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.
When success turns a man's head he faces failure
Culture is activity of thought, and receptiveness to beauty and humane feeling. Scraps of information have nothing to do with it. A merely well informed man is the most useless bore on God's earth.
A Unitarian is a person who believes in at most one God.
Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
You may not divide the seamless coat of learning.
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.
Learning preserves the errors of the past, as well as its wisdom. For this reason, dictionaries are public dangers, although they are necessities.
In every age of well-marked transition, there is the pattern of habitual dumb practice and emotion which is passing and there is oncoming a new complex of habit.
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 .
Shakespeare wrote better poetry for not knowing too much; Milton, I think, knew too much finally for the good of his poetry.
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.
I would be a billionaire if I was looking to be a selfish boss. That's not me.
But you can catch yourself entertaining habitually certain ideas and setting others aside; and that, I think, is where our personal destinies are largely decided.
Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.
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.
Nature is probably quite indifferent to the aesthetic preferences of mathematicians.
It is the business of the future to be dangerous.
The many become one and are increased by one.
Imagination is not to be divorced from the facts. It is a way of illuminating the facts.
I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming . . . and a little mad.
Nature, even in the act of satisfying anticipation, often provides a surprise.
"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.
The consequences of a plethora of half-digested theoretical knowledge are deplorable.
But harmony is limitation. Thus rightness of limitation is essential for growth of reality. Unlimited possibility and abstract creativity can procure nothing.
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