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

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

#631

Post by bredren » Thu, Jul 23, 2026, 4:09 PM UTC

Geof25 wrote: post #440

bredren wrote: post #209

Earlier quoted context omitted.

It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue." But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore. One without the other can result in frustration or worse.

It is a skill issue. Such developers have to take a problem, isolate it and explain it to someone/something else with enough context to be able to work with it. It is perplexing how many developers lack this basic skill, some of them borderline lack theory of mind and are incapable to understand that other people can't see the unspoken part in their heads.

It has been a good long while, but in my undergrad the engineering dept did not put a lot of emphasis on being a well rounded individual. It was more about passing physics and getting through all of the math.

One of my first jobs out of undergrad was technical writing on EDA tools for Mentor Graphics Calibre product. For many years, I did not appreciate that experience--it seemed orthogonal to what I'd gone to school.

Once I did become a professional SWE, something that took me too long to realize was that writing documentation and tutorials for people and looking after the quality of onboarding materials was almost never rewarded.

I came to see it more of a liability with ~all emphasis being on human code gen.

I listened to an interview with Steve Yegge earlier this year, one of his concerns was that many people just don't type very fast or read very quickly. IIRC, he expressed concern this would continue to be a big barrier to successful use of AI.

I previously commented on annotation for coding agents that an important skill was knowing the beat of a conversation.

I think that extends into critical reading. For example, it is not that unusual for a Washington Post article to put some ~explosive detail deep in the article--this is called "burying the lede."

I see this regularly reading AI output. That is: the AI will lead with the strongest claim it can state cleanly, but not the most explosive implication.

With Claude Code and Codex, you're looking essentially neverending walls of text dense with technical information.

So now you have to have high domain expertise, a good understanding of written communication, a sense for the beat of a conversation (which assists in identifying understatements or overstatements), and the ability to dense text outputs quickly and then highlight critical statements for further exploration.

I'd almost describe part of this work as that of a skilled interviewer.

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

#632

Post by jameshart » Thu, Jul 23, 2026, 4:17 PM UTC

StilesCrisis wrote: post #582

jameshart wrote: post #412

Earlier quoted context omitted.

Wow and TCP sends a stream of those microscopic memory cells?

It sends electricity or radio waves, yeah? The bits aren't the cells, they are the electricity within the cells.

Now we’re talking about bits, I thought we were talking about bytes (or was it octets?)

I’m not sure this gets us any closer to a jargon-free explanation of what TCP does.

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

#633

Post by xyzsparetimexyz » Thu, Jul 23, 2026, 4:29 PM UTC

Normal_gaussian wrote: post #366

xyzsparetimexyz wrote: post #356

Earlier quoted context omitted.

I have a pi extension that just runs the same prompt in a loop 25 times. I tried giving it a loop breaker but I found that it'd give up too readily. When the task is actually complete each iteration didn't do a lot of work so it was efficient enough. I suppose another way is to call out to a separate context to check if the task is complete?

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…

Sounds overengineered to me

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

#634

Post by theplumber » Thu, Jul 23, 2026, 4:33 PM UTC

lanstin wrote: post #602

theplumber wrote: post #464

Earlier quoted context omitted.

Indeed that’s the case but most of the programming problems are solved problems and unless you care how things are done someone with less/no knowledge can still make great use of programming to do what it needs to do. Let’s not pretend you need to be programmer to use AI for programming like mathematicians need to solve math problems. Programming (most of the time)solves real problems rather than abstract constructs.…

That seems only partially true. I have no doubt it could crank out a UI like Facebook, but having it work for high volume is less well represented in the training data. Things that seems obvious to us, like sharding, using asynchronous messaging systems to provide more durability at the cost of instant consistency, traffic shedding, intelligent clients, conscious decisions on CAP properties in the light of user exper…

I call it BS. When you are hitting real scalability issues that the AI can no longer help you are already a successful business. The non technical founder will not know about async message passing but will ask the AI why the site is loading so slowly and sure enough the AI is able to implement caching and various mechanism to improve the performance and observability. You are hitting a wall when you do novel stuff (I.e a special search engine) but something like Facebook which is more or less a CRUD app is not a problem at all. And this is the case with Today’s AI systems(GPT Sol, Kimi K3 etc).

I can’t imagine what models will be able to do next year, especially the open weights ones that are not nerfed for economic or other reasons(I.e Fable saga)

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

#635

Post by namuol » Thu, Jul 23, 2026, 4:41 PM UTC

fooo1882992 wrote: post #393

namuol wrote: post #316

Would Terrance Tao still be a renowned mathematician if he grew up with ChatGPT?

Would Kasparov still be a chess grandmaster if he grew up with Deep Blue? (Yes. Chess champions are stronger than ever, even though none of them have been able to beat a computer for decades.)

I don’t think the comparison is fair. Deep Blue was an opponent in a highly bounded game. Mathematics has no bounds, and modern agents are specifically designed and trained NOT to be opponents.

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

#636

Post by umvi » Thu, Jul 23, 2026, 6:00 PM UTC

BlackFingolfin wrote: post #273

umvi wrote: post #243

Earlier quoted context omitted.

