If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper.

If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics.

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.

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

It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.

Time makes fools of us all. Our only comfort is that greater shall come after us.

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.