In the old days when people invented a new function they had something useful in mind.

The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.

It is not order only, but unexpected order, that has value.

All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.‎

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

[T]he different branches of Arithmetic - Ambition [G]eometry is not true, it is advantageous.

There are no solved problems; there are only problems that are more or less solved.