[koito17]

A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients.

They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.

To think that Claude Fable was able to find a counterexample in degree 7 is insane to me. We are truly in a new era.

[NooneAtAll3]

you might be confusing 2 variable case (which indeed was tested to 150+ degree) and 3 variable case (this counterexample)

[zaik]

Would this counterexample not be included in their search space?

[koito17]

They were looking at 16 variables, degree of about 10 in each one. That search space is simply too big for a plain bruteforce. So they did some sort of filtering to reduce the search space to a pool containing "possible counterexamples".

There was also a paper giving a lower bound of about 100 for possible counterexamples in that particular framework. Later raised to 108 in https://arxiv.org/abs/2204.14178

I don't remember the details very well since this was back in 2015 and wasn't really involved in the research. Consider this to be some sort of telephone game between what I heard in 2015 and what I remember today.

[jasonfarnon]

based on the approach described they were probably in dimension 2, this counterexample is 3 dimensional

[inigyou]

    16 variables

[0x1ceb00da]

[flagged]

[antirez]

So many mathematicians over the years tried hard and failed, but now Anthropic just for some PR magically did it? And this after LLMs obtaining different math wins? What is your logic here really escapes my understanding.

[glimshe]

The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.

[pred_]

I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided.

Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bit much), but I think the argument is sound, so flagging seems counterproductive to the discussion.

[zaptheimpaler]

Believing that AI played a very small role while we know this problem was open for decades with at least a few people taking serious cracks at it is just not a coherent logical position.

[soerxpso]

That doesn't really even diminish the contribution from Fable, if true. Droves of grad students have been provided the same sorts of non-trivial insights and turned up no results.

[glimshe]

I'm sure they had plenty of time to think about these insights without the LLM, as well as the many other mathematicians who tried to crack it over the years. Wether the LLM was simply an assistant or solved the problem entirely isn't as important as accepting than the LLM was the essential, previously missing piece in the solution.

[witx]

That's HN. No criticism allowed of llm whatsoever

[minimaxir]

> I think the argument is sound

They got flagged because it is a literal conspiracy theory that assumes bad faith.

[latexr]

> literal conspiracy theory

The strongest words they used were “marketing stunt”. Calling that a conspiracy theory is quite the stretch.

[abc42]

If you can call it polarized when a majority of people are just happily using the technology while a minority keeps spreading delusions and hatred.

[raxxorraxor]

Those silly advertisers do everything for exposure and if that means digging yourself into a niche alleged mathematical theorem to refute it, it is what needs to be done!

Of course it would be really interesting how Claude approached this. Probably with some constraints regarding the input. And it would be interesting what these constraints were.

[dist-epoch]

Author probably doesn't want to show the prompt because they are now trying to find a bunch of other counter-examples with the same prompt

[regularfry]

The reasoning trace would be far more interesting, and that's not exposed.

[YeGoblynQueenne]

I mean we have no idea what happened exactly, how Fable was used, how many times it was run, whether earlier models were also tried, what was the prompt, how long it run for, etc etc. All we have to go by is a tweet.

Why not be skeptical about that?

[CamperBob2]

What you're asking for is exactly the sort of thing that belongs in, and will appear in, a journal article. There will likely be a preprint on arxiv, so you might keep an eye out for that.

In any case, the fact that it was found by a commercial model means that the unfiltered reasoning trace isn't available even to the original author. So there are aspects of the problem-solving process we'll never see. Even if we did get access to the reasoning trace it wouldn't necessarily be definitive, given how these things work.

Hopefully it'll be possible to get the same solution from an open-weight model like one of the 3T heavyweights that are said to be coming up for release. If so, the chain of thought can be scrutinized in-depth.

[YeGoblynQueenne]

No, I think it works the opposite way. Until there is an article somewhere that describes what happened, if there is one, all we have to go by is that some guy posted a counter-example for the Jacobian on X, with a vague allusion to using Fable and without any further information. Assuming and guessing anything about e.g. the method used at this point is just raising the noise level.

