[217]{243}

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INTRODUCTION

1. THE PROBLEM OF ATOMIC PHYSICS

2. THE MECHANICS OF THE ATOM

3. THE FUNDAMENTAL CONCEPTS OF PHYSICS IN THE NEW THEORY

4. THE REAL BASE OF THE NEW PHYSICS

5. UNSOLVED PROBLEMS

6. THE NATURE OF PHYSICAL KNOWLEDGE

7. THE FUNDAMENTAL PROBLEM

NOTE: I beg the reader to consider, first, the date of publication of these lines (1934), and second, the type of persons to whom they are directed.

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The Nobel Prizes of 1932 and 1933 have been awarded to three European physicists: Heisenberg, Schrödinger, and Dirac, who created the new mechanics of the atom. An inkling that this honorific mention signifies the consecration of a new epoch in the history of physical knowledge, rather than merely being a prize given to the labor of specialists, has attracted broad public attention to these men. And though on a smaller scale, the same thing happened to Einstein, and in the same way, when as a youth he discovered his principle of relativity. And youth is a characteristic in no way accidental to the new physics.

A few years ago, a young man was presented at a Leipzig society party. The wave of astonishment following upon the young man's entrance was quite out of proportion to his age-he was only twenty-some years old-and caused impertinent surprise in some: "But why is this student here?" It was the young Werner Heisenberg, recently named Full Professor of Physics at the University of Leipzig. Anyone who knows what this signifies in Germany can judge without further commentary the unheard-of magnitude of the case. While yet a student, or almost so, in Gottingen, he had given a first solution to one of the most recalcitrant problems of physics and thereby opened a new era for this science. A little later, in 1927, he formulated his celebrated uncertainty principle, which, if not the most radical, is certainly the most unsuspected novelty of contemporary physics.

Schrödinger, although more advanced in years, is a youthful man, younger in spirit than in body. It was not in vain {246} that he was born in Vienna, for he bears the unmistakable stamp of those who lived the Youth Movement (the Jugendbewegung) together around the motto: " Comradship: down with conventions!", full of faith and enthusiasm. When I met him, in 1930, it had been three years since he came to the University of Berlin from the Polytechnic University of Zurich to succeed Max Plank in the chair of theoretical physics. He began his lectures with a phrase of St. Augustine: "There is an old and a new theory of the quanta. And one can say of them what St. Augustine does of the Bible: Novum Testamentum in Vetere latet; Vetus in Novo patet. (The New [220] Testament is implied in the Old, the Old is patent in the New)"-a rather disconcerting beginning for that audience, accustomed to the positivism of the past century, which has served us, until recently, as science without spirit and consequently without scientific spirit. In 1926, while still teaching in Zurich, he had the idea of giving a more precise mathematical formulation to the hypothesis of the young French physicist, Louis de Broglie, also a recipient of the Nobel Prize. Since then, Schrödinger's equation has become our most powerful mathematical instrument for penetrating the secrets of the atom.

Finally Dirac, a young professor from Cambridge, has attempted a generalization of the ideas of Schrödinger, based on the Theory of Relativity, which has permitted him to obtain a more complete vision of the electron.

These lines have no other pretension than that of expounding a series of philosophical reflections to which the new physics gives rise. The new physics is, to a greater or lesser degree, just that: a novelty, and, consequently, a problem. Now, this characteristic does not so much affect the questions of which physics treats, but rather physics as such. That which is a problem in the new physics is physics itself. It has therefore touched upon a point which reverberates through the entire corpus of philosophy. So let this serve by way of justification for someone who, not being a physicist by profession, sees himself compelled to talk about subjects from physics. And bear in mind that the character {247} of the potential readers to whom this essay is directed obliges him to use expressions technically vague, though not improper; and indeed the infrequent mathematical terms occasionally employed or alluded to are but evocations and consequently can without loss of sense be passed over by readers not familiar with them. {248}

[221] {249}

In order to understand the significance of the work of Heisenberg, Schrödinger, and Dirac, we should recall the problem with which they were wrestling,. A few years earlier, Rutherford had the idea that atoms are composed of a nucleus, whose net electrical charge is positive, around which orbit other particles of negative charge, called "electrons," as the planets orbit around the sun. The nucleus contains, besides electrons, particles of positive charge, the protons. These particles mutually attract, according to Coulomb's law, and are separated only on account of the orbital motion of the electrons. This movement provokes a disturbance in the surrounding aether, which propagating in undulatory form, is the cause of all electromagnetic phenomena explained by Maxwell's equations. Moreover, each chemical element is characterized by a system of these special waves which appear in its spectrum. Hence the problem of the structure of the atom is linked to that of the interpretation of its spectrum. The model of Rutherford is a first attempt to explain these phenomena. So there is, then, an essential unity between the phenomena which occur in the world we perceive and those that occur in the interior of the atom; a single physics is that of the macrocosm and of the microcosm.

Nevertheless, there is a serious difficulty with this conception. If the energy of electromagnetic perturbations were due to kinetic energy, i.e. to the energy associated with the planetary movement of the electrons, it is evident that in virtue of the principle of conservation of energy, the emission of energy in {250} the form of electromagnetic waves would have to be accompanied by the loss of a corresponding quantity of kinetic energy, so that the electron would lose velocity, and consequently, on account of electrical attraction, begin approaching ever closer to the nucleus, until finally crashing into it. The orbit of the electron would not be circular, but spiral. At the time of collision it would cease to move and therewith to produce electromagnetic waves. Matter would quickly reach a state of total equilibrium in which there would occur neither electrical nor optical phenomena. The presumed unity of physics stumbled here upon a difficulty which menac ed it [222] at it very roots.

Something similar had occurred in the study of the distribution of temperature in the interior of a closed body which is completely isolated from the outside world; this is the so-called black body radiation problem. In order to reach agreement with experience, Max Planck was ingenuous enough to renounce the idea that radiation is a phenomenon produced in the form of continuous and insensible transitions. He thought, instead, that energy is absorbed and emitted discontinuously, in distinct jumps. For a rather absurd comparison, we might suppose that temperatures only changed in intervals of 10 degrees. If a body found itself with 12, for example, it would emit only 10 and save the remaining 2 (as if they did not exist) until it had 8 more, so as then to emit a block of 10 more degrees, and so forth. The absorption and emission of energy takes place, according to Planck, in integral multiples of a certain elementary constant, the quantum of action. The numerical determination of this constant was the great work of Planck. It carries, therefore, his name: Planck's constant (h). Energy is transmitted, then, as if it were composed of granules or corpuscles. This idea accords, in all respects, with experimental data; but it was incompatible with all physics existing up to that time, which was based essentially on the idea of the continuity of physical processes. In reality, then, the solution proposed by Planck to explain black body radiation sharply aggravated the contradiction between experience and physics as a whole.

In 1913 a collaborator of Rutherford, Niels Bohr, applied Planck's idea to the atomic model of his teacher, and his agreement with experiment {251} opened the doors of science to the absolute abyss separating it from everyday experience.

Let us, in fact, return to the atom of Rutherford. One of the reasons making it unacceptable, I said, is the possibility that the electron could collapse into the nucleus. So, let us maintain the model, postulating the impossibility of this collapse. Then, the electron would not find itself at just any distance from the nucleus, but only at those previously defined. That is to say, if we again use absurd numbers, Bohr postulates that the electron can be found at [223] a distance of 1 mm, 2 mm, or 3 mm from the nucleus, but not at 1.5, etc. Not all orbits are possible for the electron, only certain ones. Whence the possibility of collapse is eliminated. This elimination is based, as we see, on a simple postulation. But there is still more: whereas for Rutherford the atom emits or absorbs energy while moving in its orbit, for Bohr the orbits of the electrons are stationary, i.e. there is no radiation while the electron moves in them, only when it jumps from one orbit to another. The frequency of the energy emitted then is a quantity which depends on Planck's constant and has nothing to do with the frequency of rotation of the electron in its orbit. In this way the problem is aggravated still more; there is now no relation between the frequency of radiated energy and that which is derived mechanically from the stationary state of the atom. And so with this hypothesis the mechanics of electron movement no longer has anything to do with classical mechanics, which was adequate for the solar system; nor with the physics of Maxwell and Coulomb, which assumes the continuous structure of energy and admits all possible distances between the electron and the nucleus. The macrocosm obeys a physics of continuity, and the microcosm a physics of discontinuity. And the difficulty comes to a head as soon as we realize that these two worlds are not separate, but the one operates on the other. What then could be the structure of that interaction?

Such is the crucible in which physics found itself when De Broglie at first, and later Heisenberg, Schrödinger, and Dirac came to deal with it. In order to understand the magnitude of the problem, recall that {252} we are not considering the difficulties of explaining this or that phenomenon, but the difficulties of conceiving of physical happenings in general. There cannot be two physics, because there is only one nature, which either has quantum jumps, or doesn't have them. The contrast between continuity and discontinuity plays a role in this question which as we shall see later becomes more complicated when other more essential dimensions of the problem are considered. Recall a similar situation in the 19th century, with regard to the nature of light. For Newton, it was a series of corpuscles which propagated in a straight line. For Huygens, it was on the other hand the deformation of a medium which engulfs everything; and that which we call a ray of light is nothing but the line of maximum intensity of this deformation. The discovery of interference [224] phenomena seemed to give more plausibility to Huygens' theory, and in conformity with the idea of continuity it was possible to construct all the theories of optics and electromagnetism. We shall see how this allusion to optics will assume an important role in the new physics.

[225] {253}

1. In 1925, Heisenberg tackled this thorny problem via a critical consideration. The difficulty to which Bohr has led us perhaps may stem from our desire to give too detailed a picture of the atom, a picture which, for Heisenberg, is unnecessary because it contains superfluous elements and does not limit itself to the essentials.

In the first place, Bohr's model has superfluous elements. It supposes, on the one hand, that the mechanical "state" of the atom depends on the position and velocity of its electrons. But when it comes time to explain the spectrum, this stationary movement of the electrons plays no part whatever. What happens to the electron while in its stationary orbits is completely transparent to physics. The only thing that matters is the jump from one to another. And in fact Bohr postulated a jump energy which has nothing to do with the kinetic energy that from the mechanical point of view the electron must possess in any stationary state. Why should we complicate matters with this mechanical image?

