We are all familiar with Wigner's "unreasonalbe 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.