It would be foolish to pretend that we can penetrate the most obscure corners of Einstein’s theories without the aid of mathematics. I believe, however, that we can give in ordinary language—that is to say, by means of illustrations and analogies—a fairly satisfactory idea of these things, the intricacy of which is usually due to the infinitely subtle and supple play of mathematical formulæ and equations.
After all, mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediæval slowness of the syllogisms expressed in our words.
However paradoxical this may seem to people who regard mathematics as of itself a means of discovery, the truth is that we can never get from it anything that was not implicitly inherent in the data which were thrust between the jaws of its equations. If I may use a somewhat trivial illustration, mathematical reasoning is very like certain machines which are seen in Chicago—so bold explorers in the United States tell us—into which one puts living animals that emerge at the other end in the shape of appetising prepared meats. No spectator could have, or would wish to have, eaten the animal alive, but in the form in which it issues from the machine it can at once be digested and assimilated. Yet the meat is merely the animal conveniently prepared. That is what mathematics does. By means of a marvellous machinery the mathematician extracts the valuable marrow from the given facts. It is a machinery that is particularly useful in cases where the wheels of verbal argument, the chain of syllogisms, would soon be brought to a halt.
Does it follow that, properly speaking, mathematics is not a science? Does it follow at least that it is only a science in so far as it is based upon reality, and fed with experimental data, since “experience is the sole source of truth.” I refrain from answering the question, as I am one of those who believe that everything is material for science. Still, it was worth while to raise the question because many are too much disposed to regard a purely mathematical education as a scientific education. Nothing could be further from the truth. Pure mathematics is, in itself, merely an abbreviated form of language and of logical thought. It cannot, of its own nature, teach us anything about the external world; it can do so only in proportion as it enters into contact with the world. It is of mathematics in particular that we may say: Naturæ non imperatur nisi parendo.
Are not Einstein’s theories, as some imperfectly informed writers have suggested, only a play of mathematical formulæ (taking the word in the meaning given to it by both mathematicians and philosophers)? If they were only a towering mathematical structure in which the x’s shoot out their volutes in bewildering arabesques, with swan-neck integrals describing Louis XV patterns, they would have no interest whatever for the physicist, for the man who has to examine the nature of things before he talks about it. They would, like all coherent schemes of metaphysics, be merely a more or less agreeable system of thought, the truth or falseness of which could never be demonstrated.
Einstein’s theory is very different from that, and very much more than that. It is based upon facts. It also leads to facts—new facts. No philosophical doctrine or purely formal mathematical construction ever enabled us to discover new phenomena. It is precisely because it has led to such discovery that Einstein’s theory is neither the one nor the other. That is the difference between a scientific theory and a pure speculation, and it is that which, I venture to say, makes the former so superior.
Like some suspension bridge boldly thrown across an abyss, Einstein’s theory rests, on the one side, on experimental phenomena, and it leads, at the other side, to other, and hitherto unsuspected, phenomena, which it has enabled us to discover. Between these two solid experimental columns the mathematical reasoning is like the marvellous network of thousands of steel bars which represent the elegant and translucent structure of the bridge. It is that, and nothing but that. But the arrangement of the beams and bars might have been different, and the bridge—though less light and graceful, perhaps—still have been able to join together the two sets of facts on which it rests.
In a word, mathematical reasoning is only a kind of reasoning in a special language, from experimental premises to conclusions which are verifiable by experience. Now there is no language which cannot in some degree be translated into another language. Even the hieroglyphics of Egypt had to give way before Champollion. I am therefore convinced that the mathematical difficulties of Einstein’s theories will some day be replaced by simpler and more accessible formulæ. I believe, indeed, that it is even now possible to give by means of ordinary speech an idea, rather superficial perhaps, but accurate and substantially complete, of this wonderful Einsteinian structure which ranges all the conquests of science, as in some well-ordered museum, in a new and superb unity. Let us try.
We may resume in the few following words the story of the origin, the starting-point, of Einstein’s system.