It is more convenient, in the second place, to elaborate the theory of the atom using only quantities that are really measurable. And such directly measurable quantities are (among others) energy and impulse (i.e. the integrals of movement of the system), but not position and velocity of electrons.

In order to clarify the question, let us recall a problem of acoustics which it will be convenient to bear in mind during the course of this essay. We assume that we know the laws of composition of sound, {254} i.e. its structure. In order to analyze this situation we may employ the following method. It is known that sound is produced by vibrations of a medium, for example, of a cord. The problem of acoustics which has been proposed to us thus becomes a dynamical problem; if we remove a molecule of this cord from its state of equilibrium and if we know the amplitude of this deformation and the initial velocity, which with a certain force we are going to provide, we can deduce inexorably the further course of movement of the cord. A mathematical calculation will show us that this auditory vibration is composed of fundamental tones and their harmonics, and we shall thus obtain all the [226] relations of the musical scale. But there is another way of attacking the problem. Each sound is characterized by the frequency, intensity, and amplitude of its waves. There is an apparatus, called a "resonator," which serves to register sounds. It is characterized by the fact that it only emits a single frequency, and that if around it that particular sound is produced, the resonator itself will emit sound, i.e. will "resonate." If the exciting sound is not of the resonator's frequency, it will emit no sound at all in response. Let us suppose, then, that we have any sound pattern whatever. If in its proximity we arrange an ideally complete system of resonators, each one of them will extract all of the sound in its own frequency range. In this way, we will obtain a type of acoustical spectrum. Recombination of these elemental sounds will give us the structure of the whole (original) sound. The entire problem would be reduced to one of arithmetic: to determine the laws of composition of these sounds, i.e. the proportion, if I may be permitted the expression, in which each elemental sound enters into the structure of the total sound pattern. We would obtain by this route the same results obtained by the previous method: the sounds are composed, among themselves, in proportions such as one to eight, one to four, etc.

The fact that in acoustics both methods are practical, and the first is linked to the general problem of mechanics, could lead to error if one supposes that the same thing occurs in the case of light waves, and that the frequencies and amplitudes of oscillation of an electron in the spectrum must be explainable by the mechanical state of the system. This is pure fiction. In reality, the second method {255} is independent of the first and leads to the same results, but with an advantage: that of operating on quantities always directly accessible via experimental measurement, just as are tones and frequencies of sounds. It does not operate on quantities that are at times uncontrolable, such as the position and velocity of molecules in a string.

If indeed the atomic model of Bohr was incapable of yielding the intensities of spectral lines (an inherent defect), its positive merit consisted in explaining their qualitative distribution. All of the rest-the mechanical picture of the atom-was perfectly dispensable. So abandoning all this useless mechanical complication of rotating electrons, orbits, etc., Heisenberg sought a type of [227] arithmetic for the spectral rays analogous in many respects to that which exists for acoustics. Evidently, this arithmetic has to be enormously more complicated than that of the sound waves. The luminous or electromagnetic spectrum is a system of infinitely many possible frequencies and amplitudes. Since each spectral line of an atom is composed of vibrations of fixed amplitude and frequency, and is produced by the transition from one atomic state to another, then of necessity the arithmetic will have to handle a doubly infinite conjunction of elemental vibrations in order to determine all the frequencies of the spectrum. These elemental vibrations form an ordered conjunction, called an "infinite matrix." The problem then consists in establishing the laws of combination of these "numbers," i.e. of the conjunctions of elemental vibrations. All arithmetic, including that which serves for everyday use, consists of certain rules or conventions for calculating, i.e. for deducing from given numbers other new numbers. From 3 and 5, by the convention termed "adding," we deduce 8; by another convention we get 15. In the present case, matrices assume the function of numbers, and it becomes necessary to introduce {256} rules such that from them the spectral combinations shown in experience can be deduced. That is, Heisenberg proceeds in such a way that the frequencies and amplitudes are the same as they are among the corresponding quantities in Bohr's model. The quantum structure, which in Bohr's model appears as an arbitrary postulate, now appears in Heisenberg's theory as a necessary consequence of the rules of composition of the spectral quantities. But the arithmetic of Heisenberg is profoundly different from the usual: in it, the order of factors affects the product. But when we pass from the mechanics of the atom to everyday (Newtonian) mechanics, this non-commutability becomes insensible, because we are no longer dealing with quantities of the order of magnitude of Planck's constant. In the development of this theory Bohr and Jordan have actively collaborated with Heisenberg.

Heisenberg takes as his point of departure the discontinuities of atomic processes to obtain, as a first approximation, the relations of continuity of mechanics and classical physics. To do [228] this he reduces the problem of discontinuity to another more general one: the non-commutative arithmetic of infinite matrices. The fact that our everyday arithmetic, which is used in calculating forces and velocities, happens to be a particular case of this arithmetic of Heisenberg confers a radical unity on the entire edifice of physics.

2. The point of view of Schrödinger is completely different. On the surface, it seems more intuitive and less abstract than that of Heisenberg. It does not require introduction of new calculation methods and techniques, but rather makes use of the usual tools of classical physics, viz. continuous functions and differential or partial differential equations. In contrast to Heisenberg, who starts from discontinuity to obtain an explanation of continuous phenomena, Schrödinger starts from the hypothesis of continuity, and his problem is thus one of giving a complete explanation of the discontinuous phenomena of the atom.

Earlier De Broglie, studying the theory of the photoelectric effect proposed by Einstein-according to which light seems to behave as if it were composed of particles {257} called "photons"-had the idea of supposing that there was associated with every electron a wave of very small dimension, which accompanied it everywhere. That is to say, he supposed that the photon was a quantified wave, whose energy is equal to the frequency (v) multiplied by Planck's constant (h), and which fell under the laws governing electromagnetic waves. Starting from this idea, Schrödinger conceives the electron as a system of these waves which DeBroglie had associated with particles.

Let us imagine now that we have a vibrating cord. Suppose that it is fixed on only one end. If we shake it at the fixed end, a vibration will be produced which propagates along the cord, until it disappears. Now suppose the cord is fixed at both ends, and we propose to produce a sound. This sound wave will not be like the vibration in the previous case, which propagates and disappears; rather, it remains in a certain sense, i.e. is stationary, and is found to be composed of a whole number of nodes and anti-nodes related [229] in a fixed way. It we wish, then, to make a sound with a cord of fixed length, it is clear at the outset that the ends of the cord must coincide with two nodes. And consequently, the number and form of the anti-nodes along the cord is restricted. Given a cord of fixed length, the number and nature of stationary waves or elemental sounds which can be produced by it is limited. Each cord has, then, a system of vibrations-its own sounds. The physics of the macrocosmos exhibits, therefore, phenomena such as stationary waves which without diminishing its continuity offer definite discontinuities measurable in whole numbers; for example the number and distribution of nodes and anti-nodes. In less precise terminology: the general equation which permits study of all types of waves gives rise under certain restrictive conditions (the so-called "boundary values") to a definite group of stationary waves proper to each cord.

Now, Schrödinger had the idea of applying this method to the study of the atom. If it were possible to obtain the stationary states of the atom as the only permissible stationary waves {258} of a cord are obtained, the problem of the structure of the atom would be resolved without appealing to arbitrary quantum postulates or renouncing the efficacious methods which have served classical physics. Let us recall that an atom is something which, when put into a spectroscope, produces a series of spectral lines of fixed amplitude and frequency. The entire problem then reduces to that of uncovering the restrictive conditions which compel the system to produce lines unique to each atom, just as the determination of the length of the cord fixes the group of sounds it is capable of producing. Utilizing the general hypothesis that energy is equal to frequency multiplied by Planck's constant, Schrödinger succeeded in writing a wave equation which, given certain restrictive conditions (or boundary values) leads necessarily to the system of amplitudes and frequencies proper to each atom, i.e. to the quantum conditions of Bohr. This celebrated equation of Schrödinger is our most powerful instrument for studying the structure of the atom. With it, the problem of atomic structure is reduced to that of investigating the proper values and functional solutions of the wave equation. The first success of the theory was its interpretation of the spectrum of hydrogen.

But it is important not to exaggerate the similarity between the material waves" of Schrödinger and the ordinary waves we can [230] all perceive or imagine. The correspondence with vibrating strings is a tenuous one.

In the first place, ordinary waves, including those imagined in DeBroglie's hypothesis, are waves which propagate. Material waves, on the other hand, are stationary; they do not propagate.

In the second place, ordinary waves are such that at every point in space there is a certain "movement" or vibration, which is a function of position. On the other hand, speaking of an atom with various cortical electrons, the waves proper to it are functions simultaneously of as many degrees of freedom as these electrons possess. If one wishes to continue speaking of waves as functions of position, it is necessary to have recourse to a space of 3n dimensions, where n is the number of electrons in question. This is the so-called "configuration space," {259} which has nothing to do with what we understand intuitively by "space"; rather, it is part of another concept of space that is very much more abstract: the functional space of Hilbert.

But, in the third place, and above all, even when we consider an atom containing only one electron, as in the case of hydrogen, material waves still do not have the same sense as ordinary waves. Let us take an example used by Schrödinger. Suppose that we have a cork floating on the surface of the water in a pond. A rock is thrown into the pond, and a wave motion is set up which slowly propagates across the surface until it reaches the cork. It is clear that the cork will suffer a jolt of greater or lesser degree, according to the intensity which the wave possesses when it reaches the position where it encounters the cork. What we call the configuration of the wave" is nothing but the result or collective expression of what has been happening at each point of the water's surface. And what happens at each point depends on nothing but the intensity of the force acting there. This is not at all what occurs in the case of material waves. Suppose that a ray of light impinges on an electron. If this light wave acted just as the water on the cork, the jolt received by the electron would depend on the intensity possessed by the wave when it reached the electron. Now, experience shows that the electron will or will not begin vibrating according to the total configuration of the wave, totally independent of its intensity, and depending only on its color (i.e. its frequency). The electron behaves more like a resonator than a cork. The efficacy of the wave depends on its configuration prior to reaching the electron. (This is the photoelectric effect, to which I [231] alluded when discussing the hypothesis of De Broglie.) Whence it follows that the configuration of the wave proper to the electron is not the collective expression or result of what happens at each point of space; rather, on the contrary, its possible action at each point of space is conditioned by its previous wave configuration. This is the primacy of the whole over each of its parts. In acoustics, both points of view coincide. I can think of a vibration as the sum of what happens to each {260} molecule which is vibrating; but I can also characterize the vibration in terms of amplitude, phase, and frequency, through which beforehand the entire course of the wave is determined. In the case of the atom these two points of view do not coincide; rather, the only possible one is the second. We are not dealing with collective expressions, but rather expressions about: the configuration of certain stationary waves. Nothing in this recalls waves in fluid media.

