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][1]," in _Communications in Pure and Applied Mathematics_, Vol. 13, No. I (February 1960).

(2) G. H. Hardy, [_A Mathematican's Apology_][2], 1940. p.135.


  [1]: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
  [2]: http://en.wikipedia.org/wiki/A_Mathematician%27s_Apology