Post Made Community Wiki by S. Carnahan
1

I think one needs to distinguish between phenomena, expression and techniques.

Number theory and geometry are phenomena. We observe that all techniques: combinatorics (first principles), algebra (recursion), analysis (approximation) are employed to investigate number theory and geometry.

Regarding the descriptor "interesting", one could call those portions of these technical areas that are employed to investigate the phenomena of number theory and geometry as interesting.

So where then is set theory and category theory? These are expression. Set theory is a language consisting only of nouns. All functions, relations are articulated as sets, a noun. Category theory is a language consisting of verbs and nouns. Functions are modelled as morphisms, a verb. Spaces are modelled as objects, a noun.

In fact before set theory was formalized, there was still mathematics. This primitive language was used in word problems (there are 12 chickens and rabbits in a cage. There are 30 legs in total. How many chickens are there?) before set theory came and bijected the animals to some finite cardinal. It was at this primitive stage that there was no (set-theoretic) definition of natural numbers or real numbers. Yet there was arthmetic and geometry. But all constructive.

As such, one needs to be careful when saying that category theory is a technique. Category theory should be seen as a language, and there is still analysis (limits and colimits), algebra (obvious), and combinatorics (higher categories) in category theory.