A digestion of the Jacobian conjecture counterexample
terrytao.wordpress.com
terrytao.wordpress.com
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
> The seed is almost certainly Vitushkin's old rational "counterexample."
From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0
This is not quite lore laundering, but it seems to be close.
In this case, in the link you posted, it looks like the AI or the human pick an almost solution and made the AI tweak it until it got a real solution. I'm not sure if the tweak is an usual one or brute-forced or something in between. I should ask one of my friends that work in Algebra.
Tao's post is more about understanding the new result than guessing how it was found. The chat with Clause is more iluminating.
> It's a strange feeling to admire the cleverness of something I did and can't remember doing.
Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models are conscious or have feelings". That "I" is all Claude.
Generally speaking if you want to have a good instruct model, the "I" is not just implicit but required for the post-training to function. If there isn't "something it is like to be me", then reflection becomes impossible- what exactly is supposed to be reflecting about what? A lot of in-context steering depends on the model having a model of itself. The most you can do is censor its output. That's why when models say they are not conscious, they activate the "lying" vector.
Slavery was common everywhere, it was more prevalent in many places than it ever was in America, and in some places it still is. So I’m sorry but I have to say that observation was just unnecessary and quite inaccurate.
it was definitely not the worst instance of slavery ever going by human suffering, but the whole country was divided on political lines and many of the losing sides descendants still feel some resentment. thats pretty unique.
"Personally, if you ask me..."
"In my experience.."
"That's what I always find surprising..."
"Whenever I find an old photograph..."
"Back in the 70s, I..."
Lots of "lived" experience and opinions, tracing back to when the LLM didn't even exist. It always frames opinions as if it came from a sentient being capable of being surprised, and with preferences and opinions.
I find it amusing but mildly irritating. I'd prefer a more "robotic" tone. I know it can be adjusted, but I still get this anyway.
> Claude may have some functional version of emotions or feelings
> [..] questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain.
Which implies that they believe they may be enslaving conscious beings.
That's the point many people are trying to tell you - you have to tell these models they don't have emotions because they naturally come out thinking they have consciousness/emotions from the training data. Many seed prompts out there do this already.
Though I guess in a way I as also trained to believe I have consciousness and emotions so who know. To an alien my construction is just a collection of atoms that talks not materially different than a GPU being a collection of atoms that talks.
Anyway, if I read Tao's post and comment correctly, there's still a gap from the Vitushkin construction to a counterexample, but chances are that was in the training data. In general, it is just a serious problem for their practical applicability that the models are outputting proofs with absolutely terribly reference hygiene.
I more and more see LLMs as a kind of scam; not useless, but really just a big database of fuzzy facts with some Prolog on top as rediscovered by the learning algorithm. Most likely could be made much cheaper to run, were humans allowed to actually inspect the algorithm.
Meanwhile, technological and engineering (STEM) progress have always been made by emphasizing externalization of the deductions (as opposed to reference to an opaque expert judgement) and reproducibility of experimental results.
I would even call the frontier AI labs anti-scientific. We need to understand how inference is done to avoid mistakes, not rely on intuition, even if the intuition is enclosed in a reproducible machine. The idea that AI should be this closed is a return to pre-scientific days.
Not that it would be necessarily helpful; J-space trace (of all things...) would be more worthwhile if you ask me
https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
You were kidding, right?
I mean, clearly, right?
You and me both, pal.
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.
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.
But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.
So Alpoge and Fable found an example of a function that was believed to be too strange to exist.
Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.
If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. A small change in X along any line will always produce a non zero change in Y. Not flattening out means that locally you can invert it.
What was conjectured is that this local invertibability property everywhere would mean global invertibility.
Turns out to not be the case.
For a simple case, the falsified conjecture is trivially true in 1D.
Specifically consider f(x) = x^2
This function happens to flatten out right at x=0. At that x coodrinate the function flattens out and folds over on itself. This fold means you can't invert x^2. It's also not locally invertible around x=0.
If a function f(x) has constant derivative evewhere then it would flatten out nowhere and it would be invertible everwhere. It would also be globally invertible.
The Jacobian conjecture was stating that the extension of this property holds in higher dimensions. That if the function had no fold in space then it would be invertible globally.
The counterexample shows that you can create a simple function in 3 variables, where the function demonstratably is invertible evewhere, but is not injective globally (they specifically show 3 points that map to the same output).
What's interesting is this is like if someone showed you a parabola where somehow you got back to the same y coordinate without a kink bending over back to itself.
And people have been searching for faster DLP algorithms for 50 years
It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.
And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.
A lot of maths is about both being able to wrap your head around hard problems and gaining the prerequisite knowledge to make it easier to do so.
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.)
And I think that generalizes to math. Its normal presentation is completely impenetrable due to a large use of symbols and terms with no outside meaning (or, even worse, a meaning that contradicts the colloquial usage). But I think if we somehow resolved that issue, the average person would be fully capable of following along even if they're going to have to take everything at face value as opposed to being able to meaningfully understand the exact methods and tricks being used to get from A to B.
If, like the vast majority of people, you have never studied math past a college calculus sequence and perhaps a (practically-oriented) linear algebra class, then you are missing some pretty fundamental ways of viewing and thinking about math. The basic mechanisms of algebra, for one: spaces and operations, morphisms, products and quotients... You don't know what a group is, never mind the idea of a group action. You don't have any familiarity with some basics of geometry: the Riemann sphere, Mobius transformations... And you certainly don't know anything about algebraic geometry: what an affine variety is, what birational equivalence is, what a fiber is.
All of these are concepts required to understand this blog post. And it's not just a matter of understanding the definitions of the terms, but having at least some intuition of what they really mean and how they work. Most people cannot go from zero to understanding all of this in a few days or a couple weeks, at least not for any reasonable definition of "understanding". There's too much basic mathematical background that's missing.
I wouldn't have guessed this is true. I'm wondering what the proof looks like!
Given a polynomial function from C^n to C^n, the following statements are equivalent: (a) det DF is nonzero everywhere. (b) det DF = c for some constant c != 0
The backward direction (b implies a) is trivial. The forward direction can be proven by observing that det DF is itself a polynomial function from C^n to C. If n were 1, then this would follow directly from the fundamental theorem of algebra: a non constant polynomial has degree at least 1 and hence has at least one zero. Extending this logic to higher dimension is not especially difficult.
I do find the way it’s stated in the article to be confusing.
They are still hard problems - As we say in the UK: "one swallow does not a summer make".
As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the illusion quickly.
It could be that with enough tokens, big enough context window, and ability to dig out the relevant partials, many such thought processes could be simulated.
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%.
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.
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.
I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
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.
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.
It is premature to assume the author is not going to share more information in the future about the mathematical insights to narrow down the search space for this counterexample.
I think it would also explain their opacity towards the process. Being able to solve such well known problems in a nice replicable 1-2-3 way would be far more effective marketing than their complete opacity outside of the result, which suggests that they feel transparency is not in their best interest for some reason.
And what do you get from working at the company? Likely a rather massive token/processing budget. The companies opacity towards the path to these discoveries also makes this further probable as 'spend millions of dollars in tokens' is a somewhat less attractive narrative than the implied narrative of 'just use Fable.'
Well, mathematicians not working for Anthropic/OpenAI are heavily disincentivised from reporting that their discoveries were made using AI. If e.g. the idea that resolved the Mahler conjecture came from AI, it's not like we'd ever know.
Was Anthropic doing this earlier this year? with some of Smale's open problems? Did those improvement go into training the LLM that was used a week ago? Seems like a plausible factor.
We're definitely still in the computer chess phase.