Much has been written about
the "beauty and elegance" of pure mathematics,
expressed e.g. by
Proofs from THE BOOK
and in quotes like
Beauty is the first test: there is no permanent place in this world
for ugly mathematics (G.H. Hardy).
Notwithstanding the provocative title, I really have nothing against beauty and elegance, but
(cf. "the cult of the paper" elsewhere)
they strike me as part of a "small-scale" view -- it's mostly individual theorems and proofs that are
taken to be elegant.
I don't have any very crisp formulation of my views, but the thoughts below are
intended as counterpoints to the "beauty and elegance" view.
One can be elegant only on a small stage.
A Shakespearean sword fight is elegant; a real battlefield is not.
If there's only one right way of thinking about an intellectual topic,
then the topic is ipso facto rather narrow.
The latter relates to different individual styles of doing mathematics, e.g.
Dyson's birds and frogs.
What makes a topic interesting is offering scope for both!
If you choose to view mathematics as providing a pinnacle of human creative
(Sistine Chapel, Hamlet, General Relativity, Fermat's last theorem), or as
"Queen of the Sciences", please do so in private.
The profession is better served by being less full of ourselves in public, and
instead portraying mathematics as useful intellectual infrastructure, analogous to
the useful physical infrastructure of the internet.
Doron Zeilberger's 95th opinion, we need beavers to build infrastructure, as well as
birds and frogs.
Identifying "mathematics" as theorem-proof mathematics is like identifying "visual art" with oil painting.
A corollary to the "three types of mathematics" comment.