This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does.

This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does.

A typical mathematician does not actively try to be useful. Individual mathematicians are motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a whole does benefit society.

Moreover, if one selects a problem, works on it in isolation for a few years and finally solves it, there is a danger, unless the problem is very famous, that it will no longer be regarded as all that significant.

At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied.

It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers.

Here is something Category-Theorists like: it is trivial, but not trivially trivial.