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What sort of maths are LLMs good at?

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Re: What sort of maths are LLMs good at?

#121

Post by Jensson »

sdenton4 wrote: post #72
seanclayton wrote: post #68

Earlier quoted context omitted.

Would you trust a bridge built by it with zero human interference? A train? A plane? A skyscraper? What about fight in a war? Unless you can, I wouldn't consider it AGI, because you're actually trusting human intelligence to verify the bridge or train or plane or skyscraper is safe or that the robot is following orders. And even then, you're trusting human-influenced guardrails etc. I would consider it AGI when an AI…

Would you trust a bridge built by a single human? In reality, we have lots of guardrails to ensure that we don't screw up and kill a lot of people by deploying defective bridges (or cars to drive on them). Those guardrails often are written in blood, and still occasionally fail.

> Would you trust a bridge built by a single human?

But an AI doesn't replace a single human, it replaces every human in the project, so it needs to match a team of humans not a single human.

Re: What sort of maths are LLMs good at?

#122

Post by Jensson »

datsci_est_2015 wrote: post #105
seanclayton wrote: post #88

Earlier quoted context omitted.

Why not have AGI check other AGI? Peer review by fellow humans is what gives humans assurance, to the degree in which review was done by peers of equal or greater intelligence. AGI checking other AGI should give you that same trust, no? Deepseek says my ChatGPT bridge is stable, you should trust it. Claude says it's stable. The humans say it isn't, but they aren't AGI. You can trust this bridge because it's been vett…

One step further, our current generative AI is not discrete. “AGI checking other AGI” doesn’t really make technical sense. Currently, every interaction between two generative AI units is treated as agglomerative. We talk about “Claude”, not the 70 subagents that Claude spun up to achieve a task. In other words, as soon as two generative AIs interact, they become one. Our current definition of AI (generative AI operat…

Yes, and that is why you can't say "its as smart as a single human" and think its as smart as humans, because humans can get smarter by cooperating, if the AI can't do that to create a smarter AI it is not as smart as humans, meaning an AI that is not as smart as a team of humans is essentially not as smart as a single human either.

Re: What sort of maths are LLMs good at?

#123

Post by sdenton4 »

seanclayton wrote: post #88
sdenton4 wrote: post #72

Earlier quoted context omitted.

Would you trust a bridge built by a single human? In reality, we have lots of guardrails to ensure that we don't screw up and kill a lot of people by deploying defective bridges (or cars to drive on them). Those guardrails often are written in blood, and still occasionally fail.

Why not have AGI check other AGI? Peer review by fellow humans is what gives humans assurance, to the degree in which review was done by peers of equal or greater intelligence. AGI checking other AGI should give you that same trust, no? Deepseek says my ChatGPT bridge is stable, you should trust it. Claude says it's stable. The humans say it isn't, but they aren't AGI. You can trust this bridge because it's been vett…

This is really the point: We build systems and processes which help ensure that we get reliable outputs (bridges) from unreliable actors (human workers). The point is not that we will one day replace all of these processes with AGI. The point is that we will still want processes and systems in place to ensure reliability.

The importance of safety process is largely a function of the risk of failure (ie, how unreliable the production process is) and the cost of failure. If you've got a very reliable production process, perhaps you need less safety process. But if the cost of failure is measured in lives, you're still going to want to have checks in place, regardless of who/what is responsible for designing the bridge.

Re: What sort of maths are LLMs good at?

#124

Post by wyager »

TuringTest wrote: post #120
zahlman wrote: post #119

Earlier quoted context omitted.

> Sampling is what AI is good at. You might think so, but I tried asking ChatGPT to solve one of the puzzles from https://en.wikipedia.org/wiki/Countdown_(game_show) (which a Python script can brute-force on my 12-year-old hardware in half a second) and it made an elementary arithmetic error that's decidedly not human-like.

LLMs are terrible at anything systematic. They're incredibly good at anything heuristic, so it makes sense that they can explore wide mathematical spaces fast and converge towards interesting regions. But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size. You need to combine them…

> But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size.

