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Comment by intenex | Hacker News Reader

I think this is clear evidence that AI models are now at the far frontier of mathematics innovation and discovery and exceed human limits.

This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able to until LLMs came along.

Hard to claim at this point that LLMs aren't capable of novel STEM creativity and genius to a degree that will soon far surpass that of humans.

If anyone has counterpoints to this I'd love to hear them!

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Not a counterpoint per se, but I burned $50k recently on a much more modest math problem (result already known, just thought I had a sketch of a more interesting proof), and the LLM thought it had proved it within those bounds but had instead subtly fucked up the Lean definition. Take from that what you will.

Not to mention, it's still very much up in the air whether the model derived the answer of its own accord or sniped the important details from the researchers it was spying on.

But the researchers also did their research using essentially the same models, so that isn’t a counterpoint to AI models being at the far frontier…

> not a counterpoint

>> isn't a counterpoint

Where do we disagree?

How can you afford to burn $50k on an already solved problem?

Those are two separate questions:

> How can I afford?

Mostly not my money, tech makes one fabulously wealthy, I care more about math than fast cars or whatever (even as a pizza driver you can afford a fancy car if that's your primary motive), etc.

> Already solved

That's a very interesting insight into mathematics. It's absolutely just as interesting to prove that certain proof techniques are or aren't possible as it is to actually prove the main result, sometimes moreso, especially if they have any chance of improving attacks at other problems.

> will soon far surpass that of humans

To be fair, I think it's still an open question about how far it might surpass human capabilities.

I think it's clear that its speed of development will be significantly faster, but it's technically not proven that the frontier and problems don't themselves become increasingly difficult faster than any acceleration in intelligence past the point of human training, data and existing knowledge.

Should this be the case, we would see a rapid broadening of development, and a slow advance in the frontier in such a way that might surpass the collective capabilities of people, but not by very far.

Fields that allow verification, like math, will far surpass human level because they don't need human data for training. It's exactly the same as with Chess

How do you verify the ‚beauty‘ of a result. There have always been infinite provable Theorems around. Only some are interesting and beautiful.

> It's exactly the same as with Chess

I think you might be missing my point.

My point is that you are probably correct but it is not proven.

If the field that requires verification has an increase in difficulty which exceeds the increase in intelligence, than the capabilities might surpass human months but not by far.

For example, we saw a rapid decrease in space exploration progress after the discovery of the Moon because it is much much more difficult to get to Mars compared to the Moon. What we don't know is what the frontier looks like. It may have a gradual ramp up in difficulty, but it might also involve some sort of cliffs which even artificial intelligence is unable to climb.

I think the biggest hurdle remaining is that all these landmark results are generally counter-examples.

Proving something in the affirmative often requires the creation of an entire new sub-field of math, or new tools. Think of Fermat's Last Theorem or something like that.

These results, while impressive, are clever constructions using existing techniques. It isn't clear that AIs can build new machinery like this. But if/when they can, yeah it is probably game over.

Here you go:

When you read the detail the compute they are throwing at it is incredible, tens of thousands of agents with different groups competing.

It's not like a single Gauss as you imply, "just" many, many mathematicians working tirelessly in a completely ego-less way, guided by other agents and ultimately humans, built - allegedly - on recent human insights.

Stunning, undoubtedly, but this is a "brilliant autistic herd" result, not that of a singular mind.

> this is a "brilliant autistic herd" result, not that of a singular mind.

I slightly disagree. A single LLM is equally 'mindless' as a herd of them. As anyone will tell you they "simply predict the most likely next token," yet, complex solutions to difficult problems arise from them.

Many people have said that the architecture of LLMs will need to change for true ASI. I think that the herd of tens of thousands of agents can be seen as one such potential architectural extension. Whether or not a herd or a single LLM is used for a result like this is irrelevant.

To be clear, I think the orchestration of thousands of LLMs in their current form, even with ever increasing intelligence, is not the form ASI will take. There is still a major architectural breakthrough to come, in my limited, ignorant opinion.

Maybe in a few more months we'll read a blog.post about an independent wiki owner who suddenly had a spike of traffic on their site

Sure, even a 20% chance at 1 million payday after 5-6 years of fulltime work on a project with zero practical application doesn't touch the, say, 200k/year guaranteed our best mathematicians would have to forgo to devote their intellect to the problem.

Are these mutually exclusive? Why would you have to forego that salary to work on this problem? This is one of the most prestigious and meaningful problems in all of mathematics, which is why it has such a high prize amount attached to it - why would a university not support a mathematician working on such a prestigious and important problem in lieu of something else?

Publish or perish? We're talking devotion here -- so no time to do anything (like edit proofs) other than try to solve the problem.

[Edit: my only point here is that the prize is probably not driving human effort to the limit.]

i wonder how many tokens it takes to run 10,000 agents? One could argue this is simply a problem of appropriations. I find myself wondering if a corporation could spend $5M on mathmeticians and arrive at the same end result.

Thousands of some of the brightest minds have worked on this problem for over a century. The million bucks is not the big deal here.

>If anyone has counterpoints to this I'd love to hear them!

Sure. A proof without an unknown amount of human steering (and/or stolen research) would be an unquestionable achievement.

To this day there's zero (0) evidence of any result by an LLM alone (maybe I'm wrong). If I just prompt ChatGPT right now with "give me a proof of the Riemann Hypothesis" and this thing delivers, I'm sold. But anything close to "yeah ChatGPT proved X with 5 years of 24/7 work with 10x Terrence Tao level geniuses" it really doesn't cut it.

Or why's there's no new branch of mathematics invented by AI? That'd be indubitably _novel_ and _creative_. But to my knowledge (and I'm eager to be educated) there's nothing like that. What are the HARD examples of novelty, creativity and genius you claim? For how people like you talk about AI I'd expect idk, a unified theory on fundamental physics, or a novel engineering solution for material science and nuclear fusion, or at least improve itself to not need a bazillion GPUs to emulate a 20 watts wetware. Sure it would infinitely easier to make OpenAI literally print money with any of the thousand problems easier to solve with such amazing intelligence than the NSE problem right? Honest question

The amount of goalpost moving is insane. "Yeah it can solve Millenium problems, but can it do it with nothing more than a one sentence prompt?"

Also there are proofs where the only human steering was "keep going".

Parent comment is claiming creativity and genius beyond human experts, so why not ask for a fully unassisted AI novel result? Having access to the entire corpus of human knowledge, what else such amazing entity would require to solve a hard problem by its own?

Any result of such kind from an AI alone would be enough to refute my argument, yet you don't present any.

>Also there are proofs where the only human steering was "keep going".

Which ones?

https://www.anthropic.com/research/riemann-zeta

>Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.

Do you even know what a proof entails? This shit is not a proof, come on.

What? The article states:

    > Drawing on extensive prior research by mathematicians over the past decades, it 
    > [Claude] has increased this bound [for the fraction of zeros of the Riemann zeta 
    > function that satisfy the Riemann hypothesis] from 41.6% to 67.2%. Claude also 
    > produced a formally verifiable proof of its result.

How is a formally verifiable proof not a proof? You're making literally no sense.

Well, in fairness, the OP asked for counterpoints. He didn't stipulate that they need to be reasonable.

Extraordinary claims require extraordinary evidence. If OP claim superhuman genius, then they should prove superhuman genius. Simple as that.

> I'd expect idk, a unified theory on fundamental physics, or a novel engineering solution for material science and nuclear fusion, or at least improve itself to not need a bazillion GPUs to emulate a 20 watts wetware.

I think the counterpoint here is simply to look at what was being achieved with LLMs one year ago versus today, and extrapolate that trend. Sure, there may not be examples of what you've asked for yet, but Astra is literally a couple of months old, the model that solved Navier-Stokes is less than two weeks old. It appears that we're seeing the hockey stick that only the most bullish thought was possible.

See I think it’s clear demonstration that OpenAI is ethics-free

Eh... OpenAI spent significantly more than $1 million solving this...

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