TITLE: What is category theory useful for?
QUESTION [129 upvotes]: Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory?
In Wikipedia it says it is to formalize. As far as I can tell, it sort of generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and automorphisms and homomorphisms, it seems to me like it just finds a bunch of results that are true to these functions in a bunch of disciplines. But why would generalizing that be of use? I mean, wouldn't it be easier to prove them specifically in each of the important disciplines?

REPLY [8 votes]: Category theory is not something instead of other mathematical theories but an other and very interacting perspective. The search for theorems and their proofs will never end as long as there are still mathematicians left. Even the familiar system of natural numbers will always hide secrets and any theory that helps to prove or counter prove just one single conjecture is worth to be further developed.
The category language might or might not be fruitful for a certain study, but it is always a perspective worth considering. In number theory there are less and more complex functions. The less complex functions are the first to be studied but more complex functions, as the increasing prime number function $p: \mathbb Z_+\rightarrow\mathbb Z_+$, seems resist any attempt of finally survey.
There is a category with all functions $f: \mathbb Z_+\rightarrow\mathbb Z_+$ as objects and pair of commuting functions $\alpha,\beta: \mathbb Z_+\rightarrow\mathbb Z_+$, 
$(\beta f=g\alpha)$, are the morphisms $\displaystyle f\overset{(\alpha,\beta)}{\longrightarrow}g$. 
Investigating such a category (I don't know if it is done) could bring a new approach to some issues. After all, it is not only the prime number function that is unresolved. When using categories there is also a possibility to find functors to other categories, such that hard problems in the primary category is transformed to a less hard problem in the secondary category.