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

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 univers….

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

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and alm….

You can not apply mathematics as long as words still becloud reality..

The introduction of numbers as coordinates by reference to the particular division scheme of the open one-dimensional continuum is an act of violence….

One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sha….

God exists because arithmetic is consistent - the Devil exists because we can't prove it!.

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 unr….

The whole is always more, is more capable of a much greater variety of wave states, than the combination of its parts. ... In this very radical sense….

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of con….

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, bea….

Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that b….

Our federal income tax law defines the tax y to be paid in terms of the income x; it does so in a clumsy enough way by pasting several linear functio….

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, wh….

... 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 simp….

But it seems an irony of creation that man's mind knows how to handle things the better the farther removed they are from the center of his existence….

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

The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its fi….

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

No mathematician of equal stature has risen from our generation... Hilbert was singularly free from national and racial prejudices; in all public que….