When we deal, then, with an order of magnitude below Planck's constant, the problems of particle mechanics reduce to problems of wave mechanics; and reciprocally, for orders of magnitude above it, certain problems of wave mechanics can be treated as corpuscular, just as is the case with light: for orders of magnitude greater than that of a wavelength, there is an equivalence between particle and wave interpretations of light.

This equivalence is something more than a simple comparison. It was devised by Hamilton as a simple mathematical artifice for treating certain problems of mechanics. In Newtonian mechanics one begins by posing a problem in the following terms: given the velocity and initial position of a point, find the trajectory of its movement. If instead of one there are many points, the final state of the system will be the result of the trajectory of each one (bearing in mind the particular initial conditions of the system). Hamilton, though, started from another consideration. Let us take many points at some initial time. Together they determine a surface. We give to each of them an initial velocity in a definite direction. After a certain time these points will be in different places. They will also determine a surface which, in general, will not have the same shape as the first. The problem of mechanics [232] can then be interpreted as a displacement of the first surface, with or without deformation, as if it were the propagation of a wave. What happens to each point depends on what happens to the surface which drags it along, and its trajectory will be the line along which it is dragged by the surface during the {261} latter's propagation. The undulatory method of Hamilton leads to the same results as the point method of Newton: it is immaterial whether one interprets the surface in question as a geometric arrangement of points obeying Newtonian mechanics, or the movement of each point as the trajectory along which the points making up the surface are displaced. This, which Hamilton intended as nothing more than a mathematical artifice, in Schrödinger's theory acquired a definite physical meaning: the equivalence between particle and wave mechanics, and thereby conferred a unity on physics.

Heisenberg, starting from discontinuity, reduces the question to a problem of non-commutative arithmetic. Schrödinger, starting from continuity, reduces the problem of quantification to that of the investigation of waves proper to each atom. Nevertheless-and this is essential-the contraposition is more apparent than real. Schrödinger demonstrated that from his equation the arithmetic relations of Heisenberg can be obtained; and conversely with the arithmetic of Heisenberg one can obtain Schrödinger's equation. In reality, the two together constitute one single mechanics: the mechanics of the atom. And this poses a special problem, to which I shall direct my attention shortly.

3. There remain, nevertheless, profound lacunae in the construction of the new mechanics. Among others, there is the inability to explain the experiment of Stern and Gerlach, which requires that account be taken of magnetic moment. And to explain this, it is necessary to suppose that electrons, besides having translational movement around the nucleus, also possess a rotational movement about an axis. This rotational movement defines a quantified magnetic and kinetic moment, the so-called "spin." Pauli attempted a mathematical explanation of this phenomenon, but it was unsatisfactory. Moreover, in spite of an essay by Schrödinger, it has proved impossible thus far to adequately take into account the relativistic conditions imposed on electromagnetic phenomena.

Dirac addresses his efforts to this group of problems. It is difficult to give an accurate discussion of the question without [233] entering into {262} mathematical considerations, so I will restrict myself to a few allusions. Let us consider a light wave. We are already familiar with its undulatory propagation, that is, we treat the phenomenon by means of the wave equation. This was done all throughout the 19th century. But Maxwell sought to discover the forces which produce these waves. This is a completely different mathematical problem: it is not the problem of the course of movement, but rather that of the structure of a field. Fresnel had surmised that waves were due to elastic forces. Maxwell assumed that these forces were none other than the electrical and magnetic ones already known. There is an electromagnetic field. The structure of the field is such that one can deduce that any deformation introduced in it will necessarily propagate in the form of transverse waves. Light waves are nothing but a particular case of electromagnetic waves. The wireless and the radio-telephone are experimental applications of Maxwell's conception. His great creation was the discovery of the structure of the electromagnetic field. Now, it is fitting to ask what might be the structure of the field whose deformations constitute "material waves." To resolve this problem it is necessary to bear in mind relativistic considerations. The field must respect the invariance of the velocity of light and possess the same structure regardless of the observer looking at it, so long as he is moving in a rectilinear fashion. Dirac has succeeded in describing this field by means of a system of four equations, which are, with respect to the wave equation, what the equations of the electromagnetic field are with respect to light waves. The study of the movement of the electron in this field leads to Schrödinger's equation as a first approximation, if among other things we precind from the influence of magnetic fields and the variability of mass called for by relativity. But if we take the magnetic field into account, then we obtain as a second approximation an equation from which we deduce inexorably the existence of spin: the magnetic electron. {263}

But here it is necessary to return to what was said with regard to Schrödinger. In reality, this field is not comparable to the electromagnetic field of Maxwell, because in Dirac's field the waves do not propagate. And similarly, the "spin" is not a true rotation; it is a type of special orientation in space which the axis of the electron can have, but without introducing the intermediate stage of rotation-a type of rotation without rotation, a structure of configuration, but not a succession which propagates or which is [234] obtained by continuous movement whose course can be followed. It is something like-if I may be permitted a remote analogy, false in many respects-the difference between the right and left hands. It should be added, moreover, that Dirac's equations have no physical sense when applied to particles other than electrons; composite particles, such as alpha rays, do not exhibit the phenomenon of spin.

Developing these ideas in a formal and mathematical way, one arrives at a general theory, in which it is possible to obtain certain relations corresponding to those obtained in Maxwell's theory (Hartrees). But, here as there, it is impossible to deduce from field considerations the existence of particles with a unique charge. In order to achieve this, recourse was made to introducing quantum conditions, just as Bohr introduced them in Rutherford's model. But later, Dirac and others transformed the theory, introducing into the structure of the field itself relations similar to those employed by Heisenberg, with which they obtained, as a natural consequence, the quantum conditions. In this way, a general quantum theory of fields has been elaborated in which, as Klein and Jordan have demonstrated, there is absolute equivalence between the particle and wave points of view. {264}

[235] {265}

We have seen, in broad outline, the complex of ideas within which the new mechanics of the atom operates. But after detailed studies and when all the mathematics has been worked out, we return to Bohr's clear atomic model and ask ourselves anxiously, "What are the states of the atom in the new mechanics? What are the electrons? What are these waves?"

All the intuitive meaning which these terms used to have has disappeared in the new physics, whether one considers Heisenberg's or Schrödinger's version.

The state of the atom is not a state in which one encounters the electrons situated in determinant points of space and instants of time. The quantities on which the state of the atom depends are not the velocity and distance from the nucleus of the electrons, as was the case in Bohr's atom; rather, each state' is determined by the simultaneous participation of the atom in all the possible states of the classical system, in the same way that a sound is determined, at each instant and at each point of the musical instrument, by its simultaneous participation in all the elementary sounds making it up. The atom is, at one and the same time, in all possible states. Thus, the state of the atom is not a function of time and of spatial coordinates, but rather is a function of functions; or if I may be permitted the expression, a state of states. Each coordinate {266} of each spectral line measures not a spatiotemporal point, but rather the participation of the atom in the corresponding functions or proper waves. Whence it follows that the point at which an electron is located has no intuitive meaning either. The material point of quantum physics can be in various places at the same time, if the atom consists of various electrons-an essential phenomenon for the new statistical mechanics. [236]

What then is an electron? From the beginning Heisenberg maintained a generally corpuscular position, but with essential modifications, as we have seen. Schrödinger believed on the other hand (for a while) that the electron could be considered as a wave packet which propagated in space with a group velocity capable of being treated corpuscularly, but when studied microscopically it had an undulatory structure. It has not been possible to maintain this position, because the wave packet does not possess the requisite stability to constitute matter. Just as from the structure of the electromagnetic field the electron cannot be obtained as one of its singularities, neither can matter be obtained in this way from the wave theory. And there is no doubt that cathode ray experiments, for example, reveal the existence of true electrons (Jordan). But it is necessary to add that in regard to what this electron is, the meaning of "is" is nothing more than that of being the subject of a system of amplitudes and frequencies.

Finally, what are these waves? De Broglie, and at the beginning Schrödinger, thought that they were dealing with real waves. The fact of electron diffraction, experimentally demonstrated by Davisson and Germer in 1927, seemed to amount to a proof of it. This famous experiment consisted of bombarding a crystal with electrons, and inspecting the patterns on the observation surface. They were not points, which they would have been if the electrons behaved like particles, but rather patterns such as appear if the experiment is done with X-rays. But it is necessary to {267} bear in mind that this experiment is not done with one single electron, but many. On that account Schrödinger supposed that the wave function measured the density of the electrical charge. But neither is this always possible. Bohr devised another interpretation of the same experiment. In order to ascertain the place in which the electron is found, I must repeat the experiment several times. Each time I encounter the electron in a place somewhat different than the last. But if I take the mean value of the measurements, I will know the probability that the electron is found in a particular place. To each particle, then, there is associated a certain probability. This probability acquires physical meaning if we suppose that its value at each point depends on the forces acting on the particle (among other things). Thus we would have a continuous function, which leads to Schrödinger's equation, and which determines the law in conformity with which this probability propagates through space. Material waves thus would be [237] probability waves. The picture of these waves does not correspond to anything real, in the everyday sense, but rather is simply an illustration of some statistic. Seen from another point of view, a stationary state of the atom is a probabilistic cloud surrounding the nucleus, and the old orbits correspond to condensations of this probability. That is, if I want to find out where the electron is, I discover that this probability intensifies, during some states, in a certain region of space, and during others, in other regions. The same can be said of the structure of light: the amplitude of the wave represents either its intensity or the probability that at a certain point a photon will be found. Nevertheless, Schrödinger does not admit the theory of light quanta. He often used to say, "When someone begins talking to me about light quanta, I begin to understand nothing."

