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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.
I'd argue math matters -- there's a reason that people like me who need to use our knuckles for times tables still sometimes know Dijkstra -- but that's probably not going to help math PhDs choosing between a barrel of bourbon or a different type.
In the short term, this might boost math as a job, since now there's literally hundreds of lifetimes of proofs needing deep examination and translation being dumped on a weekday night... but even in the best-case, that's not what people signed up for. And worst case, the machines might be better at it than the humans, and if they aren't yet, they probably will be soon.
In theory, math PhDs are well-educated extremely smart people who should adapt best and most easily to a career change, but again, now what people signed up for, the PhD isn't cheap, and a career change to a different field that might just become the next benchmark isn't encouraging. Worst-case, this seems up there with 'moving to a new culture' or mega-dosing on THC that pegs the schizophrenia risk.
Yeah, and both anti-distillation and anti-LLM-training-assistance techniques are already known and disclosed. And given the 3SUM problem, if I'm understanding it correctly, I would be very surprised if there's zero here.
Related really fun thought: previous LLMs have been very prone to 'leaking' out to the public net as they do more advanced work or get stuck on genuinely impossible problems. Here, either OAI has really stepped up their sandboxing game, the leaked outcomes have become internalized universally without public fanfare, my search-fu has been flopping... or a lot of this stuff doesn't count as 'advanced work' or 'sticking'.
Also in my experience with Math PHDs a lot of them aren't really wired for the general workforce even if they've got unbelievable processing power. My high school ex's dad is a top tier Astrophysicist and has won a major prize for Astronomy but I wouldn't really trust him to do any job that requires particular lateral thinking and he's relatively sociable and flexible by the standards of his field. There is a certain filtering, especially these days, where only the most stubborn and pure-mathy are the ones who decide to pursue Academia (especially if they don't have good diversity points) in the first place and the ones with more lateral abilities drop off the curve when they realize how insanely difficult a slog it is to get consistent paychecks in Pure Math.
Even in applied math there's the same effect. The people with the most focused talent and interest stay in academia, in a tournament where only a fraction of entrants can win tenure. (How many grad students does the average tenured professor advise through a PhD? In a field that isn't expanding like mad, the reciprocal of that is roughly your odds of getting tenure.) Everybody else goes into industry or other research labs, trading respect for some combination of extra salary and lower stress.
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This seems entirely possible (Isaac Newton was certainly very much into alchemy, for example), but it seems more likely you meant astronomy.
And Brian Josephson took up research into psychic phenomena after he won the Nobel for his work on superconductivity. The PR required careful management when he was the only living Nobel laureate in the Cavendish.
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Always forget which is which, oops. Fixed.
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I always assumed pure math was extremely easy to get academia jobs. Even in engineering or b-school math profs tended to be foreign born with awful English schools that resulted in making any college level math class a teach yourself affair.
Being that math departments seemed stuffed with foreigners who can’t speak English I assumed that there were zero domestic demand to be math professors.
Never actually tried myself but I've got a few white male friends with Maths PHDs who've said vast majority of job openings are looking for people with more diversity points (and foreign born counts towards that) and the compensation is low enough it only really suits visa-desperates or ultra autists.
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