Henri Poincaré - Science and Hypothesis (1902)

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The man of science must work with method. Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.

Henri Poincaré
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.

Henri Poincaré
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.

Henri Poincaré
We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction.... Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry.... it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.

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

Henri Poincaré
Nothing but facts are of importance. John Lackland passed by here. Here is something that is admirable. Here is a reality for which I would give all the theories in the world." That is the language of the historian. The physicist would say rather: "John Lackland passed by here; that makes no difference to me, for he never will pass this way again."

Henri Poincaré
Every age has ridiculed the one before it, and accused it of having generalized too quickly and too naively. Descartes pitied the lonians; Descartes, in his turn, makes us smile. No doubt our children will someday laugh at us.

Henri Poincaré
It is not sufficient for a theory to affirm no false relations; it must not hide true relations.

Henri Poincaré
To observe is not enough. We must use our observations, and to do that we must generalize.

Henri Poincaré
For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rules.

Henri Poincaré
It is often said that experiments must be made without preconceived idea. That is impossible. Not only would it make all experiment barren, but that would be attempted which could not be done.

Henri Poincaré
The very name calculus of probabilities is a paradox. Probability opposed to certainty is what we do not know, and how can we calculate what we do not know?

Henri Poincaré
The firm determination to submit to experiment is not enough; there are still dangerous hypotheses; first, and above all, those which are tacit and unconscious. Since we make them without knowing it, we are powerless to abandon them.

Henri Poincaré
It matters little whether the ether really exists; that is the affair of metaphysicians. The essential thing for us is that everything happens as if it existed......some day, no doubt, the ether will be thrown aside as useless.

Henri Poincaré
If there were no solid bodies in nature, there would be no geometry.

Henri Poincaré
But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself.

Henri Poincaré
There is no science apart from the general. It may even be said that the very object of the exact sciences is to spare us these direct verifications.

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

Henri Poincaré
We must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations.

Henri Poincaré
Principles are conventions and definitions in disguise.

Henri Poincaré
It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry; the founders are hampered in their general grasp by third and fourth decimal places.

Henri Poincaré
Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet, and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?

Henri Poincaré
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant role, and without it there would be no science, because there would be nothing general.

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

Henri Poincaré
When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. For a definition to be of any use, it must teach us to measure force; moreover that suffices; it is not at all necessary that it teach us what force is in itself, nor whether it is the cause or the effect of motion.

Henri Poincaré
The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.

Henri Poincaré
The very possibility of the science of mathematics seems an insoluble contradiction.

Henri Poincaré
To doubt everything and to believe everything are two equally convenient solutions; each saves us from thinking.

Henri Poincaré
If we study the history of science we see happen two inverse phenomena... Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities.

Henri Poincaré