Physical Reality*

Max Born

The concept of reality in the physical world has, over the course of the last century, become somewhat problematic. The contradiction between the simple and obvious reality of countless instruments, machines, engines, and apparatuses of every kind, created by industry and underlying applied physics, and the unclear and abstract reality of the fundamental physical concepts, such as forces and fields, particles and quanta, is undoubtedly confused. It already exists between pure and applied natural science, between whose representatives a gulf has formed that may lead to dangerous estrangement. In order to overcome this rupture of “reality,” as it is represented in theory and in practice, physics needs a generalizing philosophy expressed in everyday language. I am not a philosopher, but a theoretical physicist. I cannot offer any well-thought-out philosophy that would properly take into account ideas of different tendencies; I would like only to set forth some thoughts that have helped me in my own explanations of these problems.

Among theoretical physicists and natural philosophers there exists a trend of thought that represents a radical, abstract point of view. This philosophy is expounded, for example, by Dingle** in his interesting lecture before the British Association in Edinburgh. I cannot make my own point of view clearer except by contrasting opposites. If I quote excerpts from Dingle’s report, I do so by no means for the sake of personal polemics. These quotations should serve only as a suitable example for developing my own views, which differ from his.

We shall begin with the following proposition: “The quantities with which physics is concerned are not numerical estimates of objective properties of parts of the external material world; they are only the results that we obtain when we perform certain operations.”

This definition looks like a denial of the existence of an objective (pre-existing) material world; it creates the impression that physics has nothing to do with the real world and that it performs experiments only for the purpose of predicting the results of one or another experiment. In general it is not explained why a physicist should take upon himself the labor of performing experiments. This question is apparently regarded as unworthy of philosophy as a science. Can we avoid the question of what role, in this system of things, is played by instruments of steel, brass, glass, and so forth, which are carefully combined and adapted for experiments? Are they not, moreover, part of an independent

*) Physical Reality, see: Max Born, Physics in my Generation, Pergamon Press, London—New York, 1956, pp. 151–168. Translated from the English by S. G. Suvorov.

**) H. Dingle, Nature 168, 630 (1951); see also Phys. Bl. 7, 481–586 (1951), where this author speaks on the topic “The New Attitude in Physics” (Der neue Standort in der Physik).

are they, like electrons, atoms, and fields, purely abstract ideas used in order to predict phenomena that can be observed in a subsequent experiment, which in its turn again represents only an assemblage of phantoms?

Before us is a view typical of extreme subjectivism, to which one might give the appropriate name “physical solipsism.” It is well known that a stubbornly defended solipsism cannot be refuted by logical arguments. However, with equal justification one may say that such solipsism does not solve the problem, but evades it. Logical consistency is a purely negative criterion; without it no system can be accepted, but no system is acceptable merely because it is logically non-contradictory. The only positive argument in favor of this abstract kind of ultra-subjectivism is a historical one. It is asserted that belief in the existence of an external world is devoid of significance and is an outright hindrance to the progress of science. Everything with which the physicist is concerned can be comprehended satisfactorily only in terms of “experience,” and not of the external world.

In reality the matter is quite otherwise. All the great discoveries in experimental physics are due to the intuition of people who frankly used models which, for them, were not products of their fantasy but representatives of real things. How could an experimenter work, and how could he communicate with his collaborators and contemporaries, if he did not use models composed of particles (electrons, photons, nucleons, neutrons), fields and waves—concepts that were condemned as inessential and useless?

There is, however, a certain reasonable basis for this extreme point of view as well. We have learned that in applying these concepts a certain caution must be exercised. The naive approach to the problem of reality, which was so successful in the classical, or Newtonian, period, has proved unsatisfactory. Modern theories require a new formulation. This new formulation is developing slowly and has probably not yet reached its final expression. I shall try to point out the present tendencies.

