From Plato to Max Planck: The Philosophical Problems of Atomic Physics

Professor WERNER HEISENBERG, who was born in Duisburg in 1901, was awarded the Nobel Prize in Physics in 1932 for his work on the quantum theory. A member of the faculty of the University of Göttingen since 1945, he is director there of the Max Planck Research Institute. His recent formulation of what is called the “unified field theorem" has caused a great stir in academic and scientific circles.


SURROUNDED as we are by hundreds of projects aimed at putting the discoveries of atomic physics to fruitful economic use, all too easily we forget that we are simultaneously struggling with questions which men have been asking themselves for a long, long time. The present-day preoccupations of the human spirit are, in fact, closely related to intellectual efforts first undertaken by men thousands of years ago. These links with the past are doubtless more interesting to the historian than to the physicist, but even the physicist can gain valuable insights into his own current problems by considering certain of the fundamental patterns that the history of the atom reveals.

Modern physics, and particularly the quantum theory — the discovery of Max Planck, the centennial of whose birth was celebrated last year —has raised a series of very general questions, dealing not only with the narrow problems of physics as such but also with the nature of matter and with the method of the exact sciences. These questions have forced physicists to wrestle once more with philosophical problems which seemed already to have found a definitive answer within the framework of classical physics.

Two questions, in particular, have been reopened by Planck’s discoveries. The first concerns the nature of matter, or, to put it more precisely, the old question that troubled the Greek philosophers: How can the manifold diversity of material phenomena be reduced to simple principles and thus rendered intelligible? The other has to do with the epistemological question which has cropped up with particular insistence since Kant’s time: the question of the extent to which it is possible to objectify scientific or any sensory experiences —the extent, in other words, to which one can go from observed phenomena to an objective conclusion independent of the observer.

Let us consider the first of these questions. When the Greek philosophers came to examine the unity of observable phenomena, they encountered the problem of the smallest units of matter. Two opposing points of view emerged from this period of human thought which were to have a profound influence on the subsequent evolution of philosophy. They have been given the names of materialism and idealism.

The atomic theory propounded by Leucippus and Democritus regarded the smallest particles of matter as existing in the most literal sense. These tiniest particles were thought to be indivisible and unchangeable; they were eternal, ultimate units, and as such they were called atoms, and could neither provide nor require further explanation. The only properties that fitted them were geometric. They were thought by the philosophers to possess form, to be separate from each other in empty space, and by virtue of their different spatial positions and movements to be capable of bringing about the manifold variety of phenomena. But they did not have color, smell, or taste, nor did they have a temperature or any other physical properties. The qualities of the things we apprehend in life were indirectly governed by the different rearrangements and movements of the atoms. Just as tragedy or comedy can be written with the same letters of the alphabet, so, according to Democritus’ theory, can very different happenings in the world be produced by the same atoms. Atoms were therefore the true, objectively real kernels of matter, and thus of phenomena. They were what existed in the truest sense, whereas the diverse multiplicity of phenomena was merely an indirect and subsidiary product of them. This view of things was therefore given the name of materialism.

In Plato’s philosophy, on the other hand, the smallest particles of matter were, in a certain sense, mere geometrical forms. Plato equated the smallest particles of the elements with regular geometric bodies. Like Empedocles he accepted the four traditional elements — earth, water, air, fire — and thus he could imagine the smallest particle of earth as being a cube, the smallest particle of water as being an icosahedron (twentysided solid); in the same way, the elemental particle of fire was thought to be a tetrahedron, and of air, an octahedron. In each case the shape was characteristic of the element’s properties.

For Plato, in contrast to Democritus, the smallest particles were not unchangeable or indestructible; they could be reduced to triangles, and out of triangles built up again. The triangles themselves would no longer be matter, for they would have no spatial extension. Thus, at the end of this series of material concepts we encounter something which is no longer material but simply a mathematical form — or, if you prefer, an intellectual construction. The ultimate root concept capable of rendering the world intelligible was for Plato the mathematical pattern, the picture, the idea. This view of things was therefore called idealism.

THE old controversy between materialism and idealism has, oddly enough, been revived recently in a quite specific way by modern atomic physics, and especially by the quantum theory. The exact sciences — physics and chemistry — of modern times were, until the discovery of the Planck quantum, materialistically oriented. In nineteenth-century chemistry, the atoms and their composite particles were thought to be the only real things, the real substratum of all matter. Atoms appeared incapable of furnishing or of demanding further explanation.

