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A digestion of the Jacobian conjecture counterexample

terrytao.wordpress.com

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Re: A digestion of the Jacobian conjecture counterexample

#81
post #59

Earlier quoted context omitted.

I was reading another source that claimed this example was inspired by an existing (rational polynomial) example from the literature (created in 1999 by a Russian mathematician Vitushkin). > The seed is almost certainly Vitushkin's old rational "counterexample." From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0 This is not quite lore laundering, but it seems to be close.

I still can't understand all the details, but it's very interesting to read that chat. Anyway, instead of close to lore laundering, for me it's "standing on the shoulder of giants".

[flagged]

Re: A digestion of the Jacobian conjecture counterexample

#82
post #44
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.

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.

The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies. I don't see any way to explain this without assuming that the limiting factor is the ability to burn a few rainforests worth of tokens in pursuit of something publishable.

I think it would also explain their opacity towards the process. Being able to solve such well known problems in a nice replicable 1-2-3 way would be far more effective marketing than their complete opacity outside of the result, which suggests that they feel transparency is not in their best interest for some reason.

Re: A digestion of the Jacobian conjecture counterexample

#83
post #44

Earlier quoted context omitted.

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.

The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies. I don't see any way to explain this without assuming that the limiting factor is the ability to burn a few rainforests worth of tokens in pursuit of something publishable. I think it would also explain their opacity to…

Would you ordinarily be able to "explain this", if a mathematician had come up with this on their own? How would that story go?

Re: A digestion of the Jacobian conjecture counterexample

#84
post #83

Earlier quoted context omitted.

The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies. I don't see any way to explain this without assuming that the limiting factor is the ability to burn a few rainforests worth of tokens in pursuit of something publishable. I think it would also explain their opacity to…

Would you ordinarily be able to "explain this", if a mathematician had come up with this on their own? How would that story go?

Bayesian probability. Were outcomes being driven by 'normal' usage of LLMs then it's extremely improbable that both big discoveries would come from the small number of people working at the companies. That suggests working at the companies is more the decisive factor than the LLMs in and of themselves.

And what do you get from working at the company? Likely a rather massive token/processing budget. The companies opacity towards the path to these discoveries also makes this further probable as 'spend millions of dollars in tokens' is a somewhat less attractive narrative than the implied narrative of 'just use Fable.'

Re: A digestion of the Jacobian conjecture counterexample

#85
post #28

reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.

The difference is your dog will never understand the Python code but you could probably understand this post in a matter of days or weeks if you really wanted to. Can we all please stop acting like this Terry guy is so special?

I'm not dumb. I'm relatively smart. I am, however, smart enough to see that people like Terry or Ramanujan are actually that special.

Re: A digestion of the Jacobian conjecture counterexample

#86
I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise: “That’s exactly the right way to think about it.”,

“Yes, you are exactly right.”

“You have gotten to the core issue.”

And non stop praise. Seems like sycophancy is still an issue lol.

Re: A digestion of the Jacobian conjecture counterexample

#87

I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise: “That’s exactly the right way to think about it.”, “Yes, you are exactly right.” “You have gotten to the core issue.” And non stop praise. Seems like sycophancy is still an issue lol.

If there's anyone that deserves praise every time he has something to say about mathematics, it's him.

Re: A digestion of the Jacobian conjecture counterexample

#88

I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise: “That’s exactly the right way to think about it.”, “Yes, you are exactly right.” “You have gotten to the core issue.” And non stop praise. Seems like sycophancy is still an issue lol.

If there's anyone that deserves praise every time he has something to say about mathematics, it's him.

Tired of this meme, honestly.

Re: A digestion of the Jacobian conjecture counterexample

#89

Earlier quoted context omitted.

I certainly do not think that the average HN reader "could probably understand this post in a matter of days or weeks". I think the level of background information you need is something like an undergraduate degree in mathematics and at many universities that's probably not enough either.

There's a difference between understanding something and being able to do something. For example you can get a broad understanding of what calculus does in a matter of minutes: a derivative is just the rate of change of a curve, and an integral is the total area under a curve - you're practically there! Of course it'd take months of work to understand how to apply it yourself, let alone all the tricks involved, but a…

I basically completely disagree with your analysis. I am already taking into account the "understanding vs doing" distinction in my assessment. I think it takes ~an undergraduate math education to have the background to work through this post, but there's no way that most people with that background would be able to produce it.

If, like the vast majority of people, you have never studied math past a college calculus sequence and perhaps a (practically-oriented) linear algebra class, then you are missing some pretty fundamental ways of viewing and thinking about math. The basic mechanisms of algebra, for one: spaces and operations, morphisms, products and quotients... You don't know what a group is, never mind the idea of a group action. You don't have any familiarity with some basics of geometry: the Riemann sphere, Mobius transformations... And you certainly don't know anything about algebraic geometry: what an affine variety is, what birational equivalence is, what a fiber is.

All of these are concepts required to understand this blog post. And it's not just a matter of understanding the definitions of the terms, but having at least some intuition of what they really mean and how they work. Most people cannot go from zero to understanding all of this in a few days or a couple weeks, at least not for any reasonable definition of "understanding". There's too much basic mathematical background that's missing.

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