Most of pure math does have practical applications - to pure maths! My favorite professors never launched into a subject without a motivating example, even if that motivation was often "Look at x, y, and z. Aren't they awfully similar?". My first exposure to Abstract Algebra started with a little number theory, moved on to rings, then to ideals, and only then to groups. Many people I've talked to about it are surprised we took axioms away rather than adding them, but the way we learned motivated each step. Indeed, groups themselves were introduced with permutations. Similarly, I found measure theory was best introduced by showing how handy cardinality was for finite sets. A "practical" application would have been probability, so perhaps this wasn't exactly application focused, but we certainly didn't start from the definition and work our way out.