I think a great example of this is cryptography. The foundations of it come from number theory (prime numbers, modular arithmetic, elliptic curves), but the subject of number theory, before the advent of computing, was possibly the most useless kinds of mathematical 'art' that could have existed. I imagine it was the mathematical equivalent of frolicking in the fields.
Mathematicians explored Fermat's little theorem starting in 1640, but they didn't do it because they knew it'd be useful several hundred years later in RSA. They did it simply because math is worth exploring in itself.
Even if you don't subscribe to the idea that we should pursue math for math's sake, history shows us that it's very difficult to know what parts of math will be useful to humanity, especially hundreds of years later. Since people work best on what they find interesting, mathematicians should continue exploring the topics that most interest them, because we really can't say with any certainty what will prove useful (or even essential) to future generations.
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.
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.
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.
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.
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.
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.
Cryptography is a pretty big deal, though. You can't run a modern economy without it.
That's like saying that we should randomly start drilling holes in the ground because sometimes we will strike oil.
people arguing for it usually ignore the silent evidence of research that lead nowhere and also, more importantly, the potential research accomplishments those people could have acheived if guided to work on different problems.
Unless you have better tools for locating the oil, that's not a bad strategy.
A crucial difference between digging for oil and doing math, however, is the nature of the externalities. In either case, you're burning some work that could be spent somewhere better, but with oil you're left with a hole that you probably want not to be there and there's no good way to put it back. In both drilling and math, "drilling" helps us refine our methods. In the case of math exploring more of the ramifications of our axioms also helps raise our confidence that they're not subtly inconsistent.
And of course, math is generally less expensive than an oil well.
I don't know where the cost-benefit analysis puts work on math when we don't yet see practical application. And I think that's often over-romanticized. However, I do think there are a lot of reasons we should expect the analysis to come out more favorably than for drilling random holes.
This reminds me of this Von Neumann quote about the importance of mathematics having an 'empirical source':
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I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.
In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.
I'm only nitpicking because I recently started compiling a list of interesting quotes and intend to save this one :)
http://arxiv.org/abs/math/9404236
Your comment could very well be its abstract.
This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important.
Incentives for researchers are mostly to push and prod at the boundaries of a field, but in my opinion mathematical ideas are only of marginal value in themselves; more important is the way they help us understand and interact with the physical universe, and for that building communities, developing effective languages and notations, codifying our understanding, and making it accessible both to newcomers and to outsiders is the most important task for a field, and perhaps for our society generally.
Just like with software projects or companies, the most “success” comes from helping a range of other people solve their problems and extend their abilities, not from making technically beautiful art projects for their own sake (not that there’s anything inherently wrong with those).
Perhaps more generally, while theorem proving has overwhelmingly dominated pure mathematics and related fields for the past 80–100 years, and has been an important tool since Euclid, theorem proving is only one way of approaching the world, and in my opinion is a mere tool, not an end in itself. Just like simulation is a tool, or drawing pictures is a tool, or statistical analysis is a tool.
I like this bit from Feynman: https://www.youtube.com/watch?v=YaUlqXRPMmY