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A digestion of the Jacobian conjecture counterexample

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Re: A digestion of the Jacobian conjecture counterexample

#41

Post by godwinson__4-8 »

foobarqux wrote: post #29
pavel_lishin wrote: post #25

Earlier quoted context omitted.

It's interesting because I like math, and typically posts like this generate a lot of interesting comments. Plus, in general, I'm wildly disinterested in the AI discourse that eats up 50-75% of the front page, so frankly anything that's slightly more interesting than that gets an upvote from me.

You seem to be saying that you would upvote an empty article (or a gibberish article) with the title "Jacobian Conjecture Counterexample", is that right?

Are you always this subtle?

Re: A digestion of the Jacobian conjecture counterexample

#42

Post by [deleted] »

jasonfarnon wrote: post #39
castedo wrote: post #34

Earlier quoted context omitted.

I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing. I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.

Not just an assumption, I saw the LLM saying it used sympy.

[deleted]

Re: A digestion of the Jacobian conjecture counterexample

#43

Post by monster_truck »

hyperhello wrote: post #8

Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?

Not much.

But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.

Re: A digestion of the Jacobian conjecture counterexample

#44

Post by tptacek »

castedo wrote: post #34
drivebyhooting wrote: post #30

Can we audit the CoT and work the AI did to generate such a remarkable cancellation?

I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing. I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.

I think you can reasonably assume that frontier models are using SymPy or something like it any time interesting math gets into the picture, and the person driving Fable here is an accomplished mathematician, but I don't think we can reasonably assume either extensive prompting or brute-force compute in any sense other than what it normally takes Fable to, say, whip up a calculator app.

Re: A digestion of the Jacobian conjecture counterexample

#45

Post by wren6991 »

minimaxir wrote: post #12
fhdkweig wrote: post #9

Earlier quoted context omitted.

Honest question. Does asking "make no mistakes" actually change the output? Does it make mistakes if you don't bother to ask for no mistakes? Is it just to make the human feel more secure?

It's a meme. Telling it to "make no mistakes" doesn't do anything because LLMs don't have an inherent concept of a mistake and they are already RLHFed to code correctly. However , if you tell it to not do particular behaviors explicitly—some of which would be considered mistakes—it will not do said behaviors and with enough checks and balances, you'll get output without "mistakes". One example of this from the OpenAI…

I like the explicit and imperative style of this prompt. Makes me feel like LLMs are just weird Turing machines and CoT is their tape.

Re: A digestion of the Jacobian conjecture counterexample

#47

Post by pavel_lishin »

foobarqux wrote: post #29
pavel_lishin wrote: post #25

Earlier quoted context omitted.

It's interesting because I like math, and typically posts like this generate a lot of interesting comments. Plus, in general, I'm wildly disinterested in the AI discourse that eats up 50-75% of the front page, so frankly anything that's slightly more interesting than that gets an upvote from me.

You seem to be saying that you would upvote an empty article (or a gibberish article) with the title "Jacobian Conjecture Counterexample", is that right?

After that kind of a stretch, you're ready for any kind of exercise.

Re: A digestion of the Jacobian conjecture counterexample

#48

Post by gerdesj »

jmward01 wrote: post #35

Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.

"problems that were hard"

They are still hard problems - As we say in the UK: "one swallow does not a summer make".

As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!

Re: A digestion of the Jacobian conjecture counterexample

#49

Post by CamperBob2 »

tptacek wrote: post #3

The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow: https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...

I like how he goes one-on-one with it like Gandalf fighting Yoda on fifteen planes at once for 80% of the transcript, and then we read "OK, I've activated Pro."
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