Later generations will regard set theory as a disease from which one has recovered.

Later generations will regard Mengenlehre as a disease from which one has recovered.

With the disappearance of the great French mathematician has disappeared the one man whose thought could carry all other thoughts, the one mind who, through a sort of rediscovery, could penetrate to its very depth all the knowledge which the mind of man can comprehend. And that is why the demise of this man at the peak of his intellectual strength is such a disaster. Discoveries will lag, groping efforts will be drawn out; for, the potent luminous brain will not be there to coordinate disjointed research, or to cast the daring plummet of a new theory into a world of obscure facts suddenly revealed by experience.

One of the enduring legacies of Napoleon was the French system of grandes écoles, the elite schools that train the country's top technocratic and managerial students even today. ... Biographies of French mathematicians often begin with awed accounts of how well they did on the entrance tests, and how they fared on various national exams and competitions. Poincaré was no exception. He obtained the first prize in several national competitions, and was among the highest-ranking applicants to the École Polytechnique and the École Normale Supérieure in Paris, schools especially famous for the quality of their mathematics.

All that is not thought is pure nothingness; since we can think only thought and all the words we use to speak of things can express only thoughts, to say there is something other than thought, is therefore an affirmation which can have no meaning.And yet—strange contradiction 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.

It is only through science and art that civilization is of value. Some have wondered at the formula: science for its own sake; an yet it is as good as life for its own sake, if life is only misery; and even as happiness for its own sake, if we do not believe that all pleasures are of the same quality...Every act should have an aim. We must suffer, we must work, we must pay for our place at the game, but this is for seeing's sake; or at the very least that others may one day see.

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.

What is objective must be common to many minds and consequently transmissible from one to the other, and as this transmission can only come about by... discourse... we are even forced to conclude: no discourse no objectivity.

All laws are... deduced from experiment; but to enunciate them, a special language is needful... ordinary language is too poor...This... is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak. And a well-made language is no indifferent thing;...the analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist....law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise...In a word, to get the law from experiment, it is necessary to generalize... But how generalize? ...in this choice what shall guide us?It can only be analogy. ...What has taught us to know the true profound analogies, those the eyes do not see but reason divines?It is the mathematical spirit, which disdains matter to cling only to pure form.

Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.