What’s pure math?
Pure math is a logical system based on assumptions, called
axioms.   It might be called a “game” of inference.  Math is
as real (and as abstract) as the game of chess.
The best book I know on the subject is WHAT IS
MATHEMATICS, REALLY?  by Reuben Hersh  (Oxford
University Press).
For a working definition of what pure math is, the best thing
I could recommend is the 65 page “NOTES” section of
Hersh’s book, which shows the core of math very cleanly,
and gives a feel for it.