In doing so, one must bear in mind from the very beginning that the word “reality” is part of our colloquial language and therefore—like most words—has no unambiguous meaning. There are subjective philosophies which teach that only the spiritual world is real and that the physical world is only an appearance, a shadow without substance. Although this point of view is of the greatest philosophical interest, it lies outside our discussion, which is concerned only with physical reality. And yet many other questions remain open. The “realities” of a peasant or an artisan, a merchant or a banker, a statesman or a soldier, obviously have little in common. For each of them the most real things are those that stand at the center of his spiritual activity, and the word “real” is used almost as a synonym for the word “important.” It is interesting to know whether any philosophy can define the concept of reality in such a way that it is not subject to the subsequent influence of subjective associations of this sort. As for us, we ask: can natural science provide such a definition?

This brings us to another point raised by Dingler, namely: can natural science, without harm to itself, dispense with the concepts and the word “reality”? My answer to this question is that,

that only those people can renounce this concept who live in isolated castles in the air, far from all experience and from all actual aims and observations; consequently, that type of person who has gone so deeply into pure mathematics, metaphysics, or logic that he has completely withdrawn from the world. Niels Bohr, who contributed more to the philosophy of modern natural science than anyone else, repeatedly and distinctly explained that real experiments could not be described without using the language of naïve realism and its concepts. Without acknowledging this, no agreement about facts is conceivable even among the most exalted minds. An essential part of this procedure consists in distinguishing between ideas, theories, and formulas, on the one hand, and real instruments and devices, which have been created in accordance with these ideas, on the other. At the same time, in actual fact, the naïve use of the word “real” is absolutely necessary: the simple belief in the real existence of material apparatuses. I believe that the abstract school represented by Dingle does not deny this, although he does not say so clearly. However, he forbids the application of the concept of reality to atoms, electrons, fields, etc.; consequently, to terms that are used in explaining observations. But where is the boundary between these two regions? A piece of crystal, belonging to the realm of macroscopic reality, can be ground into powder until its particles become too small to be perceived by the naked eye. To see them, one must use a microscope. Do the particles now become less real? Still smaller particles, colloids, under appropriate illumination in the ultramicroscope, will appear as shining structureless points. Between these particles and individual molecules or atoms there is a continuous transition. When the ultramicroscope proves insufficient, one can take an electron microscope, with the aid of which even large molecules can be seen. So where does macroscopic reality, in which the experimenter lives, end, and where does the world of atoms begin, from which the idea of reality is to be banished as illusory?

Such a boundary, of course, does not exist; if we are compelled to attribute reality to the ordinary things of everyday life—including the instruments and materials used in experiments—then we cannot deny reality to those objects which we observe only with the aid of instruments. However, the fact that we designate them as real, as part of the external world, still in no way obliges us to adopt any particular description: a thing may be real and at the same time may still differ very greatly from other things known to us.

I should like to discuss some examples that Dingle cites in order to justify the rejection of the concept of objective reality in physics.

The first example will be the kinetic theory of matter. Dingle speaks of the statistical method, in which one does not concern oneself with the motion of individual molecules, but confines oneself to the calculation of mean values necessary for depicting “observations (that is, phenomena)”; he describes this position as a “betrayal of the true mission of physics, as recognized philosophy has regarded it. They (physicists) devoted themselves to the investigation of reality, which became the investigation of the nature and behavior of molecules. Instead of following this, they occupied themselves with showing how one can use their ignorance of reality to describe pure phenomena.” I could not understand for myself whether Dingle considers the whole of kinetic theory superfluous, or whether he deprives molecules of reality by calling them “counters” (counting units) or “dummies” (tokens), for he makes no attempt to analyze the factual material which

uses kinetic theory for proofs of the existence of molecules. I shall permit myself to outline such an analysis in a few words.

The kinetic derivation of Boyle’s law established only the possibility of an atomistic explanation, but it can hardly be regarded as conclusive proof. However, formulated more precisely, this derivation leads to a definite value of the mean energy and hence of the specific heat (${}^{3}/_{2} R$ for a monatomic gas, where $R$ is the gas constant), which could not have been obtained by any phenomenological reasoning. The general formula for the mean energy contains the number of degrees of freedom of the molecule—or “dummies,” to use Dingle’s expression. The kinetic explanation of deviations from Boyle’s law leads to an estimate of the size of molecules, which is confirmed by an entirely different group of phenomena, namely the irreversible processes of thermal conductivity, viscosity, and diffusion. Many concepts that are at first introduced through theory—such as, for example, the distribution of velocities, the mean free path, and so on—are confirmed and determined by direct measurements. The fluctuations predicted by kinetic theory are observed in many ways, for example in the phenomena of Brownian motion, the blue color of the sky, and so forth.

