Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample • Hacker News Board

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#541

Post by mark_something » Thu, Jul 23, 2026, 9:33 AM UTC

daynthelife wrote: post #508

WarmWash wrote: post #224

Earlier quoted context omitted.

> https://en.wikipedia.org/wiki/Transmission_Control_Protocol compare to > https://en.wikipedia.org/wiki/Rees_algebra Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it. Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. A…

As mathematician, I actually agree. We could do a lot more to communicate ideas in a way non-experts can understand. For instance, most people are familiar with polynomials. Take a polynomial in x (with integer coefficients) and substitute x for 5t everywhere. So for instance, 2x^3 - 8x + 3 becomes 250t^3 - 40t + 3. The Rees algebra Z[5t] (here Z is the integers) is then just the set of all polynomials you can get th…

As a hobby mathematician, I agree. I know quite a bit about rings and ideals, but I don't understand much from the article on Rees algebras. Wikipedia has a lot of articles on mathematics that are explained badly, I prefer to find information elsewhere, typically I ask an AI for a good introduction on a subject (and usually that links to PDFs with over 100 pages that keep me going for hours).

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#542

Post by gamerDude » Thu, Jul 23, 2026, 9:34 AM UTC

rkuodys wrote: post #531

larsfaye wrote: post #436

Earlier quoted context omitted.

Correctamundo. That's EXACTLY what I just wrote about. And how we'll create the next generation of people who know how to evaluate responses when we're creating a generation of people who are increasingly reliant on LLMs to do the work makes for a strange paradox. https://larsfaye.com/articles/ai-coding-will-prevent-experti...

I am not sure if I'm ignorant or have too much self esteem, but in cases where domain is not my expertise I try to cross challenge using different models to get less-biased and hopefully more correct insights.

This is some work that I want to see in education. How do you validate if something is true or not.

A lot of times that will take physical tests. Or in the case of math/logic, tests to validate each line or validated sources of previously proved theorems

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#543

Post by tempfile » Thu, Jul 23, 2026, 9:34 AM UTC

eru wrote: post #459

timacles wrote: post #406

Earlier quoted context omitted.

its just as likely hallucinations will only get worse because their source data will be riddled with hallucinations

You can't hallucinate a working lean proof.

You absolutely can. How do you know your "working lean proof" actually proves the theorem you intended it to?

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#544

Post by gf000 » Thu, Jul 23, 2026, 9:43 AM UTC

eru wrote: post #450

Normal_gaussian wrote: post #366

Earlier quoted context omitted.

Commands like /goal and similar are the more complex version of this; you write a prompt like it was a singular iteration, it runs one iteration, then runs an "evaluator" to determine if the goal has been reached, then runs the prompt again with a little extra to make it go again - and so on. The evaluator is just an LLM with most of the result or context looking at the original goal and the state and answering the q…

For mathematical research, you can just run until you have a computer checkable Lean proof.

Given that it was formalized correctly, which is far from trivial in many cases

(Of course LLMs can help there, get it right etc, just a caveat that people have to keep in mind)

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#545

Post by armchairhacker » Thu, Jul 23, 2026, 9:45 AM UTC

metanonsense wrote: post #315

ecshafer wrote: post #31

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…

It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.

A lot of “advanced” concepts are simple once you find the right way to frame them.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#546

Post by aleph_minus_one » Thu, Jul 23, 2026, 9:58 AM UTC

reverius42 wrote: post #469

aleph_minus_one wrote: post #417

Earlier quoted context omitted.

Rings: * Determinant calculation: - The Samuelson–Berkowitz algorithm is best understood in terms of general rings - The Faddeev–LeVerrier algorithm and determinant calculation using Gaussian elimination work on rings with specific properties (for the Faddeev–LeVerrier algorithm the restriction is on the characteristic of the ring, for Gaussian elimination the ring must be an integral domain (ideally a field)). * Rin…

> The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name tropical is a reference to the Hungarian-born computer scientist Imre Simon, so named because he lived and worked in Brazil.[1] I'm convinced half the reason people find CS terminology more accessible and Math terminology less so, is that CS terminology tends to be named after stuff, and Math t…

> I'm convinced half the reason people find CS terminology more accessible and Math terminology less so, is that CS terminology tends to be named after stuff, and Math terminology tends to be named after people, and ... sometimes whether the place they lived is a tropical place.

