Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Claude Fable produced a counterexample to the Jacobian Conjecture

xcancel.com

Post reply on HN
Print view View original

361–370 of 562 posts

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#361

Post by nujabe »

dist-epoch wrote: post #327
Sophira wrote: post #252

Earlier quoted context omitted.

Can you please share the chat log for this? I would absolutely love to see this.

It's easy to reproduce, I've fed the example in GPT-5.6 Sol Max and it started multi-checking it in all kinds of ways, with multiple symbolic packages then manual computation, then it did extensive literature search on the subject, looked at tens of math websites, extensive arxiv research. this was soon after it was posted, it didn't find the original twits with the finding

I’m sure it’s trivial to reproduce but we don’t want to waste those tokens

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#362

Post by Certhas »

rvz wrote: post #48
kmaitreys wrote: post #42

Maybe not? https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...

I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online. Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.

This topic really is a testament to people's willingness to opine on things they have absolutely no clue about.

A first year undergraduate can completely check this counterexample in ten minutes. The original post even linked Wolfram alpha for the calculations.

And if you genuinely try you can very quickly understand using only high school math and a bit of Wikipedia that this counterexample is vanishingly unlikely to be wrong, even if you don't do the calculations yourself.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#363

Post by kevinwang »

inglor_cz wrote: post #234
laichzeit0 wrote: post #32

The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win. It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.

I used to be a young mathematician sometime at the turn of the century... "waste their time trying to prove it" is the MBA approach, where you should spit out results and articles. Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the ext…

Yeah but the Collatz probably has one of the highest man-hours of actual waste.

Ergo: > “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.

and

> “Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.

(from https://www.quantamagazine.org/mathematician-proves-huge-res... )

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#364

Post by jll29 »

slopinthebag wrote: post #179
Lerc wrote: post #146

Earlier quoted context omitted.

> "Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end. Everyone has their own experiences making it hard to judge what attitudes are prevalent. From what I have encountered the Anti-AI people have been more common and more outright scary. I have heard some ideas from the Pro AI crowd and thought to myself "Yeah, that's not going to happen anytime soon", There was enthusiasm leading to over…

I initially wanted to downvote you but I can’t because you’re right. I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea…

I understand you. As an AI research (actually, despite formally called professor of AI, I have historically always called myself more specifically NLP/IR/ML researcher rather than "AI" researcher, to avoid historic baggage).

I am intellectually curious whether "intelligence" and "consciousness" are simply layers that can be separated from the hardware they operate on or first emerged from, but I never had the urge or saw the point to practically replace human beings by machines.

Not everything that can be done should be done.

When I go to a supermarket, I always avoid self check-out. Why? First, I enjoy human interaction, even if machines may be more efficient (not yet the case). Second, I don't want the people working at the checkout to lose their jobs.

When we purchased a GPU cluster, I felt bad when I learned that the electricity bill was going to be north of 60 000 EUR every year (that number apparently assumes no jobs are running - so 100% idle time).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#365

Post by Lockal »

dist-epoch wrote: post #334
Lockal wrote: post #316

Earlier quoted context omitted.

I fed it to Google AI Studio, enabling tool execution and disabling web access. It also quickly verified it with SymPy, then went into psychosis. 5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding system…

> all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English. > "ground-breaking results that could be attributed to Gemini, if act quick" This would be such a dumb thing to do, and so easy to get caught with...

That is not dumb - erasing copyrights is just their business, and even when caught has zero consequences (for them). Providers can use any users input for improving their models, either by consent, or by flagging any dialogue for safety review (nonconsensually), or by training on whatever content they want anyways (obtained via torrents from pirate sites with U.S. court approval).

On top of that, I retried the same question + one simple question, and again, same behavior - second JC chat is loading forever. That's more just a funny observation over Gemini - today this is very likely some internal issue, tomorrow it can be used for plausible deniability against copyright accusations.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#366

Post by abernard1 »

cornholio wrote: post #357
abernard1 wrote: post #275

Earlier quoted context omitted.

I think this is actually a perfect case of an impressive thing being done exactly by mimicry. Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine? Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not qui…

The field now favors into the view that symbolic manipulation is not the mechanism of general intelligence, but rather an emergent byproduct of learning. So the fact that a connectionist machine (neural network) got so good at symbolic manipulation actually supports the view that we are closing the gap to general intelligence. Through the rote work, the machine really internalizes those rules and the symbolic manipul…

I am not confused. Because herein these forums, I predicted everything that was going to happen years ago.

And the "insane inefficiency" of deep learning is fully to be expected from how it works. As well, there are provably no—literally no—emergent properties in these models. The choice of metric was a convenient, sloppy, and embarrassing fault of the field. It should be discredited; the field should be embarrassed; expectations on messaging should have changed; and it did not.

Why? Because the industry is full of charlatans, and this is a highly profitable enterprise telling people that this would lead to AGI.

Multiply two floating point numbers without a tool call. Still can't, because it's a curve fit.

So, in summary, nothing you just said is relevant. There are no emergent capabilities, simply 1) search, + 2) the original set of learned feature vectors from throwing tons of data at this.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#367

Post by meowface »

QuesnayJr wrote: post #347
betafj wrote: post #337

Earlier quoted context omitted.

If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?

No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.

This would seem to lead to absurd implications. What if in, say, 20 years, an AI is able to independently prove nearly everything important in under an hour, including the Riemann hypothesis, with no contamination from other proofs? (Let's say it's also free to write and run arbitrary code, as well.)

Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway would be the smartest humans were never close to the theoretical intelligence and wisdom ceiling and never could've been. This will one day seem obvious in retrospect. There's no reason evolution by natural selection would've landed any species near such a ceiling.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#368

Post by jlev1 »

Davidzheng wrote: post #324
Davidzheng wrote: post #269

Earlier quoted context omitted.

I'm going to paraphrase what GPT told me: Consider the canonical degree 3 (subvariety of the trivial P1 bundle consisting of zeros) cover of the projectivization of homogenous polynomials of degree 3 in 2 variables (so it's a 3fold cover of P^3). The top space is P1 x P2 and if you take a standard affine open of the base and look at the cover over that restricted to a subset where the zero of the cubic is simple you…

here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have fo…

My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the form (p, {p, q}) and (p, {q, q}). Then it takes some calculation to identify X with A^3.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#369

Post by dooglius »

david-gpu wrote: post #314
InsideOutSanta wrote: post #298

Earlier quoted context omitted.

If I came up with this counterexample, I sure as hell wouldn't give credit to Claude.

Indeed, the incentive is the opposite: hiding the fact that they used an AI would boost their own personal brand.

He worked for Anthropic, so such a claim would be very unconvincing

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#370

Post by gmadsen »

mathisfun123 wrote: post #77
gmadsen wrote: post #70

Earlier quoted context omitted.

Why is your example SAS triangles and not something like the modularity theorem?

Because SAS triangle identity proofs are relatable and I'm trying to make a point...?

And my point is it’s a strawman. Advanced math is obviously creative to those who practice it, and it’s disingenuous to boil it down to grade school triangle proofs

That would be like saying Shakespeare isn’t creative , since grade school grammar doesn’t feel creative

Post reply on HN