Of course, all this, as Dingle says, consists of phenomena, “appearances,” while the molecules remain in the background. But a point strikes one which Dingle does not mention, namely that kinetic theory leads to definite properties of molecules—to weight, size, form (degrees of freedom), and interaction. A small number of molecular constants determines, on the basis of the molecular hypothesis, an unlimited number of phenomenological properties. Therefore every new property that is predicted is a confirmation of the molecular hypothesis. Among these predictions are such remarkable examples as Laue diagrams of X-rays on crystals and the whole field of radioactive phenomena. Here the proof of the reality of molecules has indeed convincingly broken through, and to speak of some “dummy” that leaves a track in a Wilson chamber or in a photographic emulsion seems to me, to put it mildly, inappropriate.

Let us compare this kind of reality with the following case: you see a gun being fired and a man standing a hundred meters away from it falling. How do you know that the bullet extracted from the wound really entered the body from the gun? No one saw it, and no one could have seen it, except a scientist after thorough preparation, i.e. after constructing a complex optical apparatus of the kind invented by Mach for photographing flying projectiles. Nevertheless, I am convinced that you believe that, in the brief interval of time between the firing of the gun and the wounding of the man, the bullet described a definite trajectory—and you also believe that during this interval it really was there. Or would you allow yourself to be satisfied with only the pure statement: “Oh, I do not know that; it is enough to know the phenomena of the shot and the wound. Everything that lies between them is the play of theoretical fantasy; the flying bullet is a pure ‘dummy,’ invented in order to connect the two phenomena by the laws of mechanics.” With logical arguments I cannot refute such a position. I only want to point out that one who denies the existing evidential force of the atomic trace, although it is observable, must also deny the existence of flying bullets, which are not observable, and many similar things as well.

The root of this strange denial of the reality of molecules and other similar things lies in the interpretation of the concept of “reality” as that which is “known in all details.” But this does not agree with ordinary usage—

by words. We imagine all 500 million Chinese as real people, although, perhaps, we know no separate individuals, or know only a few, and at the same time know nothing of their whereabouts, their activities, movements, reactions. We imagine the Romans of Caesar’s time or the Chinese of Confucius’ time as real, although we have no possibility of verifying this representation in the same way as Dingle requires for molecules. Are these Romans or Chinese of our time or of the past merely “dummies” invented by historians in order to connect phenomena? And what phenomena? Perhaps words that they found in newspapers, books, or on old gravestones?

But all these considerations rather remain on the surface and do not touch the fundamental difficulties that the physicist encounters and that compel us to revise our basic concepts. Dingle’s next example—the theory of relativity—brings us somewhat closer to this problem. He asserts that, “in accordance with the philosophy of our time, the real material world—regardless of whether it is considered to consist of molecules or of large bodies—has been represented in such a way that it possesses its properties by virtue of being innate; thus, its constituent elements have size, mass, velocity, etc.” Developing these thoughts, he continues: “and the chief requirement of the theory of relativity was that all these properties should become almost completely indefinite,” and as an example of this he cites the concepts of length and mass, which according to the theory of relativity depend on the velocity of the observer. One and the same lengths, measured by different observers in relative motion, may fluctuate between a maximum and zero; one and the same mass—between a minimum and infinity. He arrives at the conclusion that, “rejecting all attempts to ascribe to matter any property whatsoever in general, we increasingly learn about the dependence of phenomena.” But this is, after all, an incorrect account of the theory of relativity, which never renounced attempts to specify the properties of matter, but only refined the methods applied for this purpose, in order to adapt them to known new experiments, for example, to the famous Michelson–Morley experiment.

Indeed, this example is very apt for getting to the very essence of the problem. This essence lies in a completely simple logical distinction, which should be clear to everyone who is not prejudiced by solipsistic metaphysics: namely, that often the quantity measured is not the object itself, but a property of its relation to other objects.

