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.

What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers.

Although the prime numbers are rigidly determined, they somehow feel like experimental data.

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.

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