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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 do wonder how the reaction will be when, almost inevitably, some PhD student or postdoc gets scooped on a problem they had gone all in on and commits suicide. It's not like OpenAI seems to be following the standard process of extending feelers through the field to get in touch with any existing noosphere-homesteaders before solving these.
People still play chess even though computers have long been better at it than use humans.
I'm in programming, one of the fields that has already been most affected by AI. I can also report that I'm having some of the most fun and being more productive than I've ever been. My key has been to just embrace AI hard and use it as a tool the best that I can.
You also see the culture war starting up in that space. There are several Linux distros that have people pledge that they haven't used AI in the process of doing their contributions. My prediction is either: a) people are just lying about not using AI or b) they'll just lose and people who are using AI will simply outperform them in the marketplace of ideas.
So math people simply need to get on board. The train is leaving no matter their opinion of it.
The difference is that people largely only ever played chess for the love of the game. There are nontrivial numbers of people who do programming for the love of the game; in my experience, though, there are very few mathematicians who derive sufficient meaning from the "game"/process alone, and those who are are unlikely to find themselves in such high echelons of academia, instead gradually filtering out into other fields where you can continue pursuing it as a game (ex.: Martin Gardner, Greg Egan...).
I'm certainly in the "love of the game" category of programmer. I honestly think that you have as many in math. And unlike computers, few people ever went into math to get rich -- so my internal estimate is even higher for the math side.
Side-topic: I think the programmers who are in it for just money are the ones that are going to get selected out. I've been at this for a long enough that when I started a programmer was a good white-collar job, but not one to get you rich. My experience, and frustration, with how the field has changed over the decades is that the quality of people has become absolutely dreadful outside of a few highly selective places. And even places like Amazon (where I spent some time) are compromised.
I get joy from many things. From tweaking things big and small. I don't think I ever really got joy from the act of writing code. It was always a means to an end. And the end doesn't have to be strictly useful. Something I've been kicking around for the past 35 years (also: crap... I'm feeling old) is an implementation of the Game of Life from John Conway who would fit in with your list perfectly. I came up with what I still think is the fastest way to play it with "normal" code in the general case. Using Claude I was able to get this implemented with a front-end that I can interact with. I get joy from the algorithm. Joy from thinking of low-level optimizations. Joy from learning more about the innards of how the processor I'm on handles cache lines.
I don't get joy from setting up a build system. No joy from setting up a front-end that runs on my Mac. Very little joy from tweaking an unrolled inner loop. Less joy from setting up unit tests.
Having AI lets me get to that joy. And it makes me incredibly happy. And it lets me paper over the parts I consider ditch digging.
My gut says that the folks in math want the same thing. Come up with an idea, and have something else handle the proof. There's joy in simply making something better or pushing the frontier a bit more.
And with AI, you can make bigger leaps. Ask bigger questions. Get more joy from discovery.
In case you're interested: https://github.com/gburgyan/vlife and a technical paper in docs that's more presentable.
There are absolutely people in math for other reasons than money, but among the countless mathematicians I've known very few have been motivated by anything that did not factor through being the first to prove/discover/define something that others care about.
That's the thing exactly. If, through using AI tools, you can be the first to prove something, isn't that a good thing? Only through discovery can we reach higher to ask bigger question. It took a lot of work as a civilization to get to the point where we had the language to construct things like the Navier–Stokes equation in the first place; they don't just get written down in a vacuum.
So we've gotten to a step change in the system. The real challenge will be to be the first the ask the question in the first place. Even if that is automated away, you're still left with the pure joy of learning and discovery which is rewarding unto itself.
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