Let us give an example. Cut out of a piece of cardboard a figure, say a circle, and observe the shadows that it casts from a distant lamp onto a flat wall. The shadows of the circle will in the general case turn out to be ellipses; by rotating your cardboard figure, you can obtain any value of the length of the axis of the elliptical shadows between nearly zero and a maximum. This is an exact analogy with the behavior of length in the theory of relativity, which in various states of motion may have any value between zero and a maximum. If you wish to obtain an analogy for the behavior of mass, which, according to velocity, may have any value between a minimum and infinity, then take a long sausage and cut pieces from it at different angles of inclination; you will thereby obtain ellipses whose axes will fluctuate between a minimum and “practically” infinity. But let us return again to the shadows of the circle; it is obvious that the simultaneous consideration of shadows on many different planes is sufficient to prove the fact that the original cardboard figure is a circle, and to determine its radius unambiguously.

This radius is what mathematicians call an invariant of the transformations produced by parallel projections. In the same way, there exists an invariant for all cross-sections of the sausage, namely, the section of least area. Most measurements in physics refer not to the things that interest us, but to projections of a certain kind; moreover, this word is used in the broadest sense. One may also use the expressions coordinates or components.

A projection (the shadow in our example) is defined relative to a frame of reference (the walls on which the shadow may be cast). In the general case there exist many equivalent frames of reference. In every physical theory a rule is given that relates to one another the projections of one and the same object onto different frames of reference. This rule is called a transformation law; all these transformations have the property that they form a group, i.e., the result of two successive transformations is a transformation of the same kind. Invariants are quantities that have one and the same value for any frame of reference and therefore are independent of transformations.

And the chief progress in the structure of concepts in physics consists in the discovery that a certain quantity which had been regarded as a property of an object is in reality only a property of a projection.

An example of this is the development of the theory of gravitation. The primitive, or pre-Newtonian, conception of the force of gravity, expressed in modern mathematical language, is associated with a group of transformations for which the vertical—meaning the normal to the plane surface of the Earth—is absolutely fixed. For this transformation the magnitude and direction of the force of gravity are an invariant; this means that weight is an innate property of a body, which it carries within itself. The situation changed completely when Newton discovered that the force of gravity is only a special case of universal gravitation. And so the group of transformations expanded in such a way that space became isotropic, having no fixed direction; the force of gravity now became no more than a component of the gravitational force.

The theory of relativity continued this development. The transformations of classical mechanics—they are often also called Galilean transformations—proceed from the assumption that space and time are not connected with one another. But experiment, as reflected in the theory of relativity, showed that this does not correspond to the facts. It is necessary to apply a more general group—the Lorentz transformations—in order to introduce a close connection between spatial coordinates and time. Naturally, quantities that in the old theories were regarded as invariants, for example the distance in rigid systems, time intervals marked by clocks located in different places, and the masses of bodies, are now found to be projections, components of invariant quantities that are not directly accessible. However, as in the case of shadows, by determining a certain number of these components one can find the invariants. Thus it turns out that maximum length and minimum mass are relativistic invariants. It may be that it would have been preferable to call these invariants, which are properties of bodies, by the old names length, time, mass, and to invent new names for the projections. But natural science in such matters is, to the point of strangeness, conservative, and everyone has agreed to rename the invariants as rest length, proper time, rest mass, etc., while retaining the old expressions for the components, although they are not properties of the body but its relations to the frame of reference.

I am convinced that the idea of invariants is the key to a rational conception of reality, not only in physics but in every aspect of the world.

The theory of transformation groups and their invariants is a fundamental branch of mathematics. The great mathematician Felix Klein, as early as 1872, in his famous “Erlangen Program,” discussed the classification of geometry in accordance with this point of view; the theory of relativity may be regarded as an extension of this program to the four-dimensional geometry of space—time. From this point of view, the question of reality in relation to macroscopic matter receives a clear and simple answer.

The situation is more difficult in atomic physics. It is well known that the laws of quantum mechanics lead to indeterminacy, expressed by Heisenberg’s uncertainty relation. Is not this indeterminacy, this impossibility of answering definite questions about the position and velocity of a particle, an argument against the reality of particles and, in general, against the entire real, objective world? Here we must clarify what we understand by a particle—for example, a photon, an electron, a meson, or a nucleon—in relation to experimental indications; and we again find that these words denote definite invariants which can be unambiguously constructed by combining a certain number of observations.

