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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

#321

Post by dekhn »

JadeNB wrote: post #201
dekhn wrote: post #40

Earlier quoted context omitted.

I had a few moments of this in the past. For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature. Similarly, at some point somebody pointed out to me "the reason you're confused is that the bold on that variable means i…

For what it's worth, it's not a problem with multiple kinds of multiplication (multiplication by a scalar can be viewed as multiplication by a specific kind of matrix), but with the idea that one can cancel in a multiplication. Since you can't cancel in matrix multiplication, you run into unexpected trouble when you try to do so, even if that's the only multiplication in sight. (In fact, you can't cancel in scalar mu…

P = NP is actually one of the worst abuses of symbology I've seen in math.

"""During his own Google interview, Jeff Dean was asked the implications if P=NP were true. He said "P = 0 or N = 1." Then, before the interviewer had even finished laughing, Jeff examined Google's public certificate and wrote the private key on the whiteboard."""

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

#322

Post by metanonsense »

gowld wrote: post #133
taylorfinley wrote: post #114

Earlier quoted context omitted.

Came here to flag the same beat. It's wild to me Terrance Tao has to pay to talk to chatgpt, you would think it would be the other way around!

"Activating Pro" doesn't mean "paying". It means selecting a slower/smarter model.

Yeah. I also only use pro in very specific situations. Not (just) because it's slow, but if the question is too simple or vague, pro responses are sometimes overfitting to the noise in my question. Until there is lots of context and the basics are laid out, high or xhigh somehow work better. Pro gives you the last 10%.

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

#323

Post by chongli »

ChaitanyaSai wrote: post #22

I don't understand any of the math here, but I had two thoughts. Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit. Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.

Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.

That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.

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

#324

Post by rowanG077 »

hyperpape wrote: post #245
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…

I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree. I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field. I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concept…

[dead]

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

#325

Post by ImaCake »

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 find it helpful to think of LLMs as reflections. If you can talk like an expert mathematician at the model it will respond like one. While Terrance's first prompt looks trivial I expect a first year Uni student would be hard pressed to provide something that good.

I guess it is kind of the inverse of the "you are an expert mathematician" prompt engineering of gpt3.5. Since no one ever says that to an expert mathematician when they are doing expert math the model immediately reflects that it is not an expert mathematician.

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

#326

Post by paxys »

minimaxir wrote: post #106
napoleoncomplex wrote: post #21

This is the second ChatGPT shared conversation I've seen today that is truly fascinating. The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709 What a world we live in.

> just repeatedly saying "keep going" to ChatGPT For posterity, this indeed works for most problems where an agent might give up. LLMs don't inherently know something is impossible. The phrase I tend to use in my harder prompts to automate this with a sane loop breaker: > **REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS.** You have permission to keep iterating.

Aka /goal

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

#327

Post by rstuart4133 »

mrpf1ster wrote: post #236
umvi wrote: post #227

Earlier quoted context omitted.

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 sigm…

Your translation only makes intuitive sense to you because you are well versed in programming. I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them

> I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them

I doubt it. Greek letters convey almost no information, whereas (one hopes) the function and variable names are chosen by a programmer to help the reader. The Greek letters used by mathematicians (and physicists) weren't used to convey information, they were used because typesetting, publishing and paper were expensive. They are optimised for brevity over readability.

It was a perfectly reasonable trade-off at the time, but times have changed.

As an aside, some programming languages (such as APL, and to a lesser extent Perl) did emulate the old Greek letter style. "Line noise" is a typical description of the result. No computer language aimed at software engineers and computer scientists does that now.

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

#328

Post by paxys »

missedthecue wrote: post #128
Freedom2 wrote: post #117

Earlier quoted context omitted.

How would you envision ChatGPT paying Terrence?

OpenAI giving him a fancy schmancy title to advocate for their product, such as in the way he is inadvertantly doing here.

He doesn’t seem the kind who would sell his credibility for a few bucks.

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

#329

Post by chongli »

OJFord wrote: post #56
colordrops wrote: post #51

Earlier quoted context omitted.

I'm not sure an AI will speed things up much. You would probably still need years of layers of foundational understanding to get the advanced material. We don't go through years of school to learn math just because teachers are bad - it's because complex subtle ideas are built on countless other ideas, and aren't necessarily compressible to something every layman can understand.

The years are broad though, the nice thing with AI explanations is that they can go deep quickly, and quite precisely down the path you need for your prior experience.

It's not a path, it's a tree, and a downward-facing one at that. To understand the root, you have to understand all of the leaves first, and all of the nodes above them, all the way up.

Most people, if they haven't studied mathematics in university, would need to learn hundreds of concepts just to get to the leaves of the tree, and many of these concepts are truly difficult to understand, requiring weeks of study and practice.

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

#330

Post by mabster »

WarmWash wrote: post #261
minraws wrote: post #240

Earlier quoted context omitted.

You are comparing TCP a relatively basic topic in the grand scheme of computing with Rees_algebra which is fairly specialized, we could take a simpler topic more foundational and clearer to understand and compare them. I can understand that this feels like one is so much more complicated part of it is also how the articles were written, wikipedia is not known for quality maths explanations. But beyond that this compa…

I think you are snagging on thinking this is an observation about difficulty, time-to-mastery, or mental firepower requirements. It's not. It's a plain observation that math exists on mostly it's own path with little to zero overlap with our lived experiences. If mathematics was a vector, it would have similar magnitude to other vectors, but it's direction would be much more removed from the typical knowledge pack, f…

Engineering is very reliant on mathematics and as real world as it gets.
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