[217]{243}

[218] {244}

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.

[219] {245}

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 3*n *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}

- 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 t_{2}. 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 *t*_{1} the particle occupies a place
*x*_{1} and has an initial velocity *v*_{1}.
Newton's equations express the amount of variation which *x*
and *v* *really *undergo, from the initial time t_{1}
to when the particle finds itself at point x_{2} with
the final time *t*_{2 }velocity *v*_{2}.
Newton's equations describe, then, the trajectory leading from
*x*_{1} 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 *t*_{1} I make a measurement (in the sense
previously indicated) of the position and velocity of the particle.
Let *x*_{1} and *v*_{1} be the result
of the measurement, i.e. the observables. After a certain time,
at *t*_{2} I again make these measurements and generally
find results different from the first; i.e. at *t*_{2}
the particle is at *x*_{2} with velocity *v*_{2},
where *x*_{2} and *v*_{2} 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 *x*_{1} and *v*_{1}
to obtain *x*_{2} and *v*_{2}. 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*

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