For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.

Before one can generalize, formalize, and axiomatize, there must be a mathematical substance.

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.

Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony.

Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.

Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind.