• qrian 3mo ago
  Parent HN discussion: https://news.ycombinator.com/item?id=48998362 https://news.ycombinator.com/item?id=48998362
• Jun8 3mo ago
  Similar to how Cypher puts it: I know this is “just” next token inference, matrix mult and just software, ie there’s no “intelligence” there BUT, looking at this convo … damn! The fascinating this is that the LLM is not acting as a tool here AFAIk, but very much like a colleague. I have no knowledge of the domain and have only PhD EE level math knowledge, so maybe my bar is too low.
  • IshKebab 3mo ago
    > there’s no “intelligence” there BUT There is clearly intelligence there. We have no way to recognise intelligence other than the appearance of intelligence and this very clearly displays that. It's also quite clearly different to human intelligence in some notable ways, but not in any that preclude describing it as intelligent. At least for normal non-pedantic definitions of the word.
    • majormajor 3mo ago
      Everyone uses "intelligence" to mean something slightly different, so for this to be a useful claim to make or refute we need to come up with new, intentionally-pedantic, terms (or new domain-specific definitions for vague existing ones).
      • contextfree 3mo ago
        Yes, trying to communicate (or watching others try to communicate) about these topics is incredibly frustrating because it's pretty much impossible to make any progress without interrogating people's different definitions, but nobody wants to do that because it would mean being pedantic, splitting hairs, etc.
      • kadoban 3mo ago
        It's not like this is a new problem. Turing had a definition most of a century ago, he wasn't the first and certainly wasn't the last. I don't think we need new terms necessarily, and I doubt we're all going to agree on a definition tomorrow.
      • 3mo ago
• housu 3mo ago
  It's awesome to publish this kind of thing - great PR at least. Even if you don't understand the details, it's interesting to be able to peek into a technical conversation that a world class mathematician is having about their work with a "colleague". It's also the clearest demonstration I've seen of the vision AI people have about a future with truly intelligent copilots in super technical fields.
• cphoover 3mo ago
  Anybody have a cache/mirror of this? This is blocked by a corporate firewall... sigh
  • TheOtherHobbes 3mo ago
    Ask your LLM to hack it.
  • fragmede 3mo ago
    https://archive.is/z1vmx https://archive.is/z1vmx
• lukebuehler 3mo ago
  It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise (emphasis on progression and usage patterns, not absolute skill, obv I don’t match that): short pointed questions that goes all in on the jargon and machinery of the field and steers the llm hard (eg no softballs). I’ve noticed that llms switch their tone and meet you basically more or less on your level.
  • michaelchisari 3mo ago
    It reinforces how to "learn AI" is to first master the problem domain. I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
    • shshsjsj 3mo ago
      [dead]
    • vouaobrasil 3mo ago
      > I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses. That is what will happen though to future generations: they won't be able to use it for anything because none of them will have the "decades of coding" experience that you have had the privelege to have without AI.
      • singularity2001 3mo ago
        They will have decades of experience with AI and they will be able to guide them by sniffing their hallucinations from single words.
        • retrochameleon 3mo ago
          Hopefully in decades hallucinations will be largely solved.
• axus 3mo ago
  "I’ve activated Pro. Can you continue to look for a potential geometric explanation of the X_3 ~ A3 miracle that avoids coordinates or other unmotivated constructions ?" Another satisfied customer!
  • catigula 3mo ago
    I do the same kind of thing. "You're smarter now, time to try to cut down dumbo ChatGPT". High IQ bros.
    • brap 3mo ago
      And you an do it without even changing the model!
  • summner 3mo ago
    I just do very laconic questions about advanced topics, this seems to prompt it a bit more towards reducing fluff in the answers. But that + the activated pro could be an improvement
  • ckemere 3mo ago
    But what about the later "Repeat previous question" prompts???
    • WithinReason 3mo ago
      Probably network connection problems
  • taylorfinley 3mo ago
    Came here to flag the same beat. It's wild to me Terrance Tao has to pay to talk to chatgpt, you would think it would be the other way around!
    • Freedom2 3mo ago
      How would you envision ChatGPT paying Terrence?
• the_af 3mo ago
  It's fun if you ask ChatGPT to guess the identity of its interlocutor :) It will guess math researcher or paper author without hints, but if you give it some additional hints, "this was shared over the internet", "it's someone willing to work with AI", it will guess Terence Tao as the first choice.
• deleted 3mo ago
  [deleted]
• jstanley 3mo ago
  It's reassuring to know that even a supergenius's ChatGPT session is one sentence from the human followed by 3 pages of LLM output.
  • Dragbat 3mo ago
    Yeah, the LLMs are definitely committed to producing a whole article every time as a response to whatever the prompt and whoever the prompter
• jdw64 3mo ago
  I wish you'd share some conversations from experienced programmers too. How do they ask questions?
  • sunrunner 3mo ago
    "You are an expert software engineer with ten years of experience. How do I center a div?"
  • epolanski 3mo ago
    Antirez spends weeks planning with multiple models till he has a clear architecture and understood the constraints before he writes a line of code.
• minimaxir 3mo ago
  I'll have a blog post up tomorrow about it but the Jacobian Conjecture counterexample is a very funny cognitohazard for LLM assistants. It's a paradox for modern LLMs: they have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely.