[CamperBob2]

No one can figure out where you're coming from here. If the human solved a significant open problem without relying on AI, don't you think they'd have claimed the credit for themselves?

The suggestion that a human, working at Anthropic or elsewhere, did the hard work needed to disprove the Jacobian Conjecture yet chose to claim falsely that their AI did it, amounts to an extraordinary accusation that requires extraordinary proof.

[eru]

The original author works for Anthropic. And even without that, Anthropic might find this important enough to dig up the trace?

[jibal]

They gave their "logic", such as it is ... and it's utterly irrational.

Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

[gus_massa]

> There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

My guess is that the chat will look similar to a full transcription of a (multi month?) discussion between a few mathematicians. Full of dead ends and stupid errors (bit by the human and Claude) that would be embarrassing. We all know how bad it is, and we prefer to keep it behind the curtain.

[viccis]

Here's an example of one for a major result earlier this year: https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...

Who knows what "rewritten" means, but you can sort of see how it progresses.

[8bitsrule]

A few months ago I asked a model how many primes are divisible by 35 with a remainder of 6. It confidently replied 'none'.

Counterexample: 35 + 6.

[buzzin__]

Kimi 2.6 gives the answer ""By Dirichlet's theorem on arithmetic progressions, since gcd(6,35)=1 , there are infinitely many primes of the form 35k+6 . So the answer is: infinitely many primes give a remainder of 6 when divided by 35.

[buzzin__]

But, if the reminder is 6, they are not really divisible, are they? Try again with a sentence that actually makes sense: "How many primes, when divided by 35, give a reminder of 6?"

[jibal]

non sequitur

[8bitsrule]

Perhaps ... but the lesson in trusting AI math was worth it.

[0x1ceb00da]

> some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization

Why do you trust a random stranger so much? Will you hand over your car keys to a random stranger? Sharing the chat will take 30s of their time.

> There's no reason to think that it won't provided if asked for

But they didn't provide it.

[Liquid_Fire]

OK, so the two options are:

A) Claude really produced this counterexample

B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes

If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than solving the mathematical problem in the first place?

[fn-mote]

The concern is that there could have been expert knowledge input, whose importance/worth we are unable to evaluate.

I don’t believe a mathematician produced the counter example secretly, but how much did they contribute to the result?

AI isn’t magic, so to evaluate the value delta, you need to know the value of the input.

[Izkata]

Weirder has happened: https://mathsci.fandom.com/wiki/The_Haruhi_Problem

[InsideOutSanta]

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

[david-gpu]

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

[jibal]

> Why do you trust a random stranger so much?

Why do you make false claims and attack strawmen so much?

[dist-epoch]

> But they didn't provide it.

because they are using the same prompt to try finding other counter-examples. they are milking it

[William_BB]

There's a big difference between one shotting a counterexample using AI and using AI to find a counter-example by brute-force.

Both are impressive, of course, but they're hardly comparable.

[eru]

I'm not so sure. People had been trying to use brute force to find counterexamples before.

[William_BB]

AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.

[igravious]

"using human-like reasoning (in a brute-force way)"

that's self-contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it)

using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree

[anxoo]

if you ask the chatbots for "list of top unsolved math problems", the JC comes in at a ranking of around #10 - #20. what, a problem that's been unsolved since 1939 was cracked because anthropic has an underground sweatshop of math Phds cranking out research, just so that they can slap "made by AI" on it? hell, maybe lizard people did it.

[abc42]

This anti-AI sentiment is getting borderline insane.

[oska]

1. There is no such thing as "Artificial Intelligence" (but yes, machine learning and LLMs are real and powerful things)

2. Calling LLMs 'AI' is part of the grifting hype that has been wildly prevalent in the US corporate space. It's very natural and also very good that there is now backlash to and scepticism around all of the hype that is has been generated over the last 5 or so years

[UltraSane]

Finding a counterexample that humans have failed to find for 85 years is a marketing stunt?