PHYSICAL REGULARITY IN LIGHT OF NEW RESEARCH1

Max Planck.

I.

What do we understand by “physical regularity”? A physical law is a proposition that establishes a firm, inviolable connection between physical quantities that can be measured, and moreover such a connection by means of which one of these quantities can be calculated from the measured values of the remaining quantities. The fullest possible knowledge of physical regularity is the greatest goal toward which every physicist ardently strives, regardless of whether he values it only from the point of view of practical usefulness, seeing its true significance in the fact that knowledge of regularity eliminates the need to carry out complicated and difficult measurements, or whether, going further, he seeks in the attainment of this goal the satisfaction of a deep inner thirst for knowledge and a firm basis for a scientific worldview.

In what way, then, are individual physical laws established, and what do they represent? First of all, the very existence of physical regularity can in no case be regarded as self-evident, just as it is by no means self-evident that, if a regularity has existed up to now, it will always exist in unchanged form in the future. It is undoubtedly conceivable—and we can do absolutely nothing about this—that one fine day nature will deceive our expectations, that a completely unexpected event will occur, and that, despite all our efforts, we shall still not succeed in bringing any lawful order into the confusion that has arisen. In that case science would have no choice but to declare itself bankrupt. Therefore science is compelled to recognize the existence of regularity in nature as a fundamental postulate or preliminary

condition of all its further development, or, together with Immanuel Kant, to rank the concept of causality among the categories given from the beginning, without which knowledge in general is unattainable.

It follows from this, of necessity, that the essence of physical regularity and the content of physical laws cannot be known by reflection alone, and that the only path to this goal consists in first of all turning to nature, gathering the greatest possible number of as many-sided as possible observations and experimental data, comparing them with one another and generalizing them in the simplest and most comprehensive possible propositions—in a word, turning to the inductive method.

Since observation and experiment are the more meaningful the more exact the measurements that underlie them, the successes of all physical knowledge are, of course, connected in the closest way with the improvement of physical instruments and measuring technique. The most recent history of physics bears witness to this with particular strikingness. But measurements alone do not exhaust the matter. Each measurement is a separate, independent event, connected with quite special circumstances, above all with a definite place and time, as well as with a definite measuring instrument and a definite observer. And if in many cases the desired generalization, so to speak, suggests itself, there also exist other cases in which it is extremely difficult to find, from the diverse measurement data, a general law—either because, apparently, no possibilities at all for this present themselves, or, on the contrary, because there exist too many different possibilities of generalization, which is likewise extremely unsatisfactory.

In such cases there remains nothing else than to try to introduce a certain supposition, the so-called working hypothesis, and to see what this hypothesis can yield. An especially important sign of the usefulness of such a hypothesis is its confirmation in those domains of phenomena which were not at all taken into account in its initial formulation. For in such a case we have the right to conclude that the lawful correlation formulated by it possesses a deeper significance and opens the way to essentially new knowledge.

Thus, an expedient working hypothesis constitutes the most necessary tool of every inductive investigation. In this connection there arises the important question of where one must begin in order to arrive at the establishment of as suitable a hypothesis as possible. There is no general recipe for this. For even with the most complete and many-sided experience, logical thinking alone is in no case sufficient here. Only immediate intuition (unvermitteltes Zufassen), a lucky thought—often a leap of thought that at first seems extremely...

phenomena—a leap accessible only to a lively and original imagination, guided onto the right path by precise knowledge of the facts and by the powerful creative force of imagination.

In most cases the essence of the matter consists in introducing certain mental images, analogies, which establish a connection with lawful relations known in other fields and which help to bring greater unity into the physical picture of the world.

But precisely behind promising successes in this direction there is often a danger hidden. For if the bold step has indeed succeeded, if the fruitfulness of the introduced hypothesis has been confirmed, then a new task arises: that of further developing it, isolating its essential core, and revealing, by means of an appropriate formulation, its true content, i.e. of carefully cleansing the hypothesis of everything inessential and extraneously added. This task, however, is by no means as simple as it might seem at first glance. For a bridge created by a fortunate flight of thought and opening access to new knowledge very often, upon closer examination, proves to have only a provisional character, and it has to be replaced by a more solid structure capable of withstanding the artillery fire of critical logic. One must bear in mind that every hypothesis is the product of a groping imagination; that imagination operates with visual representations (Anschauung); and that, although one cannot dispense with visual representations in constructing hypotheses, nevertheless in constructing a rational theory—and especially in logical proofs—they constitute for physics an instrument of a very dubious character. For a natural confidence in certain visual representations and inferences which have proved their fruitfulness in certain directions easily leads to an overestimation of their significance and to unfounded generalizations. If, in addition, one takes into account that the very creators of new fruitful theories, for reasons of convenience or because of a certain feeling of piety, are usually not inclined to introduce essential changes into that train of thought which led them to success, and that they often throw their entire, deservedly acquired authority into the balance in order to support the standpoint they originally adopted, then it is easy to understand that the further healthy development of a theory often encounters considerable difficulties. Examples of this are found at every step in the history of the physical sciences right up to the present day. Allow me to mention some of the most essential of them.

The first physical laws were naturally established in that field in which the first exact measurements proved possible—namely, measurements of space and time, i.e. in the field of mechanics. It is likewise easy to understand that it was first possible to establish lawful relations in precisely those motions which proceed

independently of random external accompanying circumstances and interventions—in the motions of celestial bodies. Cultured peoples of the East, already thousands of years ago, were able to derive from observations formulas that made it possible, with great confidence, to calculate for years ahead the motions of the sun and planets. With each increase in the accuracy of measurements there was associated an improvement of the formulas. The collation and comparison of these formulas subsequently led to the theories of Ptolemy, Copernicus, and Kepler, each of which surpassed its predecessor in its simplicity and accuracy. All these theories answer the question of a regular relation between the position of a celestial body, for example a planet, and the moment of time at which this position is occupied by it. Of course, the character of this regular relation is different for different planets, although many common features can be found in the motion of the planets.

