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Vote:   | A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. |
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Vote: | A scientist worthy of his name, about all a mathematician, experiences in his work is the same impression as an artist; his pleasure is as great and of the same nature. |
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Vote: | A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. |
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Vote: | A small error in the former will produce an enormous error in the latter. |
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Vote: | A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. |
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Vote: | Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence. |
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Vote: | 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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Vote: | But a pile of stones is not a house and a collection of facts is not necessarily science. |
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Vote: | But even if it were the case that the natural laws had nolonger any secret for us, we could still know the situation approximately. |
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Vote: | But is not always so; it may happen that small differences in the initial conditions produce very great ones in the finalphenomena. |
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Vote: | Facts do not speak. |
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Vote: | Geometry is not true, it is advantageous. |
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Vote: | How is an error possible in mathematics? |
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Vote: | Hypotheses are what we lack the least. |
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Vote: | Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination. |
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Vote: | 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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Vote: | If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. |
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Vote: | If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. |
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Vote: | If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. |
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Vote: | In the old days when people invented a new function they had something useful in mind. |
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Vote: | Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority. |
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Vote: | It has adopted the geometry most advantageous to the species or, in other words, the most convenient. |
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Vote: | 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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Vote: | It is through science that we prove, but through intuition that we discover. |
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Vote: | Just as houses are made of stones, so is science made of facts. |
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Vote: | Mathematical discoveries, small or great are never born of spontaneous generation. |
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Vote: | Mathematicians are born, not made. |
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Vote: | Mathematicians do not study objects, but relations between objects. |
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Vote: | Mathematics is the art of giving the same name to different things. |
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Vote: | Need we add that mathematicians themselves are not infallible? |
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Vote: | No more than these machines need the mathematician know what he does. |
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Vote: | Now, they invent them deliberately just to invalidate our ancestors' reasoning, and that is all they are ever going to get out of them. |
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Vote: | 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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Vote: | Point set topology is a disease from which the human race will soon recover. |
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Vote: | Science is facts; just as houses are made of stones, so is science made of facts; but a pile of stones is not a house and a collection of facts is not necessarily science. |
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Vote: | Science is facts. |
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Vote: | 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. |
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Vote: | 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. |
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Vote: | The mind uses its faculty for creativity only when experience forces it to do so. |
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Vote: | The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. |
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Vote: | They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious. |
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Vote: | Thought is only a flash between two long nights, but this flash is everything. |
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Vote: | Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. |
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Vote: | Thus, they are free to replace some objects by others so long as the relations remain unchanged. |
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Vote: | 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. |
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Vote: | To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection. |
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Vote: | To invent is to discern, to choose. |
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Vote: | What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? |
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Vote: | Whether it be electrostatics or electrodynamics, the propogation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family. |