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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

#551

Post by ilaksh »

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…

I think you're mostly right, and there is still a lot of "cope" in this thread about how much he needed to guide it.

But I would say that in some ways it's already obviously superhuman. The reason I think it's lacking in some jagged ways still at the expert level is because although it's highly optimized to be incredibly capable in many domains, it still doesn't have quite the same raw capacity for complexity in understanding one problem that humans do.

I believe that LLMs (really should be called VLMs for most of them) can still get much larger, and that will push the absolute complexity level and general IQ way over human level.

They are maxing out at like 5 or 10 trillion parameters right now. I believe we will see 50 and 100 trillion parameter models and models with large portions of that active. It will round out the jaggedness and probably more than double the raw intelligence that a human can achieve. It's not a linear scaling but who knows what the limit is and can go significantly higher with the same architecture I think given continued improvements in training and hardware scale.

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

#552

Post by aleph_minus_one »

versteegen wrote: post #418
agrounds wrote: post #380

Earlier quoted context omitted.

I studied a lot of abstract algebra in college and grad school and I’m surprised that rings and algebras would come up in a CS degree. What algorithms topics used those concepts? Something about polynomials?

Algebra is useful because graphs are algebraic objects, and a lot of CS is about graphs, in particular search/planning. But no, I never saw rings mentioned except for generating functions, which are used for analysing recurrence relations. For example in search algorithms where you want to search a space without visiting state nodes twice. Each state in the search space is produced by the sequence (a product of) of o…

You are talking about (abstract) algebraic structures, and not about rings and algebras (over a ring or field)

> https://en.wikipedia.org/w/index.php?title=Algebra_over_a_fi...

> https://en.wikipedia.org/w/index.php?title=Algebra_over_a_fi...

> https://en.wikipedia.org/w/index.php?title=Associative_algeb...

The latter is what aground asked for in https://news.ycombinator.com/user?id=agrounds

> I’m surprised that rings and algebras would come up in a CS degree. What algorithms topics used those concepts?

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

#553

Post by porridgeraisin »

budsniffer952 wrote: post #536
case wrote: post #486

Earlier quoted context omitted.

I think the issue is that LLMs can be so confidently and convincingly wrong about anything. And if we (LLM tool operators) don’t know the subject matter in question, we can’t easily distinguish what they are right or wrong about.

While humans are infallible and correct.

Yes, when you use a standard theoretical physics textbook / course sequence, it is 100% correct.

More broadly, LLMs or anything at all, needs a verifier. If the task can be automatically verified, great, then anyone can use them. If you can't automate the verification, then you need to be able to verify it using your knowledge. Knowledge required to verify is lower than the knowledge required to create in very few cases. This is why you still need a fully trained human verifier. We are yet to reorganise the overall "tasks" in the economy such that verification can be done with much lesser knowledge, for no reason other than that there was never demand for this until creation became automatic few years ago. It is possible and is slowly being done, there are many many startups working on automating verification in different fields and in many cases we will see fields reorganise themselves to be more amenable to automatic verification. Note that _effort_ required to verify is much lower than what is required to create, for almost anybody, and LLMs have economic use just due to that alone, albeit in the hands of a knowledgeable human.

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

#554

Post by DiscourseFan »

michaelchisari wrote: post #12
lukebuehler wrote: post #8

It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise (emphasis on progression and usage patterns, not absolute skill, obv I don’t match that): sho…

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.

Yes but, AI can still be used for relatively common applications (coding basic pages), and furthermore you can build agentic loops to check more difficult code, or explain the conceptual order of the code, and I imagine over time the domains which will be considered “basic” for the AI will continue to grow, but perhaps so will the abilities of its users…

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

#556

Post by kakacik »

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…

We often designed various protocols and other stuff in IT as per 'lets come up with some simple chat to validate things and go ahead', or principle of least complexity (apart from classics like IBM and Microsoft but they were done specifically to make vendor lock in as big as acceptable). Its also ie in telco protocols - designed around trivial Hello handshakes, I can explain protocols like that to a child and it would grok it. Not great for advanced security but thats another topic. Or SSL handshake which is way more modern.

Math just represents this reality on most abstract level, it doesn't care if complexity for some human brains is trivial or almost fractal-like.

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

#557

Post by zq0829 »

这不是“几个系数碰巧消掉了”，而是“给三次式标记一个单根”这一 incidence correspondence 的坐标展开：resultant =1 切掉因子间的相对缩放，所以局部 étale；忘掉所标记的根，所以一般三对一；而真正低维而罕见的奇迹，是相应切片竟然等于 A 3 。常 Jacobian 只是这套几何的坐标影子。

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

#558

Post by blenderob »

This is fascinating. I'm having a very hard time wrapping my head round how a statistical model designed to predict tokens creates output that contains such deep reasoning and insight about complex ideas that we'd think only humans can understand!

Yes, I know it's not just a dumb statistical model like a simple neural network. Yes, I know all about the vectors Q, K, V and how they contain long range contexts. But still! How does all of this emerge into something that can produce such insightful output? I still can't wrap my head around this. Am I dreaming? Am I living in a science fiction?

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

#559

Post by Dragbat »

jstanley wrote: post #20

It's reassuring to know that even a supergenius's ChatGPT session is one sentence from the human followed by 3 pages of LLM output.

Yeah, the LLMs are definitely committed to producing a whole article every time as a response to whatever the prompt and whoever the prompter

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

#560

Post by codebolt »

jeef_berky wrote: post #499
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...

It's so obvious to some and may never be considered by others. If someone asks about AI I tell them step 1 is ask it how to do something you know all about. Step 2 is consider everything else you ask will be that inaccurate.

As a dev with three decades experience writing code, I use it for stuff I know all about all the time. And these days, the majority of the time it is more correct than I would've been without a few iterations of trial and error.
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