I am also greatly indebted to Bergson, William James, and John Dewey. One of my preoccupations has been to rescue their type of thought from the charge of anti-intellectualism, which rightly or wrongly has been associated with it.
In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science, as it grows towards perfection, becomes mathematical in its ideas.
I regret that it has been necessary for me in this lecture to administer a large dose of four-dimensional geometry. I do not apologize, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are.
I am sure that one secret of a successful teacher is that he has formulated quite clearly in his mind what the pupil has got to know in precise fashion. He will then cease from half-hearted attempts to worry his pupils with memorizing a lot of irrelevant stuff of inferior importance.
It is the business of the future to be dangerous.
The only use of a knowledge of the past is to equip us for the present. The present contains all that there is. It is holy ground; for it is the past, and it is the future.
Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about “any” things or about “some” things, without specifications of definite particular things.
On the ostensible exactitude of certain branches of human knowledge, including mathematics. The exactness is a fake.
The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.
Imagination is not to be divorced from the facts. It is a way of illuminating the facts.
The many become one and are increased by one.
Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.
Style, in its finest sense, is the last acquirement of the educated mind; it is also the most useful. It pervades the whole being.
To come very near to a true theory, and to grasp its precise application, are two different things, as the history of science teaches us. Everything of importance has been said before by someone who did not discover it.
The foundations of the world are to be found, not in the cognitive experience of conscious thought, but in the aesthetic experience of everyday life.
A race preserves its vigor so long as it harbors a real contrast between what has been and what may be; and so long as it is nerved by the vigor to adventure beyond the safeties of the past. Without adventure civilization is in full decay.
We must not expect simple answers to far-reaching questions. However far our gaze penetrates, there are always heights beyond which block our vision.
The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation.
The great achievements of the past were the adventures of the past. Only the adventurous can understand the greatness of the past.
Every epoch has its character determined by the way its population reacts to the material events which they encounter.