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.‎

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?

It is by logic we prove. It is by intuition we discover.

It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the later. Prediction becomes impossible, and we have the fortuitous phenomena.

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

To invent is to discern, to choose.

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