Henri Poincare Quotes

Point set topology is a disease from which the human race will soon recover.

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.

There are no solved problems; there are only problems that are more or less solved.

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.

Doubt everything or believe everything: these are two equally convenient strategies. With either we dispense with the need for reflection.

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

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

Geometry is the art of correct reasoning from incorrectly drawn figures.

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

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

One geometry cannot be more true than another; it can only be more convenient.

Every good mathematician should also be a good chess player and vice versa.

It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wished to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside. We must, example, use language, and our language is necessarily steeped in preconceived ideas. Only they are unconscious preconceived ideas, which are a thousand times the most dangerous of all.

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

Intuition is more important to discovery than logic.

Analyse data just so far as to obtain simplicity and no further.

Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty.

Thought must never submit, 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, save to the facts themselves, because, for thought, submission would mean ceasing to be.

It is by logic we prove. It is by intuition we discover.

Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing].

Tolstoi explains somewhere in his writings why, in his opinion, “Science for Science's sake” is an absurd conception. We cannot know all the facts, since they are practically infinite in number. We must make a selection. Is it not better to be guided by utility, by our practical, and more especially our moral, necessities?

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.

Mathematicians are born, not made.

Pure logic could never lead us to anything but tautologies; it can create nothing new; not from it alone can any science issue.

The mind uses its faculty for creativity only when experience forces it to do so.

So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.

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

Most striking at first is the appearance of sudden illumination, a manifest sign of long unconscious prior work.

A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.

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.

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

...the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely aesthetic feeling, which all mathematicians know

All great progress takes place when two sciences come together, and when their resemblance proclaims itself, despite the apparent disparity of their substance.

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

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

Often when works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, long or short, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind.

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.

The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things.

All that is not thought is pure nothingness; since we can think only thoughts, 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.

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

Geometry is not true, it is advantageous.

How is error possible in mathematics?

One does not ask whether a scientific theory is true, but only whether it is convenient.

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.

I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.

Guessing before proving! Need I remind you that it is so that all important discoveries have been made?

All of mathematics is a tale about groups.

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