French Mathematician, Theoretical Physicist, Engineer and Philosopher of Science

# Henri Poincaré, fully Jules Henri Poincaré

French Mathematician, Theoretical Physicist, Engineer and Philosopher of Science

## Author Quotes

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

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

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

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.

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

To invent is to discern, to choose.

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.

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.

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 infinite in number. We must make a selection ... guided by utility ... Have we not some better occupation than counting the number of lady-birds in existence on this planet?

It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.

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

Treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis.

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

Science is built of facts the way a house is built of bricks; but an accumulation of facts is no more science than a pile of bricks is a house.

What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

Science is facts; just as houses are made of stones, so is science made of facts; but a pile of stones is not a house and a collection of facts is not necessarily science.

When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind of net, web or infinitely tight mesh? One is struck by the complexity of this figure that I am not even attempting to draw.

If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision ... the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.

It may happen that small differences in the initial conditions produce very great ones in the final phenomena.

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 intimate 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.

When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.

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. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family.

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

Sociology is the science which has the most methods and the least results.