This statistical theory could not have been elaborated without amplification of classical concepts of probability. Fermi-Dirac and Bose-Einstein statistics were newly created for quantum theory. {268}

And with this statistical theory the equivalence between wave and particle points of view acquires greater precision. This equivalence Bohr enuntiated as an explicit postulate, and Dirac and Jordan have developed it mathematically in the so-called "theory of transformations."

[238] {269}

1. THE REAL BASE OF THE NEW PHYSICS

The equivalence between these two points of view is something more than a happy coincidence. It is based in reality. This is the great discovery of Heisenberg: the uncertainty principle. Let us again recall Bohr's atomic model. In order for this model to make sense it was necessary for the measurement of position and velocity of an electron at a specified moment of time to make sense too. But such measurement is impossible, not because practically it cannot be done, but because the phenomenon itself implies the radical impossibility of such a measurement. In any measurement, in fact, the measuring device should not significantly affect what it measures. But, for any measurement, it is necessary to see the object and, therefore, to illuminate it. When we deal with objects an order of magnitude or more greater than that of Planck's constant, the action of light on the object is insensible. But when we deal with electrons, the object measured is of the same order of magnitude as the light wavelength which illuminates it, and consequently is sensibly affected by it. In what sense? Compton demonstrated experimentally that when a beam of monochromatic light is directed on an electron, the velocity imparted to the electron increases as the wavelength of the incident light is decreased. Let us suppose, then, that knowing the place occupied by the electron, we desire to know its velocity. We should have to employ light of long wavelength. The way, the velocity of the electron will suffer the smallest possible change; but on the other hand, the place it occupies is now less precisely delineated. Let us then use light of short wavelength. We now precisely fix the position of the electron, but its {270} velocity will have changed considerably. it is not possible to simultaneously determine the position and velocity of the electron. Any attempt to do so will result in a total error of at least the order of magnitude of Planck's constant. Outside the atom, this measurement error can be totally disregarded; but inside it is unavoidable. It makes the concepts of wave and particle lose their meaning when we deal with quantities of the order of magnitude of Planck's constant. The equivalence between particle and wave mechanics thus turns out to be founded [239] in physical reality itself. Consequently it makes no sense to ask what the real relation is between particles and waves. DeBroglie thought at one point that this relation is such that the particle called an "electron" moves along carried by an associated wave, docilely following its laws of motion. This is the so-called "wave-pilot" theory. But De Broglie himself saw the difficulties facing such a conception, even if the wave is interpreted as a wave of probability. With the uncertainty principle, the problem of the real relation between particles and waves loses its meaning. Particles and waves are nothing more than two languages, two systems of operations for describing one single physical reality. "Waves and particles," says Dirac, "ought to be considered as two conceptual formulations which have been shown to be adequate for describing one single physical reality. We should not try to form a single 'common' image in which both play a part; and it is important not to attempt to sketch out a mechanism obeying classical laws, and with it describe the connection between waves and particles and thus determine their movements. Any such attempt goes squarely against the axioms in accordance with which the most recent physics has been developed. Quantum mechanics has no pretension other than establishing the laws which govern phenomena, in such a form that by means of them we can determine univocally what will happen under well-defined experimental conditions. Any attempt to plumb the relation between particles and waves for meanings or information beyond that necessary for the foregoing goal would be useless and senseless." {271}

Such is the general outline of the brilliant work of Heisenberg, Schrödinger, and Dirac: the formulation of a new symbolic mechanics of the quantum which, as Bohr says, should be considered as a generalization of classical mechanics. It does no violence to classical mechanics and in fact can be compared to it in beauty and internal coherence. From this standpoint, relativistic mechanics is the crowing glory of classical mechanics. The proportion and nature of the contributions of each of the three creators of the new theory has, without doubt, influenced the decision of the Jury in awarding the 1932 prize solely to Heisenberg, and dividing the 1933 prize between Schrödinger and Dirac. {272}

[240] {273}

This mechanics has been accompanied by a growing list of successes. It has succeeded in treating atoms with several electrons (the n-body problem), and through application of special mathematical theories (such as that of groups) it has been able to tackle more generally the problems of molecular structure, etc. But, even with all this, important problems recently uncovered are as yet not resolved.

In the first place, it has not been possible to take into account in a satisfactory way all the conditions required by the theory of relativity. The first efforts of Schrödinger and Dirac were limited to special relativity, but in certain respects applied also to general relativity. Recently Schrödinger, continuing the work of various physicists and mathematicians-above all that of Tetrode-has attempted to study, from the point of view of general relativity, the movement of an electron as defined by Dirac's theory in a gravitational field. And Van der Waerden has reached the same results by simpler methods. Einstein, for his part, has just dedicated an important study to this subject, which he presented to the Academy of Amsterdam a few weeks ago. But the problem remains unresolved. Without doubt the new atomic physics could reproach the theory of relativity because it does not take into account quantum conditions. But this would serve no purpose other than to underline the current lack of communication between these two worlds of physics. {274}

In the second place, Dirac's theory leads to the so-called "negative energy solutions," i.e. to electrons with negative rest mass, whose existence is inevitable if the theory is going to explain the fact of diffusion of light by electrons. But such solutions pose serious difficulties. When these new electrons enter into relation with ordinary electrons, i.e. with the only ones observed up to now, the former will experience an attraction due to the latter, which will in turn exercise a repulsive force. Whence it will result that the two will follow each other in a swift race. And besides the [241] existence of these negative energy states, a collision (according to De Broglie) with those of positive energy would produce a type of trepidation about the center of gravity of the probability (according to Schrödinger). Finally, the probability that an electron of positive mass may spontaneously become one of negative mass, or vice versa, is very great (Klein's paradox). In principle, Dirac accepted the existence of those electrons, in spite of everything, supposing that they were unobservable. When they jump to positive mass, they will become observable, i.e. normal electrons, and the hole that disappeared will be a proton. The inverse transformation would lead to the simultaneous disappearance of an electron and a proton, manifested in the form of radiation. It was difficult to admit this. But the most recent experiments have revealed the existence of positive particles of the mas s of an electron, the so-called positive electron, or positron. In a study about to be published, Dirac places the positron in relation to the negative energy solutions, and the theory acquires a plausibility which at the beginning could not have been suspected. But the matter is still full of thorny difficulties.

Finally, some new atomic phenomena fall outside the scope of quantum mechanics. The atom, in fact, is not made up solely of orbital electrons, but also contains a central nucleus, where there are particles of positive charge, such as protons, and neutrons, which are very heavy. Our nascent understanding of {275} the nucleus escapes, at present, all of quantum physics.

It seems probable that quantum mechanics can be readily applied to these heavy particles of the nucleus, prescinding from relativistic considerations. But let us not forget, as Heisenberg observes in an unpublished study devoted to this problem, that with heavy particles alone one does not obtain the whole nucleus; there are, perhaps, electrons in it as well. And their presence calls for relativisitc considerations. It seems, then, that Dirac's equations are the only adequate instrument for this study. But this [242] presents enormous difficulties. We have already seen some of the problems to which it gives rise. From Klein's paradox-which is its consequence-it would follow that there can be no electrons in the nucleus. Other difficulties are piled on top of this, which makes it seem that something more than a simple modification of wave mechanics is needed. And in fact Schrödinger has attempted such a modification. It is necessary for us to possess, besides, a complete quantum electrodynamics, something which we do not as yet have. Heisenberg notes that we are so far away from being able to interpret the physics of nuclear electrons that neither classical nor quantum physics nor the two together offer so much as a point of reference for us to orient ourselves to the problem. Let us simply bear in mind that the relations which are established between orbital electrons on the basis of their charge must be established for nuclear electrons on the basis of their mass.

Moreover, we are ignorant of the forces holding the nucleus together. Heisenberg recognizes that they are totally different from the attractive and repulsive forces of Coulomb, which maintain the connection between orbital electrons and the nucleus. The alpha particles (composed of four protons and two electrons) should be considered as independent. The neutrons, also recently discovered (masses without electrical charge), play an essential part in the structure of the nucleus. Finally, it is necessary to study the disintegration of the nucleus. And the existence of beta radiation makes Bohr proclaim, perhaps too soon, the demise of the {276} concept of energy and the principles of conservation, with regard to nuclear stability.

There are many new horizons to the remarkable work of these last ten years, to be sure. Consequently, bear in mind that the delineation of its character is, if not provisional, then at least fragmentary.

[243] {277}

In view of the foregoing, it is definitely premature to seek to philosophize too publicly on these problems, since physics finds itself almost daily in some dramatically new situation. One difficulty is no sooner resolved than new unsuspected difficulties appear on the horizon, often affecting the very roots of the science. The dizzy pace of discoveries could cause any new philosophy of science to quickly become a heap of childish relics. No more than ten years ago Bohr's model implied a curious situation: the radiation produced in a jump from one orbit to another depended not only on the initial state, but the final one as well, so that there was a type of efficacy of this latter before it was in fact realized. One could then believe in a resurgence of the concept of teleology (in the worst sense of the word) in physics. Who today would argue that way? So all of this, while not an obstacle to a philosophy of nature (which is something very different from a simple critical reflection on the conceptual pictures emerging from science), is something to beware of. Let us, therefore, do no more than sketch out a series of preoccupations and anxieties which, inevitably, the new physics awakens.

And in the first place, there is the very idea of physical knowledge. It is not merely that the so-called "crisis of intuition" (which would better be termed "crisis of imagination") has taken us quite far from what physics seemed to be prior to 1919. Apart from a few isolated and totally ignored thinkers (Duhem, above all, but also Mach and Poincaré), the physicists believed with complete unanimity that physical knowledge was this: represent {278}the things of the world to us, and therefore, imagine models whose mathematical structure leads to results coinciding with experience: waves and atomic and molecular structures. But already the Maxwell electromagnetic theory had been a rude shock to the imagination. Maxwell's waves could not be vibrations of an elastic medium. The aether ceased to signify what it used to, even for Fresnel, viz. a medium characterized by maximum elasticity; instead it became a word designating lines of forces utilized by Faraday as simply a cognitive symbol. In fact, by 1919 Einstein could say that the aether possessed no other mechanical property [244] than its immobility, nor had any mission other than that of supplying a subject for the verb "to oscillate." And the theory of relativity had just definitively taken leave of physical theories based in imagination. Correctly understood, imagination is the organ which represents, and in this sense knows, what the world is. It was apparent then that in physical theories, there were two essential and distinct elements: the image of the world, and its mathematical structure or formulation; and of these two the first is dispensable and circumstantial, only the second expresses physical truth. This much, then, appeared sufficiently clear before the new physics was systematized.

