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.

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.

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.

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.

... 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.