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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

#411

Post by staticshock »

Xcelerate wrote: post #340
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.

I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathem…

You're surprised they didn't eat the tokens to churn on lots of open problems instead of asking others to pay for those tokens? They're in the token business. If they're eating the tokens, it's in support of a marketing effort, not in support of innovation across the frontier of all the other academic disciplines. The collective frontier is way too big for them to just "solve it" without asking society to at least help them break even on such an enormous public good.

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

#412

Post by jameshart »

com2kid wrote: post #389
jameshart wrote: post #332

Earlier quoted context omitted.

A ‘Byte’ is not a concrete thing and the fact you think it is speaks to the degree to which you have immersed yourself in a mental model which thinks of ‘information’ as if it is a real concrete thing, to the extent that you don’t even realize the levels of conceptual abstraction you needed to build in order to internalize what a ‘byte’ is.

A byte is quite real. On old machines you could open them up and literally point to where bytes were physically stored in memory. Now days of course the chips are small so you have to point to where the multiple gigabyte chips are at. But they are still quite physically. Heck a C pointer points to an actual physical location on your machine, if you ignore the MMU.

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

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

#413

Post by ndriscoll »

agrounds wrote: post #380
aleph_minus_one wrote: post #336

Earlier quoted context omitted.

> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.

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?

(At least some) error-correcting codes are based on polynomials over finite fields. I couldn't say much more, but it's at least intuitively plausible since e.g. an nth degree polynomial is defined by any n+1 points, so if you know say n+1+p ("p" for "parity") points, you can lose up to p and still recover the polynomial.

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

#414

Post by LoganDark »

AgentME wrote: post #390
personalityson wrote: post #60

This was my conversation with ChatGPT 4 years ago: https://i.imgur.com/WPaWgzZ.png Where will we be in another 4 years? What a time to be alive!

Oh, GPT-3.5, recognizable in so many old screenshots by the green icon. If ever a popular model deserved the term "stochastic parrot", it was GPT-3.5. I wonder what percentage of people today still base their opinions of AI capabilities on their experiences with that model. That model was the only option for free ChatGPT users for the first year and a half of ChatGPT's existence, from November 2022 to May 2024.

GPT-3.5 was so nice because the safeguards were so easy to bypass and it wasn't lobotomized by RLHF. Newer models really don't hit the same spot and I'm not sure what I liked so much about that initial release. I look back at conversation logs I saved and I just can't replicate them nowadays.

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

#416

Post by slidehero »

nozzlegear wrote: post #242
squidbeak wrote: post #50

Earlier quoted context omitted.

I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.

I think we have wildly different conclusions about what happened here. You see the machine as instructing Terrence Tao, as if it were Plato teaching Socrates about the theory of forms; I see Terrence Tao using the machine to teach himself, like an intelligent student uses a book. In this case, it's just a book that fools us into believing it can think and reason like we do, because it generates language in much the s…

> In this case, it's just a book that fools us into believing it can think and reason like we do

that's some Harry Potter kind of "writes itself" book.

at this point, for me, any comment about LLMs that begins with "it's just ..." is hard to take seriously.

Terrence is impressed. Good enough for me.

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

#417

Post by aleph_minus_one »

agrounds wrote: post #380
aleph_minus_one wrote: post #336

Earlier quoted context omitted.

> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.

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?

Rings:

* Determinant calculation:

- The Samuelson–Berkowitz algorithm is best understood in terms of general rings

- The Faddeev–LeVerrier algorithm and determinant calculation using Gaussian elimination work on rings with specific properties (for the Faddeev–LeVerrier algorithm the restriction is on the characteristic of the ring, for Gaussian elimination the ring must be an integral domain (ideally a field)).

* Ring-learning with errors (for post-quantum cryptography and homomorphic cryptography). Here, a specific ring is the central object.

* Number-Theoretic Transform (NTT): Basically a generalization of the Fourier Transform to the ring Z_n. Important for arbitrary-precision integer arithmetic

* Chinese Remainder Theorem. Often only formulated for the ring Z, but it can be generalized to larger classes of rings. Used for example in Shamir’s scheme for secret sharing (cryptography)

* The theory of BCH and Reed-Solomon codes uses a specific ring

* The AKS Primality Test (a really deep result in computational number theory) uses the ring Z_n[X]/(x^r-1).

---

Algebras:

Very often, a ring is constructed from another ring. Examples:

* the polynomial ring R[X_1, ..., X_n]

* The ring of (square) matrices over a ring R

So, using algebras in algorithms often means: "we want to make use use of this additional structure that our (more sophisticated) ring has)". (Associative) R-algebras formalize this concept of "ring with additional structure".

To just give one algorithm for polynomials:

* Buchberger algorithm for computing a Gröbner basis

Other examples:

* Clifford algebras for a lot of geometric problems (special case: quaternions (a 4-dimensional \mathbb{R}-algebra) for rotations in \mathbb{R}^3).

* If you are willing to also consider semi-rings (in this case: tropical semi-rings): the Floyd-Warshall algorithm for finding shortest paths and the Viterbi algorithm for finding the most likely sequence of states in a Hidden-Markov Model (HMM) can very elegantly formulated using the matrix semiring over the tropical semiring.

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

#418

Post by versteegen »

agrounds wrote: post #380
aleph_minus_one wrote: post #336

Earlier quoted context omitted.

> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.

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 operators from the start state: elements of a monoid (or group if actions are invertible) which define the primitive steps. Trivial example being generating all permutations of a list. More interesting, enumerate all graphs with some property with pathwidth at most k, by adding one edge or vertex at a time. So now you want to know the structure of this group so you know which sequences of elements simplify and don't need to be tried, and you want to canonicalise each state to throw out duplicates.

And you can think in terms of orbits: if there are some symmetries then you might want to factor by the symmetry group and only visit one node in each orbit, grouping states into orbits with a single representative state. See eg. Pochter, Zohar and Rosenschein, Exploiting Problem Symmetries in State-Based Planners.

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

#419

Post by breezybottom »

overfeed wrote: post #230
breezybottom wrote: post #138

Earlier quoted context omitted.

That's something only someone who's never studied advanced math could say. Math notation and jargon can be extremely ambiguous and overloaded. "Normal" has about 20 different meanings.

Strives to minimize =/= completely eliminates

I haven't seen much of this "striving". Notation use is largely passed down from instructor to student, and there isn't much interest in the mathematical community to standardize.

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

#420

Post by pansa2 »

singularity2001 wrote: post #257
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.

Has the conversation that originally led to the counterexample been released? Surely it was similar to this one, also driven by a highly knowledgeable mathematician.

No AI would have been able to find it if asked to "Find a counterexample to the Jacobian conjecture. Make no mistakes".

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