Such was the formulation of the question until Newton took the decisive step, embracing the formulas relating to the different planets in one single law of motion, equally applicable to all planets and, in general, to all celestial bodies. He was able to achieve this success only because he formulated the law of motion independently of that special moment of time to which this law is applied; namely, he replaced the moment of time by the differential of time. Newton’s theory of planetary motion establishes a definite regular relation not between the position of a planet and time, but between the acceleration of the planet and its distance from the sun. This law, which is a certain vector differential equation, is completely the same for all planets. Consequently, from the position and velocity of a planet at some definite moment of time one can unambiguously calculate its motion for all times.

Newton’s formulation of the laws of motion is not only a new form of description of nature, but signifies a real success in the cognition of the regularities that connect things. This is evident from the results obtained in further applications of these laws. They not only surpass Kepler’s formula in their accuracy, for example in that they make it possible, in full agreement with measurements, to compute the perturbations experienced by the elliptical motion of the earth around the sun when it approaches Jupiter, but they also explain the motions of other celestial bodies, such as comets, double stars, and so forth, to which Kepler’s formulas are not applicable at all. However, the most immediate, decisive success brought to Newton’s theory was the circumstance that its application to terrestrial motions led to the same laws of the free fall of bodies and of the oscillation of the pendulum that had been established by Galileo by measurement; moreover, to the explanation of certain phenomena which, without Newton’s theory, would have been impossible.

PHYSICAL REGULARITY

...of completely incomprehensible phenomena, such as: tides and ebbs, the rotation of the plane of oscillation of a pendulum, the precession of the axis of rotation of a top, etc.

By what path, then, did Newton arrive at his differential equation of planetary motion? This question is of primary interest to us now. He arrived at it not at all by establishing a direct connection between the acceleration of a planet and its distance from the sun, and by searching for a definite numerical relation between them; no, he came to this equation by first creating a conceptual bridge linking the notion of the position of a planet with the notion of acceleration; this bridge bears the name of force. Newton arrived at the idea that, on the one hand, the position of a planet relative to the sun determines a force of attraction directed toward the sun, and that, on the other hand, this force of attraction causes a definite change in the planet’s quantity of motion. The concept of force, as the very word “force” indicates, undoubtedly arose from the idea of the muscular sensation experienced when lifting a weight or throwing a ball. With further generalization, this idea was applied to all kinds of changes of motion, including even changes so considerable that human muscular force is far from sufficient to produce them.

It is not surprising that Newton ascribed decisive importance to this concept of force, which helped him achieve such important successes, although what deserves special attention is that this concept is not found at all in the fundamental law of motion, and that he sought in it the primary cause of every change of motion. Thus it turned out that Newtonian force became the chief and fundamental concept of mechanics and not only of mechanics, but of all physics, and that in time a habit was formed, when considering all physical phenomena, always to pose first of all the question of the force determining these phenomena.

The conceptions created by the most recent development of physics stand in a certain contradiction to this. One may calmly say that at present Newtonian force has lost its fundamental significance for theoretical physics. In modern mechanics force is a secondary quantity; it has been replaced by another, higher and more comprehensive concept—that of work or potential, with force being defined as the fall, or the negative gradient, of the potential.

But how, however, one may object, is it possible to regard work as the primary concept, when, whenever work arises, there must first exist a force that performs this work? Whoever speaks thus is thinking not physically, but physiologically. Of course, in work performed in lifting a load, the primary thing is the contraction of the muscles, with its accompanying sensation, and it is the cause of the motion that arises. However, one must strictly distinguish this physiological process from the force considered by us.

of attraction with which the earth acts upon the load and which, in its turn, is uniquely determined by the gravitational potential.

The rights of the potential to precedence over force are based not only on the fact that, with the introduction of the potential, physical laws assume a simpler form, but also on the fact that the concept of potential is applicable to a far wider domain of phenomena than the concept of force, beginning with mechanics and extending to the doctrine of chemical affinity, where there can be no question at all of a Newtonian force. True, it must be admitted that the concept of potential does not possess the advantage of immediate representation (Anschauung) which belongs to the concept of force by virtue of its connection with muscular sensation, and that therefore the elimination of the concept of force does substantial harm to the intuitive clarity of physical laws. But the development of the theory in this direction lies in the nature of things. For physical lawfulness is not guided by the human organs of sense and by the corresponding capacity for visual representation.

In teaching, in my opinion, when introducing mechanics it will nevertheless always be necessary to begin with Newtonian force, just as in optics one begins with the sensation of color, and in thermodynamics with the sensation of heat, although subsequently these fundamental concepts are replaced by more precise ones. Nor must it be forgotten that the significance of all physical concepts and laws rests for us, in the final analysis, upon their relation to the human organs of sense. For precisely in this consists the characteristic peculiarity of physical investigation. In order to be able to create usable physical concepts and hypotheses, we must first of all turn to our capacity for representation, directly adapted to the specific perceptions of the sense organs. From this capacity alone do we draw all our ideas. If, however, we then wish to arrive at the establishment of physical laws, we must, as far as possible, abstract from the visual representations introduced earlier and free the basic definitions from everything adventitious and from all representations that do not stand in a logically necessary connection with measurements. Having then formulated physical laws and derived from them, by mathematical means, definite consequences, we must in the end once again translate the results obtained into the language of our organs of sense, in order to make them usable for us. Thus this path in a certain sense closes upon itself, but nevertheless it is absolutely necessary. For the simplicity and generality of physical laws become manifest only after they have been purified of all anthropomorphic admixtures.

