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We are all familiar with Wigner's "unreasonable effectiveness of mathematics" thesis (1), and of Hardy's opinion that "the great bulk of higher mathematics is useless" (2). I am wondering if there are examples of (sub)fields of mathematics that have seen fertile developments within mathematics but have not (yet) found applicability to more pragmatic concerns? Of course, the answer may be a function of time: some mathematical areas may have seen an explosion of development and yet await their first application, which may or may not ever appear. But that in itself would be interesting (to me).

Here is an attempt to formulate the question:

Are there areas of mathematics which are flourishing in terms of their internal mathematical development, and yet remain quite disconnected from applications (except perhaps to other, equally abstract mathematical concerns)?

I am wondering to what extent a mathematical area can blossom and thrive in the absence of connections to "reality." Or must such burgeoning always tether back to "reality"?

(1) "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, Vol. 13, No. I (February 1960).

(2) G. H. Hardy, A Mathematican's Apology, 1940. p.135.

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Algebraic geometry developed for a long time before its connentions with statistical problems arising in biology appeared. – Michael Hardy Apr 23 '11 at 1:08
I just noticed I used a half-dozen words indicative of Spring! :-) – Joseph O'Rourke Apr 23 '11 at 1:14
You may be interested in the following talk by Tim Gowers, on "The importance of mathematics": In other news, congratulations on your two books just published! – Andrés E. Caicedo Apr 23 '11 at 1:30
The theory of conic sections was developed in ancient Greece without regard to applications and indeed it would have been considered bad taste to consider applications (or so I've heard it said.) There may have been some uses for duplicating the cube, but certainly the great applications to gravitational motion were undreamed of. – Aaron Meyerowitz Apr 23 '11 at 2:59
Fecundity and applicability will be correlated (though perhaps not locally in time, as the relevance of G. Hardy's métier to modern cryptography amply illustrates) for a simple reason. When you have a hammer, everything looks like a nail: and on the other hand, when you have a nail, you look for a hammer. – Steve Huntsman Apr 23 '11 at 4:46

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