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.

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

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

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.

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

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