It's interesting how people will comment on LLM capabilities despite clearly not having engaged with frontier models in any meaningful way in a long time

Having models write Lean proofs of mathematical claims is standard operating procedure for any LLM math discovery!

Re: What sort of maths are LLMs good at?

#125

Post by calf »

The Vinogradov example seems curious. The "existential N" there is for encoding the intuition that the property holds eventually, e.g., that we don't know or don't care about which particular N along the number line for which P(n) is eventually true (forall n such that N n). So in this interpretation, it is like using the quantified E like a Sorities/heap/vagueness argument, it is functioning metamathematically. Mentally I am picturing a number line, and then there's a vague area on the line where for everything to the right of the line it is colored "P". So why isn't this forall-exists usage a bit pathological, isn't it more that Vinogradov's theorem is "halfway" between either type of quantification? And that seems a very special case of how counterexamples break down rather than the general case? Could someone more mathematically knowledgeable explain this?

Re: What sort of maths are LLMs good at?

#126

Post by georgeven »

root-parent wrote: post #44
scronkfinkle wrote: post #8

> A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think…

>> To me, this is more or less what I would think "Artificial General Intelligence" is So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621 "Our testing shows humans can solve 100% of the environments, in contrast to frontier AI systems which, as of March 2026, score below 1%." Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent. https://ww…

ARC-AGI-3 is actually about action efficiency not solve rate, current LLMs just make a bunch of moves that inefficient, thereby lowering their score

Re: What sort of maths are LLMs good at?

#127

Post by [deleted] »

wyager wrote: post #124
TuringTest wrote: post #120

Earlier quoted context omitted.

LLMs are terrible at anything systematic. They're incredibly good at anything heuristic, so it makes sense that they can explore wide mathematical spaces fast and converge towards interesting regions. But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size. You need to combine them…

> But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size. It's interesting how people will comment on LLM capabilities despite clearly not having engaged with frontier models in any meaningful way in a long time Having models write Lean proofs of mathematical claims is standard op…

[deleted]

Re: What sort of maths are LLMs good at?

#128

Post by gowld »

scronkfinkle wrote: post #77
root-parent wrote: post #44

Earlier quoted context omitted.

>> To me, this is more or less what I would think "Artificial General Intelligence" is So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621 "Our testing shows humans can solve 100% of the environments, in contrast to frontier AI systems which, as of March 2026, score below 1%." Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent. https://ww…

> So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621 I don't, we originally had the turing test which was designed to determine human intelligence by its ability to imitate us with natural dialogue, but we've since defeated that. I stated "to me" because it's my personal opinion on a definition whose goalpost will probably never stop being moved. > Back 1996, EQP automatically solved the Robbins…

You wrote:

> > It's the cumulative knowledge of all general human intelligence, baked into an artificial form, which can then use that knowledge to achieve novel goals.

But it (currently) can't solve puzzles like ARC-AGI-3 that children can solve.

Re: What sort of maths are LLMs good at?

#129

Post by gowld »

root-parent wrote: post #44
scronkfinkle wrote: post #8

> A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think…

>> To me, this is more or less what I would think "Artificial General Intelligence" is So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621 "Our testing shows humans can solve 100% of the environments, in contrast to frontier AI systems which, as of March 2026, score below 1%." Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent. https://ww…

One question I have about ARC-AGI-3 is how much it depends on vision ability, which has a substantial hardware component that isn't "General" "Intelligence".

What happens if the game is encoded in a non-visual logical form?

Re: What sort of maths are LLMs good at?

#130

Post by gowld »

Jensson wrote: post #121
sdenton4 wrote: post #72

Earlier quoted context omitted.

Would you trust a bridge built by a single human? In reality, we have lots of guardrails to ensure that we don't screw up and kill a lot of people by deploying defective bridges (or cars to drive on them). Those guardrails often are written in blood, and still occasionally fail.

> Would you trust a bridge built by a single human? But an AI doesn't replace a single human, it replaces every human in the project, so it needs to match a team of humans not a single human.

That's trivially solvable by deploying multiple AIs on one project, so that can't be the true obstacle.
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