Henri Poincare Quotes
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Experiment is the sole source of truth.
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It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.
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So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.
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Need we add that mathematicians themselves are not infallible?
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This harmony that human intelligence believes it discovers in nature - does it exist apart from that intelligence? No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility. A world so exterior as that, even if it existed, would be forever inaccessible to us. But what we call objective reality is, in the last analysis, that which is common to several thinking beings, and could be common to all; this common part, we will see, can be nothing but the harmony expressed by mathematical laws.
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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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All of mathematics is a tale about groups.
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Point set topology is a disease from which the human race will soon recover.
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All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.
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It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.
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A very small cause, which escapes us, determines a considerable effect which we cannot ignore, and we say that this effect is due to chance.
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But for harmony beautiful to contemplate, science would not be worth following.
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But all of my efforts served only to make me better acquainted with the difficulty, which in itself was something.
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How is error possible in mathematics?
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Mathematicians are born, not made.
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All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.
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Every phenomenon, however trifling it be, has a cause, and a mind infinitely powerful, and infinitely well-informed concerning the laws of nature could have foreseen it from the beginning of the ages. If a being with such a mind existed, we could play no game of chance with him; we should always lose.
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Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means.
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Analyse data just so far as to obtain simplicity and no further.
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One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.
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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.
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Deviner avant de démontrer! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes.
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Les faits ne parlent pas. Facts do not speak.
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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.
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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.
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If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living
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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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Guessing before proving! Need I remind you that it is so that all important discoveries have been made?
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It is not order only, but unexpected order, that has value.
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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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Euclid Mathematician -
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