But the reform in physics introduced by the new developments goes a step farther: it affects the very sense of mathematics as organon of physical knowledge. And to this delicate point I would now like to direct my attention.

What is the logical framework of the new physics?

Above all, it is necessary to recall that, as Dirac says, "the intent of quantum mechanics is no thing more than an amplification of the dominion of those questions to which an answer can be given, but not in such a way as to give answers more precise than those which can be confirmed by experience." There is, then, an attempt even more radical than that of the theory of relativity to achieve agreement with experimental truth, to create experimental concepts for actual experiences. Whence follow the distinctive internal characteristics of the facts from which it starts, of the problems based on the facts, and the meaning of the solution which it finds. {279}

Modern physics was born from the measurement of observations. This is what classical physics understood by facts. But these expressions give rise to a fundamental error in contemporary thought. What do we understand by "observation"? Whatever its structure may be in the long run, an observation is, provisionally, something which the observer contemplates. The observer does nothing, or if we desire to continue speaking of "doing," does nothing but contemplate, i.e. record. Therefore, he is disconnected-that, at least, is the idea-from the contents of what he observes. Whence it follows that, to make an observation, it suffices to realize one by one various efforts to measure the same object, excluding of course systematic or accidental errors which de facto happen to be [245] made. Nothing like this takes place in the new physics. Besides the foregoing errors, in every observation the observer by the mere fact of observing essentially modifies the nature of what is observed, because as we saw earlier it is necessary to illuminate the object. Whence it follows first, that a concrete indication of the instant in which the observation was realized is essential; and second, that to repeat an observation, a special act is necessary to recall the system to its initial state, before the first observation; i.e. the second observation is really of an object different that the first, and so forth with other observations. This is what Dirac calls observable. (It is not necessary to add that we deal here only with physical observables; hence, with magnitudes that can be measured in any observation; so that, at least at the beginning, this physics respects all the demands of the theory of relativity). This is something completely different from the fact of classical physics. And if I take the mean value of the measurements made on the same observable, I can consider this as its value. Measure has here, then, a completely different meaning too. In classical physics, "measure" signifies the really existing relation between the measuring instrument and what is measured. The measurement was a good or {280} bad approximation to the real measure, the only one which counted. But now "measure" signifies "I measure," i.e. I realize or can effectively realize a measurement This measurement is not an approximation to some true measure, but rather the measure is, in itself, the mean value of the measurements. We would, for example, call the velocity of an electron the mean value of the velocities resulting from many consecutive measurements on the same electron. If I now designate the observable by a symbol and put forth rules for combining these symbols, I will have an algebra of observables, and therewith physical happenings become dynamical variables which pose a mathematical problem.

What is the problem?

The problem of classical physics was the following: Given any system, I can measure it at two distinct times, t_l and t₂. Usually [246] I will find it in two different states. It is, then, clear that the system will have changed. I can then propose to investigate the real course of this variation, given the initial state. The symbols designating this initial state are the, expression of the real measure existing between the real quantities. And a mathematical law expresses the course of variation leading to the final state. That is to say, the mathematical equations, stripped of any imaginative allusion, are the formal expression of what really goes on in the system, without reference to any observer. The structure of the equations is the structure of reality. Let us take a simple example, the movement of a particle. At instant t₁ the particle occupies a place x₁ and has an initial velocity v₁. Newton's equations express the amount of variation which x and v really undergo, from the initial time t₁ to when the particle finds itself at point x₂ with the final time t₂ velocity v₂. Newton's equations describe, then, the trajectory leading from x₁ to x_2, {281} and the velocity possessed by the particle at each intermediate instant. The new physics takes things from another point of view. At time t₁ I make a measurement (in the sense previously indicated) of the position and velocity of the particle. Let x₁ and v₁ be the result of the measurement, i.e. the observables. After a certain time, at t₂ I again make these measurements and generally find results different from the first; i.e. at t₂ the particle is at x₂ with velocity v₂, where x₂ and v₂ signify once again the mean value of the respective measurements. I can now propose to find out what operations I have to go through with the measurements x₁ and v₁ to obtain x₂ and v₂. The conjunction of these operations is Newton's equations. In this case, the equations do not have, by themselves, any real sense: only the observations to which they lead; therefore they do not refer to what happens in the system between two observations. The sense of the equations is just this: given certain measurements at a specified time, predict future measurements made on the same object at any other time, i.e. anticipate observables. Independent of these, the equations have no meaning. Therefore, they do not express, in our example, the trajectory or the continuous variation of velocity. Neither of these concepts has the classical meaning here. So what does trajectory, [247] in fact, mean now? It is the conjunction of points at which I will encounter the particle, if I make measurements at the intermediate positions between that of departure and that of arrival. As these positions form a discontinuous succession, since they are chosen by one, two, three or more arbitrary acts of mine, it turns out that the graphical concept of trajectory lacks real meaning, through in classical physics it was a continuous line. The same can be said of velocity, as Schrödinger observes. We call "velocity" the distance separating two places which are occupied by the same body in two extremes of unit time. Therefore it is always a finite difference. But in the same way that it constructed a trajectory, classical physics constructs velocity at a point, by making the unit time infinitely small. In reality, this is something having no immediate physical meaning, i.e. measurable or operational meaning. {282}

The new physics does not pose or consider as physical problems other than those which refer to experimentally measurable quantities. This has permitted it to present itself as a natural extension of classical physics. If we Want to do all the operations necessary to reach the final state of a system given the initial state, the operations which Newton did are not sufficient; it is necessary to do others besides them, namely those of quantum mechanics, "Only when the equations of motion, along with quantum conditions, are given, " says Dirac, "only then will we know as much about the variables as classical theory, and only then can we consider that the system has been adequately characterized from the mathematical point of view."

And this is an essential innovation. Mathematics and mathematical physics are operations to be realized. Mathematical symbols are only operators: they lack any meaning other than that of being symbols of operations to realize on other symbols which designate observables. Mathematics is simply a theory of operations, it is not a theory of mathematical entities.

Of course this is no easy task, because the operators have to be defined with sufficient generality and uniqueness. Nor is fidelity to this requirement always easy. All too often anomalous cases turn up in which operators defined only for a priviledged system of coordinates are employed, though they cannot be applied to other [248] systems. It is as if, for instance, a distance were measured in meters, but could not be in kilometers. In Dirac, and also in Schrödinger, these cases are not infrequent, but are overlooked on account of their immediate success. And we do not mention Dirac's function, which has no mathematical meaning. It is true that von Neumann has managed to reach the same results as Dirac employing correct methods. But everyone recognizes that a strict foundation for all the methods and techniques employed today in the new physics would be impossible, at least right now. Of course this renunciation of truth gives rise from time to time to uneasiness, though it does let us predict experimental results There is more emphasis on the manipulation than on the understanding of reality. But, prescinding from these impurities, it would be reasonable to examine with some rigor in {283}what sense that which is called knowledge of the atom is, in reality, just that. It will be necessary then to examine the possibility that physics has renounced its status as knowledge, though I doubt-but I don't know how long I will persist in my doubt-that a theory of physical knowledge as purely operational could be viable. Mathematics has gone in such a direction. Brouwer says, "Mathematics is not a knowing, but a doing." However, the discussion of this point would carry us too far afield.

So with the problem of physics posed in the foregoing terms, what type of solution does the new physics offer? With the concept of quantity in classical physics it is clear that mathematical formulae lead from an initial quantity to a final quantity or quantities which are real; i.e. if we carry out measurements on the final state, the results will approximate more or less the true value of the quantity measured. A formula will be adequate when, among other conditions, it is such that the error of approximation is less than a predetermined limit: limit in Cauchy's sense. Only a small part of classical physics offered a different point of view, namely thermodynamics and the theory of gases. There is no [249] reason why two bodies of water with different temperatures, after being mixed for a certain time, equalize at some intermediate temperature. But the probability that this does not occur is infinitely small. The mean velocity of molecules in a gas enabled Boltzmann to explain its pressure. But always it was believed that this technique was justified only by the impossibility, in which de facto we find ourselves, of operating on individual molecules; and even if this were not the reason, on account of the enormous number of molecules with which we would have to deal. But Boltzmann did not doubt that the state of a gas was nothing more than the result of the actions of each and every one of its molecules. Very different is the situation in which the new physics of the atom finds itself. Be as it may the real activity of each molecule, from the moment it is unobservable, it lacks physical meaning. Physical laws are nothing but anticipations of experience, i.e. effective measured values, realized or realizable within the bounds of observation. Therefore, nothing has physical meaning other than that approximation which is really accessible to observation. {284} Hence, the order of magnitude of Planck's constant is a frontier, not merely de facto, but essential. Due to it physical laws, precisely because they deal with mean values of observations, have no other meaning than that of determining the distribution of these values; i.e. they are statistical laws. This does not mean that they lose their ideal character. Just as the classical laws, the laws of the new physics are ideal, limiting laws. But the reality measured by the value of practical approximations is not something independent of our observations, but rather the statistical limit of them: limit in the sense of Bernoulli. They are statistical limits. And for them, the order of magnitude of Planck's constant is a natural frontier. In classical physics the electron is at a place which perhaps I do not see, but which I believe has to exist. For the new physics the electron is where it can be found.

But this gives rise to a difficult situation. All physics attempts, in one form or another, to foretell the causal course of events, i.e. what happens independently of the observer. But the spatiotemporal scheme in which physics describes reality is itself founded on observations in whose content the observer intervenes. Whence there results an internal opposition-complementarity or [250] reciprocity Bohr called it-between causality and the spatiotemporal scheme that physics employs. Consequently, the very concept of observation is affected by an internal indetermination, on account of which it remains to be decided what things can be considered as observables or as media of observation. Whence the liberty of expounding with two different methods (waves and particles) the same reality. There is no way of escaping these difficulties, except by retaining the usual meaning of these two concepts, taken from everyday experience, and defining a posteriori the limits of their application. This is the work realized by the school of Bohr, and which led to Heisenberg's Uncertainty Principle. The problem rests, then, upon giving a unified theory of this complementarity: "Only if one attempts to create a system of concepts adequate for this complementarity between the spatio-temporal and the causal {285} descriptions, can we judge of the non-contradiction of quantum methods." (Heisenberg.)

