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.

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

... 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.

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.

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.

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.