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.
The new tinge to modern minds is a vehement and passionate interest in the relation of general principles to irreducible and stubborn facts. All the world over and at all times there have been practical men, absorbed in 'irreducible and stubborn facts'; all the world over and at all times there have been men of philosophic temperament, who have been absorbed in the weaving of general principles. It is this union of passionate interest in the detailed facts with equal devotion to abstract generalisation which forms the novelty of our present society.
Through and through the world is infested with quantity: To talk sense is to talk quantities. It is no use saying the nation is large. . . . How large? It is no use saying the radium is scarce. . . . How scarce? You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.
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.
Art heightens the sense of humanity. It gives an elation to feeling which is supernatural...A million sunsets will not spur us on towards civilization. It requires Art to evoke into consciousness the finite perfections which lie ready for human achievement.
Nature is probably quite indifferent to the aesthetic preferences of mathematicians.
Learning preserves the errors of the past, as well as its wisdom. For this reason, dictionaries are public dangers, although they are necessities.
The preternatural solemnity of a good many of the professionally religious is to me a point against them.
But harmony is limitation. Thus rightness of limitation is essential for growth of reality. Unlimited possibility and abstract creativity can procure nothing.
What the learned world tends to offer is one second-hand scrap of information illustrating ideas derived from another second-hand scrap of information. The second-handedness of the learned world is the secret of its mediocrity.
Many a scientist has patiently designed experiments for the purpose of substantiating his belief that animal operations are motivated by no purposes. He has perhaps spent his spare time in writing articles to prove that human beings are as other animals so that 'purpose' is a category irrelevant for the explanation of their bodily activities, his own activities included. Scientists animated by the purpose of proving that they are purposeless constitute an interesting subject for study.
It is the business of the future to be dangerous.
Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.
We think of the number "five" as applying to appropriate groups of any entities whatsoever - to five fishes, five children, five apples, five days... We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable feat of abstraction; and it must have taken ages for the human race to rise to it
The consequences of a plethora of half-digested theoretical knowledge are deplorable.
There will be some fundamental assumptions which adherents of all the variant systems within the epoch unconsciously presuppose. Such assumptions appear so obvious that people do not know what they are assuming because no other way of putting things has ever occurred to them. With these assumptions a certain limited number of types of philosophic systems are possible, and this group of systems constitutes the philosophy of the epoch.
I would be a billionaire if I was looking to be a selfish boss. That's not me.
"One and one make two" assumes that the changes in the shift of circumstance are unimportant. But it is impossible for us to analyze this notion of unimportant change.
There is a quality of life which lies always beyond the mere fact of life; and when we include the quality in the fact, there is still omitted the quality of the quality.
In its solitariness the spirit asks, What, in the way of value, is the attainment of life? And it can find no such value till it has merged its individual claim with that of the objective universe. Religion is world-loyalty.
The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact.
Imagination is not to be divorced from the facts. It is a way of illuminating the facts.
The defense of morals is the battle-cry which best rallies stupidity against change.
It is this union of passionate interest in the detailed facts with equal devotion to abstract generalisation which forms the novelty in our present society .
If you have had your attention directed to the novelties in thought in your own lifetime, you will have observed that almost all really new ideas have a certain aspect of foolishness when they are first produced, and almost any idea which jogs you out of your current abstractions may be better than nothing.
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