It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.

Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

...the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely aesthetic feeling, which all mathematicians know

All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.‎

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

It may happen that small differences in the initial conditions produce very great ones in the final phenomena.

Mathematicians do not study objects, but the relations between objects.