In theoretical physics there are many conceptual bridges and auxiliary intuitive concepts, similar to the Newtonian force described by me above. In this connection I shall mention only the concept of osmotic—

...of pressure, which proved so fruitful in physical chemistry and was introduced in its time by van ’t Hoff for the purpose of a vivid formulation of the physical laws of solutions, relating in particular to freezing points and to the elasticity of vapors. Osmotic pressure can be detected and measured only in a comparatively imperfect manner, or else this requires very complex devices, the so-called semipermeable membranes. All the more worthy of amazement is the intuitive penetration that led this great investigator, on the basis of rather scanty observational data, to formulate the law named after him. Yet in the modern form of this law the concept of osmotic pressure plays as insignificant a role as the concept of Newtonian force does in the laws of motion.

There exist, however, conceptual bridges of an entirely different kind, possessing a high degree of visualizability and proving very valuable in the construction of fruitful working hypotheses, but at the same time, as it turned out, creating direct obstacles to the subsequent stages of the development of science. One of them in particular deserves that we dwell on it. Just as people were accustomed to suppose that the cause of every change occurring in nature is some force, so likewise they were inclined to imagine every immutable, constant quantity in the form of a substance. From time immemorial the concept of substance has played a significant role in physics, but, as closer examination shows, not always a progressive one. It is easy to understand that every so-called “law of conservation” can be interpreted from the standpoint of the conservation of substance, and this representation undoubtedly contributes greatly to the vividness of the proposition being expressed, and, consequently, also facilitates its use. For we can scarcely form for ourselves a vivid representation of any quantity which, with all its changes, remains quantitatively unchanged, without calling upon the representation of a moving material body. In close connection with this, undoubtedly, also stands the tendency to reduce all changes occurring in nature in general to motions of substance, i.e., to mechanics. Thus, for example, the radiation and propagation of light received a vivid interpretation in the wave motion of a substantial luminiferous ether. Following this path, it proved possible to derive the most important laws of optics in complete agreement with experiment; however, in the end the time came when this substantial-mechanical theory ceased to serve and lost itself in barren speculations.

And in the field of the theory of heat, the concept of substance rendered excellent services for a certain time. The careful development of calorimetry in the first half of the last century was carried out chiefly on the basis of the idea of the transfer of an immutable heat substance from a warmer body to a colder one.

When it was subsequently shown that the quantity of heat can be increased, for example, by friction, the theory of the caloric substance tried to defend itself by resorting to additional hypotheses, and for a time it succeeded in doing so, but nevertheless not for long.

In the doctrine of electricity, even on a superficial consideration, we encounter the dangerous consequences to which an excessive exaggeration of the concept of substance may lead. True, here too the law of constancy of the quantity of electricity, and the concepts of electric current and the law of interaction of charged and current-carrying conductors that are connected with it, receive an extremely vivid interpretation on the basis of the conception of a subtle, highly mobile substance endowed with definite forces. However, this analogy ceases as soon as one takes into account the circumstance that it is necessary to admit the existence of two opposite substances, positive and negative, which, when combined, completely neutralize one another. This process of neutralization, just like the emergence of two opposite substances out of nothing, is in any case inconceivable with respect to ordinary substances.

Thus we see that visual representations and the views growing out of them are necessary for physical research, and that countless times they have furnished us with the key to the discovery of new paths of knowledge; but that, nevertheless, great caution is needed in dealing with them, even when they have justified themselves for a certain time. The only reliable guide on the path toward further development forever remains measurement and that which follows logically from the concepts immediately adjoining it. All other conclusions, and especially those among them which are distinguished by so-called immediate obviousness, must always be treated with a certain distrust. For the question of the probative force of an inference operating with precisely defined concepts is decided not by intuition (Anschauung), but by reason.

II.

Up to now we have dealt chiefly with questions concerning the way by which one arrives at knowledge of physical laws; now we shall turn to a somewhat closer examination of the content and true essence of physical regularity.

A physical law usually finds its expression in a mathematical formula that makes it possible to calculate the course in time of phenomena occurring in some physical system subject to certain given conditions. From this point of view, all physical laws may be divided, according to their content, into two large groups.

The laws of the first group are characterized by the fact that they remain valid when the sign of time in them is changed to the opposite; in other words, any process satisfying the requirements of the laws can, without entering into contradiction with them, proceed also in the reverse direction. As examples one may cite the laws of mechanics and the laws of electrodynamics, insofar as we abstract from thermal and chemical actions. Every purely mechanical or electrodynamic process can proceed also in the opposite direction. A body falling without friction is accelerated according to the very same law by which a body flying upward without friction is retarded; the conditions of oscillation of a pendulum to the right are the same as to the left; a wave can propagate in one direction just as in another, outward just as inward; a planet can revolve around the sun in one direction just as in the reverse. The question whether, and in precisely what way, the reversal of motion can be realized in actuality is an entirely different question, on which we have no need to dwell here. The matter concerns the law itself, and not those special data to which it is applied.

The laws of the second group are characterized by the fact that in them the sign of time plays an essential role. Therefore the processes subject to them are directed one-sidedly, irreversibly. To these processes belong all those phenomena in which heat and chemical affinity play a role. In friction the relative velocity always decreases and never increases; in thermal conduction the colder body always becomes warmer, and the warmer always cools; in diffusion the mixedness always increases, and not the separateness of the diffusing substances. Therefore irreversible processes always lead to a definite final goal: friction—to a state of relative rest, thermal conduction—to the equalization of temperatures, diffusion—to complete homogeneity of the mixture, whereas, conversely, reversible processes, unless there is intervention from without, have neither beginning nor end and consist in an eternal “back and forth.”

