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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

#421

Post by tyre »

pdpi wrote: post #374
Xcelerate wrote: post #340

Earlier quoted context omitted.

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…

"OpenAI's internal team solves famous maths problem" is technically impressive but dispiriting. Non-experts solving a problem by just throwing resources at it is kind of the worst possible optics for knowledge workers. It's just disempowering. "Famous mathematician uses ChatGPT to solve famous math problem" is equally technically impressive, but now you're telling those very same knowledge workers "that famous mathem…

I don't think they care if it's disempowering. They'd gain more in market share by indicating that this is God than missing out on a few subscriptions from math theory people.

My guess is they don't do this because they don't have time. They're all trying to build a company that makes them generationally wealthy before the music stops.

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

#422

Post by apt-apt-apt-apt »

CamperBob2 wrote: post #362
skeledrew wrote: post #39

Earlier quoted context omitted.

I encountered something fairly similar working with Claude a few days ago. For a current project I've been fairly hand-wavy with requirements since I was getting good results, but it seemed to be failing hard on some key points, so I started to be more strict with it. Even after the fails were resolved, I've noticed that Claude now behaves differently within that project, carefully checking and rechecking things up f…

The most powerful way to use Fable in particular seems to be to pit it against Sol as a reviewer (or, I suppose, vice versa.) After going back and forth a half-dozen times, the result tends to be pretty much bulletproof. What's somewhat disturbing is just how much Fable's code really does benefit from the review. It tends to leave a lot of low-hanging fruit, and you can see it getting kind of impatient when repeatedl…

"We've now re-explored this question three times. While there were small differences along the way, one more would be pathological. Perhaps meditation or therapy would be more fitting."

This is basically how Fable told me to get therapy.

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

#423

Post by agrounds »

ChrisClark wrote: post #404
agrounds wrote: post #318

Earlier quoted context omitted.

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 th…

My questions were mostly to agree it's hard to understand, but they were true ignorance. I'm glad you answered them. It finally makes sense to me, and now I realize I didn't even understand "over" in that context. That Ring wiki page though, um, nope... :D

> That Ring wiki page though, um, nope... :D

Fair enough! At a super high level, a ring is just a collection that has a similar structure to what you’re used to “numbers” having. That is, you can add, subtract, and multiply them. Not divide! If we restrict ourselves to just whole numbers then 2/3 is not allowed. We also require that something like 0 and 1 have to be there. “Like zero” means 0 + x = x for every x in your collection, and “like one” means 1x = x for every x. And lastly, we require that the distributive property holds.

Examples include the set of whole numbers (Z), the rationals aka fractions (Q), the reals (R), complex numbers (C). These are all infinite rings, but there are also finite rings such as the set of whole numbers modulo a fixed number n, denoted Z/nZ. For instance, Z/2Z has only two elements, namely 0 and 1, with rules like 1 + 1 = 0. There are also polynomial rings, like Z[t], whose elements are all polynomials with integer coefficients (e.g. 3t^3 - t - 2). You can add, subtract, and multiply such polynomials and the result is more polynomials, so this collection is indeed a ring.

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

#424

Post by raz32dust »

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…

The top comment has a counter example though https://x.com/DmitryRybin1/status/2079904005652893709

I think it's not even about the ability to steer the AI. Just the ability to ask the right questions

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

#425

Post by agrounds »

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…

Thanks for the reply, this is very illuminating. I never got to this depth in algorithms. I’m but a humble programmer with a math background, but no CS degree.

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

#426

Post by nozzlegear »

slidehero wrote: post #416
nozzlegear wrote: post #242

Earlier quoted context omitted.

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.

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

I'm sure people thought calculators and, indeed, computers themselves were very Harry Potter as well when they first came out. But in the fullness of time the magic and mystique has drained away, and we're left with the understanding that they're just tools.

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

Similarly I have a hard time taking seriously the people who make breathless claims of intelligence where there's only a text calculator with weights applied. It's like watching the devout cry "miracle!" at every strangely shaped piece of toast.

Terrence is impressed, but he's not a believer.

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

#427

Post by godelski »

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.

  > What a world we live in.
It's a really interesting world. You can spam GPT to get novel math results but here I am trying to scroll up to the beginning of the conversation and 5 minutes in I still don't know if I'm near the top yet.

Scroll... wait for render... scroll... wait for render... repeat...

We live in a world where there's so much crazy technology but few people use it to make products better or to improve people's lives. Most people use it to just make more money. It's funny too, because there's a million things we could use that tech for that actually reduce costs. Hell, what would be the economic impact of putting ML systems into streetlights so they properly coordinate. Don't even need LLMs for that, and I'm sure it'd save billions of dollars a year. Just a lack of will. I wonder if this will ever change. Is this how we create the high tech low life future?

(FWIW, no problems if I jump into the app. It's purely a web thing, but my point more illustrative than specific)

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

#428

Post by [deleted] »

davesque wrote: post #195
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.

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:…

[deleted]

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

#429

Post by slidehero »

nozzlegear wrote: post #426
slidehero wrote: post #416

Earlier quoted context omitted.

> 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.

> that's some Harry Potter kind of "writes itself" book. I'm sure people thought calculators and, indeed, computers themselves were very Harry Potter as well when they first came out. But in the fullness of time the magic and mystique has drained away, and we're left with the understanding that they're just tools. > at this point, for me, any comment about LLMs that begins with "it's just ..." is hard to take serious…

>I'm sure people thought calculators and, indeed, computers themselves were very Harry Potter

nobody thought that

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