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Posted by gmays 4 hours ago

Terrence Tao's ChatGPT Conversation about the Jacobian Conjecture Counterexample(chatgpt.com)
371 points | 190 commentspage 2
Jun8 3 hours 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.

tetha 3 hours ago||
I think the "But this is not intelligence because it is known math" is not a correct argument. It is unknown how the overall higher intelligence of humans works.

What I do notice however is that LLMs are becoming capable of doing an increasing part of the intellectual work I can do, and usually a lot faster.

Just today I presented an agent framework that can take an informal incident statement and propose infrastructure changes to fix it, all evidence backed. This did nothing I could not to, but it did all 5 test cases in 6 - 12 minutes each. I would have found all of the monitoring indications it did, but it would have taken me a day per test case. The LLM also included sass to silly tickets. ("This is not even worth spending monitoring resources on. It's obviously a configuration problem.")

That's how this is reading to me as well. It's just fast at slogging through a certain level of "simple" transformations.

the8472 3 hours ago|||
That argument says very little, emergent behavior is a thing in complex systems with billions of parts. Humans can also be reduced to voltage potentials propagating along of tubes of fat and synapses getting rewired.
umanwizard 2 hours ago|||
What does "predicting the next token" mean? I ask this every time people say "LLMs are just predicting the next token" and it's maddening that nobody can give a straight answer. Predicting it according to what probability distribution? Every process that produces a sequence of actions (including e.g. a human writing) can be modeled by some probability distribution and therefore their actions are indistinguishable from "predicting the next token" emitted by that distribution.
the8472 1 hour ago||
Yeah that's pretty much what gwern argues here[0]. Or to adapt another proverb: to predict the next token you first need to model the universe.

[0] https://gwern.net/scaling-hypothesis#gwern-difference--effic...

umanwizard 40 minutes ago||
> to predict the next token you first need to model the universe

Exactly. The "most likely next" series of tokens, for example, when given the first half of a correct mathematical proof, is the correct rest of the proof. I have never seen anyone define "most likely next token" in such a way that this isn't true.

IshKebab 3 hours 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 3 hours 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).
Scarblac 1 hour ago|||
At any rate, if the AI's side in this conversation were a human, that would be an extremely intelligent human indeed.

But there's no way the thinking times would have been that short, of course.

contextfree 3 hours 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 3 hours 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.
nozzlegear 3 hours ago||||
That's not clear at all. What's clear is that this is a very smart man who knows how to use this tool well.
squidbeak 3 hours ago||
I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.
nozzlegear 20 minutes ago||
I think we have wildly different conclusions about what happened here. You see the machine as instructing Terrence Tao, as if it were Plato teaching Socrates about the theory of forms; I see Terrence Tao using the machine to teach himself, like an intelligent student uses a book. In this case, it's just a book that fools us into believing it can think and reason like we do, because it generates language in much the same way we do when we think and reason.
thechao 3 hours ago||||
I'm no intelligence researcher or philosopher; but, I think LLMs make us confront the (IMO, now clear) distinction between cleverness (intuition), reasoning (rational argument), and consciousness. I suspect that we think of "intelligence" as either of the first two welded to the latter. In that vein, I'd say that consciousness may be just another emotion: happiness, sadness, egoness.
JCattheATM 2 hours ago||
> consciousness may be just another emotion: happiness, sadness, egoness.

It's clearly much more than that.

superloika 3 hours ago||||
There is no intelligence. If anything, this just shows that natural language and mathematics are both fields which are structured in a logically computable way. And if you have a machine that can compute symbolic logic, you can process both natural language and mathematics.

A second corollary is that rational consciousness and thought is less likely to be contained in language than previously thought, because if language is so simple that a machine can process it, it can't contain consciousness.

ben_w 2 hours ago||
If natural language was structured in a logically computable way, we'd have had interesting chatbots by the late 80s, basically as soon as a dictionary fit in local RAM, and for the same reason we got compilers.

Da hole raisin y nat-lang be v. hard is dat i kan rite lik dis an it be cool 4 native engrish speekrs 2 unerstand. LLMs are of course fine with this sentence in exactly the way that Zork's engine couldn't be.

superloika 2 hours ago||
The underlying structure of language, which is grammar, is obviously logical. That the symbols used to represent this grammar can be sometimes fuzzy or ambiguous, is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
ben_w 1 hour ago|||
It's not "obviously logical", it's a pattern which we mimic to avoid mockery.

example For, semi-randomise I word order can this like, Yoda worse than, and be understood.