However, the theory of transformations underlying this is rather complicated, and here I can give only a brief indication in general outline. The essence of the matter can be made clear with the aid of ordinary light.

The wave character of light was proved by Young and Fresnel with the aid of the fact that two light rays, formed by splitting one ray, produce interference rings when they meet. Almost a hundred years later Einstein explained the photoelectric effect as the action of light quanta, or photons, which knock out electrons when they strike the surface of a metal. Thus, light also has a corpuscular character—a fact that has been confirmed by numerous experiments. What is remarkable here is that between these two, apparently contradictory, concepts there exists a simple quantitative relation, which Planck had already derived five years earlier from the laws of thermal radiation, namely: \(E=h\nu\), where \(E\) is the energy of a photon, \(\nu\) is the frequency of the wave, and \(h\) is a constant. The difficulty for understanding follows from the fact that the energy \(E\) is concentrated in a very small particle, whereas the frequency \(\nu\), or, better to say, the wavelength \(\lambda=c/\nu\), requires for its determination, in practice, an infinite wave train.

This paradox can be resolved only on the condition that we sacrifice certain traditional concepts. As we now know, we must abandon the idea that particles, considered by themselves, follow deterministic laws similar to the laws of classical mechanics. The theory can predict only probabilities, and these are determined by waves (they are the squares of amplitudes). This means, of course, a decisive change in our views of nature. It calls us to a new way of describing the physical world, but not to a rejection of its reality. The essence of the new method can best be explained by a simple example.

Let a light ray pass through a Nicol prism; in doing so it will be linearly polarized. Let this primary ray, of amplitude \(A\), pass through a birefringent crystal; then two secondary rays are obtained, linearly polarized perpendicular to one another. If \(\delta\) is the angle between the directions of polarization of the primary ray and one of the secondary rays, then the amplitudes of the latter will be \(A\cos\delta\) and \(A\sin\delta\). Therefore their intensities will be related to one another as

\(\cos^2 \delta : \sin^2 \delta\). If now the intensity of the primary beam is reduced to the point where nothing can be seen with the naked eye, then nevertheless, with the aid of a sensitive photocell and suitable amplification, it is possible to observe and count the photons that pass through. It will then be found that their mean numbers in the secondary beams are in the ratio \(\cos^2 \delta : \sin^2 \delta\). This is the simplest example of the statistical interpretation mentioned above, according to which probabilities are determined by the squares of wave amplitudes. I should now like especially to emphasize that these secondary amplitudes are the projections of the primary amplitudes onto two directions determined by the apparatus. The predictions that the theory makes concerning the intensity of the split beams, or concerning the number of photons contained in them, have meaning only with respect to the entire experimental arrangement: the Nicol prism and the crystal.

This example is typical of quantum phenomena. Take, for example, the corresponding experiment with electrons, known as the Stern—Gerlach experiment, in which the Nicol prism is replaced by an inhomogeneous magnetic field, and polarization by spin orientation. And again, what can be observed here—namely, the number of electrons of a given spin—depends on the particular experimental arrangement; this dependence can be described by saying that the apparatus registers projections of the actual state.

This description is valid for any quantum effect. Observation or measurement relates not to the phenomenon of nature as such, but only to the aspect under which it is considered in the frame of reference, or to the projections onto the frame of reference which, of course, is created by the entire arrangement employed. Expressed in mathematical terms, the word “projection” is quite legitimate, since the basic operation is a direct generalization of geometrical projection, only this time in a space of many, and often infinitely many, dimensions.

