But as the work proceeded I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was not more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.

To a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal.

While I am as convinced a Socialist as the most ardent Marxian, I do not regard Socialism as a gospel of proletarian revenge, nor even, primarily, as a means of securing economic justice. I regard it primarily as an adjustment to machine production demanded by considerations of common sense, and calculated to increase the happiness, not only of proletarians, but of all except a tiny minority of the human race.

Hatred of enemies is easier and more intense than love of friends. But from men who are more anxious to injure opponents than to benefit the world at large no great good is to be expected.

Philosophy arises from an unusually obstinate attempt to arrive at real knowledge. What passes for knowledge in ordinary life suffers from three defects: it is cocksure, vague and self-contradictory. The first step towards philosophy consists in becoming aware of these defects, not in order to rest content with a lazy scepticism, but in order to substitute an amended kind of knowledge which shall be tentative, precise and self-consistent.

To acquire immunity to eloquence is of the utmost importance to the citizens of a democracy.