The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow:
> Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism {X \cong {\bf C}^3} and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian.
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.
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.
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.
> While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} coefficients, which is much larger than the {\binom{7+3}{3} = 120} degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.
Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.
I was reading another source that claimed this example was inspired by an existing (rational polynomial) example from the literature (created in 1999 by a Russian mathematician Vitushkin).
> The seed is almost certainly Vitushkin's old rational "counterexample."
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.
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?
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.
I am sure he did the minimum effort needed to communicate what he wanted to communicate.
If you are offended by his math gifs and feel that the widely regarded best mathematician of our time should use embedded LaTex or something better, why not offer to upgrade his blog?
Why are you so upset about people oohing and aahing? These may not be the fireworks you like, but don't yuck their yum. We should do more praising of each other for doing work.
You're not gonna believe what Paul Graham's blog or the discussion board related to it looks like.. straight out of the 90s. Forget LaTeX it doesn't even support images!
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.
26 comments
[ 4.6 ms ] story [ 53.7 ms ] threadhttps://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
You were kidding, right?
I mean, clearly, right?
You and me both, pal.
Claude Fable produced a counterexample to the Jacobian Conjecture
https://news.ycombinator.com/item?id=48973869
Human mathematicians are being outcounterexampled
https://news.ycombinator.com/item?id=48983382
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.
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.
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.
If you are offended by his math gifs and feel that the widely regarded best mathematician of our time should use embedded LaTex or something better, why not offer to upgrade his blog?
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
I wouldn't have guessed this is true. I'm wondering what the proof looks like!