Planck, however, discovered in the phenomena of radiation a feature of discontinuity, which seemed surprisingly linked to the existence of the atom but which could not be explained in terms of it. This aspect of discontinuity revealed by the quantum gave birth to the suspicion that not only this discontinuity but the very existence of the atom might stem from some fundamental law of nature. There might be a mathematical structure in nature the formulation of which would provide that unified understanding of the structure of matter which the Greek philosophers had looked for. The existence of the atom, far from being a final, irreducible fact, might, as Plato had thought, be traced back to the operation of mathematically conceivable laws of nature — to the effect of mathematical symmetries.

Planck’s law of radiation also differed in a quite characteristic way from the previously formulated laws of nature. Planck’s action quantum, which appears as the characteristic constant in his law of radiation, does not designate a property of things but a property of nature. It sets up a scale of measurement and shows that where the encountered effects are very big, as they are for all the phenomena of daily life, the phenomena of nature behave differently from the way they do where they are of atomic size — the realm of the Planck constant. Whereas the laws of Newtonian mechanics were equally valid for all realms of magnitude — the orbit of the moon around the earth obeying the same laws as the fall of an apple or the deviation of an alpha particle passing by the nucleus of an atom — Planck’s radiation law for the first time postulated the existence of different scales in nature and suggested that events in the different realms of magnitude need not be similar at all.

Only a few years after Planck’s discovery, the significance of a second similar constant was understood. Einstein’s special theory of relativity made it clear to physicists that the speed of light was not, as the science of electrodynamics had supposed, a property of a special stuff known as ether, which ensured its propagation, but that it was a property of space and time — a universal property of nature having nothing to do with individual objects or things in nature.

The speed of light can thus be considered a constant of nature. Our perceptual concepts of space and time can only be applied to phenomena whose speeds are small compared to that of light. Conversely, the well-known paradoxes of the theory of relativity are due to the fact that phenomena with speeds approaching that of light cannot properly be understood with our usual notions of space and time. One example is the paradox of the watch: for an observer moving at high speed, time seems to pass more slowly than for one who is at rest.

Once the mathematical structure of the special theory of relativity had been established, there was a searching analysis of the natural phenomena affected by it. Our deep-rooted, inherited notions placed many an obstacle in the way of our understanding, but the objections were quickly overruled.

It proved far harder to understand the physical implications raised by the existence of the Planck quantum. In a work of Einstein’s which appeared in 1918, it was made clear that the theoretical quantum laws must to a certain extent be dealt with statistically. The first attempt to establish the theoretical quantum laws on the basis of statistical findings was undertaken in 1924 by Bohr, Kramers, and Slater. Their interpretation turned out to be not completely correct, and the proper relationships were later accurately formulated by Born. But the theory propounded by Bohr, Kramers, and Slater contained the decisive notion that the laws of nature define not the occurrence but the probability of the occurrence of an event and that the concept of probability must be introduced into the study of wave fields in such a way as to permit the formulation of mathematical force-field equations.

At this point a decisive step was taken, away from classical physics and back to a form of concept which had once played an important role in Aristotle’s philosophy. One can interpret the probability waves of the Bohr-Kramers-Slater theory as a quantitative rehabilitation of the concept of dynamis, of potentiality, or of the later Latin concept of potentia in the Aristotelian philosophy. A decisive role is played in this philosophy by the idea that events are not necessarily determined but that the possibility of or tendency toward an occurrence constitutes a kind of reality — an intermediate layer of reality situated halfway between the bulky reality of matter and the spiritual reality of the idea or picture. This idea has won a new place for itself in modern quantum physics, and the possibility concept has been used in terms of quantitative probability for the mathematical formulation of natural laws.

THE introduction of probability into physics corresponded quite accurately to the situation which had meanwhile been reached in laboratory experiments on atomic phenomena. When a physicist seeks to determine the strength of a radioactive radiation by calculating the number of times this radiation affects a counter during a given time interval, he takes the probability of this radiation for granted. (Although the time of decay of any individual atom is uncertain, the physicist is confident that among a large collection of such atoms a certain number are sure to decay during each second.) The exact time intervals between impulses do not interest the physicist — he merely says that they are “statistically” dispersed; what interests him is the average frequency of the impulse. It has been established through numerous investigations that this statistical interpretation gives an accurate picture of the experimental situation.

