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Yes, mathematics has been perhaps the purest human intellectual pursuit. Sure, many theorems turn out to have important applications in science and engineering, but the mathematical community has mostly escaped corporate interests. And for the reasons you mentioned about not needing any resources except your brain, it has been a uniquely human activity which showed us talent can come from anywhere, with stories like Ramanujan and Galois.

I hope that pure mathematics research can retain a strongly human component forever. It would sadden me immensely for human understanding of our mathematical world to wither and die, and for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can. As far as applied research goes, I hope we will always be able to understand what we want to, but I have less qualms about becoming more scalable and efficient.

>for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can

all this fantasy books with magic artifacts should have mentally prepared us. Time to study the prompts Potter was giving to his magic wand.

After all, one of the main work the top AI companies are doing rigth now is developing AI to further develop AI. After several layers of AI developing AI we probably wouldn't be able to understand much there.

> In my opinion, I think the Clay Mathematics Institute deserves some criticism for all the drama surrounding these problems. Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama. I feel that mathematics should be free from competitions and the pursuit of glory.

Currently 0/2 Millenium problem solvers claimed the prize money so clearly money is not their motivation for tackling the problem.

Money is target but this millenium price is not enough for them.

Openai is bringing machine gun to competition that used only knife and pistol.

It's not a large prize though.

OpenAI spent many multiples of the prize money in just a few days to get there and even if one solves a problem in the traditional way, that person is most likely already an accomplished professor at a reputable university where a million dollars doesn't mean as much as the eternal fame that comes with it.

It's definitely not a large prize for OpenAI, but actually IIRC they only spend a few hundred thousand dollars.

It is a large prize if you're just an academic.

OpenAI said that at public API prices, the agents they ran would have cost $15M. I don't know what their internal pricing is, but it almost certainly cost more than $1M.

I feel like this is answered in the very second paragraph?

> to elevate in the consciousness of the general public the fact that in mathematics the frontier is open, close at hand, and abounds with important unsolved problems; to emphasize the abiding value of working towards a solution of the deepest, most difficult problems; and to recognize achievements in mathematics of historic magnitude.

Also

> These are not arbitrary puzzles akin to fiendish crosswords. Rather, they are fundamental challenges that mark the frontier of human knowledge and challenge us to develop new structures and methods. They provide foci for the continuing struggle, across generations and cultures, to deepen our human understanding of mathematics and the universe that it describes.

I agree. Technological advances can lead to a better world for sure, but I think many people underestimate the human need to create and to find meaning in their work.

If AI can do superhuman math that allows better medicines, cleaner energy etc that is great. But if AI replaces humans in all the creative and intellectual fields that is not only a loss of jobs but also a loss of deeply meaningful activities. This is waved away but I think that is mistaken.

What I fear is really the growing notion that "people shouldn't do math/art/music because machine do it better and cheaper".

Nevermind better, worse and more expensive is still on the table if you don't have to deal with a human. Cars replaced horses for a lot of reasons, but insofar as cars do have personalities, they're much less quirky than horses'

> Mathematicians know that you can make problems arbitrarily complex, and declaring problems with large prizes attached to them can lead to a lot of competition and drama.

Yes, you can make problems arbitrarily complex. But the prize problems were chosen not just because the solutions appear likely to be very complex (the problem statements aren't necessarily inherently complex--there is a way to restate the Riemann hypothesis that a junior high school student could easily understand, which I'll give below).

They were chosen because they were important problems that mathematicians really wanted solved, top people had worked on them for a long time and progress stalled a long time ago, and it seemed likely that solving them would require major breakthroughs.

Those kind of problems can be discouraging. Enough people who are probably better than you have spent enough time failing to solve them that realistically most researchers are going to focus all their efforts on something they are likely to make progress on.

A nice prize can get more people to at least work on them as side projects.

Here's that restatement of the Riemann hypothesis I mentioned.

The Riemann hypothesis is that the non-trivial zeros of the function ζ(s) occur on the line 1/2 + yi.

ζ(s) is 1/1^s + 1/2^2 + 1/3^s + ... when s is a complex number whose real part is greater than 1, and defined everywhere else except s = 1 by a process called analytic continuation. The trivial zeros are at s = -2, -4, -6, ... .

For a mathematician, or a non-mathematician who has taken complex analysis and hasn't forgotten much of that, that is not too complex a definition. For anyone else the first reaction is probably "Trivial zeros? How the heck does that thing even have zeros? And if it does how the heck can it have zeros at any negative integers! It is obviously infinity at every negative integer!!!".

Here's a different hypothesis that turns out to be exactly equivalent to the Riemann hypothesis. They are either both true of both false, so resolving one of them resolves the other.

Let H(n) = 1 + 1/2 + ... + 1/n for all positive integers n. These are called the harmonic numbers.

Let S(n) = the sum of the positive integer factors of n for all positive integers n. For example S(4) = 1 + 2 + 4, S(6) = 1 + 2 + 3 + 6, and S(17) = 1 + 17.

Hypothesis: S(n) <= H(n) + exp(H(n)) log(H(n)) with equality only when n = 1.

The proof that this is equivalent to the Riemann hypothesis is here [1].

[1] https://arxiv.org/pdf/math/0008177

and it seemed likely that solving them would require major breakthroughs

If building a machine that solves these kinds of problems isn't a "major breakthrough," I don't know what is. Is the objection merely that it came from engineers rather than mathematicians? If so, there's plenty of room for contributions from many fields.

The best thing a mathematician can do to advance their art, at this point, is to drop whatever they're doing and work on AI.

Note my comment was in response to someone questioning the very notion of prizes for mathematics problems. These prizes were created over a quarter century ago.

> It feels like the mathematical world is changing very rapidly, almost overnight.

...

>Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.

Mathematics becomes engineering. I think it is great and long overdue. Saying that as a Math PhD dropout :) Of course like manual craftsmen had to adapt to Industrial Revolution, the same would need to be done by the mathematicians. And other scientists too.