To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.

A proof is that which convinces a reasonable man; a rigorous proof is that which convinces an unreasonable man.

There are surely worse things than being wrong, and being dull and pedantic are surely among them.

Mathematics is not a separate entity. It owes a great deal of its power and its beauty to other disciplines.

In recent years we have become much more preoccupied with streamlining and organizing our subject than with maintaining its overall vitality. If we are not careful, a great adventure of the mind will become yet another profession.

... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied".

To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.