A smile happens in a flash, but its memory can last a lifetime.

In America everybody is of the opinion that he has no social superiors, since all men are equal, but he does not admit that he has no social inferiors, for, from the time of Jefferson onward, the doctrine that all men are equal applies only upwards, not downwards.

Of these austerer virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith. Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.

The human race may well become extinct before the end of the century. Speaking as a mathematician, I should say the odds are about three to one against survival.

For the learning of every virtue there is an appropriate discipline, and for the learning of suspended judgment the best discipline is philosophy.

At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.

Righteousness cannot be born until self-righteousness is dead.