French Mathematician, Theoretical Physicist, Engineer and Philosopher of Science
"Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention."
"The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why? Not to suffer is a negative ideal more surely attained by the annihilation of the world. If we wish more and more to free man from material cares, it is that he may be able to employ the liberty obtained in the study and contemplation of truth."
"Doubting everything or believing everything are two equally convenient solutions, both of which save us from thinking."
"It is often said that experiments should be made without preconceived ideas. That is impossible."
"And yet - strange contradictions for those who believe in time - geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted and will last only a moment. Thought is only a gleam in the midst of a long night. But it is this gleam which is everything."
"Perceptual space, under its triple form, visual, tactile and motor, is essentially different from geometric space. It is neither homogeneous, nor isotropic; one can not even say that it has three dimensions. It is often said that we ‘project’ into geometric space the objects of our external perception; that we ‘localize’ them. Has this a meaning, and if so what? Does it mean that we represent to ourselves external objects in geometric space? Our representations are only the reproduction of our sensations; they can therefore be ranged only in the same frame as these, that is to say, in perceptual space. It is impossible for us to represent to ourselves external bodies in geometric space, as it is for a painter to paint on a plane canvas objects with their three dimensions. Perceptual space is only an image altered in shape by a sort of perspective, and we can represent to ourselves objects only by bringing them under the laws of this perspective. Therefore we do not represent to ourselves external bodies in geometric space, but we reason on these bodies as if they were situated in geometric space."
"To doubt everything and to believe everything are two equally convenient solutions; both free us from the necessity of reflection."
"The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide."
"A scientist worthy of the 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."
"The Scientist must set in order. 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."
"Thinking must never submit itself, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, if not to facts themselves, because, for it, to submit would be to cease to be."
"Time and Space... It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient."
"Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone."
"It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness."
"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."
"Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted."
"Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility."
"If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe the moment before."
"In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer. "
"The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things. "
"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 nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp."
"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."
"If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living."
"A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations."
"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."
"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.?"
"All the scientist creates in a fact is the language in which he enunciates it. If he predicts a fact, he will employ this language, and for all those who can speak and understand it, his prediction is free from ambiguity. Moreover, this prediction once made, it evidently does not depend upon him whether it is fulfilled or not."
"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."
"As we cannot give a general definition of energy, the principle of the conservation of energy signifies simply that there is something which remains constant."
"Consider now the Milky Way. Here also we see an innumerable dust, only the grains of this dust are no longer atoms but stars; these grains also move with great velocities, they act at a distance one upon another, but this action is so slight at great distances that their trajectories are rectilineal; nevertheless, from time to time, two of them may come near enough together to be deviated from their course, like a comet that passed too close to Jupiter. In a word, in the eyes of a giant, to whom our Suns were what our atoms are to us, the Milky Way would only look like a bubble of gas."
"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."
"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."
"After the Fourier series, other series have entered the domain of anylsis; they entered by the same door; they have been imagined in view of applications."
"Everyone is sure of this [that errors are normally distributed], Mr. Lippman told me one day, since the experimentalists believe that it is a mathematical theorem, and the mathematicians that it is an experimentally determined fact."
"Guessing before proving! Need I remind you that it is so that all important discoveries have been made?"
"He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures."
"For an aggregate of sensations to have become a remembrance capable of classification in time, it must have ceased to be actual, we must have lost the sense of its infinite complexity, otherwise it would have remained present. It must, so to speak, have crystallized around a center of associations of ideas which will be a sort of label. It is only when they have lost all life that we can classify our memories in time as a botanist arranges dried flowers in his herbarium."
"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."
"How is it that there are so many minds that are incapable of understanding mathematics?... the skeleton of our understanding... and actually they are the majority... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education."
"How is an error possible in mathematics? A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. Need we add that mathematicians themselves are not infallible?"