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Ten advances in mathematics and theoretical computer science

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Re: Ten advances in mathematics and theoretical computer science

#81

Earlier quoted context omitted.

> AI has no self-awareness What is your mechanistic model of self awareness that yields this conclusion? > It's a tool Does your model suggest that tools can't have self awareness?

Dunno about the parent commenter, but I personally interpret the concept as having a hidden representation of self that is continually tended to, and influences future choices. This implies statefulness, which models are intentionally not at inference time (*). (*) Even if we hack around this and just do the usual trick of simply laundering statefulness to a higher level, in this case the context window being fed in,…

Before reading, know that I am uncertain in either direction.

> a hidden representation of self that is continually tended to

This sounds like a personality? They act like they have one of those. It may be an illusion, and even if it isn't an illusion it is unlikely to be anything like the source (us), but they act like it.

> I further fail to identify how it could be hidden or maintained, considering I control like half of it.

Indeed you control everything about a local model, and much of the context of even a remote model. But the state of activations and circuits in SotA AI is hidden in similar ways to those of synapses in your head: difficult to decipher even with probes monitoring the signals directly, and often not emitted at the normal output.

> The best you could ascribe it is a meticulous maintenance of a persona the user is talking to, but then that doesn't necessarily represent the model's internal state, the same way my own words here aren't doing so either. Difference being, I actually have one (I'm "on-line").

While we can be confident that LLMs make up personas etc., it is insufficient to go from "that doesn't necessarily represent the model's internal state" to "therefore it doesn't have one".

> You'll sometimes catch models mixing up who's who and how many who-s there even are for example.

I've, unfortunately, also experienced this with humans. Perhaps they were losing their self-awareness at the time? I do wonder if old-age dementia does that by the end, though the person in question didn't ever get diagnosed with that.

> If you know of anything like this, your turn now, would be happy to learn.

Do you mean like these, or something else?

https://researchportal.hkust.edu.hk/en/publications/decoding...

https://aclanthology.org/2026.eacl-long.165/

https://transformer-circuits.pub/2026/emotions/index.html

Re: Ten advances in mathematics and theoretical computer science

#82
post #55
post #26

Earlier quoted context omitted.

Another noteworthy difference is that Stockfish is also gpl.

If there was any real money in it Stockfish would not be the best chess engine.

Thats not the point, if there were a better proprietary engine stockfish would still be there as a baseline. Anyone can access an engine as good as stockfish to practice against. Are any open models touting mathematical breakthroughs?

Re: Ten advances in mathematics and theoretical computer science

#83
post #78
post #35

Earlier quoted context omitted.

That seems to have been more of a sensationalized joke. Even your link has a disclaimer in it now. Read this chat from the researcher who did this: https://leanprover.zulipchat.com/#narrow/channel/270676-lean...

It's not at all a joke ... that's a severe misunderstanding of the context.

There is no evidence that I can find for the claim "a bug in the Lean kernel was discovered last week by way of an LLM tricking itself and its handler into believing it had found a non-constructive proof of the existence of a nontrivial Collatz cycle."

As I currently understand it, all we know is that:

- a mathematician produced a Lean-verified counterexample to the Collatz conjecture, demonstrating a bug in the kernel

- he claims that LLMs were involved somehow but pointedly refuses to specify how

- he admits that he knew about the bug before publishing the counterexample to his repository.

Perhaps not a joke (although it sure seems to me like they discovered a bug and thought falsely disproving the Collatz conjecture would be a flashy way to announce it), but at best extremely sensationalized by the above description. If you have additional context I would be happy to hear it!

Re: Ten advances in mathematics and theoretical computer science

#84

On the token limits etc - one assumes that OpenAI et al are able to “hire expert in field, and let them spend the equivalent of a million dollars of tokens” because they are not actually selling their complete compute 24 hrs a day, so the cost internally is a negligible (ish) electricity bill. Which is very suggestive - if after everything they are not fully loaded then the next gazillion data centres being built loo…

For OpenAI, research is marketing. I’m sure they’ve got plenty of budget for that.

Re: Ten advances in mathematics and theoretical computer science

#85
post #79

Earlier quoted context omitted.

Sorry, OpenAI's take is correct here. If you're not convinced, here is how they prompted LLM: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98... [0] A slightly smarter highschooler could write these. I could write these. It's clear as day that the LLM, not the human, did the heavy lift. It'd be ridiculous to give full credit to whoever wrote the prompt. [0]: Not one of the proofs in the linked article, b…

But why can’t we prompt the LLM “just do math research”? This is what I don’t understand.

If there aren't thousands of TPUs doing that [0] right now I'd be quite surprised.

[0]: e.g. "go through wikipedia's unsolved math problem list and solve them".

Re: Ten advances in mathematics and theoretical computer science

#87
In a way the most remarkable thing about this is that it isn't even at the top of the HN homepage. Even if this is a step up from what we've seen before, we're no longer astonished by the idea that AI can make significant advances in mathematics and computer science.

Re: Ten advances in mathematics and theoretical computer science

#88
post #39
post #37

What happens when OpenAI et al stop being open about these things, and just pack it into the training?

Not much point to pure math being kept secret, in all honesty. There isn't really industrial value, its only purpose (to them) is showing off their model's capabilities. More realistically they'll just stop paying for it. Edit: Oh, are you suggesting they just use it to privately improve their models? I imagine a few more correct proofs would have a very marginal benefit, if any. Also, they'll probably just get extra…

At this stage. No doubt calculus had plenty industrial benefit.

Re: Ten advances in mathematics and theoretical computer science

#89
post #10
post #2

I don’t feel the existential dread of mathematicians is correct. It seems to me in fact these results are bringing math mainstream. I now personally look forward to the interpretations and discussions of the significance of such results by human mathematicians. Now I understand that it’s mostly the super stars benefitting from the increased attention. Folks who are less established don’t share in that glory. But on t…

Every time someone makes a comparison to chess I die inside. Chess is a spectator sport primarily funded by a few eccentric billionaires. Players artificially constrain themselves in timed environments knowing that they will never be able to produce better moves than a smartphone because a select few people find it interesting. Only ~30 top professionals actually make enough money to have a full career playing chess,…

The distinction is mathematician vs mathematics. Mathematics is going to reach new heights beyond the wildest dreams of contemporary mathematicians. But perhaps without the participation of many paid mathematicians.

Re: Ten advances in mathematics and theoretical computer science

#90
post #82
post #55

Earlier quoted context omitted.

If there was any real money in it Stockfish would not be the best chess engine.

Thats not the point, if there were a better proprietary engine stockfish would still be there as a baseline. Anyone can access an engine as good as stockfish to practice against. Are any open models touting mathematical breakthroughs?

There is money in this, so of course the closed models are far ahead. The open models will likely catch up a bit at some point, just as Stockfish caught up to AlphaZero. That being said, there are already a couple. It seems Deepseek has a claimed proof to the "Ziegler's Cross-Polytope Conjecture" [0], but I can't speak to the significance of the result.

[0] https://arxiv.org/abs/2606.31640

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