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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.
The only reason I'm not saying it's over is because I've said it before. It's been over for a while. And we've barely begun.
The good news, for mathematicians, is that their employability (in academia) did not hinge very strongly on tangible economic output. Nobody funds an algebraic geometer expecting a return on investment, which means nobody can defund one on the grounds that a datacenter in Texas now offers a better one.
The bad news? Everything else.
Just look at this fucking sweep. Just look at it. Hilbert's tenth over the rationals. Rational Hodge for CM abelian varieties. Integer multiplication below n log n, which I had mentally filed under "the floor, go home." A quasi-Riemann hypothesis thrown in like a free tote bag. Any one of these would have been the defining result of a human career, and they were released as a batch, in a GitHub repo, on a Tuesday, alongside hundreds of others.
And it wasn't brute force at absurd cost. OpenAI says the average result used roughly three hours of ChatGPT Pro-equivalent compute. I agree with your Pareto intuition, but the deeper point is that three hours is a price, and prices in this industry fall monotonically. Gwern spent 2020 arguing that neural nets would keep absorbing compute and sprouting new abilities, while noting the idea was so unpopular that it would only be accepted as a fait accompli. Well, here is the fait, and it is quite accompli. Next year the same theorem costs twenty minutes. The year after, it's a rounding error on someone's API bill.
Your own experiment deserves more attention than you gave it. As others have already done to good effect, your research methodology consisted of being a motivational poster ("you can do it!", "I believe in you") and you got a materially stronger result in half an hour. The binding constraint has shifted to elicitation and persistence.
Some might claim that human mathematicians are necessary to humanize and understand the proofs, or to come up with new frameworks and fields of research.
Maybe. I don't know. The strongest version of the argument is that mathematics is a conversation between humans about what humans find interesting, and a proof nobody understands is a tree falling in an empty forest. I'm sympathetic to that! I also notice that this position has been retreating to progressively smaller hills since the first computer-assisted proofs, and each hill gets defended with the same conviction as the last.
I do know that the current state of affairs is unlikely to last for more than six months anyway. Just get the models to come up with their own novel conjectures, or to attempt an overarching synthesis. I don't think turning abstruse Lean proofs into something human-readable will be particularly difficult, though human readability has long ceased to be a major concern for anyone. Note that OpenAI is shipping Lean formalizations for many of these proofs, so "trust me bro" is off the table; a type checker cares nothing for anyone's feelings. We're at the point where people are coping that the tsunami of new discoveries hinge on hither-to undiscovered bugs in Lean. You wish.
Every day, I feel the pace of progress accelerate, and then the practical ramifications show up in my life a few months later. I used to think of this as a lag. These days it feels more like the gap between seeing lightning and hearing thunder, and the gap keeps shrinking.
We've just had Scott Aaronson announce a new UT Austin course, CS395T: AI Alignment Theory, with Yudkowsky's AGI Ruin as the first assigned reading. To quote:
I'll give credit where it's due, but also: when the designated Reasonable Skeptic of the rationalist diaspora concedes the point in writing, the Overton window has relocated.
We've also just had an American company, Nolla Health, receive regulatory approval in Utah for AI to issue initial prescriptions. Yes, it's acne. Yes, the model can only choose from a short list of physician-approved topicals, and clinicians reportedly agreed with its recommendations in over 96% of cases. Everything starts as acne cream. Utah let Doctronic's AI handle prescription renewals back in January; nine months later, it's writing first-line scripts. You can extrapolate the line yourself. That's the harbinger of medicine's fall, as far as I'm concerned. I'm impressed we held the moat this long.
At least I've got a moat. The NHS is famously a slow ship to steer, and sclerotic at that. The future arrives unevenly distributed, and we can count on the service the last postcode on the delivery route. For once, I find that reassuring <3
Good career choice, past self_made_human. Turns out there are concrete benefits to taking theoretical but plausible concerns seriously, and preparing accordingly.
Be honest. How much of the above did you actually write? I’m reserving judgment. I’m just curious.
95% of it? I typed it out during my lunch break (I had lost my appetite), and then the only extent of LLM involvement was to dig up and then wrap in Markdown the relevant links.
Which are all real, anyway. I just didn't have the time to bother.
Hmm. Let me check closely:
Several throwaway statements had a sentence or two of context added. For example, "The binding constraint has shifted to elicitation and persistence".
If you want to be technical, it inserted the full title of Aaronson's post, or surfaced it in the first place. Initially, I just happened to come across it secondhand on Twitter and had pasted the quote in bare. Uh, now I see that it claims that I had strong opinions on the algorithmic floor of integer multiplication. I did not, beyond being vaguely aware that there was a better option than a naive n^2 based off an article I think I read on Quanta. I couldn't have told you off the top of my head that the previous SOTA was O(n log n).
Aw, dang, I was actually pretty impressed you had such specific CS knowledge!
Hey man, I read HN religiously ever day, and I once solved a LC medium in 2021.
Jokes aside, I do like maths and CS. The former all the more so when I'm not forced to for the sake of exams. And I try to learn what I can as the mood takes me. I even tried to learn Lambda calculus once, even though it provided literally no practical utility.
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