If the facts cited above are analyzed from the point of view of particles alone, one obtains the uncertainty relation, into the discussion of which I shall not enter here, since it can now be found in every textbook of quantum mechanics. Bohr introduced the concept of complementarity to express the fact that maximal knowledge of a physical entity cannot be obtained from a single observation or from a single experimental arrangement, but that different experimental devices are necessary—mutually exclusive, yet complementary. In the language adopted here, this would mean that maximal knowledge can be obtained only through a sufficient number of independent projections of one and the same physical entity, just as in the case of the circular piece of cardboard, where shadows on different planes were necessary in order to determine its shape and invariant (the radius). The observation of different shadows on two perpendicular planes, which we used above to explain the concept of an invariant, also explains very well the essence of the idea of complementarity. The final result of complementary experiments is a group of invariants characteristic of the entity under discussion. The chief invariants are called charge, mass (or, better: rest mass), spin, etc.; and in every case when we are able to determine these quantities, we conclude that we are dealing with a definite particle. I am convinced that we are entitled to regard these particles as real in a sense that does not differ essentially from the ordinary meaning of that word.

Before I give the reasons for this point of view of mine, I should like, in a few words, to touch upon the frequently repeated remark that

quantum mechanics has destroyed the distinction between object and subject, for it can describe a situation in nature not as such, but only as a situation created by a human experiment. This is entirely true. The atomic physicist is far removed from the idyllic image of the old-fashioned naturalist who hoped to penetrate the secrets of nature by watching butterflies in a meadow. To observe atomic phenomena, instruments of such sensitivity are needed that their reaction in measurements must be taken into account; since this reaction is subject to the same quantum laws as the observed particle, the factor of uncertainty, which excludes deterministic prediction, has entered together with it. It would therefore obviously be idle to discuss what the situation would be without the observer’s intervention or independently of it. But as regards the given intervention of the observer in the given experimental situation, quantum mechanics provides definite assertions concerning the maximum knowledge that can be obtained. Although we cannot know everything or even approach complete knowledge, nevertheless, by improving our instruments, we can obtain certain, limited but well-described information, independent of the observer and his apparatus—namely, invariant features of a number of suitably designed experiments. The process by which we acquire this knowledge is undoubtedly also conditioned by the observing subject, but this, however, does not mean that there is no reality in the results. For it is quite obvious that the experimenter with his instrument is part of the real world; the mental processes involved in designing experiments are also real. The boundary between the action of the subject and the reaction of the object is in any case imprecise. But this does not prevent us from applying these concepts in a reasonable way. The boundary between a liquid and its vapor is likewise imprecise, for its atoms continuously evaporate and condense, and in spite of this we can speak of a liquid and a vapor.

We now wish to return to the question of reality and clearly picture to ourselves the views of certain contemporary philosophers on this subject.

In a recently published book the American author H. Margenau developed the view that reality consists of two layers: of immediate sensory data and of constructed images (“constructs”); the latter comprise both the things of everyday life and scientific concepts, since they can be checked by numerous independent experiments. The logical positivists, who claim to possess the only exact scientific philosophy, believe, if I understand them correctly, that “constructs” are purely mental tools by means of which one can survey and order the crude sensory data, to which alone they ascribe the character of reality. These are only inessential variations on the same theme, and they seem to me unimportant because two essential points concerning reality are thereby underestimated. The first is that it is psychologically and physiologically incorrect to regard crude impressions of the senses as primary data; the second is that not every concept from the domain of scientific “constructs” has the character of real things, but only those concepts that are invariants with respect to the transformations taking place.

As for the first point, we must consider that every human being, already in earliest childhood, acquires the ability to distinguish and recognize objects. By virtue of this, the world of a normal human being is not a kaleidoscopic series of sensory impressions, but a meaningful, continuously changing arena

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of events, in which a definite thing preserves its identity despite its changing aspects. This capacity of the mind to disregard the distinction among sensory impressions and to note only their invariant character seems to me the most expressive fact of our spiritual structure.

Imagine that you are taking a walk accompanied by your dog. It sees a hare and pursues it with savage fury; soon the dog becomes only a tiny speck in your field of vision. But all the while you see only your dog, and not a sequence of visual impressions of ever diminishing size.

Modern psychology has taken this fundamental situation into account in the Gestalt psychology of Köhler, Hornbostel, and Wertheimer, to name only a few German psychologists of this school personally known to me. And I should like to translate the word “Gestalt” not by the English word “shape” or “form,” but by the word “invariant,” and to speak of “invariants of perception” (“Invarianten der Wahrnehmung”) as elements of our spiritual world. So far as I know, the physiology and anatomy of the nervous system, from the work of E. D. Adrian and J. Z. Young, are in complete agreement with the results of psychological observation.