In what way, then, can these two completely opposite kinds of laws be united under one heading, as is absolutely necessary in the interests of the unity of the physical picture of the world? In the time of the previous generation there existed in theoretical physics a tendency, naturally coming to the fore, of so-called energetics, which strove to remove this contradiction, for example, by drawing a complete analogy between the transition of heat from a higher temperature to a lower one and the descent of a weight or a pendulum from a higher position to a lower one. In doing so, however, there remained unaccounted for the essential circumstance that the weight can also rise, and that the pendulum, having reached its lowest point, ...

point, has the greatest velocity and, owing to its inertia, passes through the position of equilibrium to the opposite side, whereas, in contrast to this, the flow of heat from the warm body to the cold one becomes the weaker the smaller the temperature difference becomes, and there can be no question of passing through the state of equality of temperatures under the influence of a peculiar inertia.

Be that as it may, the contrast between reversible and irreversible processes will always remain; the question can only be to find an entirely new point of view that would reveal a certain connection between heterogeneous laws and, moreover, if possible, one that would in some way reduce the laws of one group to the laws of the other group. Which of the groups, then, should be regarded as the simpler and more elementary: the group of reversible or of irreversible processes?

To this question a certain answer is already given by a purely external formal consideration. Besides the variable quantities that are subject in each individual case to special measurement, every physical formula contains also certain constant quantities, which must be conceived as fixed once and for all and which impart a characteristic imprint to the functional relation of the variable quantities expressed by the formula. On considering these constants it is easy to convince oneself that in reversible phenomena, under the most varied external conditions, they do indeed always remain one and the same, as, for example, mass, the gravitational constant, electric charge, the velocity of light. Conversely, the constants of irreversible processes, such as, for example, thermal conductivity, the coefficient of friction, diffusion constants, depend to a greater or lesser degree on external circumstances: on temperature, pressure, and so forth.

Proceeding from these facts and circumstances, it is natural to regard the constants of the first group as simpler and the corresponding laws as elementary and further unanalyzable, while the constants of the second group and the corresponding laws should be regarded as more complex. To pass judgment on the lawfulness of this supposition, it is necessary to refine the mode of consideration; it is necessary to examine the phenomena, so to speak, under a stronger magnification. If irreversible phenomena truly have a complex character, then the laws governing them can be only roughly valid; they must have a statistical character, for they have significance only under a macroscopic, aggregate consideration, i.e. only for the mean values of a large number of different individual phenomena. The more one restricts the number of individual phenomena to which the mean values refer, the more distinctly random deviations from the macroscopic laws must appear. In other words, if the conception set forth is indeed correct, then under microscopic consideration all

laws of irreversible processes—friction, thermal conduction, diffusion—must prove to be inexact; in particular cases they must allow exceptions, which become all the more evident the more the mode of consideration is refined.

It is precisely this conclusion that, with the passage of time, has been confirmed by experiment in every respect with an ever-increasing degree of certainty, which, of course, could be achieved only with the aid of an extraordinary improvement in methods of measurement. The considerable degree of approximation within which the laws of irreversible phenomena are valid is explained exclusively by the colossal number of individual phenomena of which they are usually composed. If, for example, we take a liquid everywhere at the same temperature, then from the macroscopic law of thermal conduction it follows that within the liquid no transfer of heat occurs. In essence, however, this is not at all so. For heat is conditioned by the rapid motions of the molecules of the liquid, and thermal conduction by the exchange of velocities that takes place when molecules collide. The homogeneity of temperature therefore consists not in the equality of all velocities, but only in the equality of the mean values of the velocities in each volume of liquid containing a very large number of molecules. If, however, we take a volume of liquid containing relatively few molecules, then the mean value of the velocities of these molecules will, in the course of time, undergo fluctuations, which will be the stronger the smaller the chosen volume of liquid. This proposition may at present be regarded as a fact fully confirmed by experiment. One of the most striking illustrations of this proposition is the so-called Brownian molecular motion, which can be observed with a microscope in small dust particles suspended in a liquid. Under the influence of the impacts of the invisible molecules of the liquid, these dust particles move back and forth, and the more strongly, the higher the temperature. If we now introduce the assumption—which meets with no essential obstacles—that each individual impact is a reversible phenomenon for which the strict elementary laws of dynamics are valid, then it may be said that, by introducing the microscopic mode of consideration, the laws of irreversible phenomena, or, in other words, the rough and statistical regularities, have been reduced to an exact and absolute dynamical regularity.

The great successes achieved in recent times by introducing statistical regularities into numerous areas of physical investigation have brought about among physicists a significant change in views. Instead of, as formerly in energetics, denying the existence of irreversible processes or, at the very least, regarding them as doubtful, at present attempts are often made to bring statistical regularity to the fore, to reduce to statistical...

all laws that had hitherto been considered dynamical, including even gravitation; in other words, to exclude altogether the existence in nature of absolute lawfulness. And indeed, it is easy to understand that everything we study and measure in nature can never be expressed by perfectly definite numbers and always contains a certain indeterminacy, due to the inevitable sources of error in measurements. It follows from this that we shall never succeed, by means of measurements, in deciding with absolute precision whether a given law is valid in nature or not. We arrive at the same conclusion when we consider this question from the standpoint of the general theory of knowledge. If from the very beginning we are faced with the impossibility of proving even that lawfulness in general exists in nature, still less can we succeed in proving that this lawfulness is absolute.

Thus, from the logical point of view one must recognize as a fully legitimate hypothesis that only statistical lawfulness exists in nature. Another question is whether this supposition is useful for research, and to this question I should like to answer with a decisive negative. First of all, it must be taken into account that only a strictly dynamical lawfulness fully satisfies the requirements of our striving for knowledge, whereas any statistical law is, at bottom, unsatisfactory simply because it is not strictly valid, but admits exceptions in individual cases. Therefore the question always arises: in precisely which cases do these exceptions occur?

