...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.

Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.

... and it is probably that there is some secret here which remains to be discovered.

It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones.

Mathematics is purely hypothetical: it produces nothing but conditional propositions.

The pragmatist knows that doubt is an art which hs to be acquired with difficulty.

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

It has never been in my power to study anything, mathematics, ethics, metaphysics, gravitation, thermodynamics, optics, chemistry, comparative anatomy, astronomy, psychology, phonetics, economics, the history of science, whist, men and women, wine, metrology, except as a study of semeiotic .

Another characteristic of mathematical thought is that it can have no success where it cannot generalize.

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.

This branch of mathematics [Probability] is the only one, I believe, in which good writers frequently get results which are entirely erroneous.

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