[Houshalter]

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc...

The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC.

Stuff like prime numbers have eaten up millions of brain hours of highly intelligent people. I remember thinking it was weird that so many project Euler problems were about prime numbers. And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.

And they seem to have been chosen for cryptography simply because it was a well studied problem with certain properties. Not because cryptography inherently needs prime numbers and would be impossible without centuries of previous work studying them.

[blintzing]

> Stuff like prime numbers have eaten up millions of brain hours of highly intelligent people

I think the idea that brilliant minds have been 'wasted' on prime numbers is nonsense. Don't 'highly intelligent people' have the right to pursue what interests them, and even disregarding that, won't they do their best work on problems that interest them?

Even further, is learning anything that is not practical or useful a 'waste'? Certainly not. Calculus might not be of the utmost importance career-wise for an aspiring musician, but learning it helps us think in new ways.

> The vast majority of "useless" mathematics really do turn out to be useless.

That's fine! So long as we strike gold every once in a while (cryptography, which is pretty essential to the internet functioning as anything more than a bulletin board), math is doing it's job.

[coliveira]

> Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC.

That's the worst example you could find, because Einstein didn't develop the mathematics for general relativity. He relied on the math invented in the XIX century for non-Euclidian geometry. If nobody had though about such a "sillY' geometry with "no practical value" it would probably take much longer because the necessary results would be out of the reach for Einstein.

[sdenton4]

Riemann's contribution is overlooked far far too often. The early non-Euclidean geometries were spaces of constant curvature - spherical and hyperbolic - and Riemann brought the idea of a manifold, and the notion of having a geometry that changes as you move around the space. And he did it in a fantastic lecture with only one equation in 1854, a good 50 years before special relativity.

Einstein was also definitely familiar with the work of Helmholtz, who did some fascinating work on non-Euclidean geometry in the context of ophthalmology: Lenses change the amount of curvature we perceive in space (think of fish-eye lenses), and provide a great jumping off point for the notion that the universe might not be as flat as it appears.

The Dover book 'Beyond Geometry' collects a bunch of the major papers in non-Euclidean geometry leading up to relativity, and is a fantastic read.

[Houshalter]

He figured out that spacetime may be non-Euclidean before he was aware that the math had been extensively studied. Then he learned about the preexisting work.

It's true it probably would have been much harder for him to work out the math on his own. But he and/or others eventually would have done it.

[shas3]

I think you are needlessly generalizing the personal argument of the LessWrong post to apply to general epistemology. As a person who is a utilitarian, you cannot fully justify studying pure math. However, at a societal level, you do need critical mass in terms of enough people working on math for practically useful insights to emerge. So 'don't do pure math' (or art, or music, or theoretical CS) is good as a personal goal for someone with utilitarian aspirations, but is inappropriate 'public policy'.

Consider this: a vast majority of mutations are useless, but for this reason, if there were no mutations at all, and mutations somehow willed themselves out of existence, then there would only be primitive lifeforms on earth.

[goolulusaurs]

That is nonsense. We have have no idea what math will turn out to be useful in the future. To say that it has already turned out to be useless presupposes that we already know all uses we might put it to in the future, which we clearly don't.

[Houshalter]

Well empirically most math developed in the past turned out to be useless up until now. Are you suggesting that it will suddenly become useful in the (near) future? Are the past few centuries not enough time for you to generalize from?

[goolulusaurs]

I think it is very plausible that a large portion of mathematics that is not very useful presently could be very useful for problems we have yet to tackle. As science progresses it will be less and less able to make grand unified theories and increasingly focus on the manifold particular. I can imagine much of math being useful only for problems we haven't even identified yet, like algebraic topology being used to study social dynamics, or engineering problems at strange scales. Even beyond that I think it may be the case that much of mathematics will become useful for reasons unforeseen. Unknown unknowns always seem to be where new science pops up.

[tsotha]

>And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.

Cryptography is a pretty big deal, though. You can't run a modern economy without it.