Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing].
It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
Tolstoi explains somewhere in his writings why, in his opinion, “Science for Science's sake” is an absurd conception. We cannot know all the facts, since they are practically infinite in number. We must make a selection. Is it not better to be guided by utility, by our practical, and more especially our moral, necessities?
Often when works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, long or short, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind.
Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.
All of mathematics is a tale about groups.
It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wished to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside. We must, example, use language, and our language is necessarily steeped in preconceived ideas. Only they are unconscious preconceived ideas, which are a thousand times the most dangerous of all.
Doubting everything and believing everything are two equally convenient solutions that guard us from having to think
One geometry cannot be more true than another; it can only be more convenient.
One does not ask whether a scientific theory is true, but only whether it is convenient.
Mathematicians do not study objects, but the relations between objects.
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.
A small error in the former will produce an enormous error in the latter.
Sociology is the science with the greatest number of methods and the least results.
A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds...how does it come about that so many persons are here refractory?
Thought is only a flash between two long nights, but this flash is everything.
The mind uses its faculty for creativity only when experience forces it to do so.
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means.
We also know how cruel the truth often is, and we wonder whether delusion is not more consoling.
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.
Astronomy is useful because it raises us above ourselves; it is useful because it is grand; .... It shows us how small is man's body, how great his mind, since his intelligence can embrace the whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony.
Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.
Analyse data just so far as to obtain simplicity and no further.
All that is not thought is pure nothingness; since we can think only thoughts, and all the words we use to speak of things can express only thoughts, to say there is something other than thought is therefore an affirmation which can have no meaning.
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