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

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.

The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.

One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sharp but cold ... thus we are clearest where knowledge matters least: in mathematics, especially number theory.

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

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.

It is impossible to discuss realism in logic without drawing in the empirical sciences... A truly realistic mathematics should be conceived, in line with physics, as a branch of the theoretical construction of the one real world and should adopt the same sober and cautious attitude toward hypothetic extensions of its foundation as is exhibited by physics.