The laws of physics are generally looked upon as a paradigm of exactitude. Therefore one would naturally take it for granted that probably no other science would be able to give such a clear and definite answer when asked what is meant when we speak of a law of nature.

What is a Law of Nature? The answer does not really seem to be very difficult. When man's higher consciousness first awakens he finds himself in an environment whose changing elements are of the highest significance for his weal or woe. Experience —first the unsystematized experience of his daily struggle for life and afterwards the experience derived from systematically and rationally planned scientific experiments—shows him that the natural processes which take place in his environment do not follow one another in an arbitrary, kaleidoscopic manner, but that they present a notable degree of regularity. He eagerly strives to become acquainted with the nature of this regularity, because such knowledge will be of tremendous advantage to him in his struggle for life. The order of nature thus perceived by man is of the same type throughout. Certain features in the succession of natural events always and everywhere show themselves to be connected with certain other features. Of special biological significance is that case in which the one group of characters precedes the other group in time. The circumstances preceding a certain happening (A) which is often observed in nature, divide themselves into two typical groups: (1) circumstances that are always present — the invariable, and (2) those which are only sometimes present — the variable. When it is further discovered that conversely the unchanging group is always followed by A, this discovery gives rise to the statement that this invariable group of circumstances is the cause that brings about the phenomenon A. Thus, hand-in-hand with the discovery of special regular connections, we come to the idea of a general necessary connectedness between one phenomenon and others as an abstraction from the mass of connections as a whole. Above and beyond our actual experience, the general postulate is laid down that in those cases in which we have not yet succeeded in isolating the causal source of any specific phenomenon, such a source must surely exist —in other words, that every natural process or event is absolutely and quantitatively determined at least through the totality of the circumstances or physical conditions that accompany its appearance. This postulate is sometimes called the "principle of causality." Our belief in it has been steadily confirmed again and again by the progressive discovery of causes that specially condition each event.

Now, what we call a "law of nature" is nothing else than any one of the regularities observed in natural occurrences, insofar as it is looked upon as necessary, in the sense of the above-mentioned postulate. Is there still some obscurity here, some occasion for doubt? And, if so, where? Since about the actual facts there can be no doubt whatever, the only questionable feature is the justifiability or universal applicability of the causal postulate.

Within the past four or five decades physical research has clearly and definitely shown — strange discovery — that chance is the common root of all the rigid conformity to Law that has been observed, at least in the overwhelming majority of natural processes, the regularity and invariability of which have led to the establishment of the postulate of universal causality.

In order to produce a physical process wherein we observe such conformity to Law innumerable thousands, often billions, of single atoms or molecules must combine. (For professional physicists I may say here in parenthesis that this is also true of those phenomena in which, as we often say to-day, the effect produced by a single atom can be successfully studied; because in truth the interaction of this atom with thousands of others determines the observed effect.) In a very large number of cases of totally different types, we have now succeeded in explaining the observed regularity as completely due to the tremendously large number of molecular processes that are cooperating. The individual process may, or may not, have its own strict regularity. In the observed regularity of the mass phenomenon the individual regularity (if any) need not be considered as a factor. On the contrary, it is completely effaced by averaging millions of single processes, the average values being the only things that are observable to us. The average values manifest their own purely statistical regularity, which they would also do if the outcome of each single molecular process were determined by the throwing of dice, the spinning of a roulette wheel or the drawing of sweepstake tickets from a drum.

The statistical interpretation of the laws is illustrated in the simplest and clearest manner by the phenomena of gases, from which, by the way, the new ideas started. In this case the individual process is the collision of two gas molecules, either with one another or with the wall of the container. The pressure of the gas against the walls of the container was formerly attributed to a specific expansive force of matter in the gaseous state; but according to the molecular theory it is due to the bombardment of the molecules. The number of collisions per second against one square centimeter of the surface of the wall is tremendous. For atmospheric pressure at zero degrees Centigrade it runs into twenty-four figures (2.2. X 10"). Even in the most complete terrestrial vacuum and for only one square millimeter and only one-thousandth of a second the number still runs into a figure of eleven places. Besides giving a complete account of the so-called gas laws, that is, of the dependence of pressure on temperature and volume, the molecular theory also explains all other properties of real gases, such as friction, heat conduction, diffusion — and this purely statistically, as a consequence of the molecules being exchanged between different parts of the gas by individual processes of the utmost irregularity. In performing the corresponding calculations and discussing the relevant considerations we generally assume the validity of the mechanical laws for the single happening, the collision. But it must be stated that this is not at all necessary. It would be quite sufficient to assume that at each individual collision an increase in mechanical energy and mechanical momentum is just as probable as a decrease, so that taking the average of a great many collisions, these quantities remain constant in much the same way as two dice cubes, if thrown a million times, will yield the average 7 whereas the result of each single throw is a pure matter of chance.

