In formal logic, a contradiction is the signal of defeat, but in the evolution of real knowledge it marks the first step in progress toward a victory.

It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

We must produce a great age, or see the collapse of the upward striving of our race.

It is the business of the future to be dangerous; and it is among the merits of science that it equips the future for its duties.

Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.

Life is complex in its expression, involving more than percipience, namely desire, emotion, will, and feeling.

Nature is probably quite indifferent to the aesthetic preferences of mathematicians.