Find some development which has been important in human history, and then working backwards argue that broader and broader theoretical contributions were valuable insofar as they facilitated this development.
Your previous comments indicate that you have a ready-made formula for rejecting any such example, which is to deny that the theoretical contributions were "necessary". That is, you will argue that the important development "could have" happened without them. For instance, you addressed the classic example of non-Euclidean geometry paving the way for general relativity by saying that Einstein could have just invented non-Euclidean geometry on his own (!) when he needed it.
Such an argument is difficult to take seriously and seems to me only to indicate that, whatever your past history, you simply have, at the moment, a strong distaste for abstract mathematics. (Or, alternatively, perhaps you're playing an intense game of Devil's Advocate with yourself.) That's fine, tastes differ. But again, that's hardly enough reason to jump to the conclusion that nobody should be doing things like inventing non-Euclidean geometry.
You've characterized my position as motivated cognition in defense of the status quo, but in my view that is both wrong and unfair. It's wrong, first of all, because I don't necessarily think the status quo is optimal -- in fact I think mathematical research could probably be done a lot more efficiently. I just don't happen to favor changes in the particular direction that you advocate (basically the adoption of a concreteness heuristic for deciding what's "useful"). But even more importantly perhaps, it's unfair: even if my ideal mathematical world looks more like the current one than yours does, the fact remains that we are living in an exceptional period in human history. Most societies have not conducted extensive amounts of "theoretical research", but instead have occupied themselves almost exclusively with what they found to be "practical" and "relevant" within their local world; your position, rather than mine, is closer to the human default. In my view, the last few centuries of Western civilization are an exceptional instance of people finally beginning to almost start getting things right with respect to this question.
I don't actually think the issue here is exclusively empirical; I can sense important disagreements that may be better characterized as value differences (conceivably, these may ultimately also reduce to empirical disagreements, but if so it would be at several more inferential steps' remove). But I will concede that there is a substantial empirical component. On my analysis, what it boils down to is that you think you have a good way of predicting the long-term impact of mathematical work, whereas I don't think you do, because I don't think the heuristics you're using are any more powerful than (or even particularly different from) the most common human default.
you addressed the classic example of non-Euclidean geometry paving the way for general relativity by saying that Einstein could have just invented non-Euclidean geometry on his own (!) when he needed it.
My claim was that Einstein explicitly considered gravity as curvature of space before being aware of any formal developments in non-Euclidean geometry. I inferred that, if Einstein had lived before any formal developments in non-Euclidean geometry occurred, then non-Euclidean geometry would have become an object of study not because it was interesting bu...
For the past few days I've been pondering the question of how best to respond to paulfchristiano's recent posts and comments questioning the value of mathematical research. I don't think I can do it concisely, in a single post; bridging the inferential distance may require something more like a sequence of posts. I may end up writing such a sequence eventually, since it would involve ideas I've actually been wanting to write up for some time, and which are actually relevant to more than just the specific questions at issue here (whether society should sponsor mathematics, and given that it does, whether paulfchristiano or anyone else in the LW readership should pursue it).
However, as the preceding parenthetical hints at, I'm actually somewhat conflicted about whether I should even bother. Although I believe that mathematical research should be conducted by somebody, it's not at all clear to me that the discipline needs more people beyond those who already "get" its importance, and are out there doing it rather than writing skeptical posts like paulfchristiano's. It seems perfectly plausible to me that those who feel as paulfchristiano does should just leave the profession and do something else that feels more "important" to them. This is surely the best practical solution on an individual level for those who think they have a better idea than existing institutions of where the most promising research directions lie, at least until Hansonian prediction markets are (ever) implemented.
Nevertheless, for those interested in the society-level question of whether mathematics (as such) may be justifiably pursued by anyone, or any community of people, as a professional occupation (which is quite distinct from the question of whether e.g. paulfchristiano should personally pursue it), I recommend, at least as a start, grappling with the arguments put forward by the best mathematicians in their own words. I think this essay by Timothy Gowers (a Fields Medalist), titled "The Importance of Mathematics", is a good place to begin. I would particularly draw the attention of those like paulfchristiano, who think they have a good idea of which branches of mathematics are useful and which aren't, to the following passage, from pp.8-9 (unfortunately the illustrations are missing, but the point being made is pretty clear nonetheless):