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Poincare, Jules Henri (1854-1912)
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.]
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Poincare, Jules Henri (1854-1912)
Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered.
[Whether or not he actually said this is a matter of debate amongst historians of mathematics.]
The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991.
Poincare, Jules Henri (1854-1912)
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Poincare, Jules Henri (1854-1912)
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Poincare, Jules Henri (1854-1912)
Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Poincare, Jules Henri (1854-1912)
Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
La Science et l'hypothese.
Poincare, Jules Henri (1854-1912)
A scientist worthy of his name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
In N. Rose, Mathematical Maxims and Minims, Raleigh, NC: Rome Press Inc., 1988.
Poincare, Jules Henri (1854-1912)
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Poincare, Jules Henri (1854-1912)
Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
Poincare, Jules Henri (1854-1912)
Thought is only a flash between two long nights, but this flash is everything.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

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