It is this insistence upon the æsthetic value of science that caused him to shrink from being called a "pragmatist", although those who accept that name have always laid unusual stress upon the æsthetic factor in thinking. But in his theory of knowledge Poincaré is decidedly pragmatic, and no one has given a clear exposition or stronger expression to the practical mode of thought by which the natural sciences have made their progress and which is now being extended to the fields of metaphysics, religion, ethics, and sociology. Poincaré's favorite word is "convenient" (commode). Theories are strictly speaking not to be classed as true or false. They are merely more or less convenient. For example:
Masses are coefficients it is convenient to introduce into calculations. We could reconstruct all mechanics by attributing different values to all the masses. This new mechanics would not be in contradiction either with experience or with the general principles of dynamics. Only the equations of this new mechanics would be less simple.—"Science and Hypothesis", p. 76.
We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them. But what is the nature of these rules? No general rule, no rigorous rule; a multitude of little rules applicable to each particular case. These rules are not imposed upon us, and we might amuse ourselves by inventing others; but they could not be cast aside without greatly complicating the laws of physics, mathematics, and astronomy. We therefore choose these rules, not because they are true, but because they are most convenient, and we may recapitulate them as follows: "The simultaneity of two events or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible; in other words, all these rules, all these definitions, are only the fruit of an unconscious opportunism."—"Value of Science", p. 35.
Time should be so defined that the equations of mechanics may be as simple as possible. In other words, there is not one way of measuring time more true than another. That which is generally adopted is only more convenient. Of two watches, we have no right to say that one goes true, the other wrong: we can only say that it is advantageous to conform to the indications of the first.—"Value of Science", p. 30.
Behold then the rule we follow and the only one we can follow: when a phenomenon appears to us as the cause of another, we regard it as anterior. It is therefore by cause we define time.—"Value of Science", p. 32.
Experience does not prove to us that space has three dimensions. It only proves to us that it is convenient to attribute three dimensions to it.—"Value of Science", p. 69.
It has often been observed that if all the bodies in the universe were dilated simultaneously and in the same proportion we should have no means of perceiving it, since all our measuring instruments would grow at the same time as the objects themselves which they serve to measure. The world, after this dilatation, would continue on its course without anything apprising us of so considerable an event. —"Value of Science", p. 39.
But Poincaré goes farther and shows not only that two such worlds of different sizes would be absolutely indistinguishable, but that they would be equally indistinguishable if they were distorted in any manner so long as they corresponded with each other point by point. This conception of the relativity of space may be thought a little hard to grasp, but M. Poincaré is kind enough to suggest a way by which any one may see it for himself if he has ten cents to admit him to one of those hilarious resorts where life-size concave and convex mirrors are to be seen.[1] You may think yourself a gentleman of proper figure, that is to say, somewhat portly, and you look upon the tall slim shape that confronts you in the cylindrical mirror as absurdly misshapen. But you would find it difficult to convince him of his deformity. His legs, as well as yours, fulfill the requirement that Lincoln laid down as their proper length; that is, they reach from the body to the ground. If you touch your chin with your thumb and your brow with your forefinger, so does he. It occurs to you that here is a case where your knowledge of geometry would, if ever, prove useful, but when you appeal to it, you will find that the geometry of his queer-looking world is just as good as yours; in fact, is just the same. You get a foot rule and measure yourself; 70 inches high, 14 inches in diameter at the equator, ratio 5:2. But meanwhile the mirror man is also measuring himself, and his dimensions come out exactly the same as yours, 70 and 14 and 5:2, for when he holds the rule perpendicular it lengthens and when horizontal it shrinks. Lines that in your world are straight are curved in his, but you cannot prove it to him, for when he lays his straightedge against these curves of his, behold it immediately bends to correspond. By this time, finding it so difficult to prove to the mirror man that you are right and he is wrong, it occurs to you that perhaps he isn't, that he may have just as much reason as you for believing that his is the normal, well-proportioned world, and yours the distorted image of it. Since, then, you have no way of perceiving the absolute length, direction, or curvature of a line, your space may be as irregularly curved and twisted as it looks to be in the funniest of the mirrors, and you would not know it. Now the principle of the pragmatist is that anything that does not make any difference to anything else is not real. The reason why we have not been able to discover any differences between the mirror space and our space, each considered by itself, is because there is none. Or to return to the language of Poincaré, "space is in reality amorphous and the things that are in it alone give it a form." Why do we say that space has three dimensions instead of two or four or more? Why do we stick to an old fogy like Euclid when Riemann and Lobachevski proffer us new and equally self-consistent systems of geometry wherein parallels may meet or part? Because:
by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
Such language may pass without notice in university halls, for all scientists are more or less clearly conscious of the provisional and practical nature of the hypotheses and conventions they employ. But to the outside world it sounds startling. To some it seemed that the foundations of the universe were being undermined. Others saw in it a confession of what Brunetière had called "the bankruptcy of science" and openly rejoiced over the discomfiture of the enemy of the Church. Now Poincaré had chanced to use in discussing the relativity of motion the following illustration:
Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence. Hence this affirmation "the earth turns round" has no meaning, since it can be verified by no experiment; since such an experiment not only could not be either realized or dreamed by the boldest Jules Verne but cannot be conceived of without contradiction. Or rather these two propositions: "The earth turns round" and "it is more convenient to suppose the earth turns round" have the same meaning; there is nothing more in the one than in the other.—"Science and Hypothesis", p. 85.
This remark was at once seized upon by the Catholic apologists, and the Galileo case, once closed by the voice of Rome, was reopened for the admission of this new evidence. If the Ptolemaic and the Copernican theories are equally true, and the choice between them is merely a matter of expediency, was not the Holy Inquisition justified in upholding the established theory in the interests of religion and morality? Monsignor Bolo, an eminent and sagacious theologian, announced in Le Matin of February 20, 1908, that M. Poincaré, the greatest mathematician of the century, says that Galileo was wrong in his obstinacy. To this Poincaré replied in the whispered words of Galileo:
"E pur si muove, Monseigneur"?
In a later discussion of the point, he explains that what he said about the rotation of the earth could be equally well applied to any other accepted hypothesis, even the very existence of an external world, for "these two propositions, 'the external world exists' or 'it is more convenient to suppose that it exists' have one and the same meaning." The Copernican theory is the preferable because it has a richer, more profound content, since if we assume the earth is stationary we have to invent other explanations for the flattening at the poles, the rotation of Foucault's pendulum, the trade winds, etc., while the hypothesis of a revolving earth brings all these together as the effects of a single cause.
M. Le Roy, a Catholic pragmatist and a disciple of Bergson's, goes much further than Poincaré in regard to the human element in science, holding that science is merely a rule of action and can teach us nothing of truth, for its laws are only artificial conventions. This view Poincaré considered to be dangerously near to absolute nominalism and skepticism, and in his controversy with Le Roy[2] he showed that the scientist does not "create facts as Le Roy said, but merely the language in which he enunciates them." Of the contingence upon which Le Roy and Boutroux insist, Poincaré would admit only that scientific laws can never be more than approximate and probable. Even in astronomy, where the single and simple law of gravitation is involved, neither absolute certainty nor absolute accuracy can be attained. Therefore we cannot safely say that at a particular time Saturn will be at a certain point in the heavens. We must limit ourselves to the prediction that "Saturn will probably be near" such a point.