Join the discussion
Write your take first — we'll ask for email only when you're ready to publish.
- Hacker News
- Can we audit the CoT and work the AI did to generate such a remarkable cancellation?
- by netvarun
- I 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 tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
by castedo - What’s a chance the counterexample was in the training?by dumpstate
- Extremely 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.
by TeriyakiBomb - I'd wager close to 0%by zeusdclxvi
- Close 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.
- The 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%.
by imtringued - From a comment by j2kun 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.
by gus_massa - Related:
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
- Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.by jmward01
- "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!
by gerdesj - > 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!
- I’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.
by amluto - reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.by zzzeek
- The difference is your dog will never understand the Python code but you could probably understand this post in a matter of days or weeks if you really wanted to. Can we all please stop acting like this Terry guy is so special?
- Some 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.
by tgrowazay - Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?by hyperhello
- There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...
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.
by QuesnayJr - No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.by sfpotter
- Not much.
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.
- It overturns the Jacobian conjecture (i.e., speculation) for dim >= 3, which we now know was an overgeneralization. Tao characterizes it as "can be viewed as an assertion that local invertibility implies global invertibility". It was already widely suspected to be false. Assuming that it was true was never warranted, so this really doesn't change anything. The significance is that an AI was able to find a relatively simple counterexample. Its "chain of thought" would be very interesting to see.by jibal
- For the Jacobian determinant to be constant is a massive coincidence, in general it is some complicated and messy polynomial. The conjecture was that this coincidence couldn't happen, except for simple special cases.
So Alpoge and Fable found an example of a function that was believed to be too strange to exist.
by impendia - it doesn't overturn much. For example, here is a post from 2004
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.
by mswphd - I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation:
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.
by kingstnap - After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?by aayushdutt
- Not really. I've found that they often believe that they understand the code. They obviously don't. But they do feel like they do.by clarionbell
- I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise: “That’s exactly the right way to think about it.”,
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
by sashank_1509 - Huh? I went through the first ten messages in his thread and there's no praise there at all.by square_usual
- If there's anyone that deserves praise every time he has something to say about mathematics, it's him.by welferkj