Where it afforded quantitative evidence, as with the wave lengths of spectrum lines or with the binding energy of molecules, quantum mechanics provided a precise confirmation through experiments, and no one could cast further doubt on the validity of the theory. More troublesome, however, was the question of harmonizing this statistical method of explanation with the large body of experience amassed through classical physics. All experiments rest on the assumption that there is a meaningful connection between the act of observation and the basic physical event. When, for example, we measure the spectrum line of a certain frequency with a refraction grid, we automatically assume that the atom of radiating matter must have emitted light of this same frequency. Or, when a photographic plate shows dark patches, we assume that it has here been hit by rays or particles of matter. Physics uses the meaningful determination of events for the collection of experimental evidence, and thus obviously falls into a contradiction when dealing with atoms or the quantum theory, where the precise determination of such events appears to be called into question.

The inner contradiction which here appears is avoided in modern physics by asserting that the determinacy of events extends only so far as the events can be described with the concepts of classical physics. The application of these concepts, on the other hand, is limited by the relations of indeterminacy, which provide quantitative data over and beyond the limits imposed on the use of classical concepts. The physicist knows when he should consider events as determined and when not; he can thus use for observation and its physical interpretation a method which is free of internal contradictions. The questions remain: Why should it still be necessary to go on using the notions of classical physics? Why can we not rewrite the entire description of physical events in a new quantum-theory system of concepts?

It should be noted here, as von Weizsacker has emphasized, that the concepts of classical physics play a role in the interpretation of the quantum theory similar to that played by the a priori forms of intuition in Kant’s philosophy. Just as Kant establishes the concepts of time and space and causality as being a priori, since they provide the grounds for experience and thus cannot be regarded as derivative of it, so the notions of classical physics provide an a priori foundation for the investigations of quantum physics, since we can carry out experiments in the atomic field only with the aid of concepts from classical physics.

While Kant could still assume that our a priori intuitions of time and space must forever furnish the unchangeable basis for physics, we now know that this is in no wise the case. Our forms of intuition, though they are a priori, are not suited to the apprehension of events whose speed approaches that of light, events which can only be captured with the most subtle technical instruments. There are large areas of experience which cannot be even approximately described with the concepts of classical physics.

In these areas of atomic physics, a great deal of the earlier intuitive physics has gone by the board — not only the applicability of its concepts and laws but the entire notion of reality which underlay the exact sciences until our present-day atomic physics. By “notion of reality” I mean the idea that there are objective occurrences which somehow take place in time and space quite independently of whether or not they are observed. Observations in atomic physics can no longer be objectified in this simple manner; they can no longer be related to an objective, describable interval in time and space.

Here again we are brought up sharply before the rock-bottom truth that in science we are not dealing with nature itself but with the science of nature — that is, with a nature which has been thought through and described by man. This is not to introduce an element of subjectivity into science, for it is in no way asserted that events in the world of nature depend on our observation of them; it is simply to say that science stands between man and nature and that we cannot renounce the application of concepts that have been intuitively given to or are inborn in man.

This aspect of the quantum theory has made it difficult to adhere absolutely to the program of materialistic philosophy and to designate the smallest particles of matter as being “truly real.” For if the quantum theory is correct, these elemental particles are not real in the same sense as the things in our daily lives — for example, trees or stones — are real; they appear as abstractions derived from observed material which in a literal sense is real. Now, if it is impossible to ascribe existence in the strictest sense to these elemental particles, it is difficult to regard matter as truly real. This explains why, in recent years, reservations have been expressed in the camp of dialectical materialism about the now generally accepted implications of the quantum theory.

We cannot escape the conclusions that our earlier notions of reality are no longer applicable in the field of the atom and that we are dealing with weighty abstractions when we set out to define the atom as what is truly real. Modern physics, in the final analysis, has already discredited the concept of the truly real, so that it is at the very starting point that the materialistic philosophy must be modified.

THE development of atomic physics in the last two decades has led it even further away from the basic assumptions of the materialistic philosophy of the ancients. Experiments have shown that the elemental particles, which we assume to be the smallest units of nature, are not eternal and immutable, as Democritus supposed, but can be transmuted into one another. Now, there must be some reason for our considering these subatomic particles as being the smallest particles of matter. It could, nonetheless, be true that the elemental particles are made up of still smaller ones, themselves eternal and unchangeable. How can the physicist exclude the possibility that these still smaller units have so far eluded our powers of observation?