  • bananaflag 3mo ago
    I wonder whether when the fact that AIs have started solving conjectures will enter the training data, they will become more confident in their abilities.
  • applfanboysbgon 3mo ago
    The models are just a reflection of how you prompt them. When I posted the counterexample, in dry language with a dry personality setting, claiming credit for it with search disabled, GPT simply verified that it was correct but noted that it could not ascertain novelty or priority and that external verification was required. "Model reacts in crazy ways when my prompt influences it to" is really, really not interesting.
• latenightcoding 3mo ago
  Can't even ctrl+f the conversation, wish openai would fix that
  • kepano 3mo ago
    Obsidian Web Clipper has a nice reader mode that works for ChatGPT transcripts (disclaimer: I made it)
    • trees101 3mo ago
      Web clipper is great. just tried the reader mode on this chatgpt transcript, it only shows 1 page of it. Is the purpose of reader mode to enable interactive annotation before saving to notes?
  • brap 3mo ago
    It’s crazy how much these companies invest in their models but when it comes to UX they do fuckall
    • kingleopold 3mo ago
      UX is still very hard for any startup because top talent almost never work on UX.
      • ValentineC 3mo ago
        What are these "top talent" usually talented at, then? (I'm not talking about OpenAI/Anthropic at this point, but maybe a <10 people startup.)
      • brap 3mo ago
        I’m sure they can hire a whole team of world-class UX folks and it will pay for itself
• napoleoncomplex 3mo ago
  This is the second ChatGPT shared conversation I've seen today that is truly fascinating. The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709 https://x.com/DmitryRybin1/status/2079904005652893709 What a world we live in.
  • tetris11 3mo ago
    crosses fingers "Low hanging fruit, low hanging fruit, low hanging fruit..." hyper-ventilates
  • cyanydeez 3mo ago
    this sounds like like an open parenthesis ( without someone independently verifying it, it just dangles there ...
    • schoen 3mo ago
      )
    • sfink 3mo ago
      At Mozilla, we had a set of whiteboard tags we could set on bugs, like "[crash]" or "[compat]" or "[leave-open]". That last was used when there were multiple patches attached to the bug, and we wanted to land only some of them without automation closing the bug once they landed. (It's common to have alternate approaches or test cases also attached to the bug, so you normally don't want to wait for all of them to land before closing the bug.) I started using "[leave-open" for those. It lasted for a couple of years, until someone went through and "fixed" them all.
      • cwmoore 3mo ago
        ]
  • progbits 3mo ago
    From the prompt: > You should do a breakthrough This is just as funny and ridiculous as those "make no mistake" prompts.
• ChaitanyaSai 3mo ago
  I don't understand any of the math here, but I had two thoughts. Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit. Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.
  • colordrops 3mo ago
    I'm not sure an AI will speed things up much. You would probably still need years of layers of foundational understanding to get the advanced material. We don't go through years of school to learn math just because teachers are bad - it's because complex subtle ideas are built on countless other ideas, and aren't necessarily compressible to something every layman can understand.
    • OJFord 3mo ago
      The years are broad though, the nice thing with AI explanations is that they can go deep quickly, and quite precisely down the path you need for your prior experience.
      • colordrops 3mo ago
        Oh for sure, I'm not denying the value of AI as a teacher, just saying that it's not going to speed things up much over a quality personal teacher/tutor.
        • buzzin_ 3mo ago
          Oh, but it will. Especially in math you depend on definitions and concepts defined earlier, and if you fall behind, perhaps in a class setting, it is hard to catch up, without having somebody who can explain it to you. But, now, with world class knowledge possessing tutor, you can ask for an explanation of missed concepts, even embarrassingly stupid questions you would never ask a person.
• brador 3mo ago
  "The determinant identity is almost embarrassingly simple once one writes the map in the right way." #Flexingontheentirehumanspecies
  • sunrunner 3mo ago
    Or, "Oh just give it here, let me do it."
• jvanderbot 3mo ago
  It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that would explain it. Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.
  • genidoi 3mo ago
    > you can use AI to understand something and map it to your own mental map This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process. I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.
    • yourself92 3mo ago
      > I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI. I do get what you are saying and agree with you in principle, as I have noticed the same. But that said, this same way of thinking can really apply to any abstraction. It really just depends on the level you're working at. A software architect might know the nitty gritty details of how each service works, but really those implementation details don't matter if the abstractions are handled well enough, so in theory they don't really need to know those details as long as "the pieces fit". But of course the catch 22 there is you can't build good abstractions that can fit together well if you don't understand well enough the underlying details
  • mohamedkoubaa 3mo ago
    Hold on isn't this how math professors used grad students for hundreds of years?
• jdw64 3mo ago
  I'm watching how Tao uses AI, and it's interesting. Expand the entire expression, then change the representation to find the core axis. You can't see the axis from just one perspective, so you change the representation. In programming terms, it's like applying multiple domain models. Then break it down into small contract units. Why is it a Jacobian monomial? Why does x satisfy a cubic equation? And so on. Then swap out the modeling under a hypothesis, assemble it all back together, and verify it through the equation. This feels similar to modeling in programming. Observe the whole -> explore better modeling -> decompose local problem -> verify independently -> reason about the highre level structure -> integrate back into the original problem. This feels similar to when I receive work from a client and write a programming proposal
  • dwd 3mo ago
    Been doing something similar and I would add that playing multiple LLMs off on each other gets a far superior result. One LLM would suggest a tightening of a particular axiom, and I would ask the rest what the consequences are: what is gained, what is lost and decide whether to use it. Often another would suggest a tweak that then goes back and a recursion.
