Il <i>n’y pas d’infini actuel</i>; les Cantoriens l’ont oublié, et ils sont tombés dans la contradiction. Il est vrai que le Cantorisme a rendu des s… - Henri Poincaré

We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them.
...We ...choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: "The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions, are only the fruit of an unconscious opportunism."

Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better