Questions of just this kind are the strongest stimulus to the expansion and refinement of methods of investigation. If statistical lawfulness is recognized as the last, most profound form of lawfulness, then in principle there are no grounds for seeking the causes of fluctuation deviations from any known statistical law. In reality, however, it is precisely the effort to find behind every statistical lawfulness a dynamical lawfulness, one strictly causal in character, that has led to the most important successes in the study of atomistic phenomena.

On the other hand, if there is a law which until now has always proved strictly valid within the limits of errors of measurement, then, of course, one must admit that with the aid of measurements it will never be possible finally to establish whether it nevertheless has a statistical character. It is, however, of essential importance whether theoretical considerations lead to recognizing it as statistical or as dynamical. For in the first case there will be persistent efforts, by the continuous refinement of methods of measurement, to determine the limits of applicability of the law, whereas in the second case such efforts will be deemed fruitless, which will spare much useless labor. Already too much

...so much effort has been expended in physics on the solution of such imaginary problems that these considerations may be regarded as of little importance.

Proceeding from this, I believe that, in any case, in the interests of healthy further development, one must number among the postulates of the physical sciences not only the existence of regularity in general, but also the strictly causal character of this regularity—which, in essence, has always been done up to now—and regard the aim of investigation as attained only when each of the observed statistical regularities has been reduced to one or several dynamical ones. This should by no means diminish the great practical significance of statistical regularities. Like meteorology, geography, and the social sciences, physics too in many cases must work with statistical laws. But just as no one doubts that the so-called chance fluctuations in climatological curves, in population statistics, or in mortality statistics have, in each individual case, a strictly causal origin, so for physics the question will always have a strictly lawful meaning: why, of two neighboring uranium atoms, does one decay many millions of years earlier than the other?

The science of spiritual life, too, will never be able to dispense with the presuppositions of the existence of strict causality. Opponents of this view often put forward the argument that man possesses freedom of the will. I have already once before had occasion to give a detailed justification of the fact that there is no contradiction here at all; that, on the contrary, the freedom of the human will is fully reconcilable with the universal dominion of the strict law of causality. Since the considerations I developed were, by some, understood quite incorrectly, and since this question is undoubtedly of considerable interest, I shall ask permission to dwell on it briefly.

The law of causality requires that both actions and mental phenomena, and in particular the volitional motives of every person at every moment of time, should be completely determined by the state of his entire inner world at the preceding moment and by the influences of the external world. We have no grounds whatever, in any respect, to doubt the correctness of this proposition. For the question of freedom of the will concerns not at all whether such a determinate relation exists, but whether this relation is accessible to the knowledge of the subject himself. The solution of the question whether a person can feel himself free or not depends solely on this circumstance. Only in the case that someone were able, on the basis of the law of causality alone, to foresee his own future, would it be necessary to deny him the presence of consciousness of freedom of the will. Such a case, however, is impossible, since it contains a logical contradiction. For perfect knowledge of the pred...

assumes that the object known does not change under the influence of internal phenomena in the knowing subject; and this premise is incorrect if object and subject are identical with one another. Or, to speak more concretely: since the cognition of some inner volitional motive of one’s own is an experience from which a new volitional motive may arise, this cognition thereby increases the number of possible volitional motives. The establishment of this proposition constitutes a new cognition, which in turn may condition the emergence of a new volitional motive. This chain of consequences may be continued as far as one likes, and we shall still never succeed in establishing the final, decisive motive of our own future action—that is, we shall not succeed in attaining a cognition which would not in turn lead to the emergence of a new volitional motive.

Whoever doubts the sense of this reasoning and cannot understand why a sufficiently intelligent mind cannot fully grasp the causal determination of its present self, should, in essence, also fail to understand why a giant, so tall that he looks down upon everyone from above, is not able to look down upon himself from above as well. No—from the law of causality alone even the most intelligent human being will never be able to derive the decisive motive of his own conscious actions. For this purpose he requires another guiding thread—namely, the moral law, which can be replaced neither by the highest reason nor by the most subtle self-analysis.

III.

But let us return to physics, in which there is no place for such complications. I should like also to describe to you the most important characteristic features of the contemporary physical picture of the world, in which there has been reflected the striving to bring, in the manner we have described, all physical phenomena into a strict causal connection. Even a superficial glance reveals the enormous changes that have taken place in this picture since the beginning of our century. We are entitled to say that there has been no such rapid development since the times of Galileo and Newton, and we are proud that this time German science has taken a very substantial part in it. The impulse to this development was naturally given by the progress of technology, most closely connected with it, and by the extraordinary refinement of methods of measurement, which in turn led to the establishment of new facts and, thanks to this, to a revision and expansion of theory. In particular, two new ideas give contemporary physics its characteristic stamp. These ideas are expressed, on the one hand, in the theory of relativity, and on the other—in the quantum hypothesis. Each of them is fruitful in its own way, but nevertheless they are entirely alien to one another and

in a certain sense even opposite. Allow me to speak a little about them, since the time remaining at my disposal permits it.

At one time the theory of relativity was, one may say, on everyone’s lips. Arguments for and against it spread through the broadest circles, right down to the daily press, where the dispute about it was conducted both by specialists and, to an even greater degree, by non-specialists. At the present time a certain calm has set in on this question, which cannot but arouse the most sincere satisfaction, just as it does in the founder of the theory himself. The interest of the broad public, apparently, has to a certain extent been satiated and has now turned to other fashionable themes. Perhaps many would be inclined to conclude on this basis that at the present time the role of the theory of relativity in science has already, to a considerable degree, been played out. As far as I can judge, precisely the opposite is the case. The theory of relativity has now become so firm a constituent part of the physical picture of the world that it, like everything obvious, is no longer discussed. And indeed, however revolutionary and new the effect produced by the fundamental idea of the special and general theory of relativity on the whole physical world may have been at the first moment of its appearance, all its assertions and all the attacks upon it were directed, in essence, not at all against the basic, recognized, and confirmed laws of physics, but only against certain views which, it is true, were deeply rooted, yet nevertheless bore only the character of habits of thought; among those which, as I tried to explain above, are very useful for the first understanding of physical relations, but which must be cast aside if it proves necessary to generalize and deepen these relations.

