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Comment by somesortofthing | Hacker News Reader

Taken on its premises at least, I think this piece inadvertently fails to make a case for why "pure mathematics" should persist as a field of human or machine activity. A real-world system with the abilities of a robustly superintelligent mathematician can, when some practical problem requires it, formulate a problem statement, churn for a bit, spit out a formalized answer, and continue whatever outer-loop task it was doing without a human even finding out.

If you have a system that gives you arbitrary on-demand math results, dedicating any resources, be they human labor or compute, to producing them for their own sake just seems like a waste. Why catalog the Library of Babel?

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There's a substantial gap between "the machine is better than human mathematicians" and "the machine can formalize a solution to any problem immediately." Much of the important mathematical theory behind modern science was mapped out in advance by mathematicians, and scientists were able to find and take advantage of this mathematics where applicable. Mapping out our understanding of e.g. physics and our understanding of mathematics in parallel and finding connections later has historically been a highly successful approach, and I see no reason to believe the same would not be true for advanced machines. Would Einstein have been able to formulate GR without the existing theory of differential geometry? Maybe, but maybe not. Would an advanced machine be able to develop the theory of differential geometry in response to gravitational measurements? Maybe, but maybe not.

Edit: Perhaps I could have been terser here. Really the idea is that small jumps are easier to make than big jumps. Continually developing pure mathematics is a way to make small jumps in logic and intuition add up to big jumps in applications.

I don't see a reason this dynamic couldn't continue with zero dedicated effort toward "pure math" as such. Fundamentally, the function of math in practical application is producing deterministic(or stochastic, but obviously not in the LLM sense) models that, given initial observations/conditions, are capable of predicting some aspect of the future with reasonable accuracy. In the same way that human society does, a machine would grab the physics rung before the theory rung every time, but why does that matter? If progress is stuck, the machine turns the gauge away from "exploit" and toward "explore" and put together a bag of new tools it can sequentially try on the real-world problem.

I still don't see a role for humans in this process. They might direct the practical/physical aspect(if AI turns out to be less superhuman there) but they'd likely turn the hard conceptual problems over to the machine and never look inside the box - no human alive could understand even the smallest part of what's going on in there in less than a thousand lifetimes anyway.

> If progress is stuck, the machine turns the gauge away from "exploit" and toward "explore"

It sounds like you've answered your own question of:

> why "pure mathematics" should persist as a field of human or machine activity.

And you're really just quibbling about scheduling.

Whether humans will be involved or not is an entirely different debate.

I doubt many pure mathematicians would be okay with totally subordinating all exploratory mathematical investigation to specific practical problems, and the field would look very different both in terms of how it's organized and what output it produces were it run that way.

There are some parallels here to AI coding where even if the AI can write all the code, it's still valuable to have a human who can read and understand the code to verify correctness. The same is true of AI generated math proofs that humans will want to verify are around before they feed them back into the AI and build new insights.

Proof verification can be done automatically via Lean (or similar), so the parallel breaks down unfortunately. The only thing humans really need to verify is the translation of the theorem statement and the correctness of the proof checker.

The purpose of having a human look at LLM output is not just to verify correctness but also to understand a system. The parallel to math is that the value of a proof is not just the verification of a statement but also the working out of the mathematics required to establish that truth.

> A real-world system with the abilities of a robustly superintelligent mathematician can, when some practical problem requires it, formulate a problem statement, churn for a bit, spit out a formalized answer, and continue whatever outer-loop task it was doing without a human even finding out.

Without anyone ever finding out.

In the long run this means almost no progress past what we already have, because there's no shoulders of giants to stand on. Not even AI shoulders.

Anything generally useful/novel probably does end up in some kind of central catalogue(or maybe just the weights). After a while though, you're tall enough yourself that the giant's shoulders don't make much of a difference.

The whole point is to generate understanding and frames of reference which can be applied to other fields as well. Autogenerating without ever sharing solves nothing at all.

Isn't it widely accepted that many useful results were just the result of exploring math that felt interesting for the mathematician?

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