The new physics has taken seriously this concept of probability and of observation. In contrast to the old physics, it has the virtue of audaciously accepting probability and moving therein without dissimulation. This is a task which has cost humanity centuries; more, perhaps, than that of acclimating itself to necessity. It has not been a whim or conceptual game-that is its great significance-but a requirement of the evolution of science, which began with Einstein and here has reached its maximum degree: the subordination of theory to experience. Probably the union of theoretician and experimenter in the very same person of the physicist has wider significance than the purely methodical one of erasing the isolation in which theoretical and experimental physics used to live. That union has a constructive sense for physics as such: the creation of experimental concepts, translatable into conceptual experiences. Both of them pertain essentially to the new physics. And by "experimental concepts" we do not mean concepts with which experience is in agreement, as if experience were something external to them and limited to "suggesting," proving, or rejecting them; rather in the experimental concept experience is an essential part of the concept itself. In classical physics almost all concepts are substitutions for experience. In the new physics the concepts are experience itself made into a concept. The meaning of a physical concept is to be in itself a virtual experience. Reciprocally, experience has a conceptual structure. Experience is the actuality of the concept. But this is no [251] longer a question of logic; rather, it is one of ontology. And this is the definitive point. Heisenberg touched on this problem when he spoke of complementarity. It is the problem of what should be understood by physical reality, i.e. what is nature in the physical sense. At the bottom of the evolution of contemporary physics there is taking place the elaboration of a new idea of physical reality, of nature. For this reason, and in this precise sense, I call the new physics "a problem of philosophy." {286}

[252] {287}

This problem of complementarity is what impelled Heisenberg to formulate the Uncertainty Principle: any simultaneous determination of position and velocity of an electron results in an essential error of an order of magnitude no smaller than Planck's constant (). As we have said, for any measurement I must illuminate the object measured, and in the case of electrons, the light modifies their position and velocity. The concepts of wave and particle lose their meaning when we deal with atomic magnitudes, so that the Uncertainty Principle supplies the real foundation of this new concept of the physical universe. And "real foundation" is precisely what must be clarified, because it could well happen that this expression is ambiguous.

Uncertainty or indetermination seems to be what is most opposed to the character of all scientific thinking. Planck, therefore, indignantly rejects this concept; to renounce determinism would be to renounce causality, and with it, everything that has constituted the meaning of science from Galileo up to the present day. If our measurements on the atom are indeterminate, it would seem to say that our manner of investigating it is likewise indeterminate. Indeterminism, if it exists, would be for Planck a characteristic of the present state of our science, but in no way a characteristic of things themselves.

But regardless of the ultimate fate reserved for physics, Planck's attitude' categorically denies the anomaly to which Heisenberg's principle grants a place. {288}

Above all, it is unnecessary to interpret the said principle as a negation of determinism. It is possible that things are interrelated by determinate links, i.e. that the state of the electron in an instant of time univocally determines its later course. But what Heisenberg's principle affirms is that such a determinism has no physical meaning, on account of the impossibility of knowing exactl y the initial state. If this impossibility were accidental, i.e. if [253] it depended on the subtlety of our means of observation, Planck would be right. But if it is an absolute impossibility for physics, i.e if it is founded in the very nature of measurement as such, the presumed real determinism escapes physics. It no longer has physical meaning. In such case, the Uncertainty Principle would not necessarily be a renunciation of the idea of a cause, but rather of the idea that classical physics formed of causality. This, and nothing more, is the scope of the Uncertainty Principle. It is not a statement about things in general, but rather about things as objects of physics. And precisely for this reason, because the new physics is pure physics, it renounces everything earlier which is a mixture of what is physics and what is not.

And secondly, in response to Planck, it is not true that the idea of nature, in the physical sense, is the idea of the nature of things simpliciter. In fact, Galileo's great work consisted in distinguishing these two ideas and attempting to give physical sense to physics. This task had been fully prepared in the ontology of Duns Scotus and Ockham, but was only realized explicitly and in mature form in the work of the Pisan thinker. For Galileo there is a radical distinction between nature in the sense of nature of things and nature in the sense of physics; and analogously, a distinction between causality as an ontological relation, and physical causality. The latter seeks to measure variations; the former, to discern the origin of the being of things. This distinction has sufficed to the point that an uncontrollable variation, i.e. something which does not vary at all in our experience, has no physical meaning; such, for instance, is the supposition that the universe is characterized by uniform rectilinear motion. Physics cannot occupy itself with {289} the origin of things, but only with the measurement of their variation; it is not an etiology, but a dynamics. Force is not the cause of being, but the reason for changes in state. In this sense, inertial movement does not require any force. So, not only is it untrue that the idea of cause gave rise to modern science, but in fact modern science had its origin in the exquisite care with which it restricted this idea. That renunciation was for the representatives of the old physics the great scandal of the epoch. How is it possible for physics to renounce explanation of the origin of all movement? This heroic renunciation, nevertheless, engendered modern physics. Hence it is not permissible to whisper of scandals in the face of Heisenberg's principle; it is rather necessary to faithfully examine the situation and see if it [254] does not give to physics its ultimate stroke of purity.

Summarizing:

1. Like every science, physics utilizes certain methods to discover truths about things. Such, for example, is the use of differential equations or various practical methods of measurement. The methods, thus understood, are an aspect of the cognitive activity of man, and every affirmation about them is an affirmation of logical character. But the methods, in plural, are diverse with a certain unity: they attempt to move us closer, in the most efficacious manner, to the things present to us. Therefore they presuppose that these things are in fact present to us. If one desires to continue using the word "method" for this primary presence, it will be necessary to understand by "method" something different than what is understood when we speak of the "diverse methods" of physical science. "Method" here will be the primary discovery of the physical world, as opposed to the other methods, which discover to us some of the things that there are in this world. All methods, then, are possible thanks to a primary method, the method whose result is not knowledge of what things are, but rather to put things before our eyes. Only in this sense can it be said that science is defined by the world of objects to which it refers. This operation is by no means insignificant. After Aristotle we had to wait for Galileo to put {290} before our eyes a world different than that which Aristotle discovered to us: the world of our physics. Galileo has instructed us to see what we call "world" with a different vision, viz. the mathematical one. All the other methods presuppose that "the great book of nature is written with mathematical characters." A mathematical vision of the world is the work of Galileo. Affirmations dealing with method thus understood are no longer affirmations about human knowledge and therefore, not logical affirmations-but affirmations about the world, real affirmations.

2. These real affirmations do not constitute affirmations about what things are, simpliciter. I can, for example, say that things have always existed, or that they have been created by God; that none has in itself the principle of movement, or that some move themselves; that their essence is extensio (Descartes) or vis (Leibniz), etc. Correctly viewed, none of these affirmations is a physical truth. They are, it is true, affirmations which refer back to bodies'. But it is not quite true to proclaim without further qualification that physics is the science of bodies. Physics does not [255] consider bodies insofar as they are. It is not to them that the methods I alluded to above are applied.

3. Physics is directed to natural things. (Let us leave aside the complications that biology would oblige us to introduce into this problem if we wanted to be completely rigorous). Physics begins not when we deal simply with things, though they be corporeal; but rather when the meaning of the adjective "natural" is made precise. What do we understand by "natural"? What is "nature"? An answer to these questions has to be an affirmation that will mark off, within the world of what there is, those entities which fall within the region of the natural. Hence, it will have a double dimension. On the one hand, it will look at the whole world of things that are; on the other, it will look at the interior of a region of it. In its primary aspect such an affirmation will be a methodical negation of everything which is not this new region; consequently, within its negativity, it will pose for ontology the problem of discerning the regions of being. But with respect to {291} the second aspect, it will be an affirmation giving primary meaning to what there is in this new region. It will be, then, what permits establishing or placing things in the region; it will be the principle of their positum, of positivity, a positive principle; i.e. it will permit giving univocal meaning to the verb "to exist" within the region; it will have given rise to a positive science. To these principles Kant gave the name "Original metaphysical principles of natural science." And science has always had the impression that such principles are, in fact, philosophical. It suffices to recall the title of the mechanics of Newton: Mathematical Principles of Natural Philosophy.

Now, the Uncertainty Principle is not primarily a logical one. It is not an affirmation about the scope of our means of observation, but rather about observable things. It has nothing to do with the subjectivity or objectivity of human understanding. The relation found between light and matter is perfectly real, just as the vision of a cane submerged in water is not less real or more illusory than that which we have of it when it is out of the water. In both cases we are far removed from any subjectivity. The relation between a photon and an electron is just as real as the law of gravity or the principle of inertia.

But neither is the Uncertainty Principle a principle of ontology in general, as if it pretended to deny the existence of causality. Whatever may be the verdict on that, it does not affect the [256] Uncertainty Principle at all. Causality is not synonymous with determinism; rather, determinism is a type of causality.

The Uncertainty Principle is one of those principles of regional ontology which seeks to define the primary sense of the expressions "natural" and "nature." Or in other words, to define the meaning of the verb "to exist" in physics. And this is the question which must be analyzed with some precision.

