The real danger is not that computers will begin to think like men, but that men will begin to think like computers.
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something else.
Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.
Order is heaven's first law.
My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.
He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.
Everything of importance has been said before by somebody who did not discover it.
Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes.
A human being is a part of the whole called by us universe, a part limited in time and space. He experiences himself, his thoughts and feeling as something separated from the rest, a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty.
Know what you are talking about.
If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’ ’The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.
Few men speak humbly of humility, chastely of chastity, skeptically of skepticism.
If my theory of relativity is proven successful, Germany will claim me as a German and France will declare me a citizen of the world. Should my theory prove untrue, France will say that I am a German, and Germany will declare that I am a Jew.
Everything that is written merely to please the author is worthless.
Seek simplicity, and distrust it.
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures.
If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
Only mathematics and mathematical logic can say as little as the physicist means to say.
Pure mathematics is on the whole distinctly more useful than applied... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.
The moving power of mathematical invention is not reasoning but imagination.
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