My example was contrived, I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax. At a certain point though, your layman has to know the "atomic" (as in, you can't break them down further) mathematical concepts like "functions" and "infinity": `sum function(x) from x=0 to x=infinity`

I am a research mathematician. In my field (abstract algebra, computational group theory), a sum or some such notation is like the most trivial of trivial things in terms of notation. There are a few things like sums that could in theory be made to "look more like what a computer programmer would expect", but that'd be just a tiny corner of it. And if you think about how summation would look in Lisp or APL (which som…

yeah you'd have to make a "mathematical grammar" to cover all the cases. one problem with current mathematical notation is that it is optimized for writing by hand, not with keyboards. If you want to write mathematical notation with keyboards you have to use a complex typesetting tool like LaTeX to position all the super/sub-scripted, nested, etc. symbols.

And it all seems arbitrary anyway. Why are trigonometric functions written in english ("tan" being short for "tangent") but other stuff uses greek letters and other stuff still uses esoteric/abstract symbols? Why is the integral symbol shaped the way it is, and why use super/sub symbols for the bounds versus `integral [0,Inf] ...`?

In my ignorance, I'm assuming math is the way that it is because that's the way it's been for centuries, and messing with it harms its ubiquity. Math notation is not the way that it is because it's particularly well thought out. It's centuries of legacy tech debt that can't be changed. Kind of like how English is a crappy language in a lot of ways, but we can't change it now because too many people use it and you'd never get enough momentum to switch.

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

#637

Post by slartibardfast0 » Thu, Jul 23, 2026, 6:01 PM UTC

mkl wrote: post #589

slartibardfast0 wrote: post #196

Earlier quoted context omitted.

hi, would you mind linking to the book? https://www.jsoftware.com/help/learning/23.htm is the closest i've found, but wondering if i'm missing something perhaps, Julia? tyvm

Probably this: Kenneth E. Iverson - Calculus https://www.jsoftware.com/books/pdf/calculus.pdf

makes sense! thanks very much

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

#638

Post by ecshafer » Thu, Jul 23, 2026, 6:32 PM UTC

lanstin wrote: post #616

rlt wrote: post #443

Earlier quoted context omitted.

I don't understand. You will ask it for something you want, and instead of needing to pay a the team of people with expertise, it will just do it for you. Someone will still need to ask for something (or maybe it will just run autonomously making things it thinks we'll like) but it won't require any expertise in the field.

I can’t help but believe that reality will keep the interactions more useful when coming from a person with expertise in the field, e At least for technical questions. For things like sickness, old age and death, generally the lacks are not knowledge but the consistent will of the group. Like global warming, there is no real profound knowledge gap except how to get everyone to agree to mitigating the carbon burn. Per…

Global Warming does have a knowledge gap of sorts. Yes there is a viable solution to Global Warming, drastically cut back carbon and methane emissions. Where the knowledge gap is: drastically cut back on carbon and methane emissions, whilst maintaining or even improving quality of life in developed countries, having no economic issues, maintain industry, and keep developing countries developing. That is the hard part.

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

#639

Post by yourself92 » Thu, Jul 23, 2026, 6:49 PM UTC

genidoi wrote: post #246

jvanderbot wrote: post #26

It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that would explain it. Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what…

> you can use AI to understand something and map it to your own mental map This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process. I could probably have worded this better but I'm…

> I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.

I do get what you are saying and agree with you in principle, as I have noticed the same. But that said, this same way of thinking can really apply to any abstraction. It really just depends on the level you're working at. A software architect might know the nitty gritty details of how each service works, but really those implementation details don't matter if the abstractions are handled well enough, so in theory they don't really need to know those details as long as "the pieces fit". But of course the catch 22 there is you can't build good abstractions that can fit together well if you don't understand well enough the underlying details

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

#640

Post by lanstin » Thu, Jul 23, 2026, 6:57 PM UTC

ecshafer wrote: post #638

lanstin wrote: post #616

Earlier quoted context omitted.

I can’t help but believe that reality will keep the interactions more useful when coming from a person with expertise in the field, e At least for technical questions. For things like sickness, old age and death, generally the lacks are not knowledge but the consistent will of the group. Like global warming, there is no real profound knowledge gap except how to get everyone to agree to mitigating the carbon burn. Per…

Global Warming does have a knowledge gap of sorts. Yes there is a viable solution to Global Warming, drastically cut back carbon and methane emissions. Where the knowledge gap is: drastically cut back on carbon and methane emissions, whilst maintaining or even improving quality of life in developed countries, having no economic issues, maintain industry, and keep developing countries developing. That is the hard part…

There might be a few industries where a technical fix is needed, but having to replace all our infrastructure is always a huge decades long win for the economy. Solar and batteries are made from abundant materials and cheaper than anything else, and almost all the sticky technical issues have good solutions with large amounts of cheap energy. Fuck we could desalinate and pump sea water back onto Greenland and Antarctic but instead we are arguing that windmills cause cancer or whatever the fuck the oil companies are paying morons to be persuaded by. Before the morons took over in the US, banks and insurance companies were salivating to fund electrification and selling 5 billion electric vehicles. The leadership there apparently has the moral fiber of a petty crook instead of a vanquishing hero, so they are ditching a lot of that, but none of these are prima facie cases where GenAI / LLM will help, unless you think "everyone having a personal physics tutor" will transform the world the way my friends and I thought "everyone being able to chat with a physicist" would transform the world 35 years ago.