Like
applied mathematics except without the beards, the
arm waving and the real world utility. Pure mathematicians wear
cardigans as a uniform and like playing
chess.
The subject is concerned with all the 'interesting' results that can be obtained by pure thought alone. What they define as interesting depends on their particular field of study; mathematical
anaylsts find
pain and extreme
confusion interesting, whereas
geometers are interested in
beauty.
People have different approaches to how the subject should be taught and understood. To see three completely different points of view, look at the first three write ups for the
Gram-Schmidt Theorem.