1. Since Aristotle, the conjunction of items grouped under the name of physics has been understood as referring to things which change, or, as he said, {292} which move. (Aristotle's Physics is not a physics in our present-day sense, but the difference only comes to mind when we recall the double ontological and positive dimension of the work.) The word "nature" signifies movement, actual or virtual, which emerges from the very depths of the being which moves. To emerge from the depths this way is essential to the movement. For this reason the physis is properly the arkhe, the principle of kinesis. But to describe the meaning that nature has for Aristotle in its entirety, we must see how he views movement. Without the necessity to enter into commentary on his definition, or even to quote it, it suffices to say that for Aristotle movement always involves a coming to be; he considers movement from the point of view of being. It is also true to say that he looks at being from the point of view of movement. And Aristotelian physics rests squarely on the internal unity of both of these viewpoints. Now, what a thing is becomes patent to me when I consider it as something determinate among all the rest; hence, when I regard it from the point of view of metron, measure. "Measure" does not here signify anything primarily quantitative, but rather the internal unity of being as such, the hen, the one. Measure, in a quantitative sense, is based on this more general concept of measure as ontological determination. When I regard things from the point of view of measure, they appear in their proper figure, in their eidos, their idea. In it, then, is contained what the thing truly is. The idea is therefore its form, where form has as little to do with geometry as measure with arithmetic. What a thing is, its idea, is thus what is seen in a certain special vision, in the noein, which gives us its measure and its form. In what a thing is, therefore, its being and the being of man are linked in a radical unity. Treating movement from the point of view being is treating it from the point of view of measure. And the principles which give ontological reality and precision to movement are consequently principles of [257] being, i.e. causes. This then is the meaning of Aristotelian physics. Nature is taxis, order, measure of causes. {293}

For Aristotle, this point of view is common to every class of movement, including local movement. Suffice it to recall that place, for Aristotle, is an ontological category, and that therefore change of place is change of mode of being. But he realized that it is precisely in local movement where this dimension most easily escapes us. Whence his opposition to mechanical explanations, not because he considers them necessarily false, but because they do not affect the being of things. On this point, Aristotle has almost always been misunderstood, because it can be said that he goes against common sense, which is not very flexible with respect to ontology. And, if the truth be told, it must be recalled that Aristotle is the first in the history of human thought (Plato is very confused) and the last to have conceived movement ontologically.

2. In fact, the natural propensity of the mind is just the opposite. Man inexorably seeks to elude non-being. Hence he eludes all true becoming, because all becoming is coming to be from what was not. We tend, then, to disguise the real significance of this non-being, thinking that movement is simply an appearing of what already was, but was obscure; or a disappearing, i.e. continuing to be in a hidden way what before was patent. Since Democritus, for example, atoms have served to skirt the abyss of not-being. The atoms are invariable, indestructible, eternal; things are, for Democritus, aggregates of atoms; hence their generation is a simple combination of what already exists, but not a true generation, i.e. a becoming. Aristotle emphasized on various occasions the difficulties encountered by the atomistic concept of generation. Consequently, the movement preferred by all atomists is local movement, not only because it is clearer and more distinct, as Descartes said, but because it is, as Aristotle realized, that in which it is easiest to elude the problem of the origin of being. Indeed, local movement is the clearest because it makes least reference to non-being. There is no coming to be of what was not, but a mere variation of what already is. {294} When regarding movement from the point of view of being in general was renounced, quantity and movement thus became the interpretive principles of reality. The distinction between movement as becoming and movement as simple variation is essential not only to physics, but to ontology as well. It implies a [258] radical reform of the Aristotelian meaning of nature. But only a reform, because the conceptual scheme in which we move derives squarely from Aristotle. In this sense, modern physics could not have been born without Aristotelian ontology, even though the latter had to be reformed in some of its points.

What things are, said Aristotle, is in effect present when I look at them from the point of view of their measure. But while for him measure was ontological unity, it now has been converted into quantitative determination. Hence, the nous, the mens, sees the being of all things from the quantitative point of view. It is in measurement now that man and the world are linked. Measure is now the meaning of mens and the meaning of things. For this reason Nicholas of Cusa said, repeating a phrase of St. Thomas, that every mensura is the work of a mens. This is the consecration of the mathematical method. And, reciprocally, the thing seen by the mens is a measurable determination: Aristotelian form is turned into material configuration. And from antiquity the idea has been gaining strength that in the metron as quantity (materia signata quantitate) is contained the explanation of individual things. Reality is quantitative measure. Thanks to Aristotelian ontology, mathematics now acquires the rank of an ontological character of reality. With it the meaning of the verb "to exist" is circumscribed: only the measurable has physical existence. Movement, as pure variation, is seen from the mathematical point of view as a function of time. Therefore all movement is, at bottom, just like local movement: a function; it is stripped of connotation of generation and destruction. The "always" of nature is its mathematical structure. Nature is no longer order of causes, but norm of variations, lex, law. And every law is the work of a legislator. Nature is thus a law imposed by God on the course of things. Our concept of natural law has this double ontological and theological origin. The course of things is such that {295} the state they possess at each instant determines univocally any later state. Nature is, in this sense, a habit of God. That is, the formal character of law is the determinatio, the determination. Thus it can be captured with security and certainty by man in a mathematical function. It is essential to record here these too often forgotten connections. With them it is easy to understand the sense of the expression "phenomenon:" as aspect of nature; hence, not a thing, as for a Greek, but a happening, an event. This happening will be understood when we know its place in the course of nature. [259]

This is obtained by measurement. And here we have the origin of modern physics: measuring the variations of phenomena. Modern physics is anything but the invention of a new special method; it is the enthroning of the ontological and constituent character which mathematics has acquired as the interpretation of reality. In this physics, there is no question of the origin of either things or movement, but only of the variations of initial states. Every body tends to remain in its state of rest or uniform rectilinear motion as long as there is no force acting on it. Such is the principle of inertia and its double ontological and positive significance.

But this does not mean that the Aristotelian concept has been abandoned, only that it answers another problem, viz. the problem of being in general. It is possible to interpret determinism as causality, admitting that causes act determinately. But even so, it would not do us any good, not because causes are not real, but because they have no physical meaning.

Analogously, the objects of physics arc not seen from the point of view of being: they are not entia, things, but simple phenomena, that is, manifestations of what already is, just as movement is simple variation of what is. The phenomena of nature are not the things of the world. Hence, concepts of mass, material, etc. which up to now have been assimilated to the idea of thing, henceforth signify something different. They correspond to different problems. Mass, for example, is nothing more than the quotient of force by acceleration, and so forth. But just as variation neither excludes nor includes causality, {296} phenomenon neither includes nor excludes entity in the sense of thing. It should be pointed out that this concept of phenomenon has nothing to do with the phenomenalism currently under discussion in the theory of knowledge.) The problem of nature is not, for Galileo, a problem of entity and causality sensu stricto. The cardinal difference which makes a being, besides existing, to be natural, is not that its movement is caused in a certain way, but that it is determined as phenomenon, i.e. measured in the course of nature: nature = measure of a course = phenomenonal law.

The development of this idea is the history of physics from Galileo up until our time. It is a history which is nothing but the labor of refining this concept of "nature." It explains why the formation of natural concepts is in no way similar to a simple abstraction, but is, on the contrary, a construction, and more concretely, the construction called passing to the limit. And by this [260] I do not only refer to the infinitesimal method, but to every application of mathematics to physics; a simple measurement is already, in this sense, a passing to the limit.

Now, this pass to the limit and all other mathematical operations, independently of their utilization in physics, have a meaning proper to mathematics by itself. And the result is that physics has had a propensity to define physical existence as a simple case of mathematical existence. A physical reality is existent when it is determined as a mathematical function. Whence it follows that measure is a relation between mathematical magnitudes. What has happened then to the phenomena? The true reality is the mathematical relations; a phenomenon is something which remains outside of them and only acquires physical meaning, i.e. only is properly a phenomenon when it is submitted to mathematical laws. Nature in the sense of physics, and experience have grown farther apart until now they are almost completely separated. In fact the latter acquires physical meaning, physical usefulness, only insofar as it is submitted to this other world which is nature properly so-called: mathematical laws. Consequently, the only physical meaning of experience is to be an approximation. That is, understanding experience {297} is nothing more than determining which systems of mathematical relations we have to substitute for it.

As long as mechanics dominated all of physics in despotic fashion, the success of such a conception could not be doubted. But physics has to give explanations of things which apparently are not movements: temperature, colors, sounds, etc. And it is easy to understand that physics would devise a subterfuge so as to avoid speaking about the origin of colors as if we were dealing with a generation ex nihilo: such was ;he establishment of a bijective correspondence between the facts of color perception and certain quantities submitted to mathematical laws. With it, the coming to be of colors turns into a simple modification of what already is: particles or elastic media. Once again, the sensible happenings or facts corresponding to these quantities have been relegated to the border of physics; they are, in the last analysis, approximations which suggest, corroborate, or contradict the truth of mathematical laws. But in and of themselves they are nothing, they do not form part of nature.

But the time came when these sensible happenings began to force a change not in this or that law, but in the very concept of [261] law. At that moment science, just as in Galileo's time, had to ask itself about its proper world and inquire, "What is the physical world"? This is where it finds itself today. Let us look at it.

3. This uneasiness began with the study of electrical phenomena. Since Maxwell, electricity has not been governed by mechanical laws. It has its own laws. An abyss separated these two regions of the physical world, the world of motion and the world of electromagnetism. There was only one possible point of contact: Hamilton's principle. But this principle is not purely and exclusively mechanical in the usual sense of the word; it is a variational principle of much greater scope. Hence, within mechanics itself a breach was opened for a possible radical reform. To obtain the equations of mechanics starting from the invariant integral of Hamilton is to concede the subordination of mechanics to more general principles. Physics was no longer mechanical, {297} but mathematical. Not every function of time was necessarily local movement.

But things did not stop here. Electromagnetic laws are not only distinct from, but in a certain way, opposed to the mechanical laws. The velocity of light is constant, not only in vacuo (i.e. measured with relation to the aether), but also referred to any observer in an inertial system (that is, characterized by uniform rectilinear motion). Now, no one dared put his hands on Maxwell's laws, which are such an admirable theoretical and experimental work, about which Helmholtz used to ask if "some god had written them." It was Einstein who had the genial audacity to reform mechanics, setting himself the question of the meaning of measurement, and with it, of physical nature.

