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

Besides language and music, it [mathematics] is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.

... numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.

Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold.

In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry.

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.

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