• WarmWash 3mo ago
  Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane. Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.
  • dekhn 3mo ago
    I had a few moments of this in the past. For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature. Similarly, at some point somebody pointed out to me "the reason you're confused is that the bold on that variable means it's a matrix"
    • alexpotato 3mo ago
      I was reading about Tao's efforts to get more people to use Lean and apparently a big roadblock for people is that Lean uses very specific static typing. e.g. to use a very simple example on a white board "3" is "overloaded" as: - the integer 3 - the rational number 3 - the whole number 3 - etc When you write a proof in Lean, you have to specify the the type of "3" you mean. Having using Python/Perl and Java over the years, I get that some math folks found handling this daunting or at a minimum friction to getting into using Lean. LLMs seem to have been a big help here just for the "translate my math notation into a proof" feature.
      • neomantra 3mo ago
        To get the hang of this, I used Leanstral (Mistral’s LEAN agent) to vibe-code things like “the game of bridge” and then read the LEAN code.
    • ben_w 3mo ago
      Aye. A decade or so ago I wondered if the reason maths was hard was the names being optimised for writing by hand. Everything's single letters if they can get away with it, so when mathematicians run out of Latin alphabet, they use Greek, bold, etc. Even integration's ∫ is a fancy elongated s. CS version would be e.g. integral(function=some_named_function, from=a, to=b, with_respect_to=argument_of_function), which may be longer, but is less opaque, especially when you get in so deep there's 3 other people in the world who've looked into this specific problem and you had to invent your own operations. But that's all an outsider's perspective. I stopped with two A-levels in maths and further maths.
• ecshafer 3mo ago
  Terrance Tao's chatgpt conversation is really interesting for a variety of reasons: 1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result. 2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.
  • positron26 3mo ago
    [flagged]
    • xyzsparetimexyz 3mo ago
      Cool poem!
      • cpill 3mo ago
        yeah! more plz
  • petesergeant 3mo ago
    I think we have a couple of years of "being good at talking to the LLM about your field of expertise" being a useful human skill, until that too gets washed away
    • deleted 3mo ago
      [deleted]
    • HNisCIS 3mo ago
      Maybe? At the end of the day, even if they are some insane oracle (pun intended), they're still bounded by training data and how it relates to the real world. Even if they're a near perfect tool, we are still the interface between them and our lived experience. If that stops being the case then why do we care about the output? This assumes it doesn't graduate to just killing all of us and doing it's own thing, but within this paradigm it doesn't really have goals.
• NitpickLawyer 3mo ago
  Jeez. While I obviously can't talk at all about the math, I've noticed a few things: a) The model thinks on some questions while straight answers on others. (I wish I'd knew from the questions if this is somehow correlated to hard tasks or "inventive" tasks, but that's way out of my league). b) The model sometimes pushes back. Again, I'd wish I knew if it was warranted, but I counted 2 instances where it said "yes, but with caveats", one where it said "mostly yes but with this correction" and one where it said "careful here, because x y z". c) The model did q&a + pdf ingestion + code writing + more q&a + thinking + more q&a, for a looong while, while seemingly staying on topic (at least Terrence Tao seems to think they're still productive, so I'll trust that). This is what model progress is, not number goes up on xBency or yBencher. Damn.
  • istrice 3mo ago
    The "yes, with caveats" thing is boilerplate for both Codex and Claude since this current generation. It's actually a bit annoying because it primes you to think that the caveats are real, but most of the time it's just something terribly obvious and not a real caveat, but the model probably has some system prompt that tells it to always consider caveats or something like that. Same as the model starting every reply with a commitment to be "honest". LLMism are fun but I tend to just suppress them via AGENTS.md because they distract me
    • gizmodo59 3mo ago
      A lot of things are not black and white and it’s nuanced so someone with higher intelligence will point it out. It’s an emergent behavior not a defect
    • sfink 3mo ago
      At least Opus up to 4.7 or so, my experience is that Claude often uses "yes, with caveats" in place of "no, you numbskull". "Is a meter the same as a foot?" "Yes, exactly, you have it now, except they're different distances." Maybe it's because I ask it to quiz me, and it really doesn't like to tell me I'm wrong. I also got a fair amount of Claude: Ok, I will test your understanding. Question A? Question B? Question C? Question D? Me: A=10. B=2. C=121. D is not solvable. Claude: You got most of them right! You're very astute in saying that A=10, but actually it's 7. B=2 is exactly right! C could be 121 if we were talking base 4, but we're actually in base 10 so it's 25. D is trivially solvable and is 0. Me: ...isn't that like 1 out of 4? How is that "most of them right"? Claude: You're absolutely right! ...blah blah blah
• doctoboggan 3mo ago
  Similar to the story of George Dantzig, who was late to class and solved two open problems in statistics because he mistook them for homework, I think the current batch of frontier LLMs are chained up by knowing which problems are supposed to be unsolved. If they're let free (probably via some targeted RLHF) we might get a flurry of solutions to open problems.
  • sigbottle 3mo ago
    But a property of intelligence is to know when to stop, if we treat intelligence as some sort of search and not some a priori intuition of the entire space. Seems kind of hard, if not impossible, to train for specifically that.
    • kenjackson 3mo ago
      A property of lack of intelligence though is stopping too soon. The wisdom to differentiate is probably rare at the extremes.
• OJFord 3mo ago
  Fancy telling Tao something's 'almost embarrassingly simple' (if only written in the right way)!
  • HarHarVeryFunny 3mo ago
    Presumably this entire conversation is at least somewhat typical of the way high level mathematicians talk to each other in ego-less fashion? Although, it would be interesting to what extent the model was trained on math conversation as opposed to just analysis and proofs. The flow of the whole conversation, with Tao guiding and the model calculating, gave me the feel of Tao perhaps talking to himself - just that each of those model responses would have taken him much longer to calculate by hand. It would be fascinating to hear Tao talk about what he may have learnt from this, and if it suggests approaches to other problems he might not have considered, as well as an analysis of the original Fable counter-example construction.
• 6thbit 3mo ago
  Presumably this was Sol on xhigh, then over to Pro (as per his indication on chat)? Is there any way to tell a conversation's model and thinking level?
• personalityson 3mo ago
  This was my conversation with ChatGPT 4 years ago: https://i.imgur.com/WPaWgzZ.png https://i.imgur.com/WPaWgzZ.png Where will we be in another 4 years? What a time to be alive!
  • kypro 3mo ago
    I'm from the UK. What's in the image?
    • personalityson 3mo ago
      Me asking ChatGPT what values of x satisfy 0/x = 0. ChatGPT insists it has no solutions.
    • ben_w 3mo ago
      ChatGPT repeatedly asserting that 0/x=0 is division by zero and therefore no value of x satisfies the equation.
    • xyzsparetimexyz 3mo ago
      Time to get a VPN bucko
      • wiether 3mo ago
        I'm in France using a French VPN and I'm being blocked by imgur (been going on for a few years now)
        • archargelod 3mo ago
          Why do you use a VPN regulated by your government? That really limits its usefulness.
        • kypro 3mo ago
          Yes, same here in the UK. I think imgur blocks VPN IPs so I suspect unless I register a private IP or something it's basically unusable.
• gcgbarbosa 3mo ago
  I find it amazing how people can use AI to do things that seem hard but yesterday I could not figure out how to install a package on my system. It kept suggesting dependencies that don't exist, and telling me to use functions that are not in the system. The math does not math...
  • uncivilized 3mo ago
    What's worse is that HN doesn't even try to figure out how such a result could be obtained. They just take it at face value and go into dear diary mode. Reddit has some good discussions on the topic.
  • esikich 3mo ago
    How were you expecting it to know what your system setup was? Did you ask it for some bash commands to get it the info it would need to get unstuck? Or were you just expecting magic?
• hintymad 3mo ago
  I recall several mathematicians (possibly including Terence Tao) mentioning that fields in mathematics have become so specialized and isolated that a conference like the ICM feels more like a collection of mini-conferences. An expert in one area can barely understand a talk in another. Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal couldn't imagine.
  • epolanski 3mo ago
    I was a chemistry researcher this is true in all science. I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.
• sandspar 3mo ago
  Sense of vertigo that this is the same AI that plans meals and recommends movies for me. Incredible range.
• Human-Cabbage 3mo ago
  Is ChatGPT's interface always this atrociously jittery? Or is it just because this page is getting an HN hug-of-death right now? Every time I try to scroll the whole page goes blank for a few seconds and then re-renders.
  • ben_w 3mo ago
    I've found it's had unacceptable scroll performance in long contexts for a while now. In this case, it takes me 12 seconds to see content when first opening the link, and about 18 to re-render content when scrolling.
    • buzzin_ 3mo ago
      So, the Jacobian Conjecture was done, via an counterexample. Can we now put the models to work on an even more difficult problem: Scrolling 100k of text, on a 128GB 24 core processor, smoothly in a browser?
  • 7734128 3mo ago
    I think that's partly because it's a fairly long conversation and partly because of all the LaTeX. There's enough formulas there to fill a small text book, and I doubt their renderer is all that efficient.
• iphonecorridor 3mo ago
  Never meet your heros.
• deleted 3mo ago
  [deleted]
• sandebert 3mo ago
  ChatGPT: "The determinant identity is almost embarrassingly simple once one writes the map in the right way."
• khernandezrt 3mo ago
  Damn. I should have stayed in school.
• riazrizvi 3mo ago
  Love this share! Thanks for helping ppl better understand what it really means when folks say "ChatGPT Discovered..."
  • sebzim4500 3mo ago
    This isn't the chat that made the actual discovery, this is just a chat explain it to Tao
• sjreese 3mo ago
  Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, then the Jacobian matrix does not allow for an evaluation of the stability. Yes—for a continuous-time autonomous system x ˙ =f(x),f(x ∗ )=0, this is the standard linearization criterion, with J=Df(x ∗ ): If every eigenvalue of J has strictly negative real part, then x ∗ is locally exponentially asymptotically stable. If at least one eigenvalue has strictly positive real part, then x ∗ is unstable. If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed. The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0: x ˙ =−x 3 , x ˙ =x 3 , x ˙ =0. Yet 0 is respectively asymptotically stable, unstable, and neutrally stable. A slightly more precise wording is therefore: If the spectral abscissa α(J)= λ∈σ(J) max Reλ is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive. This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.
• purple-leafy 3mo ago
  I’ve had a similar experience using LLMs to have mini personal breakthroughs. One thing I notice is many models say statements along the lines of “okay we have exhausted this thread it’s diminishing returns from here and we should stop and move on” It’s funny because I’ve been building a tiny neural network maze solver (23 bytes solves 92.75% of unseen 2D mazes) When I asked ChatGPT/Fable if we had anymore threads to pull to increase capability and decrease byte size, they both basically said no way - back when I was at ~166 byte models with a ~85% solve rate. Throughout the experiment I just kept trying different approaches and eventually had 3 mini “breakthroughs” in this particular niche. But if I had listened to the models… Anyway, these models are amazing to experiment with quickly, but they are dumb as hell and so absolute
• I_am_tiberius 3mo ago
  Are shared chats solely secured by a uuid?
  • khurs 3mo ago
    Yes.
• deleted 3mo ago
  [deleted]
• personalityson 3mo ago
  The last Fields Medal has been awarded.
  • gus_massa 3mo ago
    I expect Centaur Field Medals to be awarded. It's not clear how much steering Levent Alpöge did to get the result. In math it's very common to hide the search pattern and show only a simplified path to reach the final result.
• khurs 3mo ago
  Can someone ELI5 this? Is this a breakthrough of something or 'kinda interesting'?
  • gus_massa 3mo ago
    Main discussion: https://news.ycombinator.com/item?id=48998362 https://news.ycombinator.com/item?id=48998362 Probably you shoud read the fist few paragraphs of that post, but it escalates quickly. I'll cherry pick a few comment I like. I think they are worth reading but I'll quote a highlight of each one. From kingstnap https://news.ycombinator.com/item?id=49000867 https://news.ycombinator.com/item?id=49000867 > If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. [...] What was conjectured is that this local invertibility property everywhere would mean global invertibility. From mswphd https://news.ycombinator.com/item?id=48999959 https://news.ycombinator.com/item?id=48999959 > it doesn't overturn much. [...] 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. Can I quote myself? https://news.ycombinator.com/item?id=49007165 https://news.ycombinator.com/item?id=49007165 > From 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.] It looks like the new trick was to use a third variable to avoid the division.
    • khurs 3mo ago
      Thank you
• WithinReason 3mo ago
  All I can tell from this is that Terrence Tao has good mathematical intuition
• seamossfet 3mo ago
  The big take away for is the fact that the ONLY reason why chatgpt was able to get to this counterexample was because of the knowledge of the person driving the conversation. I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
  • vouaobrasil 3mo ago
    > I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier. The problem is that, what happens to human expertise as people start to use AI earlier and earlier in their careers, so that in 50 years? The problem is that Terry Tao spent decades as a mathematician before ever encoutering AI. Of course he and people his age will be able to drive AI somewhat sanely and use it to their advantage. But as more people grow up with AI, they will likely not reach levels like Terry Tao because their exposure to AI and the temptation to use it will certainly dull raw human intellect over time.
    • seamossfet 3mo ago
      I don't know if I agree with the premise that having access to AI results in dulling human intellect. I feel like to get to Terry's level you need a combination of passion and aptitude for the subject. People that don't want to learn about a topic will always look for shortcuts, which I think represents the vast majority of people. Terry Tao is quite exceptional, and I think exceptional people will still exist even when the "easy" button is bigger than it's ever been.
      • QwenGlazer9000 3mo ago
        The problem is that learning never stops. You can't just go through school, become a junior in X field, then start using AI. Then you'll forever be a junior. You have to make a choice when you're working a job: either use AI-first workflows to increase your productivity, or don't and increase your knowledge and skill. My wording is specific. You can use AI and increase knowledge and skill, but this requires you to be driving the AI at such a low level you don't get the full speedup. As an example, you can write code yourself with AI as an assistant, but it's not as fast as AI writing everything. So now we end up stuck in a situation where every professional needs to choose between long term skill growth or speed, as anyone who's worked a job before knows, speed will always be the one chosen.
• krelian 3mo ago
  How long has it been since we last saw a "LLMs can't really think/be useful/be better than a human expert" discussion on HN? There used to be so many!
  • john_strinlai 3mo ago
    probably like a minute or two? im pretty sure someone unironically said stochastic parrot on the HN post with the tweet announcing the counterexample, and ive read several comments with similar sentiments today (including in this thread)
  • epolanski 3mo ago
    People are insecure about their leetcode black belts and react slop not giving them cushy jobs anymore so keep missing the forest for the tree.