In my opinion: a lot of math terminology is much older than computer science terminology, so the origin of the names of many concepts in math is much more obscure for today's people than CS terminology currently is (and least if you are not into history of science/math).

On the other hand, in my observation a lot more terms in computer science are based on obscure (often pop-cultural) puns. I guess in 50-100 years these CS terminology might seem even more obscure for then-contemporary people than math terminology is today.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#547

Post by mkagenius » Thu, Jul 23, 2026, 10:11 AM UTC

tempfile wrote: post #537

mkagenius wrote: post #491

Earlier quoted context omitted.

They are solving jacobian conjectures, I don't think they will hallucinate till masters level of any subject. Edit: do give counter examples if you have any in maths, physics, chemistry, biology etc

This deduction is baseless, there is no reason to think an LLM will only hallucinate on complex topics. They still get the "number of es in seventeen" question wrong regularly. The kind of mistakes an LLM makes has no clear resemblance to the kind of mistakes a human makes, because they are not doing the same thing.

Specific quirk of llms, also pretty much solved by existing thinking models.

I think you guys are misunderstanding me, you can still talk to a llm to learn everything about physics or maths.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#548

Post by buzzin_ » Thu, Jul 23, 2026, 10:15 AM UTC

singularity2001 wrote: post #250

HNisCIS wrote: post #190

Earlier quoted context omitted.

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 wi…

they're still bounded by training data and how it relates to the real world. Yes and no. They can extrapolate and build upon the training data, as was the case with the last dozens of math proofs

Of course they can't just talk to themselves and improve the knowledge of the world. They are just stochastics parrots that give you an average answer.

LLMs giving you something novel would be like if you let a model play chess against itself and become the best player in the world this way. Totally impossible.

/sacrasm

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#549

Post by aleph_minus_one » Thu, Jul 23, 2026, 10:15 AM UTC

versteegen wrote: post #409

aleph_minus_one wrote: post #336

Earlier quoted context omitted.

> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.

Having studied CS and maths to post-grad, I think you exaggerate. Although a CS course might use these tools, they didn't in my experience go into explaining or defining them. The only use of linear algebra I can remember was in analysis of recurrence relations for algorithms, and for some graph theory. And I had one CS course on multivariate generating functions (formal variables) but most CS students would have bee…

> Having studied CS and maths to post-grad, I think you exaggerate. Although a CS course might use these tools, they didn't in my experience go into explaining or defining them.

I studied computer science (and mathematics) in Germany. I am very certain that this was taught to computer science students, even though (compared to the lectures for math students) the lecturer did not get very deep into these topics.

> most CS students would have been terrified of that.

This is a feature, not a bug. :-)

Seriously: In Germany, the "math for ..." lectures often are intended to be "weed-out lectures" so that students who simply are not qualified for their major get to quit their degree course fast (either by realizing that the degree course is too hard for them, or by (typically) failing math exams so that they get exmatriculated), so that they don't waste many semesters on a degree course which they simply are not suited for.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#550

Post by Kon5ole » Thu, Jul 23, 2026, 10:19 AM UTC

larsfaye wrote: post #436

michaelchisari wrote: post #12

Earlier quoted context omitted.

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.

Correctamundo. That's EXACTLY what I just wrote about. And how we'll create the next generation of people who know how to evaluate responses when we're creating a generation of people who are increasingly reliant on LLMs to do the work makes for a strange paradox. https://larsfaye.com/articles/ai-coding-will-prevent-experti...

Good writeup, and certainly food for thought.

I wonder if this key point actually holds though:

>The skills to do so, however, are a function of someone who has experienced the friction and challenges over time that culminate in "good taste".

It's certainly sometimes true, but I don't think it's a general rule. Sometimes friction is just friction and sometimes you spend 1000 hours learning something that disappears and becomes obsolete or at least irrelevant to the goal.

Programmers used to need to know the instruction set of the CPU, assembly language and so on. Some still do but for most developers today that's not useful. Everything you know about 6800 assembly will not make your note-taking app any better.

I think we are in a state where AI tools so easily mimic what we used to do by hand that we think the friction is gone, but that's because we haven't raised the bar yet. One day we'll look at the Fable one-shot that's better than anything we ever made by hand ourselves and say "It could be even better".

And then the friction is back, just on a new level.