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

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, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road we want to understand the idea of the proof, the deeper context.

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

Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years. [Said in 1950]

Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory.

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

No mathematician of equal stature has risen from our generation... Hilbert was singularly free from national and racial prejudices in all public questions, be they political, social or spiritual, he stood forever on the side of freedom.

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

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 constraint and freedom that is the very essence of human nature.

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. Thus we are cleverest where knowledge matters least.

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