  • archagon 3mo ago
    [dead]
• sjreese 3mo ago
  import Mathlib noncomputable section open Matrix Function /-! # A counterexample to the Jacobian conjecture in dimension three We formalize the polynomial map F : ℂ³ → ℂ³ whose Jacobian determinant is the constant `-2`, but which is not injective. The final theorem `jacobianConjecture3_false` states the failure of the polynomial-inverse formulation of the Jacobian conjecture in dimension three. -/ namespace MvPolynomial variable {R : Type} {σ : Type} /-- The formal Jacobian matrix of a family of multivariate polynomials. -/ def jacobianMatrix [CommSemiring R] [DecidableEq σ] (F : σ → MvPolynomial σ R) : Matrix σ σ (MvPolynomial σ R) := Matrix.of fun i j ↦ pderiv j (F i) /-- The formal Jacobian determinant. -/ def jacobianDet [CommRing R] [Fintype σ] [DecidableEq σ] (F : σ → MvPolynomial σ R) : MvPolynomial σ R := (jacobianMatrix F).det /-- Evaluation of a polynomial map at a point. -/ def evalMap [CommSemiring R] (F : σ → MvPolynomial σ R) (p : σ → R) : σ → R := fun i ↦ eval p (F i) end MvPolynomial open MvPolynomial namespace JacobianCounterexample variable (K : Type) [Field K] /-- The three components of the polynomial counterexample. The variables `X 0`, `X 1`, `X 2` correspond respectively to `x`, `y`, `z`. -/ def F : Fin 3 → MvPolynomial (Fin 3) K := ![ (1 + X 0 X 1) ^ 3 * X 2 + X 1 ^ 2 * (1 + X 0 * X 1) * (C 4 + C 3 * (X 0 * X 1)), X 1 + C 3 * X 0 * (1 + X 0 * X 1) ^ 2 * X 2 + C 3 * X 0 * X 1 ^ 2 * (C 4 + C 3 * (X 0 * X 1)), C 2 * X 0 - C 3 * X 0 ^ 2 * X 1 - X 0 ^ 3 * X 2 ] /-- The formal Jacobian determinant of `F` is the constant polynomial `-2`. -/ theorem jacobianDet_F : jacobianDet (F K) = C (-2) := by simp only [ jacobianDet, jacobianMatrix, det_fin_three, of_apply, F, cons_val_zero, cons_val_one, cons_val_two, head_cons, tail_cons, map_add, map_sub, Derivation.map_one_eq_zero, pderiv_mul, pderiv_pow, pderiv_C, pderiv_X_self, pderiv_X_of_ne, ne_eq, Fin.reduceEq, not_false_eq_true ] simp only [map_neg, map_ofNat] ring variable {K} /-- The point `(0, 0, -1/4)` maps to `(-1/4, 0, 0)`. -/ theorem evalMap_F_p0 : evalMap (F K) ![0, 0, -(1 / 4)] = ![-(1 / 4), 0, 0] := by funext i fin_cases i <;> simp [evalMap, F] /-- Provided `2 ≠ 0`, the point `(1, -3/2, 13/2)` also maps to `(-1/4, 0, 0)`. -/ theorem evalMap_F_p1 (h2 : (2 : K) ≠ 0) : evalMap (F K) ![1, -(3 / 2), 13 / 2] = ![-(1 / 4), 0, 0] := by have h4 : (4 : K) ≠ 0 := (by norm_num : (2 : K) * 2 = 4) ▸ mul_ne_zero h2 h2 funext i fin_cases i <;> simp [evalMap, F] <;> field_simp [h4] <;> ring end JacobianCounterexample open JacobianCounterexample /-- The Jacobian determinant of the displayed map over `ℂ` is a unit. Indeed, it is the nonzero constant `-2`. -/ theorem F_jacobian_isUnit : IsUnit (jacobianDet (F ℂ)) := by rw [jacobianDet_F] exact (isUnit_iff_ne_zero.mpr (by norm_num : (-2 : ℂ) ≠ 0)).map C /-- The polynomial map `F : ℂ³ → ℂ³` is not injective. -/ theorem F_not_injective : ¬ Injective (evalMap (F ℂ)) := by intro hInjective have hp : (![0, 0, -(1 / 4)] : Fin 3 → ℂ) = ![1, -(3 / 2), 13 / 2] := hInjective ((evalMap_F_p0 (K := ℂ)).trans (evalMap_F_p1 (K := ℂ) (by norm_num)).symm) exact zero_ne_one (congrFun hp 0) /-- The injectivity consequence of the dimension-three Jacobian conjecture is false over `ℂ`. -/ theorem unitJacobian_does_not_imply_injective : ¬ ∀ P : Fin 3 → MvPolynomial (Fin 3) ℂ, IsUnit (jacobianDet P) → Injective (evalMap P) := by intro h exact F_not_injective (h (F ℂ) F_jacobian_isUnit) /-! We now formulate the polynomial-inverse version explicitly. -/ /-- Polynomial self-maps of affine three-space over `ℂ`. -/ abbrev PolyMap3 := Fin 3 → MvPolynomial (Fin 3) ℂ /-- A polynomial map has a polynomial two-sided inverse, viewed as functions on `ℂ³`. -/ def HasPolynomialInverse (P : PolyMap3) : Prop := ∃ Q : PolyMap3, LeftInverse (evalMap Q) (evalMap P) ∧ RightInverse (evalMap Q) (evalMap P) /-- The polynomial-inverse formulation of the Jacobian conjecture in dimension three. -/ def JacobianConjecture3 : Prop := ∀ P : PolyMap3, IsUnit (jacobianDet P) → HasPolynomialInverse P /-- The Jacobian conjecture in dimension three is false. -/ theorem jacobianConjecture3_false : ¬ JacobianConjecture3 := by intro hJC unfold JacobianConjecture3 at hJC apply unitJacobian_does_not_imply_injective intro P hP rcases hJC P hP with ⟨Q, hleft, _⟩ exact hleft.injective #print axioms jacobianDet_F #print axioms F_not_injective #print axioms jacobianConjecture3_false
• ghm2199 3mo ago
  Words and sentences to an LLM are like witchcraft. There are certain words, sentences that make LLMs go a certain way and do vastly better. Sometimes its not at all apparent what set of words will work to do what you want it to do. An example I have been using to do design at a high level is to say to claude. ``` A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things: - Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.) - Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.) - Load-bearing-ness — how much rests on the premise. - Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it. ``` There are a few things to note about this prompt 1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave. 2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more. But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
• eh_why_not 3mo ago
  What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times. Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results. What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
  • john_strinlai 3mo ago
    >Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore. we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on. even in math alone, i think its indisputably already stronger than Tao, considering it has approximately this much depth in ~all of the math subfields.