> is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.

We had to invent Transformers to be able to do that with reliability anything close to being worth caring about. Transformers have to learn from examples, not be pre-programmed.

EnergyAmy 1 hour ago|||
The idea that grammar is all it takes to process natural language is absolute beans.
Diogenesian 3 hours ago|||
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IshKebab 22 minutes ago|||
> There is about 150 years of cognitive science experimentation in animals

Yeah that would be relevant if AI were an animal...

As I said, it's clearly intelligent, but a quite different intelligence to that shared by animals.

anthonypasq 2 hours ago|||
> If AI researchers cared about scientific thinking, they would be intensely focused on the brains of bees.

Basically every academic AI researcher in history was doing what you described. The AI industrialists stopped caring 6 years ago once they realized LLMs seem to have been the only thing in 80 years that actually seems to work at any useful level.

There are plenty of pioneering scientists who are either returning to actual AI research (Yann Lecun, Ilya, etc), and plenty who never left (Richard Sutton) who are doing exactly what you are talking about.

Diogenesian 2 hours ago||
> Basically every academic AI researcher in history was doing what you described.

That is not true. Alan Turing did not view things that way, his test would say that a dog has zero intelligence. Neither did any of the MIT Lispers. And neither do Lecun or Sutskever or Sutton! They are all focused on human intelligence. None of them are even slightly concerned about an AI which is intelligent before it learns any language.

> the only thing in 80 years that actually seems to work at any useful level

This isn't true either! Mathematica / Maple / etc are "old-fashioned AI" and they obviously work. The Lisp expert systems were also useful, though less so than an LLM.

anthonypasq 1 hour ago||
> And neither do Lecun or Sutskever or Sutton! They are all focused on human intelligence. None of them are even slightly concerned about an AI which is intelligent before it learns any language.

??? https://www.youtube.com/watch?v=GvibIstOn_E his arguemtn here is clearly built around using some sort of sensory data to build a model of the world like humans (animals) do. also you clearly decline to mention Lecun who has made this point ad-infinitum

> This isn't true either! Mathematica / Maple / etc are "old-fashioned AI" and they obviously work. The Lisp expert systems were also useful, though less so than an LLM.

i personally find it very strange that non-deep learning AI approaches which essentially boiled down to a giant bundle of if statements, or some very simple statistical modeling were called AI in the first place.

peheje 1 hour ago||
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minimaxir 3 hours 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 3 hours 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.
doctoboggan 3 hours 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 3 hours 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.
ChaitanyaSai 3 hours 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.

brotchie 3 hours ago||
GPT 5.6 already is an explainer agent.

Fork Tao’s convo and prompt this (with your own math level described).

GPT did a great job of translating Tao’s questions and concepts (e.g. “pre image”) into a progression I could understand.

“Ok I have a PhD in financial math and undergrad in engineering math. I have almost zero knowledge of polynomial algebra / geometry, I know what a polynomial is and what roots are but not much beyond that. Could you try and explain to my level what questions the user I the conversation has asked and what the agent has responded with, we can probably go user query by user query to build up”

ValentineC 1 hour ago|||
One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.
epolanski 3 minutes ago||
They lack the empathy to understand where and why you're struggling.

I've given private math lessons and seen students struggle with ai, even though ai gave the right answers.

My intuition is that humans spot xy problems easier when teaching (user ask x but really needs y), whereas llms will oblige writing about x.

thechao 3 hours ago|||
I want to be able to take a conversation and ask for subconversations as red pen annotations "on the side". The linear nature of the context tends to frustrate this.
charcircuit 2 hours ago||
There are already browser extensions similar to this.
colordrops 3 hours 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 3 hours 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.
purple-leafy 2 hours 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

krelian 1 hour 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!
epolanski 1 minute 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.
john_strinlai 1 hour 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)
6thbit 3 hours 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?

qrian 3 hours ago||
Parent HN discussion: https://news.ycombinator.com/item?id=48998362
seamossfet 1 hour 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.

furyofantares 1 hour ago||
Maybe true, I'm not sure, but this isn't the conversation where the counterexample was found.
vouaobrasil 1 hour 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 1 hour 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.

WithinReason 1 hour ago|
All I can tell from this is that Terrence Tao has good mathematical intuition
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