Einstein, the searcher

by Alexander Moszkowski

If we call to mind some of the most valuable statements made by modern investigators about the nature of natural laws, we recognize that they are all connected by a single thread of thought, namely, that even in the most certain law there is left a remainder that has not been accounted for, and that obliges us to consider a greater approximation to the truth as possible, even if a final stage is not attainable.

Mechanics furnishes us with the expression of its laws in equations, whose importance Robert Kirchhoff explained in 1874 by a definition that has been considered conclusive by scientists. According to him, it is the object of mechanics to describe completely (and not to explain) in the simplest manner the motions that occur in Nature.

The postulate of simplicity is derived from the fundamental view of science as an economy of thought. It expresses the will of man's mind to arrive at a maximum of result by using a minimum of effort, and to express the greatest sum of experience by using the smallest number of symbols. Let us consider two simple examples quoted by Mach. No human brain is capable of grasping all the possible circumstances of bodies falling freely, and it may well be doubted whether even a supernatural mind like that imagined by Laplace could succeed in doing so. But if we take note of Galilei's Law for Falling Bodies and the value of the acceleration due to gravity, which is quite an easy matter, we are equipped for all cases, and have a compendious formula, accessible to any ordinary mind, that allows us to picture to ourselves all possible motions of falling bodies. In the same way no memory in the world could retain all the different cases of the refraction of light. Instead of trying to do the impossible task of grasping this infinite abundance, we simply take note of the sine law, and the indices of refraction of the two media in question; this enables us to picture any possible case of refraction, or to complete it, since we are free to relieve our memories entirely by having the constants in a book. Thus we have here natural laws that give us a comprehensive yet abbreviated statement of facts, and satisfy the postulate of simplicity to a high degree.

But these facts are built up on experiences, and it is not impossible that some new unexpected experience will reveal a new fact, which is not sufficiently taken into account in the law. This would compel us to correct the expression for the law, and to seek a closer approximation for the enlarged number of facts.

The Law of Inertia, according to our human standard, seems unsurpassable in simplicity and completeness; it seems to us fundamental. But this law, which prescribes uniform rectilinear motion to a body subject to no external forces, selects only one possibility out of an infinite number as being valid for us. It does not seem evident to a child, and it is easy to imagine a good scholar in some branch of knowledge other than physics, to whom it would likewise not seem evident. For it is by no means necessary a priori that a body will move at all when all forces are absent. If the law were self-evident, it would not need to have been discovered by Galilei in 1638. Nevertheless, it appears to us, now, to be absolutely self-evident, and we can scarcely imagine that it can ever be otherwise. This is simply because we are bound to the current set of ideas that cannot extend beyond the sum of sense-data and experiences that have been inculcated into us by heredity and environment. At a very distant date in the future the average mind may surpass that of Galilei to the same extent as Galilei's surpasses that of a child, or of a Papuan native. And of all the infinite possibilities one may occur to a Galilei of the distant future, which, when formulated as a law, may serve to describe motions of a body subject to no forces better than the law of inertia, proposed in 1638.

These reflections are not mere hallucinations, but have to do with scientific occurrences that we have observed in the twentieth century. Newton's equation that gives the Law of Attraction is beyond doubt a model of simplicity, and it would have occurred to no thinking person of even the last generation to doubt its accuracy. The easily grasped expression k (m.m¹⁄r²) apparently expresses truth in a law which is valid for all eternity. In this expression, he denotes a gravitational constant, that is, a quantity which is invariable in the whole universe; m and m¹ are two masses that act attractively on one another; and r is the distance between them. But Newton has been followed by Einstein, who has proved that this expression represents only an approximate value, that leaves a small remainder as an error that may be detected if the greatest refinement be made in our methods of observation. The equations that have been set up by Einstein represent the approximation that is to be considered final for the present, and that may remain valid for thousands of years. They are certainly very complicated, being included in a system of differential equations of awe-inspiring length, and we may feel tempted to object with the question: how do they agree with Kirchhoff's postulate that the simplest description of the motions must be sought? But this objection falls to the ground if we look carefully into the question. For simplicity consists not merely in being brief or in excluding difficulty from a formula, but rather in asserting the simplest relation to the universe as a whole, which is independent of all systems of reference. When this independence is proved—and in Einstein's case it is so—the complicated aspect of the formula disappears entirely in the light of the higher simplicity and unity of the world-system that presents itself—a world-system that is directed in conformity with the one fundamental law of general relativity as well in the motion of the electrons as in motion of the most distant stars. With regard to the other postulate, that of completeness, i.e. absolute accuracy, we have been furnished with proofs that have rightly excited the wonder of the present generation. But are we then to recognize the Principle of Approximation in every direction? Is there then nothing that can be proved rigorously, nothing that is unconditionally valid in the form of knowledge that corresponds exactly to truth?

We are led to think of mathematical theorems, which, when they have once been proved, are evident to the same degree as the axioms from which they have been derived, by virtue of logic which cannot be disputed since a contradiction leads to absurdity. It has been said that mathematics est scientia eorum, qui per se clara sunt, that is, is the science of what is self-evident.

But here again doubts arise. If we should get to know only a single case, in which the self-evident came to grief, the road to further doubts becomes open. Such a case will now be quoted.

As we know, a tangent is a straight line, which makes contact with a curve at two coincident (or infinitely near) points without actually cutting the curve. The simplest case of this is the perpendicular at the extremity of a radius of a circle. And it agrees fully with what our feeling leads us to expect when it is stated that every curved line that is "continuous," that is, which discloses no break and no sudden bend, has a tangent at every point. Analysis, which treats plane curves as equations in two variables, gives the direction of the tangent in terms of the differential coefficient, and declares accordingly that every continuous function has a differential coefficient, that is, may be differentiated, at every point. The one statement amounts to the same as the other, since there must be an equivalent graphical picture corresponding to every functional expression.

But this apparently rudimentary theorem involves an error, which was not discovered before the year 1875. The theory of curves has been in existence for centuries, but it occurred to no one to doubt the general validity of this theorem of tangents. It was regarded as self-evident, as a mathematical intuition. And certainly neither Newton, nor Leibniz, nor Bernoulli, not to mention the mathematicians of olden times, even dreamed that a continuous curve without a tangent, or a continuous function without a differential coefficient, was possible.