... the atlas is a manifold. This is a typical mathematician's use of the word "is", and should not be confused with the normal use.

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

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.

... the atlas is a manifold. This is a typical mathematician's use of the word "is", and should not be confused with the normal use.

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.

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

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.