The question at issue is this: given any physical system, is it possible, at any rate in theory, to make an exact prediction of its future behavior, provided that its nature and condition at one given point of time are exactly known? It is assumed, of course, that no external and unforeseen influences act upon the system from without; but such influences can always be eliminated, at least theoretically, if all bodies, fields of force and the like capable of acting upon the system are included within it. It is assumed, in other words, that the condition of these external elements, too, is exactly known at the initial moment of time. It is possible, and indeed if we argue rigorously it is certain, that in order to do so, the system under consideration has to be extended to comprehend the entire universe. Yet it is possible to imagine a finite, self-contained system, and in practice this abstraction is invariably made use of whenever a law of physics is enunciated. The question therefore is whether it is possible exactly to predict the behavior of such a system provided its initial condition be exactly known.

Some fifteen years ago this was never doubted: absolute determinism was, in a manner, the fundamental dogma of practical physics. The clearest example, which had given this direction to physics, was classical mechanics: given a system of mass points, their masses, positions and velocities at an initial point of time, and given the laws of force in accordance with which they act upon one another, it was possible to calculate in advance their movements for all future time. And when applied to the celestial bodies, this theory had been triumphantly confirmed.

Today many physicists assert that such a strictly determinist view cannot do justice to nature, and that this applies equally whether mass points, fields of force or waves are used as the bricks from which we build our system. They make this assertion on the strength of the experimental results obtained in physics during the last thirty years — results which relate to measurements of every kind; on the strength of the long-continued failure of all attempts at comprehending satisfactorily the totality of these experiments through the medium of a deterministic model; and finally on the strength of the very creditable success which has been reached by a departure from a strict determinism.

Evidently such success and failure cannot in itself determine so grave a question. However firmly we may be convinced that it was determinism which was the stumbling block in all the attempts that had been made hitherto, and however strongly we may believe that it is the obstacle preventing a completely satisfactory explanation of all the observed phenomena; however considerable finally the successes achieved by the employment of an undeterministic picture may be, it is unlikely that we shall ever be able to demonstrate the impossibility of finding any deterministic model of nature capable of doing justice to the facts.

The modem attempts to relinquish determinism are rendered particularly interesting by the fact that their claims with regard to the absence of determinism, far from being vague and inaccurate, are quantitatively quite definite and can be expressed in centimeters, grams and seconds. As a simple example, we may take a mass point in motion either in a state of isolation from others or as a member of a system of many mass points exerting force upon each other. The claim which is made is that its movement cannot be foretold with complete accuracy because, among other things, it would be necessary to know its position and velocity at the initial point of time; and it is claimed that it is impossible in principle to determine both of these exactly.

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According to classical physics, and especially mechanics, it would be necessary to undertake certain operations in order to take a mass point to a given place at the initial point of time and in order to impress upon it a given velocity. Thus we might take it between nippers, carry it to the place in question and push it in an appropriate direction. Quantum mechanics teaches us that if such an operation is undertaken with a mass point a great number of times, the same result does not invariably come about even if the operation is always exactly the same. But it further teaches that the result obtained is not entirely a matter of chance. What is claimed is that if you repeat the same experiment a million times and register the frequency with which the different possible results occur, they will in a second million experiments repeat themselves with exactly the same frequency. It is assumed, of course, that all the experiments are exactly identical.

It will be seen that this claim approximates closely to the so-called law of trial and error governing actual measurements. What is peculiar in this theoretical assertion is the fact that there is a rigid limit to the accuracy of observation, a limit which in its turn is determined by a constant of Nature. Hitherto in all our theoretical considerations we had quite unconsciously assumed, that at any rate in principle, observations could be carried out with any degree of accuracy; nobody had dreamed that a correlation of the kind mentioned between the accuracies of the different measurements (in the present instance position and velocity) did in fact exist. The other assertions made by modern physics in support of indeterminism are essentially of a similar kind, although they are less easily comprehensible, especially to non-physicists; and a discussion of them would not promote our present argument.

