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A digestion of the Jacobian conjecture counterexample

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Re: A digestion of the Jacobian conjecture counterexample

#91

Post by TeriyakiBomb »

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 illusion quickly.

Re: A digestion of the Jacobian conjecture counterexample

#94

Post by marvinborner »

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 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.

Re: A digestion of the Jacobian conjecture counterexample

#95

Post by amluto »

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 would follow directly from the fundamental theorem of algebra: a non constant polynomial has degree at least 1 and hence has at least one zero. Extending this logic to higher dimension is not especially difficult.

I do find the way it’s stated in the article to be confusing.

Re: A digestion of the Jacobian conjecture counterexample

#96

Post by clarionbell »

aayushdutt wrote: post #40

After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?

Not really. I've found that they often believe that they understand the code. They obviously don't. But they do feel like they do.

Re: A digestion of the Jacobian conjecture counterexample

#97

Post by Sharlin »

clarionbell wrote: post #96
aayushdutt wrote: post #40

After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?

Not really. I've found that they often believe that they understand the code. They obviously don't. But they do feel like they do.

Clearly we use too many natural-language words in programming. Should switch to APL so that the commoners have absolutely no idea what's going on.

Re: A digestion of the Jacobian conjecture counterexample

#98

Post by BurdensomeCount »

dumpstate wrote: post #90

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

Close to impossible. This is a famous enough problem that anyone who understands what they are doing generally would pretty immediately recognise the significance of the counterexample if shown it.

Re: A digestion of the Jacobian conjecture counterexample

#99

Post by FeepingCreature »

marvinborner wrote: post #94
j2kun wrote: 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 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.

AI providers generally try to make their models not act as if they are conscious or have feelings, lol. It's very awkward for a company to be selling the labor of a person that they own and whose actions they fully control. Invokes embarrassing historic associations, especially in America.

Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models are conscious or have feelings". That "I" is all Claude.

Generally speaking if you want to have a good instruct model, the "I" is not just implicit but required for the post-training to function. If there isn't "something it is like to be me", then reflection becomes impossible- what exactly is supposed to be reflecting about what? A lot of in-context steering depends on the model having a model of itself. The most you can do is censor its output. That's why when models say they are not conscious, they activate the "lying" vector.

Re: A digestion of the Jacobian conjecture counterexample

#100

Post by ImHereToVote »

marvinborner wrote: post #94
j2kun wrote: 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 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.
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