Mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics.

When an image is said to be singular, it is meant that it is absolutely determinate in all respects. Every possible character, or the negative thereof, must be true of such an image.

Theoretically, I grant you, there is no possibility of error in necessary reasoning. But to speak thus "theoretically," is to uselanguage in a Pickwickian sense. In practice, and in fact, mathematics is not exempt from that liability to error that affects everything that man does.

A true proposition is a proposition belief which would never lead to such disappointment so long as the proposition is not understood otherwise than it was intended.

A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody,that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen.

Law is par excellence the thing that wants a reason. Now the only possible way of accounting for the laws of nature, and for uniformity in general, is to suppose them results of evolution.

Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.