You are confusing that. The biggest advancements in science are the result of the application of leading-edge pure math concepts to physical problems. Netwonian physics, relativistic physics, quantum field theory, Boolean computing, Turing notions of devices for computability, elliptic-curve cryptography, and electromagnetic theory all derived from the practical application of what was originally abstract math play.

Among others.

Of course you never know which math concept will turn out to be physically useful, but clearly enough do that it's worth buying conceptual lottery tickets with the rest.

There is a difference between inventing/axiomatizing new mathematical theories and proving conjectures. Take the Riemann hypothesis (the big daddy among the pure math conjectures), and assume we (or an LLM) prove it tomorrow. How high do you estimate the expected practical usefulness of that proof?

That's an odd choice, because prime numbers routinely show up in important applications in cryptography. To actually solve RH would likely involve developing new mathematical tools which would then be brought to bear on deployment of more sophisticated cryptography. And solving it would be valuable in its own right, a kind of mathematical equivalent to discovering a fundamental law in physics which permanently changes what is known to be true about the structure of numbers.

Ironically this example turns out to be a great object lesson in not underestimating the utility of research based on an eyeball test. But it shouldn't even have to have any intuitively plausible payoff whatsoever in order to justify it. The whole point is that even if a given research paradigm completely failed the eyeball test, our attitude should still be that it very well could have practical utility, and there are so many historical examples to this effect (the other commenter already gave several examples, and the right thing to do would have been acknowledge them), and besides I would argue they still have the same intrinsic value that any and all knowledge has.

> To actually solve RH would likely involve developing new mathematical tools which would then be brought to bear on deployment of more sophisticated cryptography.

I doubt that this is true.

It already has! The progress that's been made thus far, involved the development of new ways to probabilistically estimate density of primes, which in turn have already been used in cryptography for secure key based on deeper understanding of how to quickly and efficiently find large prime numbers.

Just to throw in another one, string theory was practically nothing but a basic research/pure research program unearthing new mathematical objects which drove physics research and vice versa. And unfortunately for the haters, string theory has borne real fruit with holography, producing tools for important predictions in plasma physics and black hole physics among other things. I feel like culture hasn't caught up to the fact that holography is now the gold rush frontier that has everyone excited that it might be our next big conceptual revolution in physics.