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Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

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

#251

Post by keeda »

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…

Computer science is not a great example for this. I could ELI5 most of the terms you listed (and I actually have done so for many of them!) This is because it's pretty easy to map these concepts to everyday physical objects. Once a child understands any of those objects in their lives, it's pretty easy to explain in those terms.

Like, just the concept of "books" gets you very far. E.g. a file is a like a book, a folder is like a shelf to keep books, a stack is literally a stack of books, a heap is just a place you can pile books in willy-nilly, a database is like a library, a cache is books on your desk versus books in the library, replication is having multiple copies of a book so we can afford to lose some copies, indexing/sharding is like arranging books alphabetically, and so on.

Others are trickier but not much: a process is an app that is running on your device, a socket / tcp / http / websocks is a way to exchange information between devices, a namespace is how the name "Tom" in Tom Sawyer is different from "Tom" in Tom & Jerry, DNS is a way to get an address from a name, etc. etc.

You'll also notice that many of the terms you mentioned are already derived from well-known real-world concepts like pool, stream, channel, stack, queue, worker, transactions. You can mix those with other everyday concepts to make useful analogies.

But I could not even begin making analogies for most topics in Mathematics. I guess this is because advanced topics in Mathematics are just too abstract to map to everyday things.

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

#252

Post by epolanski »

ValentineC wrote: post #202
brotchie wrote: post #64

Earlier quoted context omitted.

GPT 5.6 already is an explainer agent. Fork Tao’s convo and prompt this (with your own math level described). GPT did a great job of translating Tao’s questions and concepts (e.g. “pre image”) into a progression I could understand. “Ok I have a PhD in financial math and undergrad in engineering math. I have almost zero knowledge of polynomial algebra / geometry, I know what a polynomial is and what roots are but not…

One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.

They lack the empathy to understand where and why you're struggling.

I've given private math lessons and seen students struggle with ai, even though ai gave the right answers.

My intuition is that humans spot xy problems easier when teaching (user ask x but really needs y), whereas llms will oblige writing about x.

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

#253

Post by epolanski »

krelian wrote: post #165

How long has it been since we last saw a "LLMs can't really think/be useful/be better than a human expert" discussion on HN? There used to be so many!

People are insecure about their leetcode black belts and react slop not giving them cushy jobs anymore so keep missing the forest for the tree.

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

#254

Post by eh_why_not »

john_strinlai wrote: post #248
eh_why_not wrote: post #234

What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times. Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao…

> Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore. we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on. even in math alone, i think its indisputably already stronger…

Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.

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

#255

Post by math_dandy »

breezybottom wrote: post #247
math_dandy wrote: post #244

Earlier quoted context omitted.

Certainly overloaded but rarely ambiguous. Context will determine which notion of “normal” applies.

"Context dependent" is basically the definition of ambiguity.

I guess what I’m saying is that since there’s always a context, ambiguity stemming from uninspired naming is never an issue in practice.

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

#256

Post by epolanski »

hintymad wrote: post #65

I recall several mathematicians (possibly including Terence Tao) mentioning that fields in mathematics have become so specialized and isolated that a conference like the ICM feels more like a collection of mini-conferences. An expert in one area can barely understand a talk in another. Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal coul…

I was a chemistry researcher this is true in all science.

I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.

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

#257

Post by singularity2001 »

seamossfet wrote: post #160

The big take away for is the fact that the ONLY reason why chatgpt was able to get to this counterexample was because of the knowledge of the person driving the conversation. I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.

You are badly informed the counter-example was found shortly before. Terrence just tried to replicate how it was found.

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

#258

Post by cindyllm »

eh_why_not wrote: post #234

What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times. Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao…

[dead]

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

#259

Post by steve_adams_86 »

WarmWash wrote: post #224
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…

> 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…

"The Rees algebra is an algebra over Z[t^−1]"

Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small statement that seems perfectly cromulent, but there's a lot packed in there that someone like me is totally missing.

I suppose there may be similar concepts in computer science, but nothing comes to mind that ever stumped me. To be frank, the field has been relatively accessible to me because it hasn't been too challenging. Not sure if that's a personal aptitude thing or it is genuinely simpler.

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

#260

Post by QwenGlazer9000 »

seamossfet wrote: post #169
vouaobrasil wrote: post #164

Earlier quoted context omitted.

> I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier. The problem is that, what happens to human expertise as people start to use AI earlier and earlier in their careers, so that in 50 years? The problem is that Terry Tao spent decades as a mathematician before ever encoutering…

I don't know if I agree with the premise that having access to AI results in dulling human intellect. I feel like to get to Terry's level you need a combination of passion and aptitude for the subject. People that don't want to learn about a topic will always look for shortcuts, which I think represents the vast majority of people. Terry Tao is quite exceptional, and I think exceptional people will still exist even w…

The problem is that learning never stops. You can't just go through school, become a junior in X field, then start using AI. Then you'll forever be a junior. You have to make a choice when you're working a job: either use AI-first workflows to increase your productivity, or don't and increase your knowledge and skill.

My wording is specific. You can use AI and increase knowledge and skill, but this requires you to be driving the AI at such a low level you don't get the full speedup. As an example, you can write code yourself with AI as an assistant, but it's not as fast as AI writing everything.

So now we end up stuck in a situation where every professional needs to choose between long term skill growth or speed, as anyone who's worked a job before knows, speed will always be the one chosen.

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