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" It is through science that we prove, but through intuition that we discover. "

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" A small error in the former will produce an enormous error in the latter. "

" Mathematicians are born, not made. "

" To invent is to discern, to choose. "

" It has adopted the geometry most advantageous to the species or, in other words, the most convenient. "

" Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination. "

" Need we add that mathematicians themselves are not infallible? "

" Geometry is not true, it is advantageous. "

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

" Hypotheses are what we lack the least. "

" 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. "

" It is far better to foresee even without certainty than not to foresee at all. "

" To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection. "

" If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. "

" 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. "

" Mathematics is the art of giving the same name to different things. "

" Science is facts. "

" 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. "

" No more than these machines need the mathematician know what he does. "

" Facts do not speak. "

" Science is built up of 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. "

" 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. "

" 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. "

" Just as houses are made of stones, so is science made of facts. "

" In the old days when people invented a new function they had something useful in mind. "

" A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. "

" 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. "

" 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. "

" Mathematical discoveries, small or great are never born of spontaneous generation. "

" Thus, they are free to replace some objects by others so long as the relations remain unchanged. "