Henri Poincaré

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.

The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile or vain.

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.

I think I have already said somewhere that mathematics is the art of giving the same name to different things.

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.

The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.

The greatest chance is the birth of a great man. It is only by chance that two reproductive cells, of different sexes, met, each containing precisely those mysterious elements whose mutual reaction was to produce genius. We will agree that these elements must be rare and that their meeting is even rarer. That it would have taken little to deviate the spermatozoon that carried them from its course; it would have been enough to deviate it by a tenth of a millimeter and Napoleon would not have been born and the destiny of a continent would have been changed. No other example can better explain the true nature of chance.

Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.

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.

The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.

Or... they observe an astronomic phenomenon... an eclipse of the moon, and... suppose... this... is perceived simultaneously from all points of the earth. That is not altogether true, since the propagation of light is not instantaneous; if absolute exactitude were desired, there would be a correction... according to a complicated rule.

Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship. Science, in other words, is a system of relations. ...it is in relations alone that objectivity must be sought. ...it is relations alone which can be regarded as objective.

External objects... are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation.