On your second point: there is a more plausible explanation which David Bessis calls the "overhang". The short version is that there is a large amount of relatively low hanging fruits in mathematics, because no human has broad enough knowledge and enough time to try them all. AI is not constraint by that, and therefore can systematically pluck all those low hanging fruits.
Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.
LLMs can be trained on the entirety of the mathematical corpus. Thanks to their phenomenal memorization and pattern-matching abilities (without always being able to map out their associative logic and attribute due credits), they are in a unique position to harvest the Overhang. By contrast, professional mathematicians have typically read a few hundred articles in their career, out of millions of existing references, less than 0.1% of the total.
This will lead to great discoveries, which is unambiguously exciting. But it could also lead to a sad new deal, where human slaves painfully curate the Overhang while AIs systematically beat them at the finish line."
>Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward.
I've made an entire career out of being 'jack of all trades, master of none'. Being able to synthesize connections from relatively trivial knowledge in a bunch of domains is SOP for many humans as well. I think AI just has deeper knowledge and better pattern matching to make up for it's (at least now) lack of strength in cognition and 'ex nihilo' creativity.
(Which probably isn't 'ex nihilo' at all, and has more to do with the plethora of modalities that humans live in vs. large language models. For example, why do we pick the color red for notating important things and why do we say a schedule 'slips'...these are informed by a shared human experience borne of distinct physical sensation deep in our wiring that LLMs can only infer from what we write.)
A college advisor I had 20 years ago was a firm believer that interdisciplinarity was the future, that generalist skills and the ability to make connections between different fields would be paramount in advancing science. I suppose he was right in the big picture, even if the career prospects for human generalists aren't looking so rosy.
I'm actually still quite bullish on generalists. Specialists advance every front but build the supply lines between them.
In favor of the generalist, I think AI is also quite limited in its scope of how it generalizes. I'm mowing through hundreds of mythos-generated security findings right now for work and while it's amazing that it can build an exploit chain 20 steps deep, it's completely lacking in all of the external layers that render it's speculation moot.
How do you thrive in an environment of specialists? That's is the problem I seem to have. I'm spread a little across a few of the domains involved with what I do. Because of that, I have a bit more insight, so am very often the person pointing out relatively fundamental problems, usually caused by either not understanding the problems from a "first principles" perspective, resulting in, or being caused by, categorical type errors, where they've boxed a problem into a tiny space it doesn't belong.
I've been trending "quiet" lately, because I don't like the "friction"/convincing aspect of it all. It's hard to get people to see things from a different angle, or even convincing them there's a problem to begin with!
The last project required a complete redesign from a problem I pointed out during the first review, and second, and third, but now I'm seeing even more friction.
Maybe this is just corporate life, after a group gets large.
Any tricks/advice?
> Quote: "The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.
That overhang seems like a precious resource for AI companies. They can exploit that overhang to inflate the impression of AI's capabilities, and hopefully that exploitation will discourage the next generation of mathematicians from pursuing math. If they play their cards right, OpenAI and Anthropic can dominate the field even if they ultimately can't replicate the creativity of human mathematicians, because they'll have driven their competition out.
What we should be trying to achieve is a ladder-breaking maneuver: knock out the lower rungs so no person can reasonably climb to the top-reaches of mathematical skill anymore. That may ultimately result in stagnation, but it's what's best for AI, so it's what should be done now.
We need to do everything we can to create the greatest-possible dependence on AI tools.
/s, I hope?
Great essay, thanks for sharing.
When I was a software library developer, I came to resent application developers. I noticed a pattern. Libraries solved hard problems and did so carefully, thoughtfully, in a way that others could reuse. Apps would come along and carelessly, recklessly glue together several high quality libraries into a piece of software targeting a general audience. The apps would then harvest all the credit.
What's happening in mathematics right now feels similar. Applications (theorems) were always how one built objective reputation, but libraries (concepts, definitions, boring lemmas) were also rewarded socially within the mathematics community. And individual mathematicians often managed to both build their own libraries, and use them to prove an important result. And then those libraries were sometimes of use in other results.
Bessis asks whether AI Lean proofs will land in Mathlib or Mathslop. Or in my framing: will they be libraries, or applications?
At present they're mostly Mathslop. The proven result is perhaps useful, but the methods employed aren't novel or reusable. I worry that this trend will only worsen, because applications make headlines, and the libraries they used do not. We are not properly incentivizing library development in OSS, or in math, or in infrastructure writ large. There's a serious credit assignment problem here.
What might change this? Once the low hanging fruit is picked, will citation count rise in relative status again? Will we get result fatigue and start to reward legibility — no one cares unless the paper has an accompanying ELI5 tiktok video? A labeling regime that certifies the proof was produced sustainably, organically, by local artisans with no AI additives?
There is also "sexy proof", people want nice math that can be printed in t-shirt. Not super hard grind, where you need several years of studying, just to understand the question (that is before even trying to solve it).
Many problems are solvable, but require months of work, and thousands of pages of proof. So people do not even try to create or verify the proof. AI changes that, it can verify and perhaps even simplify it, to more digestible form.
The overhang, being defined as the Cartesian product of existing knowledge — randomly combining existing knowledge.
(I mean actually randomly, not asking an LLM to do the randomness.)
Most of the output would be incoherent (like many dreams), but occasionally you would get a gem.
> no human has broad enough knowledge and enough time to try them all.
The other part is, humans don’t really want to fund other humans doing this.
Very few want to be a math major; and of those that do, fewer complete a grad degree; and for those that do get grad degrees, there’s scant few research jobs; and for those who do get jobs there’s hardly any research funding to go around.
There does seem to be unlimited money for ai researchers to use ai to solve these problems though.
We’ve turned education into job training, so because there’s no jobs in solving math problems, few aspire to do it. If there were more opportunities for people, more people would do it, and more low hanging fruit would be plucked.
I’m assuming the reported 22 million dollars worth of tokens used to solve this particular problem is far more than what humans have paid to solve it previously. So I think you’re correct.
22x more to be exact
Could "superintelligence" arrive as basically applying this overhang to all other domains?
It already did.
That is not "superintelligence" but string concatenation of stored data. Anyway, the marketing succeeded.
It's like AlphaGo but playing against all living mathematicians. (Overhang being low hanging fruit is what allows this comparison, of course the general moot point is the skepticism that LLMs are also innovative etc.)
We are not seeing those incredible moves yet. The approach used in N-S was conjectured to work after B&L’s initial breakthrough. See a post by Tao. So on one hand the proof is an amazing accomplishment. On the other hand, humans have not yet discovered any superhuman moves in the proof. Just $MM grind.