From what has been said it follows that the statistical interpretation of the gas laws is possible, perhaps also that it is the most simple; yet we cannot conclude that it is the only possible interpretation. But a crucial test is furnished by the following experiment. If the pressure of a gas is really only a statistical average value it must be subject to fluctuations. These must become all the more obvious the more the number of cooperating elementary processes is reduced by reducing (1) the surface against which the pressure is exerted and (2) the inertia of the body which experiences the pressure, in order to allow a prompt reaction to a fluctuation that occurs within a short period of time. Both these conditions can be attained by suspending tiny, ultra-microscopical particles in the gas. These actually show a zig-zag movement of extreme irregularity, long known as the Brownian movement, which never comes to rest and agrees in all particulars with the theoretical predictions. Although the number of molecules which hit the particle during a measurable period of time is still very large, it is yet not large enough to produce an absolutely uniform pressure from all sides. Through a chance preponderance of the impact in a chance direction, which changes quite irregularly from moment to moment, the particle will be driven hither and thither on quite an irregular path. Here, therefore, we see a law of nature — the law of gas pressure - losing its exact validity in proportion as the number of cooperating individual processes decreases. One cannot easily imagine a more convincing proof of the essentially statistical character of at least this law.

I could here mention numerous other cases that have been experimentally and theoretically investigated, such as the uniform blue of the sky, which results from entirely irregular variations of atmospheric densities (consequent upon their molecular constitution), or the strictly law-governed decay of radioactive substances which results from the disintegration of the individual atoms, whereby it appears to depend entirely on chance whether an individual atom will disintegrate immediately or to-morrow or in a year's time. But however many examples are considered, they scarcely suffice to render our belief in the statistical character of physical laws as certain as does the fact that the Second Law of Thermodynamics, or Law of Entropy, which plays a role in positively every physical process, has clearly proved to be the prototype of statistical law. Although this matter would justify a closer examination, on account of its extraordinary interest, I must confine myself here to the very cursory remark that empirically the Law of Entropy is very intimately connected with the typical one-directional character of all natural processes. Although the Law of Entropy by itself is not sufficient to determine the direction in which the state of a material system will change in the next instant, it always excludes certain directions of change, the direction exactly opposite to the one which actually occurs being always excluded. By virtue of the statistical method the Law of Entropy has taken on the following content: namely, that every process or event proceeds from a relatively improbable--that is to say, more or less molecularly ordered—state to a more probable one--that is to say, to a state of increasing disorder among the molecules.

In regard to what I have said up to now there is no essential difference of opinion among physicists. But the case is otherwise in regard to what I shall have to say from now on.

Although we have discovered physical laws to be of a statistical character, which does not necessarily imply the strictly causal determination of individual molecular processes, still the general opinion is that we should find the individual process — for instance, the collision of two gas molecules - determined by rigid causality, if we could trace it. (In a similar way the result of a game of roulette would not be something haphazard for any one who could measure exactly the impetus given to the wheel, the resistance of the air, the friction on the axis, etc., etc.) In some cases, among which is also the one of colliding gas molecules, it is even claimed that quite definite features of the individual process can be ascertained; viz., the conservation of energy and momentum at every single impact, not merely in the average.

It was the experimental physicist, Franz Exner, who for the first time, in 1919, launched a very acute philosophical criticism against the taken-for-granted manner in which the absolute determinism of molecular processes was accepted by everybody. He came to the conclusion that the assertion of determinism was certainly possible, yet by no means necessary, and when more closely examined not at all very probable.

