After you understand about the sun and the stars and the rotation of the earth, you may still miss the radiance of the sunset.

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilised of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.

In the conditions of modern life the rule is absolute, the race which does not value trained intelligence is doomed. Not all your heroism, not all your social charm, not all your wit, not all your victories on land or at sea, can move back the finger of fate. To-day we maintain ourselves. To-morrow science will have moved forward yet one more step, and there will be no appeal from the judgment which will then be pronounced on the uneducated.

True courage is not the brutal force of vulgar heroes, but the firm resolve of virtue and reason.

The art of progress is to preserve order amid change, and to preserve change amid order. Life refuses to be embalmed alive. The more prolonged the halt in some unrelieved system of order, the greater the crash of the dead society.

The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.

The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit.