Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
A Unitarian is a person who believes in at most one God.
In a living civilization there is always an element of unrest, for sensitiveness to ideas means curiosity, adventure, change. Civilized order survives on its merits and is transformed by its power of recognizing its imperfections.
It is not paradox to say that in our most theoretical moods we may be nearest to our most practical applications.
We must produce a great age, or see the collapse of the upward striving of our race.
Nobody has a right to speak more clearly than he thinks.
Our habitual experience is a complex of failure and success in the enterprise of interpretation. If we desire a record of uninterpreted experience, we must ask a stone to record its autobiography.
Religion increasingly is tending to degenerate into a decent formula wherewith to embellish a comfortable life.
There is only one subject matter for education, and that is Life in all its manifestations.
In a certain sense, everything is everywhere at all times. For every location involves an aspect of itself in every other location. Thus every spatio-temporal standpoint mirrors the world.
Civilizations can only be understood by those who are civilized.
The art of progress is to preserve order amid change, and to preserve change amid order. Life refuses to be embalmed alive. The more prolonged the halt in some unrelieved system of order, the greater the crash of the dead society.
Nature is probably quite indifferent to the aesthetic preferences of mathematicians.
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.