The measurement to which physics prior to Einstein referred was a relation between mathematical quantities in space and time. Consequently, physical existence had the same meaning as mathematical existence. After Einstein, this is not true. Physical existence is mentally distinct from mathematical existence. Or, as seen from the mathematical point of view, mathematics as the meaning of nature, physics, must not be confused with pure mathematics, To physics belongs light, i.e. the entire electromagnetic field and all matter. Hence the quantities from which physics, including mechanics, starts are cosmic quantities, i.e. the indivisible complex space-time-matter (including fields). Measurement is not a relation among mathematical quantities, but among cosmic quantities. The world of the so-called sensible things and [262] the physical world are not two distinct worlds; the former is part of the latter. To this the name "geometrization of physics" has been given. Also, and perhaps with more propriety, it could have been called "physicalization of geometry." And at this point the interpretation of movement as pure variation reached its perfection; so much so, that Weyl believed it possible to eliminate any reference to real movement of bodies, in order to speak instead of a simple variation of the field in which they are located. It is impossible to go farther away from the idea that movement, in the sense of our physics, has anything to do with becoming. {299}

That is to say, the so-called geometrical structure of the universe depends-this is essential-on what used to be called reality. And, reciprocally, nothing has physical meaning unless it is a cosmically measurable quantity. Now, the physics of Galileo-Newton-Lagrange has quantities which are not measurable in this sense, e.g. absolute space, absolute time, bodies independent of space and time, etc. Whence the physics of Einstein is in many respects the culmination of classical physics: physical nature is real measurability.

But this word 'real' involves an ambiguity which must be clarified. One could think that this expression alludes to observations of an observer. Then, the meaning of Einstein's work would be to give a description of the universe valid for every observer from whatever point of view. That is, Einstein's physics would be, not a physics without an observer, but a physics with any observer whatever. This is true. But it is not the whole truth, nor even the essential or primary truth. The condition of invariance of physical laws does not refer primarily or fundamentally to the picture which an observer acquires of the universe, but rather to the structure of the universe, relative to any system of coordinates. But to this it is necessary to reply first, that the reciprocal is not true, and second, that then the system of coordinates is not to be interpreted as a point of observation, but on the contrary the point of observation as a system of coordinates. That is, the "human" mediation of physical quantities does not enter at all into the concept of measurement. The measurement is a relation which exists, i.e. is defined, among "cosmic" unities, but just as independently of the existence of the physicist as mathematical proportions exist independently of the mathematician. Mathematics is still, therefore, in Einstein's physics the formal structure of nature. Mathematics and matter have been fused [263] together in a world, but man is left out of it.

But quantum physics takes the decisive step. In it, too, nature is real measurability. Here 'real' does not mean simply cosmic, as for Einstein, but effectively observable. {300} Measure does not mean only existence of a relation, but I can "make" a measurement. Nature = real measurability = measurement of observables. Here we have what Heisenberg had to clarify for us when he enutiated the Uncertainty Principle, or in other words, when he inaugurated a new epoch in the history of physics.

Provisionally, 'observable' signifies for him visible, in a concrete sense; positions and velocities cannot be effectively measured without being seen. Visibility does not refer then, to subjective conditions, but to the presence of things in light. But one then speaks of light in two radically different senses. In the first place, it is something which acts on things. In this sense, it is apart of what nature is. But if this action is supposed to give rise to an Uncertainty Principle, then I am considering light from the second point of view, not as something which acts on things, but as something which permits them to be seen, which makes them visible, i.e. makes them patent. These are two completely different meanings. In the first, light is a part of nature; in the second, it totally envelopes it: it is what constitutes the very meaning of what should be understood by nature, what separates it from everything which is not nature. In the first acceptation, light is a fragment of nature, an electromagnetic and photon phenomenon which occurs in it. In the second, light is simply clarity, and therefore is not so much a phenomenon as what constitutes phenomenality as such. Dislodged from physics at the end of the Middle Ages, light as clarity reenters it. And if the first function of light is independent of man, the second makes essential reference to him. From the coincidence of both points of view the Uncertainty Principle is born, and this coincidence is purely human. The indeterminancy between position and velocity on account of the action of light does not arise unless there is a being who desires or needs to make use of light to determine the position occupied by bodies and the velocity with which they are animated. This does not occur in the theory of relativity. In it the existence of the physicist is necessary for there to be physics; but in the meaning of physics the nature of the physicist plays no part; what he does, does not pertain to physics, or at least, {301} does not pertain to it in the same sense as in quantum theory. In relativity theory, the physicist is limited to [264] putting some things in relation to others; but in the content of this relation man does not intervene. In quantum theory not only does man put some things in relation to others, but nothing beyond that which is visible in such a relation has any meaning. Only then does it make sense to speak of indeterminancy. And this indeterminancy arises because light has both functions: it is, at the same time, a part of nature and that which envelopes it. Every being which physics deals with has to be referred, ultimately, to sight; if I handle temperatures, it is necessary to see the height of the mercury column in the thermometer, etc.

In other words, classical physics occupied itself only with the relative localization of some bodies with respect to others in the course of time, as measured by a periodic movement. Whence follows that the supposition-the condition, Kant called it-of every physical phenomenon, i.e. the formal structure of what is called 'nature,' is the spatio-temporal scheme, regardless of whether one considers it something a priori, as Newton and Kant maintained, or something a posteriori, as Leibniz and Einstein believed.

But the new quantum physics realizes that this is not sufficient: something is not a phenomenon, primarily, through its localization in a simple spatio-temporal structure, but through "visibility," if I may be permitted the expression. So it turns out that the supposition or condition of all phenomenality, the formal structure of nature, is light in the sense of clarity.

Hence, while for classical physics a law enuntiates the character of the articulation of a phenomenon with spatio-temporal structure, in the new physics a law enuntiates, in a certain way, the articulation of a phenomenon in the field of clarity, in which it is visible, and thanks to which it is "observable."

But this second point of view clearly involves the first: what is seen' is the spatio-temporal 'localization' of the material (in the broadest sense, including energy). Through this implication Heisenberg's indeterminancy is inexorably produced, {302} and what the Uncertainty Principle effectively expresses is this new idea of nature.

In fact, if this attempt is successful-it is not yet time for a decision, nor do I feel myself qualified to make it-we should say that not only mathematics and matter, but the mathematical, the material, and the visible enter into the concept of nature, in a compact unity. That is to say, 'Space-Time-Matter-Light' (in the [265] sense of clarity), the observable: this is Nature (this sense of observable' does not exactly coincide with Dirac's usual meaning). Physics, even more than in the case of Einstein, has nothing more than a human meaning. Strictly speaking, for God not only is there no physics, there is no Nature in this sense, either.

Thus, phenomena are not approximations to ideal objects of physics; rather they are the very objects themselves. The phenomena of Galileo become observables. Therefore the atoms, electrons, etc. rapidly lose their old meaning and become words designating a system of phenomenological relations. Let us recall, once again, that since Galileo the object of physics is not things, but phenomena. Consequently, when contemporary physics speaks of the equivalence between waves and particles, it does not imply that material things soften and become diluted in some vague and formless reality, but that this equivalence is, in fact, a purely phenomenal equivalence. The concepts of particle and wave are interpretations of observables. For this physics does not have to take leave of observables and substitute objects of thought for them. The new physics does not substitute some beings for others. It must certainly pass to the limit; this is a pass to the limit within the phenomena, the limit of Bernoulli. The mathematical expression, considered as a law, has no meaning other than that of being a conjunction of virtual observations; consequently (given its concept of measure) it is the probability of an observation, not the real determination of a state. Or in other words, for physics the real state of something is only that in which I see it. Whence mathematics, which since Galileo has served to define the metron of what things are, now becomes a purely symbolic operation. It is not a geometrization or an arithmetization, but {303} a symbolization of physics. Movenient not only is not a coming to be, or even a variation of things, but an alteration of observables.

Summarizing, for Aristotle nature is a system of things (material substances) which come to be though causes; for Galileo nature is mathematical determination of phenomena (happenings) which change; for the new physics nature is distribution of observables. For Aristotle physics is etiology of nature; for Galileo it is mathematical measurement of phenomena; for the new physics it is probability calculation of measurements of observables. In the crisis faced by the new physics, whatever its resolution [266] may be, we are not dealing with a problem internal to physics or with a problem of logic or theory of physical knowledge; we deal, ultimately, with the problem of the ontology of nature. The intent of this brief essay is to show that that is so.

It is scarcely necessary to say that with respect to a complete system of physics, we have not yet left a more or less purely pragmatic phase. Nor is this program, in the opinion of everyone, even realizable. I cannot forget what Einstein told me on one occasion: "There are among the physicists those who believe that science is only weighing and measuring in a laboratory, and regard everything else (relativity, unification of fields etc.) as extra-scientific labor. They are the Realpolitiker of science. But with only numbers there is no science. A certain religiosity is required Without a type of religious enthusiasm for scientific concepts there is no science.... Others abandon themselves to statistics. An electrical phenomenon has associated with it a value of probability. Very well, but a probability that something will be present obeying Coulomb's law. And this law? Also a probability. I do not understand it. It is conceivable that God could have created a different world. But to think that at each instant God is playing dice with all the electrons in the universe, this, frankly, is 'too atheistic'."

In this problem, positive science is nothing other than the obverse of ontology. That is to say, we have an ontological and scientific problem at the same time. Science will only be able to ask for a new concept of {304} nature, and later discard it; but, by itself, cannot create it. Without Aristotle there would have been no physics. Without medieval ontology and theology Galileo would have been impossible. "The adaptation of our thought and our language," says Heisenberg, "to the experience of atomic physics, is indubitably accompanied, just as in relativity, with the greatest difficulties. In the theory of relativity earlier philosophical discussions about space and time were very useful for this adaptation. Analogously benefits can be reaped for atomic physics from the fundamental discussions of the theory of knowledge about difficulties inherent in a split of the world into subject and object. Many abstractions characteristic of modern theoretical physics have been dealt with already in the philosophy of past centuries. And while these abstractions were discarded then as though games by scientists, attuned only to realities, the refined experimental art of modern physics compels us to discuss them in [267] depth.

The fact that this physics is provisional is not a reproach, but a eulogy. A science which finds itself in the situation of being unable to advance without going back and revamping its principles is a science which lives at every moment. It is living science, and not simply an office, That is, it is science with spirit. And when a science lives, i.e. has spirit, the scientist and the philosopher meet in it, as we have seen, because philosophy is nothing but intellectual spirit and life.

"The physicists," wrote Heisenberg in 1929, and his words are even more apropos today, "will not see themselves, in the decades to come, compelled to stay within the bounds of a domain which has already been completely explored, rather, they will have to leave it behind and seek adventures in unknown territories."

We hope that in this adventure, in which the entire human intellect emotionally accompanies them, the physicists will not lose themselves, but that they will find themselves there where spirits always find themselves: in the truth.

Cruz y Raya, 1934