Protagoras did not know if the gods exist, but he held in any case they ought to be worshiped. Philosophy, according to him, had nothing edifying to teach, and for the survival of morals we must rely upon the thoughtlessness of the majority and their willingness to believe what they had been taught.

We shall say that we have acquaintance with anything of which we are directly aware, without the intermediary of any process of inference of any knowledge of truths.

It seems to me a fundamental dishonesty, and a fundamental treachery to intellectual integrity to hold a belief because you think it's useful and not because you think it's true.

The true spirit of delight...is to be found in mathematics as surely as in poetry.

Aristotle, in spite of his reputation, is full of absurdities. He says that children should be conceived in the Winter, when the wind is in the North, and that if people marry too young the children will be female. He tells us that the blood of females is blacker than that of males; that the pig is the only animal liable to measles; that an elephant suffering from insomnia should have its shoulders rubbed with salt, olive-oil, and warm water; that women have fewer teeth than men, and so on. Nevertheless, he is considered by the great majority of philosophers a paragon of wisdom.

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.

The process of philosophizing, to my mind, consists mainly in passing from those obvious, vague, ambiguous things, that we feel quite sure of, to something precise, clear, definite, which by reflection and analysis we find is involved in the vague thing that we start from, and is, so to speak, the real truth of which that vague thing is a sort of shadow.