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