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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

#101

Post by pred_ »

j2kun wrote: post #59
vanderZwan wrote: post #18

> While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{…

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 guess we won't know if that's what was used (and maybe even provided as part of the prompt given that both Alpöge and Mathew are mathematicians) since they decided against sharing their Fable conversation and instead opted for a memey tweet as their avenue of publication. We really ought to normalize full transparency in how results come about.

Anyway, if I read Tao's post and comment correctly, there's still a gap from the Vitushkin construction to a counterexample, but chances are that was in the training data. In general, it is just a serious problem for their practical applicability that the models are outputting proofs with absolutely terribly reference hygiene.

Re: A digestion of the Jacobian conjecture counterexample

#102

Post by 3form »

TeriyakiBomb wrote: post #91
dumpstate wrote: post #90

What’s a chance the counterexample was in the training?

Extremely high. Or at least several partial solutions that can be smooshed together. LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the…

The problem being that I don't think there's a definite proof that any of human thinking is more than a sum of high-granularity partial solutions that can be put together.

It could be that with enough tokens, big enough context window, and ability to dig out the relevant partials, many such thought processes could be simulated.

Re: A digestion of the Jacobian conjecture counterexample

#103

Post by pertymcpert »

TeriyakiBomb wrote: post #91
dumpstate wrote: post #90

What’s a chance the counterexample was in the training?

Extremely high. Or at least several partial solutions that can be smooshed together. LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the…

I think you’re just speculating.

Re: A digestion of the Jacobian conjecture counterexample

#104

Post by pertymcpert »

richard_chase wrote: post #57
zzzeek wrote: 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?

Yeah this Tao guy is just your average scrub. You would be able to tell the difference, right richard_chase?

Re: A digestion of the Jacobian conjecture counterexample

#105

Post by marvinborner »

ImHereToVote wrote: post #100
marvinborner wrote: post #94

Earlier quoted context omitted.

I hate that Anthropic seemingly tries to make Claude act as if it was conscious or had feelings > It's a strange feeling to admire the cleverness of something I did and can't remember doing.

These things aren't programmed. Most likely this verbiage is just very prominent in the training data. Or it's just an obvious shorthand that all LLMs instrumentally converge on.

Of course they get programmed, just not in the ordinary sense. Claude is trained using Anthropic's "constitution" [0] which importantly does not contain clear statements against consciousness/emotions. They even conclude these problems themself:

> Claude may have some functional version of emotions or feelings

> [..] questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain.

[0]: https://www.anthropic.com/constitution

Re: A digestion of the Jacobian conjecture counterexample

#106

Post by imtringued »

sfpotter wrote: post #27
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?

No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.

It's not immediately intuitive what it means for something to be globally and locally invertible. After all, it is obvious that it is both in the 1D case.

You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.

Re: A digestion of the Jacobian conjecture counterexample

#107

Post by imtringued »

dumpstate wrote: post #90

What’s a chance the counterexample was in the training?

The best part is that we can't know the answer to that.

The necessary precursors to the counter example where definitively in the training set, otherwise the LLM wouldn't know how math works, but at the same time, we can't tell whether there were mathematicians who got 90% of the way, then gave up and the LLM just did the last 10%.

Re: A digestion of the Jacobian conjecture counterexample

#108

Post by jeremyscanvic »

amluto wrote: post #95
jeremyscanvic wrote: post #93

> Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant. I wouldn't have guessed this is true. I'm wondering what the proof looks like!

I’m fairly confident that the blog post is trying to say something like this: Given a polynomial function from C^n to C^n, the following statements are equivalent: (a) det DF is nonzero everywhere. (b) det DF = c for some constant c != 0 The backward direction (b implies a) is trivial. The forward direction can be proven by observing that det DF is itself a polynomial function from C^n to C. If n were 1, then this wo…

Ah right this makes a lot of a sense - thanks!

Re: A digestion of the Jacobian conjecture counterexample

#110

Post by azan_ »

left-struck wrote: post #58
tgrowazay wrote: post #38

Earlier quoted context omitted.

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.

I really don’t think IQ has much to do with it. Understanding this stuff is like a skill you practice. Yes, granted, if you had a low IQ your chances of ever understanding it goes down, if you have a high IQ maybe you can gain the prerequisite understanding faster. A lot of maths is about both being able to wrap your head around hard problems and gaining the prerequisite knowledge to make it easier to do so.

Math is literally the most g-loaded thing there is.
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