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Presumably I may take it as known that some fifty years ago it was grasped that a very large number of so-called natural laws were statistical laws which were fulfilled with extreme accuracy only because the number of individual entities concerned was extremely great. Thus, for example, the pressure exerted by a gas on the walls of the container is taken to be the resultant of a very large number of individual impulses exerted by molecules striking against the container and rebounding from it. Now the kinetic energy of an individual molecule at a given temperature is far from being exactly determined; all that is determined is its average value, while the individual values vary somewhat considerably (their law of distribution being exactly known both theoretically and experimentally). The direction in which the molecules strike the container is wholly contingent and the number of molecules striking it in any unit of time is also, of course, subject to variations. Nevertheless, the average value of the pressure is a well-defined physical quantity. Its casual fluctuations are far beyond the limit of experimental accuracy, provided that the surface of the body, which experiences the pressure, and the time which is physically involved in the "process of averaging" is not too small. If, however, a very light and small body is subjected to pressure these conditions are not fulfilled and, as might have been expected, the purely contingent variations in pressure cause it to execute a trembling motion known as Brownian movement.

But not only the laws governing the stationary equilibria have disclosed their statistical nature: the same holds, in most cases, for the dynamic evolution of physical happening. To put it briefly, all the laws relating to irreversible natural processes are now known definitely to be of a statistical kind; and this means, of course, the great majority of laws, since in the main the course of events in nature is irreversible. As an example I may quote the conduction of heat in a gas. An arbitrary distribution of temperature gradually approaches uniformity in a definite manner, governed by the law that the current of heat runs in the direction of the steepest fall of temperature and is proportional to the thermal gradient. To explain this on a statistical basis, let us imagine a surface within a gas, its left-hand side being warm and its right-hand side relatively cold; in other words, having relatively fast and slow-moving molecules on its left and right-hand side respectively. (Fig. 3) In accordance with the calculus of probability, approximately equal numbers of molecules will move from left to right and from right to left. The former, however, transport more energy than the latter, with the result that the thermal current flows in the direction of the gradient. The degree of exactness with which the law is fulfilled is once again due to the great number of molecules concerned. Theoretically, indeed, it would be easy to imagine cases in which the exact opposite would arise. In order to construct such a case, let us imagine that the process towards the thermal equilibrium has been going on for some time; and let us now assume that by some conjuring trick all the various velocities were exactly reversed: this conjuring trick would leave the distribution of temperature unaltered and would produce a perfectly possible state of the system. But from this initial state onwards the differences in temperature would be increased through the action of thermal currents opposed to the fall of temperature until finally the original initial stage would be reached. Fortunately it can be shown by calculation that such a happening is unlikely in the extreme.

Since the time of Ludwig Boltzmann this view has come to be applied to the vast majority of the laws determining the events in our organic and our inorganic surroundings. All chemical transformations, the velocity of chemical reactions and their variation according to temperature, the processes of melting and evaporation, the laws of vapor pressure, etc., everything, in fact, with the possible exception of gravitation, is governed by laws of this kind, and all the "predictions" derived from these laws are of a statistical nature and are true only within limits, although these limits can be determined with complete accuracy. Now surely we have here a striking resemblance to the modern statements concerning "indeterminateness," and it may be worth while asking why similar statements made at that earlier time did not cause quite the same degree of excitement (though they did evoke quite a little stir!) Why did nobody say, forty or fifty years ago, that modem physics (modern as it was then) was compelled to give up causality and determinism? Why was this sort of thing being said only five or six years ago?

