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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

#31
post #13

Earlier quoted context omitted.

I believe it's a reference to a joke meme that goes something like "Write Windows 12 from scratch. Make no mistakes." At least that's the first context I heard it in.

Precisely

You are "not allowed" to make that joke here. [0] and you probably upset a bunch of grumpy meanies.

[0] https://news.ycombinator.com/item?id=48838228

Re: A digestion of the Jacobian conjecture counterexample

#32
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…

There really was a time when “make no mistakes” was thought to increase response quality. Maybe it steered the content of the thought channel?

Re: A digestion of the Jacobian conjecture counterexample

#33

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

it doesn't overturn much. For example, here is a post from 2004

https://www.math.columbia.edu/~woit/wordpress/?p=105

it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.

Anyway, in that post it says

> It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*

so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.

Re: A digestion of the Jacobian conjecture counterexample

#34

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.

Re: A digestion of the Jacobian conjecture counterexample

#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.

Re: A digestion of the Jacobian conjecture counterexample

#36
post #21

Earlier quoted context omitted.

He’s famous, but I don’t believe he is a celebrity, as he is not famous for his persona.

I just mean, anything Tao writes that is related to AI will get on the front page, because Tao represents the authoritative voice of reason using AI tools. And this website is primarily for AI news.

[dead]

Re: A digestion of the Jacobian conjecture counterexample

#37
post #34

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.

> a human prompting with deep math expertise

The original tweet implied that the whole thing was done while the author was watching the World Cup final.

I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.

Re: A digestion of the Jacobian conjecture counterexample

#38
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.

Some people downvoting you, but I think it is a valuable illustration of IQ gap.

And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.

Re: A digestion of the Jacobian conjecture counterexample

#39
post #34

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.

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