Every individual nerve fiber, whether motor or sensory—and in the latter case regardless of whether it carries tactile, visual, auditory, or thermal signals—transmits a series of regular impulses that have not the slightest resemblance to the physical stimulus. The brain receives nothing but a series of such impulses, each of which is conveyed by various fibers to a definite place in the cerebral cortex. The brain possesses the astonishing ability to recognize these encoded signals almost instantaneously. In this way it solves an extremely difficult algebraic problem, determining the invariant form in the confusion of constantly changing signals. These invariants thus determine not a blurred series of impressions, but recognizable things.

If one tried to construct a philosophy of natural science on the assumption that our initial material consists of disordered sensory sensations, then we could not even describe our actions and simple instruments. As I have already said, natural science must accept the concepts of everyday life and the expressions of spoken language. But, by employing amplifying devices, telescopes, microscopes, electromagnetic amplifiers, and so forth, it goes beyond these concepts. As soon as such new situations arise, in which ordinary experience deceives us, we find ourselves at a loss as to how the perceived signals should be explained. You will understand what I mean if you have ever looked into a microscope in which a kind physician was showing his remarkable cells or microbes; namely, you saw nothing but a tangle of indefinite lines and colors, and had to believe him that some yellow oval image was of interest to him. The situation is exactly the same in all branches of physics in which instruments are used. We must cast our gaze into the unknown, and this brings us into confusion; for now we are no longer children; we have already lost the capacity unconsciously to decipher the incoming nervous signals and must call upon the technique of our conscious thought—mathematics, with all its artifices. The exceptions to this are only a few geniuses, such as Faraday, who, like a child, was able intuitively to see the inner connection of nature. Thus we apply analysis in order to find in the stream of phenomena something constant,

which is precisely an invariant. Thus, invariants are concepts about which natural science speaks in the same way that ordinary language speaks of “things,” and to which it assigns names just as though they were ordinary things.

Of course, they are not ordinary things. If we call the electron a particle, we know very well that it is not at all the same as a grain of sand or a speck of flower pollen. For example, under certain circumstances it has no definite individuality: if an electron is knocked out of an atom by another electron, the two electrons that fly away can no longer be distinguished. Nevertheless, the electron has certain properties in common with an ordinary “particle,” which justifies the name given to it. Such an extension of nomenclature, both in life and in natural science, is a common matter; it is systematically developed by mathematics. For example, a number originally denotes an integer by means of which one can count a series of objects. But this word is also used for fractional numbers, such as \(2/3\), roots, such as \(\sqrt{2}\), transcendental numbers, such as \(\pi\), and imaginary numbers, such as \(\sqrt{-1}\), although one cannot count with the aid of these numbers. We justify this by the fact that these numbers have certain formal properties in common with integers, although each kind does so to a somewhat lesser degree, yet still sufficiently to apply the familiar word to it. The same principle is valid in analytic geometry, when we speak of straight lines at infinity in the plane, or of a four-dimensional sphere, and so on. It is exactly the same in physics. We speak of infrared or ultraviolet light, although we cannot see it, or of ultrasonic waves, although we cannot hear ultrasound. We have become so accustomed to extrapolating into regions lying beyond our capacity for sensation that we no longer realize that in doing so we are extending concepts beyond the original domain of their application. Yet we always follow one and the same principle. We once consider the concept of a wave. We regard waves on a lake as real, although they are not anything material, but only a certain form of the surface of the water. We can justify this because we can characterize their spectrum by means of known invariant quantities, such as frequencies, wavelengths, and so on. But the same is true of light waves. Why, then, should we discard this attribute of “the real” for waves in quantum mechanics, if we represent them only as probability distributions? What brings them closer to reality here is always their peculiar invariant character of structure, independent of aspect and of projections. This character, however, is common both to everyday life and to natural science, and this connection—though remote—between the things of everyday life and natural science compels us to use one and the same terminology. In that case, this is also the prerequisite for preserving the unity of pure and applied natural science.