The answer which present-day physics gives to this question brings out clearly the nonintuitive character of modern atomic physics. If someone wants to decide experimentally whether an elemental particle is simple or compound, he clearly must set out to shatter it with the most powerful means available. The huge atomic accelerators which are today being used, or being built, in various parts of the world serve precisely this purpose. With such machines one can accelerate elemental particles — usually protons — to extremely high velocities, have them intercepted by other tiny particles of matter, and then make a detailed study of what happens in the subsequent collisions. Although a great deal of experimental material on the details of such collisions still must be collected before we can have a truly clear picture of this area of physics, we are already in a position to describe roughly what takes place. What has been discovered is that a shattering of the elemental particles can certainly take place. Sometimes a great many particles are produced by such collisions, but the surprising and supreme paradox is this: the particles resulting from these collisions are not smaller than the shattered particles they spring from. They are themselves elemental particles all over again.

The paradox is explained by the theory of relativity, according to which energy can be converted into mass. The elemental particles, which are given a huge kinetic energy with the aid of accelerators, can with the use of this energy, which is convertible into matter, produce new elemental particles. The elemental particles are therefore the ultimate units of matter — precisely those units into which matter decomposes under the impact of external forces.

This state of affairs can be summed up thus: All elemental particles are made of the same stuff—namely, energy. They are the various forms which energy must assume in order to become matter. Here once again we encounter that old pair of concepts — content and form or matter and form — which has come down to us from Aristotle’s philosophy. Energy is not only the force which keeps everything in motion; it is also, like fire in Heraclitus’ philosophy, the basic stuff of which the world is made. Matter exists because energy assumes the form of the elemental particle. We now know that there are no less than twenty-five different kinds of subatomic particles, and we have good grounds for supposing that all these forms are patterned on fundamental mathematical constructions. They are, therefore, the consequence of a fundamental law that can be expressed in mathematical language and from which the subatomic particles can logically be derived in much the same way as the different energy states of the hydrogen atom could be deduced from Schrödinger’s differential equation.

This fundamental law, which present-day physics is seeking, must fulfill two conditions, both of them stemming from experimental evidence. The investigations that have been made on elemental particles with the aid of huge accelerators have produced certain rules of selection for the transformations occurring from collisions or in radioactive fall-out. These rules of selection can be formulated mathematically on the basis of collected quantum figures, and they are the immediate expression of properties of symmetry bound up with the fundamental equation of matter. The fundamental law must in one way or another embrace these observed symmetries; it must represent them mathematically.

In the second place, the fundamental equation of matter, assuming that such a simple formulation exists, must contain at least another constant of measurement similar to the one for the velocity of light and to the Planck quantum. Observations of the atomic nucleus and of the atomic particles lead us to believe that this constant can be represented as a universal length, whose magnitude should be around ten centimeters.

The real conceptual core of this fundamental law, however, must be based on the mathematical symmetries exhibited in it. The most important of these symmetries are already empirically known to us (the Lorenz group, describing the properties of time and space needed by the special theory of relativity; the quantum number, derived from twenty years of observation, that allows us to differentiate between neutrons and protons and which is now generally known as isotopic spin; mirror symmetries of time and space; and so forth).

One proposal for such a fundamental equation of matter which satisfies the above requirements has already been made. This is a most simple and highly symmetrical equation of nonlinear waves for a field operator known as a spinor. Whether or not this is the correct formulation of the fundamental law of nature will be known only after searching mathematical analysis.

One can already say that, irrespective of the exact nature of this evidence, the final answer will more closely approximate the views expressed in Plato’s Timaeus than those of the ancient materialists. This realization should not be misunderstood as an all-too-facile rejection of the modern materialistic thinking of the nineteenth century, which contributed many important elements of knowledge that were lacking in ancient science. But it is true that the elemental particles of present-day physics are more closely related to the Platonic bodies than they are to the atoms of Democritus.

The elemental particles of modern physics, like the regular bodies of Plato’s philosophy, are defined by the requirements of mathematical symmetry. They are not eternal and unchanging, and they can hardly, therefore, strictly be termed real. Rather, they are simple expressions of fundamental mathematical constructions which one comes upon in striving to break down matter ever further, and which provide the content for the underlying laws of nature. In the beginning, therefore, for modern science, was the form, the mathematical pattern, not the material thing. And since the mathematical pattern is, in the final analysis, an intellectual concept, one can say in the words of Faust, “Am Anfang war der Sinn“ — “In the beginning was the meaning.”

The task of present-day atomic physics is to explore this meaning in all its details and with all the complex apparatus at our command. It seems to me fascinating to reflect that today, in the most varied parts of the earth and with the most powerful instruments available, men are seeking to wrest solutions from problems posed by the Greek philosophers more than two thousand years ago; and it is exhilarating to think that we may know the answer in a few years, perhaps — or, at the latest, in a couple of decades.