Why math?

If we really want to evaluate this argument, it is important to understand the conditions under which the important math of today was done. In the case of calculus, differential equations, statistics, functional analysis, linear algebra, group theory, and numerical methods, the important results for modern work were in fact developed after their usefulness could be appreciated by an intelligent observer. There is very little honestly compelling evidence that pushing math for the sake of pushing math is likely to lead to practically important results more effectively than waiting until new math is needed and then developing it. Perhaps the most compelling case is number theory and its unexpected application to cryptography, which is still not nearly compelling enough to justify work on pure math (or even provide significant support).

Consider the actual marginal utility of advances in your field of choice, honestly. In the overwhelming majority of cases, the bulk of research effort is directed grotesquely inefficiently from a social perspective. In particular, a small number of largely artificial applications will typically support research programs which consume an incredible amount of intelligent mathematicians' time, compared to the time required to make fundamental progress on the actual problem that people care about. Here you have to make a different argument for every research program, which I would be happy to do if anyone offers a particular challenge.

I agree with you. For the most part, applied souls dream up their advances and make them without relying on the mathematical machine. They invent the math they need to describe their ideas. Or perhaps they use a little of the pure mathematician's machine, but quickly develop it in ways that are more important to their work than the previous mathematical meanderings.

I think you underestimate the role of mathematics as the grand expositor. It is the tortoise that trails forever beyond the hare of applied science. It takes the insights of applications, of calculus for example, and digests them. It reworks them, understands them, connects them, rigorizes them.

The work of mathematics is not useful in your mind because a mathematician does not make a truly new applied advance. A mathematician invents and connects notations to ease the traversal, the learning, and most importantly the storage in working memory of past insights.

What is the purpose of a category? An operad? A type theory? A vector bundle? The digit 0? When these languages were introduced, it could always be claimed they were worthless because the old languages could express the same content as these new languages. But somehow the new language makes it easier to conceptualize and think about the old ideas; it increases the working human RAM.

And what of the poor student? He who must learn so many subjects is grateful when it is realized that many of those subjects are in fact the same: http://arxiv.org/abs/0903.0340 . Mathematics digests theories and rewrites them as branches of a common base. It makes it possible to learn more insights quickly and to communicate them to the next generation.

So young applied scientists, perhaps generations later, benefit by more compactly and elegantly understanding the insights of their forebearers. Then, the mathematician dreams, they are freer to envision the next great ideas: http://arxiv.org/abs/1109.0955

So why the mathematician's focus on solving specific problems? Why so much energy to characterize finite groups? It is not that these problems are important. It is that they serve as testbeds for new languages, for new characterizations of old insights. The problems of pure math are invented as challenges to understand an old applied language, not to invent a new one.