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

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

#221

Post by Jtarii » Wed, Jul 22, 2026, 9:07 PM UTC

minraws wrote: post #206

Jtarii wrote: post #194

Earlier quoted context omitted.

None of what you listed is even 1% as intense as the mathematics in the link. Learning anything in maths requires weeks of hard effort, learning enough to be broadly comfortable in how an 8086 CPU works can be done in a weekend.

Again, you are underestimating how much effort it takes to understand how an 8086 CPU works. There are a lot of foundational concepts that you are simply assuming the person already understands. That may be a reasonable assumption for a CS undergraduate, but the average person does not understand binary arithmetic, registers, memory addressing, instruction execution, calling conventions, or even what a CPU is doing a…

> There is an enormous amount of computer science that most programmers are completely unfamiliar with, especially within academic CS: programming-language theory, type theory, formal semantics, compiler theory, algorithmic research, complexity theory, distributed computing theory, verification, cryptography, computational geometry, numerical methods, and so on. Being proficient in one narrow area does not automatically give you the prerequisites for another.

And the difficult part of all those areas of computing is the mathematics part. Which I think is what I am arguing, mathematics is a fundamentally different type of "difficult" to any other subject.

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

#222

Post by epolanski » Wed, Jul 22, 2026, 9:07 PM UTC

WarmWash wrote: post #30

Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane. Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same…

Mathematicians emphasize definitions not labels/names.

It doesn't matter if natural numbers include 0 or not, what matters is how you define them, not how you call them.

This makes them also bad at naming things because...there's a definition anyway.

Most other fields do not have or can't have the same luxury, so naming might be more thoughtful.

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

#223

Post by B-Con » Wed, Jul 22, 2026, 9:07 PM UTC

Jtarii wrote: post #194

minraws wrote: post #151

Earlier quoted context omitted.

This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane. Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, dea…

None of what you listed is even 1% as intense as the mathematics in the link. Learning anything in maths requires weeks of hard effort, learning enough to be broadly comfortable in how an 8086 CPU works can be done in a weekend.

I think one of the key differences is that math is abstract whereas CS is relatively concrete.

CS examples are often easy to picture and understand the motivation for. You can use tools to visualize or play around with them and test them.

Math gets abstract so fast you have to spend a week of research to even understand the problem statement. The the motivations themselves can be completely unclear until you have a lot of context.

I majored in math (B.S.) and upper level math is completely foreign to me.

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

#224

Post by WarmWash » Wed, Jul 22, 2026, 9:08 PM UTC

minraws wrote: post #151

WarmWash wrote: post #30

Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane. Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same…

This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane. Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, dea…

>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. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.

It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.

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

#225

Post by minraws » Wed, Jul 22, 2026, 9:09 PM UTC

Jtarii wrote: post #213

ghshephard wrote: post #204

Earlier quoted context omitted.

"tcp" can take roughly 3-4 weeks of heads-down dedicated study to have some reasonable familiarity with. Same is true with most of the other concepts. Being able to speak with expertise on that list of topics is 3-4 years of really focused study and work. I think what happens is that people often have passing familiarity with a word or topic and presume knowledge, and years (decades) later they realize they knew almo…

You could get a surface level understanding of TCP, or 99% of topic areas in computing, in less than 4 weeks of study. You could not get a surface level understanding of literally any of the maths in the link in less than 4 weeks of study. The vast majority of what computers do just isn't that complex. I'm not saying it isn't "complex" just that any reasonably smart person can understand how a computer works and stil…

This is not true, 99% is very exaggerated claim, but yeah you can learn 50-60% of the field at a surface level in months rather than years.

But you can have a surface level understanding of mathematical topics as well, ofc some topics might require deeper understanding, but that's true for both.

Any claims of being able to learn 99% of computing in a just 4 weeks even at surface level, is greatly underestimating your own knowledge built over the years perhaps, or perhaps underestimating your own ignorance.

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

#226

Post by Razengan » Wed, Jul 22, 2026, 9:12 PM UTC

minraws wrote: post #151

WarmWash wrote: post #30

Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane. Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same…

This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane. Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, dea…

old.reddit.com/r/vxjunkies

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

#227

Post by umvi » Wed, Jul 22, 2026, 9:13 PM UTC

minraws wrote: post #151

WarmWash wrote: post #30

Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane. Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same…

This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane. Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, dea…

No, the problem with mathematics is that it is basically its own language separate from your native tongue. You have to learn dozens of symbols and greek letters and such and memorize what their meaning is in the context of mathematics in order to "follow" a mathematical conversation.

Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature. But on the flip side, mathematics being its own language means that a mathematician from any country can read and understand mathematics from a different country without needing to translate words such as "sum" and "infinity"

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

#228

Post by alpinisme » Wed, Jul 22, 2026, 9:15 PM UTC

minraws wrote: post #225

Jtarii wrote: post #213

Earlier quoted context omitted.

You could get a surface level understanding of TCP, or 99% of topic areas in computing, in less than 4 weeks of study. You could not get a surface level understanding of literally any of the maths in the link in less than 4 weeks of study. The vast majority of what computers do just isn't that complex. I'm not saying it isn't "complex" just that any reasonably smart person can understand how a computer works and stil…

This is not true, 99% is very exaggerated claim, but yeah you can learn 50-60% of the field at a surface level in months rather than years. But you can have a surface level understanding of mathematical topics as well, ofc some topics might require deeper understanding, but that's true for both. Any claims of being able to learn 99% of computing in a just 4 weeks even at surface level, is greatly underestimating your…

The claim isn’t knowing 99% of the field but that 99% of topics are ones you could do a quick-and-dirty crash course and come out with some understanding.

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

#229

Post by traes » Wed, Jul 22, 2026, 9:15 PM UTC

hyperhello wrote: post #41

sdwr wrote: post #15

Earlier quoted context omitted.

It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.

To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.

This is not true at all. The parameter space is absolutely MASSIVE. The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There's no particular way to bound these coefficients or even the degree or number of variables apriori, but assume you somehow did. Also assume you were confident that it would work with integer coefficients bounded from -12 to 12. Now you have to iterate over 360 degrees of freedom, verify that the Jacobian is a nonzero constant, and somehow show uninvertibility of the transformation, which is not a particularly simple task.

If you searched for coefficients from -12 to 12, this would be 25^360 = 2 * 10^503 different possibilities. A common reference point is that there are 10^80 atoms in the observable universe. Sure you could probably reduce this a bit with clever tricks, but the starting point makes the method completely unviable, even with the knowledge: A) a counterexample exists, B) it's in 3 variables, C) it's in degree 7 or less, D) it's in integer coefficients, E) those coefficients are 12 or lower.

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

#230

Post by overfeed » Wed, Jul 22, 2026, 9:15 PM UTC

breezybottom wrote: post #138

overfeed wrote: post #120

Earlier quoted context omitted.

Math strives to minimize ambiguity, which other fields don't do as much. Non-math fields tend to reuse regular words as jargon (i.e. with specificity of meaning that may fly over the laymen's heads). Social sciences and humanities are most notorious for this, often resulting in non-practitioners not realizing they are out of their depth because they are not looking at symbols from non-Roman alphabets.

That's something only someone who's never studied advanced math could say. Math notation and jargon can be extremely ambiguous and overloaded. "Normal" has about 20 different meanings.

Strives to minimize =/= completely eliminates