Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.

A number of aspects of mathematics are not much talked about in contemporary histories of mathematics. We have in mind business and commerce, war, number mysticism, astrology, and religion. In some instances, writers, hoping to assert for mathematics a noble parentage and a pure scientific experience, have turned away their eyes. Histories have been eager to put the case for science, but the Handmaiden of the Sciences has lived a far more raffish and interesting life than her historians allow.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.

I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.

The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes.