TITLE: How does one justify funding for mathematics research?
QUESTION [142 upvotes]: G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be beautiful, but unlike music, visual art or literature, much of the beauty, particularly at a higher level, is only available to the initiated. And while many great scientific advances have been built on mathematical discoveries, many mathematicians make no pretense of caring about any practical use their work may have. So when the taxpayer or a private organization provides grants for mathematical research, what are they expecting to get in return?
I'm asking not just so I can write more honest research proposals, or in case it comes up in an argument, but so I have an answer for myself.

REPLY [4 votes]: Since this question has popped up to the front page, and at the risk of interpreting it over-broadly, I think it is worth reproducing a famous quote by Carl Gustav Jacob Jacobi in a letter to Legendre dated 2 July 1830:

Il est vrai que M. Fourier avait l'opinion que le but principal des mathématiques était l'utilité publique et l'explication des phénomènes naturels ; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c'est l'honneur de l'esprit humain, et que sous ce titre, une question de nombres vaut autant qu'une question du système du monde.

(Carl Gustav Jacob Jacobi, Gesammelte Werke, band 1, Reiner,
Berlin 1881, Correspondance mathématique avec Legenre, 454–455, scan here.)
Translation from Wikiquote (ever-so-slightly edited):

It is true that Mr Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of numbers is as important as a question of the system of the world.

The reference to the "honor of the human mind" was used, in particular, by Dieudonné for the title of his book, Pour l'honneur de l'esprit humain (Les Mathématiques d'aujourd'hui) (1987).