Wisdom is rooted in watching with affection the way people grow.
Logic is invincible because in order to combat logic it is necessary to use logic.
Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.
Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined; it is as limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer`s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life.
No mathematician can be a complete mathematician unless he is also something of a poet.
Music is the pleasure the human soul experiences from counting without being aware that it is counting.
Formalized mathematics, to which most philosophy has been devoted in recent years, is in fact hardly to be found anywhere on earth or in heaven outside the texts and journals of symbolic logic.
Poets do not go mad ; but chess players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination.
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight!
A mathematician is a blind man in a dark room looking for a black cat which isn`t there.
On the basis of my historical experience, I fully believe that mathematics of the twenty-fifth century will be as different from that of today as the latter is from that of the sixteenth century.
`I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me.`