The answer is easy. At that time the negation of determinism would have been a practical negation: to-day it is supposed to be a theoretical one. Fifty years ago it was held that, if the position and velocity of every molecule was completely known at the beginning, and if the trouble were taken to make an exact mathematical calculation of all the collisions between the molecules, then it would be possible to predict exactly what would happen. It was believed that what forced us to content ourselves with average laws was merely the practical impossibility (1) of finding out exactly what was the initial condition of the molecules and (2) of pursuing the fate of the molecules with complete mathematical accuracy. Nor was any regret felt at this confinement to average laws, because average values were all that our crude senses enabled us to observe; therefore the laws calculated on this basis proved sufficiently accurate to predict our observations with all desirable precision. To sum up: it was held that the individual atoms and molecules were subject to a rigid determinism which formed a kind of background to those statistical mass laws which in practice were alone available empirically. And the majority of physicists con sidered this deterministic background to be a most essential foundation for the physical universe. They considered it a logical contradiction to surrender such a belief, and held it necessary to assume that in such an elementary event as the collision of two atoms, the result was predetermined by the preceding conditions fully and with complete accuracy. It was said (and continues to be said) that an exact knowledge of nature is impossible on any other basis, that all the foundations would be lost, that without a determinist background our view of nature would become wholly chaotic and that accordingly it would not fit the nature actually given to us, since this nature is not a complete chaos.

Now this view is certainly erroneous. It is quite certain that the view of the events within a gas as held by the kinetic theory of gases may be modified to the effect that the future trajectory of two molecules, after they have collided, is determined, not by the well-known laws of impact, but by an appropriate law of chance. All we have to do is to see that the laws of chance which we admit should, with reasonable accuracy, take care of certain "bookkeeping" laws (or "laws of conservation," to use the technical term); e.g., that the sum of the energies before and after the collision shall be approximately the same. For this much has been empirically demonstrated even for individual molecules. These bookkeeping laws do not, however, determine the result of the collision unequivocally: and it might be the case that, apart from them, there predominated a "prior" contingency. For this would not introduce a further degree of uncertainty into the result of the collision than there already is from the determinist view. We do not know whether, e.g. in the case of a given collision, the one molecule hits the other a little further to the right or to the left, which affects the result of the collision immensely (though not the conservation laws, of course). Whether we regard the result of the collision as being determined by this "a little further to the right or left" or whether we regard it as primarily undetermined (the "conservation laws" at the same time remaining unaffected) is a matter of indifference. Fifty years ago it was simply a matter of taste or philosophic prejudice whether the preference was given to determinism or indeterminism. The former was favored by ancient custom, or possibly by an a priori belief. In favor of the latter it could be urged that this ancient habit demonstrably rested on the actual laws which we observe functioning in our surroundings. As soon, however, as the great majority or possibly all of these laws are seen to be of a statistical nature, they cease to provide a rational argument for the retention of determinism.

We may briefly summarize this second footnote as follows: Long before modern quantum mechanics made its quantitative statements with respect to the degree of inaccuracy, it was possible, although it was not necessary, to doubt the justification of determinism from a far more general point of view. In fact, such doubts were raised in 1918 by Franz Exner, nine years before Heisenberg set up his relation of indeterminacy. Little attention was paid to them, however, and if support was given to them, as by the author in his inaugural dissertation at Zurich, they met with considerable shaking of heads.

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I should like finally to revert to our original question of determinism as against indeterminism. The question was whether, given complete knowledge of the state of an isolated system, it is possible to predict its future behavior accurately and unequivocally. Is nature of such a kind that this might be possible, at any rate theoretically, even if we are practically unable to obtain the necessary data?

Let us now consider the question from the phenomenological standpoint previously mentioned. From this point of view the number of answers possible to any question addressed to nature must be finite: in fact we may safely say that there can only be two answers, yes or no. If there are more they can be analyzed into a series of consecutive questions. Now in practice we can inform ourselves on the condition of a system at any given moment only by a number of individual observations: in principle any other method is impossible. And if we have made a merely finite number of observations our information on the initial state must consist in a finite series of ayes and noes. In writing, the series might be expressed as a succession of 0's and 1's.

001011110...0110100001

If nature is more complicated than a game of chess, a belief to which one tends to incline, then a physical system cannot be determined by a finite number of observations. But in practice a finite number of observations is all that we can make. All that is left to determinism is to believe that an infinite accumulation of observations would in principle enable it completely to determine the system. Such was the standpoint and view of classical physics, which latter certainly had a right to see what it could make of it. But the opposite standpoint has an equal justification: we are not compelled to assume that an infinite number of observations, which cannot in any case be carried out in practice, would suffice to give us a complete determination.

This is the direction in which modern physics has led us without really intending it.