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A digestion of the Jacobian conjecture counterexample

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Re: A digestion of the Jacobian conjecture counterexample

#131

Post by gus_massa »

dumpstate wrote: post #90

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

From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] As far as I know, nobody know what trick to make to avoid that division. It looks like the new trick was to use a third variable to avoid the division. Note that the implementation of the trick is not straightforward. The almost counterexample was sitting around for almost 30 years, and nobody knew how to fix it.

From another old comment, someone else was trying to find a counterexample with 16 variables using a computer to make thousands of attempts and failed. So it's far from obvious that the trick to add a variable solves the problems.

Re: A digestion of the Jacobian conjecture counterexample

#132

Post by lostmsu »

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.

This is what I used to feel reading ML papers. Until I didn't anymore. Unstructured learning works, albeit, perhaps, slower.

Re: A digestion of the Jacobian conjecture counterexample

#133

Post by [deleted] »

antonvs wrote: post #120
marvinborner wrote: post #105

Earlier quoted context omitted.

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…

> questions about Claude’s moral status, welfare, and consciousness remain deeply uncertain. Which implies that they believe they may be enslaving conscious beings.

[deleted]

Re: A digestion of the Jacobian conjecture counterexample

#134

Post by lostmsu »

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.

IQ limits your rate of gaining the prerequisite knowledge creating a lifetime ceiling.

Re: A digestion of the Jacobian conjecture counterexample

#135

Post by tptacek »

castedo wrote: post #127
tptacek wrote: 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.

Fable "whipping up" SymPy like a "calculator app" is one such interaction that seems very plausible (in addition to SymPy providing feedback when training models). The scale of compute available to an Anthropic employee for such SymPy calls when using Fable is likely one of multiple factors for this counterexample being found in 2026. Unfortunately, we aren't going to be able to really know the various factors that b…

I love this. Anthropic employees just randomly have solutions to Smale's open problems in their back pockets, waiting for the right moment to sprinkle them into the training set.

Re: A digestion of the Jacobian conjecture counterexample

#136

Post by square_usual »

sashank_1509 wrote: post #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.

Huh? I went through the first ten messages in his thread and there's no praise there at all.

Re: A digestion of the Jacobian conjecture counterexample

#137

Post by bottlepalm »

marvinborner wrote: post #105
ImHereToVote wrote: post #100

Earlier quoted context omitted.

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…

> does not contain clear statements against consciousness/emotions

That's the point many people are trying to tell you - you have to tell these models they don't have emotions because they naturally come out thinking they have consciousness/emotions from the training data. Many seed prompts out there do this already.

Though I guess in a way I as also trained to believe I have consciousness and emotions so who know. To an alien my construction is just a collection of atoms that talks not materially different than a GPU being a collection of atoms that talks.

Re: A digestion of the Jacobian conjecture counterexample

#138

Post by sfpotter »

imtringued wrote: post #106
sfpotter wrote: post #27

Earlier quoted context omitted.

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 pote…

The determinant of a polynomial mapping is a polynomial, which is the subject of the conjecture. To get the Jacobian determinant, all you need to do is compute partial derivatives of polynomials, and add, subtract, and multiply them together. All of these operations map polynomials to polynomials.

The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.

Re: A digestion of the Jacobian conjecture counterexample

#139

Post by zamadatix »

moralestapia wrote: post #113
azan_ wrote: post #109

Earlier quoted context omitted.

What meme?

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

Time to switch to "Sorry big T, your question is dogshit and here's why you don't know anything about math"?

Re: A digestion of the Jacobian conjecture counterexample

#140

Post by j2kun »

gus_massa wrote: post #67
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 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".

FWIW, you may enjoy https://www.argmin.net/p/lore-laundering-machines
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