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Mathematicians do not study objects, but the relations between objects.

Henri Poincare

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Henri Poincare
Age: 58 †
Born: 1854
Born: April 29
Died: 1912
Died: July 17
Astronomer
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Jules Henri Poincaré
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Henri Poincare
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More quotes by Henri Poincare

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Pure logic could never lead us to anything but tautologies it can create nothing new not from it alone can any science issue.
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Geometry is not true, it is advantageous.
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If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
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Just as houses are made of stones, so is science made of facts.
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A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.
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Why is it that showers and even storms seem to come by chance, so that many people think it quite natural to pray for rain or fine weather, though they would consider it ridiculous to ask for an eclipse by prayer.
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. . . by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
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It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
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I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.
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Zero is the number of objects that satisfy a condition that is never satisfied. But as never means in no case, I do not see that any progress has been made.
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Thought is only a flash between two long nights, but this flash is everything.
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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.
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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?
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It is through science that we prove, but through intuition that we discover.
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Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered.
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For a long time the objects that mathematicians dealt with were mostly ill-defined one believed one knew them, but one represented them with the senses and imagination but one had but a rough picture and not a precise idea on which reasoning could take hold.
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Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose.
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The mind uses its faculty for creativity only when experience forces it to do so.
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How is error possible in mathematics?
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Every good mathematician should also be a good chess player and vice versa.
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