Einstein, the searcher

by Alexander Moszkowski

Moreover, a proof of the theorem had been accepted. It appeared in text-books, and was often to be heard in lecture rooms; nor was a shadow of a doubt suggested. For it was not merely a demonstratio ad oculos, but it appeared directly to our sense of intuition. And we may safely say that up to the present day no one has ever been able to imagine a continuously curved line which has no tangent; no one has been able to picture even one point of such a curve at which no tangent could be drawn.

Nevertheless, scientists appeared who began to entertain doubts. In the case of Riemann and Schwarz these doubts assumed a concrete form, in that they proved that certain functions are refractory at certain points. But Weierstrass was the first to make a real breach in the old belief that was so firmly rooted. He set up a function that is continuous at every point, but differentiable at no point. The graphical picture would thus have to be a continuous curve having no tangent at all.

What is the appearance of such a configuration? We do not know, nor shall we presumably ever get to know. During a conversation in which this problem of Weierstrass arose, Einstein said that such a curve lay beyond the power of imagination. It must be remarked that, although the mathematical expression of the Weierstrass function is not exactly simple, it is not inordinately complex. Moreover, seeing that one such function (or curve) exists, others will soon be added to it (Poincaré mentions that Darboux actually gave other examples even in the same year that the first was discovered); there will, indeed, be found an infinite number of them. We may go still further, and say that, corresponding to each curve that has tangents, there are an infinite number that have no tangents, so that the former form the exception and not the rule. This is an overwhelming confession that shakes the foundations of our mathematical convictions, yet there is no escape.

How may we apply the principle of "approximation" to these considerations? May we say that the theorem that was believed earlier is an approximation to a mathematical truth?

This is possible only conditionally, in a certain extremely limited sense, namely, if we picture to ourselves that point in the development of science at which the conception and properties of tangents first began to be investigated. Compared with this stage of science, the above theorem denotes a first approximation to the truth, in spite of its incorrectness; for it makes us acquainted with a great abundance of curves that are very important for us and that exhibit tangents at every point. This knowledge brings us a step nearer to the more approximate truth given by Weierstrass's example. In the distant future, the earnest student will learn this theorem only as a curious anecdote, just as we hear of certain astrological and alchemistic fallacies. He will learn, in addition, other theorems that are looked on as proved by us of the present day, although actually they were proved only approximately. For what does it mean when Gauss, for example, repudiated certain proofs of earlier algebraists as being "not sufficiently rigorous," and replaced them by more rigorous proofs? It signifies no more than that, in mathematics, too, what appears to one investigator as flawless, strict, and evident, is found by another to have gaps and weaknesses. Absolute correctness belongs only to identities, tautologies, that are absolutely true in themselves, but cannot bear fruit. Thus at the foundation of every theorem and of every proof there is an incommensurable element of dogma, and in all of them taken together there is the dogma of infallibility that can never be proved nor disproved.

It must appear extremely interesting that, at first sight, this example of the tangent has its equivalent in Nature herself, namely, in molecular motions the investigation of which is again largely due to Einstein.

Jean Perrin, the author of the famous book, Atoms, describes, in the introduction, the connexion between this mysterious mathematical fact and results that are visible and may be shown by experiment, to which we have been led by the study of certain milky-looking (colloidal) liquids.

If, for example, we look at one of those white flakes, which we get by mixing soap solution with common salt, we at first see its surface sharply outlined, but the nearer we approach to it, the more indistinct the outline becomes. The eye gradually finds it impossible to draw a tangent to a point of the surface; a straight line which, viewed superficially, seems to run tangentially, is found on closer examination to be oblique or even perpendicular to the surface. No microscope succeeds in dispelling this uncertainty. On the contrary, whenever the magnification is increased, new unevennesses seem to appear, and we never succeed in arriving at a continuous picture. Such a flake furnishes us with a model for the general conception of a function which has no differential coefficient. When, with the help of the microscope, we observe the so-called Brownian movement, which is molecular by nature, we have a parallel to the curve which has no tangent, and the observer is left only with the idea of a function devoid of a differential coefficient.... We find ourselves obliged, ultimately, to give up the hope of discovering homogeneity at all in studying matter. The farther we penetrate into its secrets, the more we see that it, matter, is spongy by nature and infinitely complex; all indications tend to show that closer examination will reveal only more discontinuities.

I have not yet had an opportunity of seeing these Brownian movements under the microscope, but I must mention that Einstein has repeatedly spoken to me of them with great enthusiasm, of an objective kind, as it were, for he betrayed neither by word nor by look that he himself has done research leading to definite laws that have a recognized place in the history of molecular theory.

As soon as we approach the question of molecular irregularities we recognize that, when we earlier spoke of the figure of the earth in discussing the principle of "approximation," we were still very far from the limit that may be imagined. We had set up the three stages: plane—sphere—ellipsoid of revolution, as relative geometrical steps, beyond which there must be still further geometrical approximations. If we imagine all differences of level due to mountains and valleys to be eliminated, for example, and if we suppose the earth's surface to consist entirely of liquid, undisturbed by the slightest breath of wind, even then, the ellipsoid is by no means the final description. For now the discontinuities from molecule to molecule begin, the infinite number of configurations without tangents, the macroscopic parallels of what the white flake soap solution showed as microscopically, and no conceivable geometry would ever be adequate to grasp these phenomena. We arrive at a never-to-be-completed list of functions which can never be described either in words or in symbolic expressions of analysis.