There is a particular kind of open-ended idea which I call the blue-sky idea. Namely, someone sitting at his desk with his feet up and looking out the window has come up with some concept, and now hopes that he or someone he knows can come up with a good way of using that idea. At conferences one sometimes encounters mathematicians who specialize in this sort of idea, button-holing everyone who can't manage to avoid them and inflicting their most recent blue-sky idea on them.
...
When one looks at the history of mathematics, it may seem that a lot of the most important developments have come out of blue-sky ideas. But in fact, from what I know, this is almost never the case. Good ideas always arise out of some existing line of thought, and it is only after when has put a lot of effort in that one realizes that there is some gradiose general concept that underlies all one's work.
For instance, I am pretty sure that Eilenberg and Steenrod didn't sit down together over a beer one day and say, "Wouldn't it be neat to draw up a set of axioms for a thing called a category and then invent a concept called a functor, and then see if these would be useful for anything?" It's pretty clear that in fact they noticed that in algebraic topology the same sorts of situations keep coming up over and over again, and one keeps seeing different theorems with different subject matter, where somehow the proofs always turned out to be more or less the same. So they saw (I believe) the need for a vocabulary and a conceptual framework that would enable mathematicians to talk about all this in a unified way. Ergo: category theory.
