All discussion of the ultimate nature of things must necessarily be barren unless we have some extraneous standards against which to compare them.

Humanity is at the very beginning of its existence-a new-born babe, with all the unexplored potentialities of babyhood; and until the last few moments its interest has been centred, absolutely and exclusively, on its cradle and feeding bottle.

We may as well cut out group theory. That is a subject that will never be of any use in physics.

...nature seems very conversant with the rules of pure mathematics, as our own mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.

In this model, the sun is a very tiny speck of dust indeed-a speck less than a three-thousandth of an inch in diameter ... Think of the sun as something less than a speck of dust in a vast city, of the earth as less than a millionth part of such a speck of dust, and we have perhaps as vivid a picture as the mind can really grasp of the relation of our home in space to the rest of the universe.

Sciences usually advances by a succession of small steps, through a fog in which even the most keen-sighted explorer can seldom see more than a few paces ahead. Occasionally the fog lifts, an eminence is gained, and a wider stretch of territory can be surveyed-sometimes with startling results. A whole science may then seem to undergo a kaleidoscopic rearrangement, fragments of knowledge sometimes being found to fit together in a hitherto unsuspected manner. Sometimes the shock of readjustment may spread to other sciences; sometimes it may divert the whole current of human thought.

The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures.