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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

#311

Post by appplication »

markfsharp wrote: post #158
minimaxir wrote: post #106

Earlier quoted context omitted.

> 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.

What I am thinking is the way you make it 'keep going' and when you have people of the calibre of Tao doing it I kept thinking how many breakthroughs is he going to cause the LLM to find with his targetted questions :D Amazing that we have the privilege of witnessing a true expert in such a way question the LLM.

Do notice he is quite Socratic, the approach works well for LLMs they love to please so you have to be careful in how you lead them.

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

#312

Post by chongli »

reinitctxoffset wrote: post #272
steve_adams_86 wrote: post #259

Earlier quoted context omitted.

"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…

It's a class with an array of integers in it with .length() == t - 1 and the same methods as Matrix. In lean4, even without mathlib4, TCP/IP is way more code than a Rees algebra. Math uses dense notation that is gigaoverloaded, and the disambiguating context was historically the leisure and proximity to have someone explain what the lexemes even mean. lean4 is proving to be very revealing as an uncorruptible referee…

That's false. Z[n] in rings does not mean "an array of integers of length n", it means the subring generated by Z union with {n}, where n is an element of some other set. For example:

Z[i], the Gaussian integers, is the subring (of C) generated by Z union {i} where i is the imaginary unit in C, the complex numbers. The Gaussian integers correspond to the integer grid-points of the complex plane, if you want to visualize them.

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

#313

Post by appplication »

blargey wrote: post #215
davesque wrote: post #195

Earlier quoted context omitted.

Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example: LLM: This package hasn't made it to production. ME:…

There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the small…

There’s also probably a lot less noise from casual internet conversations. I imagine a nontrivial amount of what LLMs know about certain technologies comes directly from forums like reddit where quality of response isn’t guaranteed.

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

#314

Post by ftjftjft »

jongjong wrote: post #301
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…

>> Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on. ... communication protocol, method signatures, web components, ssh, CSS media queries, HTTP headers, WebSockets, timeouts,…

a janitor could do the same though

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

#315

Post by metanonsense »

ecshafer wrote: post #31

Terrance Tao's chatgpt conversation is really interesting for a variety of reasons: 1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result. 2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some…

It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.

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

#317

Post by hyperpape »

AlienRobot wrote: post #290
hyperpape wrote: post #245

Earlier quoted context omitted.

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…

I agree, and I don't even know math. Most of things that are impenetrable in programming aren't about... programming. They are about some actually complex field like math being applied to programming. For example, a library that does stuff with geometry. You need to know geometry to understand the program, but the program itself will never be complicated. It's the geometry that is complicated. In cryptography, it's n…

I'd slightly disagree. Here's an example of a programming task that is incredibly complicated:

Your job is to maintain and modernize a system while delivering a constant stream of new features. Your system is several million lines of code, with some multi-thousand line classes, a rulesengine that can trigger nearly unlimited effects at any time, a persistence system that's weirder than anything any of your friends have ever worked with, and hundreds of customers delivering tens of millions in revenue who use the system in incredibly varied ways.

You can't stop to rewrite the thing, you can't just throw features out there and pray, because you'll cause regressions and your existing customers will hate you. You have to fix the thing as you're building on top of it.

But where I agree is that there's no single deep concept that unlocks it all, it's not like you'll fix it by reading a textbook about it. It's complicated, and it's going to stay complicated, no matter how long you work on it.

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

#318

Post by agrounds »

ChrisClark wrote: post #303
agrounds wrote: post #281

Earlier quoted context omitted.

Z is the ring of integers, t is a formal variable allowing us to discuss polynomials whose coefficients are in some ring. That’s what R[t] means: the ring of polynomials of the formal variable t with coefficients in R. Adding in t^-1 lets us include inverted terms like 2t^-3. An algebra over a ring (call it S so we don’t confuse it with R from the previous paragraph) is a like a vector space over S, with the added st…

But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? His point is the terms are dense too

Absolutely agree. All formal statements (like mathematical ones) are going to have some level of assumed background. And as the assumed background expands, the language naturally becomes more information dense.

As for your specific questions, I believe Wikipedia does a great job of answering two of them for a layperson:

https://en.wikipedia.org/wiki/Ring_(mathematics)

https://en.wikipedia.org/wiki/Vector_space

For the others, I’ll say that a formal variable is just a symbol (literally, like the letter t). With such a symbol, we can construct polynomials like 2t^2 - t + 3. Also, there’s no need to only use integers as the allowed coefficients; you can use any ring you like instead.

An “algebra over the ring R” is what I was attempting to define in my comment above. The algebra is “over” R if we can multiply an element of the algebra by an element of R. The useful analogy here is scalar multiplication in a vector space: you can multiply a vector by 2 to double it or -1/2 to reflect and shorten it. More generally, it makes perfect sense to consider some more general version of vectors which can be scalar multiplied by elements of any ring R.

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

#319

Post by lpd59 »

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…

To be fair, Tao's specialty is not algebraic geometry.

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

#320

Post by chongli »

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.

No, I'd say ambiguous means "context dependent" and the context is unclear.

Pronouns like you/me/he/she/they/them are context dependent in everyday English writing but they're only ambiguous when the context is unclear, otherwise most people have no trouble dealing with them at all!

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