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Notes -
A Big Day in the Culture War
Apologies for any incoherency or grammatical errors; today's events led me to down more than my usual share of booze.
Earlier today, big things happened in math. No, not Claude providing a sub quadratic 3SUM. OpenAI released hundreds of notable math results, in a GitHub repo.
Fun results:
Hilbert's tenth problem over (\mathbb{Q}) is undecidable
The quasi-Riemann hypothesis
The rational Hodge conjecture holds for every CM abelian variety
Integer multiplication can be done in sub-log-linear time. Oh, there's also a sub-log linear DFT.
Thoughts and observations:
Mathematicians are big mad. View the relevant subreddit on our progenitor site. Most there are probably, at best, adjuncts at community colleges desperately coping with the downward trajectory of already marginal careers, but it's fair to say that the writing is on the wall for mathematicians. There's probably a double digit number of grad students staring into a glass of whiskey tonight and thinking of hanging themselves.
My immediate question was about whether this closer to the current peak of performance, or just a lazy demonstration of OpenAI's power. So, I took one of the particularly interesting preprints to me (memory and precision in Gaussian models) and tested whether it's at the edge of capabilities or not. 30 minutes of back and forth with Astra (itself behind OAI's internal model) resulted in a significantly stronger result, on multiple dimensions. While I finished my first bottle, I spent a fair amount of time convincing myself of the result; I was convinced the strengthened results were plausible. Write this off as AI psychosis if you want, but try it yourself; I'm genuinely curious for what you get. (The back and forth, here, was entirely me saying "you can do it!", "keep at it, I believe in you", and "you've got this, finish it!")
Probably the most important line in OAI's announcement post is "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." This isn't a case of OAI spending millions for a marketing bump. My bet is a kind of Pareto distribution: most took minutes, not hours, with a long tail around Riemann-level results pulling up the average significantly.
Notably, ML related results are nearly entirely absent from this batch of proofs; maybe a half dozen touch on it, distantly. Some problem indices are skipped in overview.md. Conspiratorially, my inclination was to think they filtered them out for competitive advantage. I can't find any evidence of that in the GitHub repo or any of the preprints (equally plausible: deduping), so maybe they judiciously decided not to point their mathematical ballista at ML. ML also doesn't have a meaningful bank of rigorous conjectures, so given the conjecture sources, maybe they're not yet digging into ML math. But color me skeptical.
Does math matter? Is it something to advance civilization and technology, or an artistic pasttime for humans to create logical beauty? Likely both, today, but this is an almost nuclear detonation against the latter.
Can you expand a bit on how this conversation went for you? You put the paper in, turned Pro mode on, and then gave the verbal equivalent of a 'continue' when the model turn came to an end? As a researcher, I'm curious to know how others are working.
Here, yes, just that. Used higher diction than my comment, but you could put it in a loop.
For my more usual interests, it's more involved; oftentimes the model will go off on a tangent or latch onto something I don't care about, and I need to tell it to stop going in a certain direction. There's also specifically generating artifacts that're useful for later sessions to build on (OAI has a library tool that does something similar now, but I don't love the implementation.) But for this math paper extension, it was just the dumbest elicitation repeatedly.
Thanks, that's interesting.
This is more what I'm used to. I wondered if I was crippling it by stifling it, and would get better results from direct elicitation.
I guess worth trying both depending on the problem :)
I am pretty torn about it; originally I had a fairly sophisticated research ontology. But I've gradually simplified it to just files that can be discovered in a hierarchy of progressive disclosure, and trusting the agent is smart enough to find what it needs when it needs it.
I get that. I have moments of thinking, "is the AI a useful tool implementing my ideas, or am I increasingly a useful tool allowing the AI to work around the bullshit that human-designed systems put in its way?".
It's why I ask the kind of questions I asked you: to see if I'm slowing things down by insisting on participation to justify my existence, rather than just solving the problem. I think still largely no, but it's good to be aware.
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