I know that the great Hilbert said "We will not be driven out of the paradise Cantor has created for us," and I reply "I see no reason for walking in"!

Richard Hamming

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

In mathematics we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.

David Hilbert
The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.

George Pólya
In mathematics you don't understand things. You just get used to them.

It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way.. Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.

John Von Neumann
Last modified: Wed Jul 16 20:43:34 CEST 2014