As to the non-necessity, I have already given my opinion; and I believe with Exner that it can be upheld, even despite the fact that most physicists claim quite definite characteristics for the elementary laws which they postulate. Naturally we can explain the energy principle on the large scale by its already holding good in the single events. But I do not see that we are bound to do so. In like manner we can explain the expansive force of a gas as the sum of the expansive forces of its elementary particles. But this interpretation is here decidedly incorrect, and I do not see why there it should be looked upon as the only possible one. I may further remark that the energy-momentum theorem provides us with only four equations, thus leaving the elementary process to a great extent undetermined, even if it complies with them.

Whence arises the widespread belief that the behavior of molecules is determined by absolute causality, whence the conviction that the contrary is unthinkable? Simply from the custom, inherited through thousands of years, of thinking causally, which makes the idea of determined events, of absolute, primary causalness, seem complete nonsense, a logical absurdity.

But from what source was this habit of causal thinking derived? Why, from observing for hundreds and thousands of years precisely those regularities in the natural course of events which, in the light of our present knowledge, are most certainly not governed by causality; or at least not so governed essentially, since we now know them to be statistically regulated phenomena. Therewith this traditional habit of thinking loses its rational foundation. In practice, of course, the habit may safely be retained, since it predicts the outcomes satisfactorily. But to allow this habit to force upon us the postulate that, behind the observed statistical regularities, there must be causal laws, would quite obviously involve a logically vicious circle.

Not only are there no considerations that force this assumption upon us, but we should realize, still further, that such a duality in the laws of Nature is somewhat improbable. On the one hand we should have the intrinsic, genuine, absolute laws of the infinitesimal domain; while on the other there would be that observed macroscopic regularity of events which in its most essential features is not due to the existence of the genuine laws but is determined rather by the concept of pure number, the most translucent and simple creation of the human mind. Clear and definite intelligibility in the world of outer appearances, and behind this a dark, eternally unintelligible imperative, a mysterious Kismet! The possibility that this may be in reality the case must be admitted; but this duplication of natural law so closely resembles the animistic duplication of natural objects, that I cannot regard it as at all tenable.

It must not be supposed, however, that I consider it a simple and easy matter to carry through and defend this new, a-causal (i.e., not necessarily causal) point of view. The ruling opinion to-day is that at least the laws of gravitation and electro-dynamics are of the absolute, elementary type, that they also govern the world of atoms and electrons and are perhaps at the basis of everything as the primary and fundamental Law. You are all familiar with the amazing success of Einstein's gravitation theory. Must we conclude from this that his gravitational equations are an elementary law? I hardly think so. In no case of a natural process is the number of single atoms which must cooperate in order that an observable effect may be produced so vast as in the case of gravitational phenomena. This would explain, from the statistical point of view, why we can attain such extraordinary accuracy in forecasting movements of the planets centuries ahead. Moreover I shall not deny that Einstein's theory yields powerful support to the belief in the absolute validity of the energy and momentum principles. With reference to the particle, these principles actually involve nothing more than a tendency towards absolute perseverance. For Einstein's gravitation theory is not really anything more than the reduction of gravitation to the law of inertia. That under certain conditions nothing changes is surely the simplest Law that can be conceived, and hardly falls within the concept of causal determination. It may after all be reconcilable with a strictly a-causal view of Nature.

In contradistinction to gravitation, the laws of electro-dynamics are quite generally applied to-day to processes within the atom itself, and indeed with amazing success. These positive results will be considered the most serious objection that can be advanced against the a-causal view. The space at my disposal does not allow of my going further into this question. I must confine myself to the following general remark, which at the same time briefly sums up the conclusions we have reached —

Exner's assertion amounts to this: It is quite possible that Nature's laws are of thoroughly statistical character. The demand for an absolute law in the background of the statistical law — a demand which at the present day almost everybody considers imperative — goes beyond the reach of experience.¹ Such a dual foundation for the orderly course of events in Nature is in itself improbable. The burden of proof falls on those who champion absolute causality, and not on those who question it. For a doubtful attitude in this respect is to-day by far the more natural.

The electro-dynamic theory of the atom appears unsuited to furnish the proof, because this theory itself is universally recognized to be suffering from serious intrinsic incoherences which are often felt to be of a logical character. I prefer to believe that, once we have discarded our rooted predilection for absolute Causality, we shall succeed in overcoming these difficulties, rather than expect atomic theory to substantiate the dogma of Causality.