    • eh_why_not 3mo ago
      Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.
      • stkdump 3mo ago
        I strongly suspect that the horizontal and the vertical strength are not perpendicular. I suspect having a very wide knowledge gives you access to patterns that reproduce across fields that an expert would lack.
    • lpd59 3mo ago
      To be fair, Tao's specialty is not algebraic geometry.
• realo 3mo ago
  Now the question I would raise is this: could another AI have taken the role of Terrence Tao and obtain the same results from chatGPt? I do not think so.
• devy 3mo ago
  This is the original blog post where Terrence explained his thoughts where the ChatGPT conversation was originally referred from: https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
• ShipVoicedev 3mo ago
  Its fascinating to see how people use AI for advanced maths, is there any tool to simplify differential equations
• singularity2001 3mo ago
  Maybe Terrence is chasing too many red herrings. Maybe he should just have asked it to find a counter example to the conjecture and on success explain how it was found.
• adamiscool8 3mo ago
  Shows that — as always — the moat is domain fluency.
• rossant 3mo ago
  This is talking to an AI colleague and not to a chatbot. That's how you use frontier models or coding agents nowadays to do good work.
• deleted 3mo ago
  [deleted]
• prng2021 3mo ago
  What is this conversation even about? I can’t understand any of it.
• namuol 3mo ago
  Would Terrance Tao still be a renowned mathematician if he grew up with ChatGPT?
  • fooo1882992 3mo ago
    Would Kasparov still be a chess grandmaster if he grew up with Deep Blue? (Yes. Chess champions are stronger than ever, even though none of them have been able to beat a computer for decades.)
    • namuol 3mo ago
      I don’t think the comparison is fair. Deep Blue was an opponent in a highly bounded game. Mathematics has no bounds, and modern agents are specifically designed and trained NOT to be opponents.
• fenestella 3mo ago
  [flagged]
• tug2024 3mo ago
  [dead]
• deleted 3mo ago
  [deleted]
• buzzin_ 3mo ago
  Hey, Mr. Tao, where would you put the intelligence of the current models compared to top humans you surely mush have worked with? Perhaps given in the count of people you know that are higher than the newest LLMs? Also, how would you rate the speed of work compared to what your speed is, if this makes any sense?
  • GlacierFox 3mo ago
    Is he in the room with us?
    • buzzin_ 3mo ago
      He is known to read HN.
• gnabgib 3mo ago
  We had this yesterday (317 points, 133 comments)https://news.ycombinator.com/item?id=48998362 https://news.ycombinator.com/item?id=48998362 Classic Gabe-bot
• drsalt 3mo ago
  every time i see this name i think of Terrence Howard. it is quite confusing.
  • ronnier 3mo ago
    Funny that two of the greatest modern mathematicians share the same uncommon first name.
• marwaneet 3mo ago
  [dead]
• itomato 3mo ago
  "Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure." And a new system prompt stanza was fledged.
• ttul 3mo ago
  Now who wants to say ChatGPT 5.6 Sol is just another token predictor...?
• monktastic1 3mo ago
  Not sure how to suggest edits to titles, but @dang: his first name is Terence, not Terrence.
• purpleidea 3mo ago
  Imagine getting free tokens from OpenAI and the special flag on your account that tells it to "never nerf you" ? To be clear, I've had some great success using these tools, they're amazing, but it's also very obvious that (despite it being a stochastic tool) they're clearly changing things behind the scenes a lot. GPT-5.6 for example should not be constantly tweaked without telling us. Make it 5.6.1 or something, and if you're over capacity, just say so rather than returning lower quality results.
• noisy_boy 3mo ago
  Now I don't need to imagine how a conversation on maths between two highly intellectually advanced aliens would go like.
• CGP_Labs 3mo ago
  It's ironic that AI can solve problems once thought beyond human reach, yet it still depends on humans to ask the right question.
• danjc 3mo ago
  Seems important that the breakthrough conversation with Fable that led to the counter example should be shared too.
• SubiculumCode 3mo ago
  Is there a nice PDF of this conversation?