As an especially instructive example I shall cite only the concept of simultaneity. To the naïve observer nothing seems more natural than the presence of a definite meaning in the assertion that two events occurring in two places remote from one another, for example on the Earth and on Mars, are simultaneous. For no one is forbidden, in his thoughts, to fly instantaneously across arbitrarily great distances and to compare both events directly with one another in his inner contemplation. It must, however, be emphasized once again that the theory of relativity has in no way altered this truth. On its basis, anyone, provided he has sufficiently accurate measuring instruments at his disposal, can establish with complete certainty whether both events are simultaneous. And if he correctly carries out measurements of time by different methods, with different instruments controlling one another, he will each time arrive at identical results. Thus everything, consequently, remains as before.

However, according to the theory of relativity, he cannot take it for granted that another observer, moving with respect to him, must also regard both events as simultaneous. For the thoughts and conceptions of one person do not always coincide with the thoughts and conceptions of another. If both observers share with one another the content of their thoughts and conceptions, then each of them will refer to his own measurements, and it will then become clear that, in interpreting their measurements, they both proceeded from entirely different premises. The question of which of the premises is correct likewise cannot be decided, any more than the dispute as to which of the two observers is at rest and which is in motion. This question, however, is essential, for the rate of a clock—and in this there is nothing surprising—depends on the speed with which the clock is moving away from the given place, and it follows from this that the rates of the clocks of the two observers are different. The final conclusion, then, is that each of the observers may with equal right assert that it is he who is at rest and that his measurements of time are correct, while, however, one of the observers regards as simultaneous events which, according to the other observer, are not simultaneous. Reasoning of this kind, of course, places heavy demands on our faculty of representation; yet this sacrifice of vivid intuitiveness is exceedingly small in comparison with the inestimable advantages connected with the grand generalization and simplification of the physical picture of the world.

Anyone who still cannot free himself from the opinion that the theory of relativity ultimately suffers from some internal contradiction should consider that a theory whose content can be completely embraced by a single mathematical formula can no more be internally contradictory than two different conclusions drawn from one and the same formula can contradict one another. Our views must be guided by deductions from formulae, and not the other way around.

Of course, the last word on the question of the admissibility and significance of the theory of relativity belongs to experience, and the most important indication of the fruitfulness of a theory must be regarded as the very possibility of testing it by experience. Up to the present time no contradictions with experience have been established, a fact which I should especially like to emphasize in contrast to certain reports that have lately found their way even to the general public. But even one who, for whatever reasons, considers it possible or probable that contradictions with experience will be discovered can, from the point of view of his own interests, do nothing better than take part in the development of the theory of relativity and in the further development of the consequences following from it. For therein lies the only way to refute it

with the aid of experiment. This work is facilitated by the fact that the propositions of the theory of relativity are distinguished by unambiguity and comparative clarity, and that they lend themselves excellently to inclusion in the system of classical physics.

If considerations of a historical character did not stand in the way of this, then I, for my part, would not hesitate for a minute to count even the theory of relativity among classical physics. For it alone, in a certain sense, crowned this physics, at the same time, with the fusion of space and time, uniting from a more elevated point of view the concepts of mass and energy, and also of gravitation and inertia. The fruit of these new views is that irreproachably symmetrical form which the laws of conservation of energy and momentum have henceforth acquired as equivalent consequences of the principle of least action—this most unifying of all physical laws, reigning equally in mechanics and in electrodynamics.

Alongside this majestic structure of wondrous beauty and harmony, there stands, as an alien and threatening explosive projectile, the hypothesis of quanta, which has already succeeded in driving a gaping breach from top to bottom through the whole building. The quantum hypothesis did not appear in the form of a monolithic, simple thought, closed in upon itself and with a clear content, which, like the theory of relativity, introduced into the physical concepts and relations known until then changes that were in principle extremely significant, but for the most part practically barely noticeable. No, the quantum hypothesis first arose in an entirely special field of physics as the only saving way out of the serious difficulties encountered by the classical theory in attempting to understand the laws of thermal radiation. When it subsequently turned out that this hypothesis, in addition, directly and as if in jest, solves—or at least extraordinarily advances the solution of—a number of completely different problems, for example, the problems of the photoelectric effect, specific heat, ionization, and chemical reactions, which presented a number of definite difficulties for the classical theory, it very soon became clear that it had to be regarded not only as a working hypothesis, but also as a new physical principle of fundamental significance, whose influence manifests itself in all those cases where the matter concerns subtle and rapid phenomena.

It is necessary, however, to reckon with the fact that the quantum hypothesis not only contradicts the views that had existed up to that time—on the basis of what has been set forth above, one could still have reconciled oneself with this comparatively easily—but that it, as becomes ever more evident with the passage of time, directly denies some of the basic premises that are absolutely necessary for the construction of classical

theory. Therefore, the introduction of the quantum hypothesis is equivalent to the collapse of classical theory, and not to a mere modification of it, as in the case of the theory of relativity.

