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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

#181

Post by famouswaffles »

don_esteban wrote: post #93
mathisfun123 wrote: post #66

Earlier quoted context omitted.

I don't disagree with you but there's no need for exaggeration; ain't no high school student writing this: > 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 compl…

the fact that such things have to be explicitly in the prompt points to the fact that the underlying system is still far from where it needs to be (basically, lacks basic understanding what a proof is)

They don't have to be. At this point, we have multiple results from 3rd parties where the prompts are very basic.

To name a few:

- https://xcancel.com/DmitryRybin1/status/2079904005652893709

- https://archive.ph/2w4fi (https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...)

Re: Ten advances in mathematics and theoretical computer science

#182

Post by brighteyes »

einpoklum wrote: post #14
aabhay wrote: post #6

My main gripe here is the lack of transparency around the total experiment and construction. I doubt that they simply pointed their model at these ten specific problems alone and gave the model one shot; therefore the $2000 number could be completely misleading, similar to P-value hacking by not disclosing the total experimental setup. I want to know: 1. How many total problems were given to the model, and what perce…

Also, have there been examples of researchers not affiliated with OpenAI (or another LLM creator), who have done something similar? 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.

Yes, here is another example of major work in this area:

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

Re: Ten advances in mathematics and theoretical computer science

#183

Post by Chance-Device »

antirez wrote: post #100
robinhouston wrote: post #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.

This is not at the top as it is actively flagged by people that can't psychologically cope with the advances of AI. Hacker News is no longer a web site of an elite.

> people that can't psychologically cope with the advances of AI

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.

Re: Ten advances in mathematics and theoretical computer science

#184

Post by dgacmu »

uh_uh wrote: post #119
dist-epoch wrote: post #95

Earlier quoted context omitted.

I don't think you want to bring cost into this argument. Even if the cost was $1 mil for these 10 problems, that's maybe 10-20 math researchers for a year. Do you really think that if you paid that to humans, they will deliver the same results?

It is comical at this point. Some people just can not stand the thought of AI actually delivering and are trying to find whatever ways to discredit it.

This isn't really about delivering - it's more about helping to understand the shape of problems that AI can solve right now. If they took 1000 problems and threw the model at it and it solved these ten, is there something we learn about these ten problems and the kinds of things that current AI is good at? That's very different from picking ten problems _at random_ and solving all of them successfully, which would suggest a much less bumpy capability surface. It's interesting and it would be good science to release it.

Re: Ten advances in mathematics and theoretical computer science

#185

Post by einpoklum »

brighteyes wrote: post #182
einpoklum wrote: post #14

Earlier quoted context omitted.

Also, have there been examples of researchers not affiliated with OpenAI (or another LLM creator), who have done something similar? 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.

Yes, here is another example of major work in this area: 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

The actual quote:

> Our full-featured agent autonomously solved 9 Erdős problems out of 353 attempted, including two questions that had been open for 56 years

Note _had_ been open, not _have_ been open. Can you clarify?

Re: Ten advances in mathematics and theoretical computer science

#186

Post by patcon »

kcexn wrote: post #158

Not being an expert in any of the fields OpenAI has "advanced" I don't want to prematurely downplay the significance of this contribution. However, I am worried that the language they are using in this blog post is exaggerating for the sake of marketing. It is true there hasn't been a reliable computational approach to solving these problems before. But do these proofs contribute new ideas to the mathematical corpus,…

"Breakthrough research" can be defined (in the citation record) as research that both (1) becomes highly cited, and (2) brings together citation chains that were previously not showing up together.

Mundane incremental research is cobbled from existing citations that already appear nearby in the record.

Basically, innovative research is a measure of bridging thought and domains that were previously not bridged. It's quite concrete as a measure in the citation record.

So we can know pretty conclusively.

Puja Ohlhaver gave a talk on this[1], and ran some experiments (that I had the pleasure to support on)

[1]: https://www.youtube.com/watch?v=guLDNMAOn24

Re: Ten advances in mathematics and theoretical computer science

#187

Post by [deleted] »

antirez wrote: post #100
robinhouston wrote: post #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.

This is not at the top as it is actively flagged by people that can't psychologically cope with the advances of AI. Hacker News is no longer a web site of an elite.

[deleted]

Re: Ten advances in mathematics and theoretical computer science

#188

Post by jsnell »

einpoklum wrote: post #185
brighteyes wrote: post #182

Earlier quoted context omitted.

Yes, here is another example of major work in this area: 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

The actual quote: > Our full-featured agent autonomously solved 9 Erdős problems out of 353 attempted, including two questions that had been open for 56 years Note _had_ been open, not _have_ been open. Can you clarify?

The original was an actual quote?

But "had" still doesn't mean what you are implying: once the model solved the problems and the solutions were verified, the problems weren't open any more, so a later description using the past tense is totally consistent.

Re: Ten advances in mathematics and theoretical computer science

#189

Post by ianm218 »

curt15 wrote: post #111
robinhouston wrote: post #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.

What about AI research itself? Is OpenAI close to automating its human staff out of a job?

They and Anthropic have indicated that the models are substantially augmenting the research and doing large amounts of work autonomously at this point. Here is one of the many blog posts on it [1]. Many people would dismiss this as "marketing" so take it for what you will.

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...

Re: Ten advances in mathematics and theoretical computer science

#190

Post by ianm218 »

emil-lp wrote: post #138
kingstnap wrote: post #114

Earlier quoted context omitted.

Why would you factor in salary unless they had to baby it through. You would only count the hours for setting up the harness and prompt and checking the result. Training the model is going to be amortized over other uses.

> Why would you factor in salary 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…

It feels like the real cost might be negative though.. They use frontier math as a way to test improvements in their model. So solving the problems is like a positive externality, but the important thing is they can verify that the model is improving instead of looking at useless benchmarks. Plus it is good for marketing and attracting talent.
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