Ten advances in mathematics and theoretical computer science

The cost of running a model is not only $/token, but the salaries of the people managing/orchestrating the models, deciding what theorems to try, etc. Once we factor that in, how much are we really paying per theorem?

The other factor is the subjective component of the value of a theorem. Not all theorems are created equal, and the only way to really measure the value is to ask professional mathematicians for their opinion, or publish the results and look at citations over months/years.

Once we have both of these nailed down, then we can start to do the cost/benefit analysis. To be fair, we should actually compare three groups: human experts, hybrid agent/human expert teams, and fully autonomous agents.

https://arxiv.org/html/2605.22763v1

> Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures

A couple small ones that I've seen (example here [0]), but not anything of the magnitude that OpenAI and Anthropic have put out. Likely just related to token limits.

> Another question I have is whether or not OpenAI 'simply' hired capable combinatorics researchers to work on problems, and they have, and the use of the model is incidental / secondary to their work.

I think their output has reached a level that precludes this possibility, but I of course don't have any hard proof.

[0]: https://www.reddit.com/r/math/comments/1uxj3cy/after_openais...

I have no affiliation whatsoever with any AI company, nor any formal education outside high school, for what it's worth. Simply being curious and persistent can get you quite far in my anecdotal experience.

Yes. And there are many of them. I wonder what would help them come to terms with it. Seriously, people are going to be grieving over this. Loss of identity, loss of social standing, ideas of entire future lives that will now never happen. The greatest crime people may hold AI guilty of is taking away their dreams.

Moreover, mister elite, you don't know why this press release was flagged.

I'm not sure why we should privilege your bitter speculation over more mundane possibilities.

Never was.

It was always mainly a website for employees of an elite.

hah, sorry, we are plebs out here.

https://developers.openai.com/cookbook/examples/vector_datab...

How are the sales going?

My guess from following this stuff quite closely is that these companies are still a couple years away from fully autonomous research staff.

[1]. https://www.anthropic.com/institute/recursive-self-improveme...

I’m not a mathematician so I have zero clue what “ New upper bounds on sphere-packing density down to the Cohn–Elkies thresholds” means.

That's not what people mean when they say "moving the goalposts". It means that people are adamant that something wasn't important/hard/impressive once the "AI" solves it. And then they come up with another thing that needs to be solved in order to prove it is important/hard/impressive. And once that happens, they do it again. And again. That's what "moving the goalposts" means.

It's also very much not a new phenomenon. It's been happening since the 1980s. As you can see from this quote from GEB by Hofstadter:

> There is a related "Theorem" about progress in AI: once some mental function is programmed, people soon cease to consider it as an essential ingredient of "real thinking". The ineluctable core of intelligence is always in that next thing which hasn't yet been programmed. This "Theorem" was first proposed to me by Larry Tesler, so I call it Tesler's Theorem: "AI is whatever hasn't been done yet."

They get to make cool 3D plot visualizations of functions so obscure to me that they’re named after someone who is still alive - and/or get to work on cryptography for the NSA - I think?

https://leanprover.zulipchat.com/#narrow/channel/270676-lean...

Say that it turned out that the total cost of the proof of the Erdős unit-distance conjecture was $50 million.

Then the question really becomes: yes, these models are capable of proving important mathematical results, but at a very high cost. Is it worth it?

If a mathematician applied for a research grant of $50M USD for proving the same thing, they would have been laughed out of the bank.

What's more is that when you have a research grant, you train PhDs and postdocs, you hire new staff, and you disseminate. That is, you get much more value for the money spent.

I'm just curious what the cost is.

Are you arguing there is not an AI bubble, and that all the DC buildout is fine, going to be profitable etc?

I am not looking for a online slanging match - just looking for a different point of view

Was this different before chess computers were invented?

Stop coping and deluding yourself mate.

To begin with, whether AI is the one doing the discovering or not makes no difference. Any "pure math" person would aim to understand regardless - and would be quite glad that they have a longer paved path.

Any mathematician in academic or industry is more than likely not a "pure math" person (tainted by capitalism).

The non-sofic group one is definitely a big deal - would have been a Fields medal if discovered by a human.

I think you're over-estimating what a smarter highschooler could write.

A "finite loopless undirected multigraph" could have been explained to me at that age if we'd taken Discrete rather than Mechanics and Pure (and one module of Stats) in my two A-levels* in maths and further maths; but from what I saw of the Discrete module, neither:

  Every finite loopless multigraph with no bridge possesses a cycle double cover, without additional assumptions such as cubicity, planarity, connectivity, or higher edge-connectivity.

  repeated-edge closed trails masquerading as cycles

* For those not from the UK: A-levels are between secondary school and university, when aged 16-18. Functionally they are university entrance qualifications: https://en.wikipedia.org/wiki/A-level_(United_Kingdom)

Provability is just going the way of computation. John Napier had to manually compute logarithm tables over decades and was recognised for his work; now that same work could be performed by a 10 year old with a calculator in an evening.

> In particular, proofs for special graph classes, constructions of cycle covers with some edges covered other than twice, bounded-length or prescribed-cycle variants, reductions to another unproved conjecture, computational verification through any fixed graph size, and candidate counterexamples without a complete nonexistence certificate are insufficient.

which is infact a very important part of the prompt.

A language model (or an image model or whatever) cannot even be sentient, and I think sentience is a prerequisite for awareness.

Even if we express a lot of our subjective experience with words, the language is just a symbolic representation of those experiences. The qualia themselves, even those that are quite abstract, are rooted in our physical presence and evolution.

You can't have an understanding of what hunger or physical pain feel like if you have no need for food or a sensory capacity for feeling pain. You can't understand what loneliness or pride at an achievement mean if you don't have a neural network wired to value social connection or status. We value connection because we're social animals that have needed each other for survival.

Even the more abstract of our subjective experiences are in some way rooted in our physical evolution.

I see no reason to believe that a neural network built entirely based on the symbolic level of language could have the features needed for the subjective experience itself.

AI awareness might actually be more believable if that awareness manifested itself in an entirely different way than in humans. But if we assume awareness because outputs resemble what we consider meaningful as humans, yet the neural network has had no inputs or evolution that could form the actual basis of human-like experience, I think we're seeing something that isn't actually there.

(*) 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, I fail to identify (**) a representation of its own state in these bodies of text that it'd be meticulously maintaining. I further fail to identify how it could be hidden or maintained, considering I control like half of it. 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").

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

(**) I did wish for something hidden though, so maybe it's just concealed? The same way people can encode a lot more of their emotional and mental state than normal into text if they read and write a lot of it, I'm aware of research that suggested the same for LLMs, albeit I cannot cite it. Maybe those phrasing signatures are just alien to me and will never pop out. Either way, I'd expect researchers to stumble upon this during interpretability studies, and either they haven't, they have but it wasn't popsci adopted, or they're keeping awfully tight lipped about it. If you know of anything like this, your turn now, would be happy to learn.

I do wonder how reasonable it is to expect e.g. a single maintained identity though. Maybe it isn't?

(*) Another way to hack around this of course is to just precompute some internal "self-awareness states" and hop around between them. Probably the closest to what the models are actually "doing".