Of course, if the quantum hypothesis in all questions truly surpassed classical theory, or at least were its equal, then nothing would prevent us from sacrificing the whole of classical theory; more than that, such a sacrifice would have to be made. However, in this respect matters are by no means so, for in physics there are domains—especially the broad domain of interference phenomena—in which classical theory has been confirmed in all its details by the most precise measurements, whereas the quantum hypothesis, at least in its simplest form, completely refuses to serve in these domains, and moreover not only in the sense that it is not applicable to them, but also in that it leads to certain results not agreeing with experiment.

Thus it has turned out that at the present time each of the two theories rules, so to speak, in its own proper domain, in which it can feel invulnerable, and that in the intermediate regions—for example, in the phenomena of dispersion and scattering of light—a struggle between rivals is being waged with varying success, in which both theories prove to be approximately equivalent, so that, depending on their personal inclinations, physicists make use now of one and now of the other of these theories. Of course, this situation is exceedingly unpleasant, and in the long run altogether intolerable for anyone who seriously strives to discover the true relation of things.

To illustrate this peculiar situation, I shall, with your permission, select from the exceedingly abundant available material—obtained by both experimental and theoretical investigations—only two quite special questions connected with two simple facts. Let us imagine two thin beams of violet light, obtained with the aid of a point source of light and an opaque screen with two small openings. If, by means of a suitable reflection, the beams of rays passing through the openings are deflected so that they meet on a distant white wall, then the light spot produced on the wall by the two beams will prove not to be uniformly bright, but to be traversed by dark bands. This is one of the facts. The second consists in this: from any photosensitive metal placed in the path of one of these beams of rays, electrons will continuously fly out with a perfectly definite velocity independent of the brightness of the illumination.

If we begin to diminish the intensity of the source of light, then, according to all existing observations, in the first experiment the appearance of the bands will remain completely unchanged; only the brightness of the illumination will decrease.

will correspondingly decrease. In the second experiment, however, the velocity of the emitted electrons remains entirely unchanged; only their emission will occur more rarely.

How, then, are these facts accounted for by theory? The first of them is excellently explained by the classical theory in the following way: at each point of the white wall, illuminated simultaneously by both beams, the light rays meeting in it are either weakened or strengthened, depending on the path difference of the corresponding light waves. The second fact is just as excellently explained by the quantum theory in the following way: the energy of radiation falls upon the light-sensitive metal not in a continuous stream, but in jolts, in the form of more or less numerous identical indivisible quanta, and each incident quantum calls forth one electron from the metal. On the other hand, all attempts to explain interference bands from the theory of quanta, or the photoelectric effect from the classical theory, have so far ended in failure. For if the energy of radiation really propagates in the form of indivisible quanta, then a quantum emitted by a light source can fly either through one or through the other aperture in an opaque screen, and, consequently, with a sufficiently weak intensity of light two different rays can in no way meet simultaneously at one point of the white wall; consequently, the possibility of interference is excluded. Indeed, if one of the rays is completely closed off, the interference bands always disappear completely.

On the other hand, if the energy emitted by a point source of light propagates continuously throughout an ever-increasing volume in all directions, then it must be correspondingly rarefied, and one cannot understand how an electron, under very weak illumination, can acquire the same velocity of emission as under very bright illumination. Various attempts have, of course, been made to eliminate these difficulties. The most natural of them consists, perhaps, in the supposition that the energy of the emitted electron is borrowed not at all from the radiation falling on the metal, but from the interior of the metal, so that the radiation merely releases the energy enclosed in the metal, like a spark in a barrel of gunpowder. However, it has not been possible either to discover this source of energy or even to make its very existence plausible. According to another supposition, the energy of motion of the electron is indeed borrowed from the incident radiation, but the emission of the electron occurs only after the illumination has lasted long enough for all the energy needed to impart a definite velocity to the electron to accumulate. In many cases, however, this would require minutes and hours, whereas in reality the action of radiation is often manifested considerably sooner.

The extraordinarily serious character of these difficulties was expressed with particular vividness in the fact that recently a proposal was even made by the most authoritative persons to sacrifice the postulate of the strict validity of the principle of conservation of energy. This way out of the impasse may with some justice be called desperate; true, before long its inadmissibility was proved by specially designed experiments.

Whereas all attempts to explain, from the standpoint of the classical theory, the laws of electron emission had until now ended in failure, these laws, and many other regularities relating to the interaction of radiation and energy, at once become intelligible and even seem necessary if one assumes that separate, smallest light quanta fly through space independently of one another and, when striking matter, behave in the same way as real substantial atoms.

Since, however, we must take up some definite point of view, the whole problem is, in essence, sharpened into the question whether, on leaving a source of light, the energy emitted by it is split up in such a way that one part of it passes through one, and another part through another, aperture of an opaque screen, or whether this energy penetrates in the form of indivisible quanta alternately through now one, now another of the apertures. This question confronts every quantum theory, and every theory is compelled to take a definite position with respect to it; up to the present, however, no physicist has been able to give a satisfactory answer to it.

The opinion has sometimes been expressed that the difficulties of quantum theory arise, in essence, not in the propagation of radiation in free air space, but only in the interaction of radiation with electrically charged matter. I cannot agree with this opinion. For in the question formulated above, the matter concerns only the propagation of radiation, and it has no relation whatever to the phenomena caused by radiation or generating it.