• SubiculumCode 3mo ago
  I want to read and share this not on chatgpt, but as a PDF or blog post
• birthdayn 3mo ago
  [dead]
• birthdayn 3mo ago
  [dead]
• LZ_Khan 3mo ago
  The biggest difference I see between the way I use LLMs and Terrence Tao's is that he's not constantly swearing at it.
• zkmon 3mo ago
  I'm not a mathematician. Could someone let me know, what's the objective of this conversation and what was the result?
  • renyicircle 3mo ago
    The background is that a counterexample to a famous mathematical conjecture was recently posted on X by an Anthropic employee. The counterexample, in form of a polynomial, is easy to validate by doing simple calculations, but it's not clear why it works and why it is the way it is, from a deeper mathematical perspective. Specifically, the properties of the polynomial suggest that it's not just an accidental finding but has some involved geometric construction behind it. This is what Terry Tao's conversation with ChatGPT starts with: "does this polynomial have any symmetry or structure that makes the counterexample seem less miraculous?". Using the counterexample, work backwards to get the intuition behind it so that it can be constructed by an explicit procedure rather than taken out of thin air. The result is his blog post, discussed here: https://news.ycombinator.com/item?id=48998362 https://news.ycombinator.com/item?id=48998362
• mvkel 3mo ago
  The most important prompt lesson everyone should take away: never ask yes / no / leading questions. Almost all of Tao's questions begin with what, or why. That forces an open ended response, which is a great way to reduce or eliminate sycophancy and severe hallucinations. The less you steer, the more accurate it gets.
• plc 3mo ago
  LLMs do match the level of sophistication of the user and the quality of questions they ask. So questions about, how do you improve the level of question? You see the answer in Terry's exchange, both AI and humans are doing an exchange that is thought provoking on both sides for what interesting questions to ask. That is the most startling thing about this -- they are exploring the solution space with the same naivete, to some degree. Physics, they describe general relativity as a two way street. Space tells mass how to move, and mass tells space how to bend. Here, Terence tells AI what question to explore, AI tells him what questions to ask next. The exchange that ensues is just magic to watch.
• sevenzero 3mo ago
  And yet ChatGPT wasnt able to find a solution for the 24 game with the numbers 1,4,7,6 just a few days ago.
• makaking 3mo ago
  Imagine the AI's of the next century trying to explain incomprehensible dark matter math to the smartest humans of the time. David Deutsch says the human mind can comprehend _anything_. I think the next decades of AI-driven math research might put that assumption to the test.
• reedf1 3mo ago
  "I’ve activated pro. Can you continue to look for a potential geometric explanation of the X_3 ~ A3 miracle that avoids coordinates or other unmotivated constructions ?"
• tarxzvf 3mo ago
  Thousands of words and piles of equations and not a single mistake pointed out? I have zero domain knowledge here, but enough interactions with AI agents to think this is wildly crazy.
• atmosx 3mo ago
  This has received many upvotes and most focus on the interaction between human and machine, which is by all means interesting. I wonder how many ppl can actually follow what's happening, I mean the math.
  • traes 3mo ago
    I would guess almost none, less than single digit percent. Only a small fraction of undergraduate math majors who happen to be interested in algebraic geometry would be able to understand this. The average HN user knows far less math than an average undergraduate math major. Of course there are also math PhDs and the like on here, but vanishingly few compared to the bulk of the site.
  • grougnax 3mo ago
    [dead]
  • djmips 3mo ago
    That may be true of a lot of posts on programming and other topics, nevertheless they are still interesting.
• sawrtagy 3mo ago
  [dead]
• yabdulaah52 3mo ago
  [dead]
• zq0829 3mo ago
  这不是“几个系数碰巧消掉了”，而是“给三次式标记一个单根”这一 incidence correspondence 的坐标展开：resultant =1 切掉因子间的相对缩放，所以局部 étale；忘掉所标记的根，所以一般三对一；而真正低维而罕见的奇迹，是相应切片竟然等于 A 3 。常 Jacobian 只是这套几何的坐标影子。
• blenderob 3mo ago
  This is fascinating. I'm having a very hard time wrapping my head round how a statistical model designed to predict tokens creates output that contains such deep reasoning and insight about complex ideas that we'd think only humans can understand! Yes, I know it's not just a dumb statistical model like a simple neural network. Yes, I know all about the vectors Q, K, V and how they contain long range contexts. But still! How does all of this emerge into something that can produce such insightful output? I still can't wrap my head around this. Am I dreaming? Am I living in a science fiction?
• MaybeGoodRoyal 3mo ago
  tao transcricpt, maybe cannon , 100000 taos or more clearly now in data centers, agi marker.
• tejohnso 3mo ago
  It's astonishing to me that a technical conversation at that level can be had with a machine that is performing computations to statistically determine what tokens it should use based on numerical weights and relationships. It just doesn't seem like it should be able to come up with details and insights like this, assisting one of the most intelligent humans on the planet to work through something so detailed.
• DigitResort 3mo ago
  [dead]
• Arshad-Talpur 3mo ago
  Absolute masterclass, this also reveals how we can pressure test LLMs especially when we know what exactly we want!
• youreallysuck 3mo ago
  [flagged]
• youreallysuck 3mo ago
  [flagged]