Can one, however, in such a case speak at all of the energy of free radiation, since all measurements relate, after all, to phenomena in material bodies? If we truly wish to adhere to the strict meaning of the principle of conservation of energy, to which, in particular, the most recent observations lead, then to every field of radiation there must be assigned a perfectly definite, more or less exactly calculable quantity of energy, which decreases upon absorption and increases upon emission. The question concerns only the behavior of this energy. And there is no doubt that, in order to find a way out of this very difficult dilemma, we shall have to resolve upon a certain expansion and generalization of the most fundamental premises from which we have been accustomed to proceed in theoretical physics and which until now

have always justified themselves. This conclusion, at least at first, undoubtedly contains something that does not satisfy our thirst for knowledge. But if the possibility of solving the riddle seems open in some direction, then that alone already has a calming effect; in view of this I cannot refrain from the temptation to touch in a few words on the question of the precise direction in which, perhaps, a way out might be found. The radical means of circumventing all the difficulties would undoubtedly be to renounce the usual assumption that the energy of radiation is localized in one way or another, i.e. that in each spatial part of a given electromagnetic field at a given time there is a definite amount of energy. For if this assumption is abandoned, then the solution of the whole problem consists simply in denying any definite physical meaning to the question whether a light quantum flies through the first or through the second aperture of an opaque screen. Nevertheless, this way out of the dilemma is connected, in my opinion, at least at the present time, with too considerable sacrifices. For the total amount of the energy of radiation has a quite definite value that can be specified; the electromagnetic vector field of the ray, in all its space-time relations, is taken into account by classical electrodynamics down to all optical details in full agreement with reality; finally, energy arises and disappears simultaneously with the field, and therefore one cannot so simply brush aside the question of precisely how the detailed structure of the field determines, in detail, its energy.

If we venture to enter, perhaps more deeply, into the consideration of this question, then, in order to avoid the indicated alternative, it seems natural, while preserving the lawful relation between the ray—or, more precisely, between the electromagnetic wave—on the one hand, and the energy carried by it, on the other, to give this relation a less simple and less close character than is assumed by the classical theory. According to the classical theory, every arbitrarily small part of an electromagnetic wave contains the corresponding amount of energy, proportional to its magnitude, which propagates together with it. If this connection is weakened, i.e. if it is admitted that the energy of the wave is not connected in so direct a manner with its smallest elements, then the possibility is created of splitting the wave emitted by the source of light into an arbitrarily large number of parts, in the sense of the classical theory, while, however, the energy of the wave is still concentrated in definite places in the sense of the quantum theory. The first circumstance makes it possible to explain interference phenomena by the fact that even the weakest wave passes partly through one, partly through another aperture of an opaque screen; the second circumstance makes it possible to explain the photoelectric effect by the fact that the wave gives up its own

energy to electrons only in whole quanta. How, however, is it possible to conceive of a part of a light wave without the corresponding amount of energy? Undoubtedly, this is sufficiently difficult, but in my opinion it is, in essence, no more difficult than conceiving of a part of a body without the corresponding density of its mass. We are, however, as is known, compelled to accept the latter assumption in view of the fact that, upon successive volumetric subdivision, matter loses its simple properties; its mass ceases to be proportional to the volume it occupies and becomes concentrated in a multitude of separate molecules of definite size. Something quite similar may also take place with respect to electromagnetic energy and the momentum corresponding to it.

Until now the elementary laws of electromagnetic phenomena have usually been sought exclusively in the domain of the infinitely small. All electromagnetic fields were divided in space and in time into infinitely small parts, and all their regular properties and processes were expressed by means of space-time differential equations. In this respect we shall evidently have to retrain ourselves anew. For it has become clear that simple regularities have a limit at a certain degree of subdivision, and that in still finer or smaller phenomena there arise certain complications which compel us to atomize the space-time quantity of action, i.e. compel us to accept the existence of an elementary quantity of action, or atom of action. And indeed, it is highly remarkable and significant that not one of the laws in which the universal quantum of action plays a role is expressed by a differential equation, and that they all relate to a finite spatial extent and to a finite time, such as: to a definite period of oscillation, to a complete revolution, to a finite jump, and so on. In order properly to take account of this circumstance, we shall apparently have, at least in part, to replace relations between infinitely close quantities by relations between finitely separated quantities. In that case the differential is replaced by the difference, continuity by discontinuity, analysis by arithmetic.

A promising beginning in this direction is the foundation of so-called quantum mechanics, which has recently already led to considerable successes in the hands of the Göttingen physicists Heisenberg, Born, and Jordan. But only further development will show how far we can approach the solution of our problem along the path opened by quantum mechanics. For even the finest mathematical speculations hang in the air until definite experimental facts provide them with a firm foundation, and we must believe and hope that the art of the experimental physicist, which has already brought an indisputable solution to so many tangled questions, will dispel the darkness in this difficult case as well.

Then there will remain no doubt that the part of the edifice of classical physics blown up by the assault of the quantum hypothesis will be discarded as useless rubbish and replaced by a more suitable and more durable structure.

We have seen that physics, which only a generation ago was counted among the oldest and most mature of the sciences of nature, has now entered a period of storms and onslaughts, promising to become the most interesting of all that have ever existed. Overcoming this storm and onslaught will lead us not only to the further discovery of new phenomena of nature, but undoubtedly also to an entirely new penetration into the mysteries of the theory of knowledge. In this latter domain, perhaps, many surprises await us, and it may happen that some of the former views, now consigned to oblivion, will thereby be revived and acquire new significance. Therefore, the attentive study of the views and ideas of our great philosophers may in this respect prove very useful.

There were times when philosophy and the natural sciences stood opposed to one another, estranged and hostile. Those times have long passed. Philosophers have understood that one must not prescribe to the natural scientist the methods and aims of his work, and natural scientists have made clear to themselves that the starting point of their investigations lies not only in the perceptions of the sense organs, and that the natural sciences cannot do without philosophy. It is precisely the newest physics that, with complete definiteness, brings us to the recognition of the old truth: there exist realities independent of the perceptions of the sense organs, and there exist problems and conflicts in which these realities are of greater value to us than the most precious treasures of our entire sensory world.

1. Report at the academic courses in Düsseldorf on February 14, 1926. Die Naturwissenschaften 14, 249, 1926. In the Russian translation the first two paragraphs, which are not related to the main topic, have been omitted. Translated by Ig. Tamm. ↩