Toward a Consistently Materialist Interpretation of the Foundations of Mechanics

S. G. Suvorov and R. Ya. Shteinman

The struggle against bourgeois ideology, which developed especially actively after the well-known decisions and instructions of the Central Committee of the VKP(b) on questions of literature, art, philosophy, and biology, has brought a number of urgent problems to the fore also for physicists. Naturally, the fundamental questions of quantum mechanics and, in part, the theory of relativity came to occupy the center of attention of Soviet physicists and philosophers. The reactionary views of the so-called Copenhagen school, which has exerted the greatest influence on foreign physicists and, to a certain extent, on some Soviet physicists, were subjected to serious criticism.

However, the tasks of combating reactionary tendencies in physics, and of upholding a consistently materialist worldview in science, are by no means limited to the physics of the microworld. This struggle must also be waged over the fundamental problems of classical physics, which contemporary bourgeois ideologists are trying to interpret perversely, in an idealist manner*.

It would be mistaken to think that classical physics, in particular mechanics, as something “long since established and settled,” lies outside the bounds of ideological struggle. This is not so, first of all because science is not reducible to a sum of empirical observations expressed by mathematical equations, as the positivists maintain. It also includes definite conceptions that reflect the nature of material objects and the laws of motion inherent in them. Therefore every natural science, including classical physics, contains in an implicit form a theory of knowledge. The same facts or equations may be interpreted

* An example of such an idealist treatment of the foundations of physics, beginning with the laws of classical mechanics and ending with the theory of elementary particles, is the book by one of the present-day leaders of Machism, the physicist P. Frank: Foundations of Physics; this book appeared in 1946 in Chicago; it is a component part of the encyclopedia of logical empiricism, International Encyclopedia of Unified Science, whose advisory council also includes the head of the Copenhagen school, N. Bohr.

from various theoretical-cognitive positions, and this interpretation enters into physics itself as a constituent part and influences the further paths of its development.

Our Party teaches us the necessity of evaluating every theory in the light of the theory of knowledge of dialectical materialism. Such are our tasks also in the sphere of interpreting the foundations of mechanics. That in this sphere not everything is satisfactory is attested by much evidence. Thus, in recent years, in organizations of Moscow University, in the Physical Institute of the Academy of Sciences of the USSR, and also in a number of reviews, the book by Prof. S. E. Khaikin, Mechanics, has more than once been subjected to criticism. The author was justly accused of hushing up the scientific investigations of Russian scholars and of errors of a Machist kind.

In this article we set ourselves the task of showing that a correct interpretation of the foundations of mechanics can be given only by being guided by a consistently materialist theory of knowledge, i.e., by adhering to that understanding of the foundations of mechanics which, in its general features, was already outlined by F. Engels. At the same time we consider it necessary to show what harm is done to the correct understanding of science by present-day idealism. This can most easily be done by criticizing the interpretation of mechanics that was given in its time by Mach and that is adhered to by present-day Machists.

We hope that such an analysis will contribute to a fuller disclosure and elimination of the fundamental errors that have been made on questions of mechanics by certain Soviet scholars, in particular by S. E. Khaikin, and will also be of some assistance in the work of creating a new textbook on the basis of a consistently materialist world outlook*).

The development of physics in the second half of the nineteenth century led to the necessity of a deeper understanding of the foundations of Newtonian mechanics. In order to clarify what influence the development of physics of this period had on the interpretation of the laws and concepts of mechanics, it is necessary at least briefly to consider the basic ideas of Newtonian mechanics.

It is known that Newton made a considerable step forward in comparison with the Cartesians, formulating the fundamental law of motion of a material point and of the interaction of a pair of material points. He expressed them in the form of three axioms of mechanics—the axiom of inertia, the axiom of proportionality of force and change in the quantity of motion, and the axiom of equality of action and reaction.

*) We consider that, above all, questions connected with the understanding of the foundations of mechanics must be discussed; this is the subject of the present article. As for questions of the methodology of teaching courses in physical mechanics on a consistently materialist basis, they deserve special consideration; in this article we shall not touch upon them.

The fundamental law of motion of a material point is expressed by the equation

\[ \frac{d\bar p}{dt}\equiv m\frac{d\bar v}{dt}=\bar F. \]

In Newtonian mechanics this equation was regarded as the initial proposition of mechanics. The basic concepts of mechanics were taken to be: force as a characteristic of the interaction of bodies, mass as the basic parameter of a body, and momentum as a characteristic of the state of a moving body. The principal problem of mechanics was formulated as the problem of finding the motion of a body from a given action of a force and, conversely, of finding the force from a given motion.

Newtonian mechanics is based on the assumption that, in the general case, the action of a force is independent of the state of motion (the velocity of the body), that the mass of a body is independent of its velocity, and also on the assumption that the actions of forces are independent of one another (the principle of superposition of forces). These assumptions imply that the action of bodies is transmitted with instantaneous velocity, i.e. they exclude the idea of the transmission of action as a certain physical process occurring with finite velocity. The very motion of a body is defined as its transfer through empty space.

Newtonian mechanics contains the important proposition of the conservation of momentum of a closed system, which is nothing other than a further generalization of the law of inertia. It is precisely this proposition that was preserved in subsequent generalizations of classical mechanics.

However, Newtonian mechanics did not exclude the possibility of the traceless disappearance of motion in nature under the action of dissipative forces. Only under the assumption that exclusively central forces act in nature was it possible to pass to the formulation of the law of conservation of the sum of kinetic and potential energies in a closed system.

Newtonian mechanics was a major step forward in comparison with the Cartesian mechanics that preceded it; it freed one from the need to invent, for each problem, a concrete model of the interaction of bodies; without going into the nature of the interaction of bodies, Newtonian mechanics provided a more general method for solving mechanical problems than the ad hoc devices that existed in Cartesian mechanics.

The laws of the mechanics of material points discovered by Newton retain their significance even today for a certain domain of mechanical processes, a practically very broad one.

However, the extremely abstract and limited content of the concept of force in mechanics served as the basis for the absolutization of forces by the Newtonians. Force, among the Newtonians, was regarded not as

some characteristic, aspect, moment of a physical process, but as the cause of motion, existing independently of the motion of matter, as a special principle, which, even if attributed to bodies and particles, nevertheless acts independently of the properties of matter. Forces were endowed with absolute properties; among such properties is the instantaneous action of force through empty space (action at a distance).

In this conception matter appears as something mobile, passive, inert. The properties of a body in motion appear as independent of the state of motion.

But if force is a special object that causes a change in the motion of a body, then the conclusion is inevitable that Newton’s equation, which determines the action of force, is not only the initial but also the most general equation in mechanics, embracing any mechanical processes, however complex. Physicists invariably drew this conclusion up to the last quarter of the nineteenth century.

The law of the conservation and transformation of energy, discovered in the middle of the nineteenth century—this universal law of nature, establishing the connection among all forms of motion of matter—led to results that in essence proved to be in contradiction with the old notions of the interaction of bodies. The physical theories that developed on the basis of this law created a new understanding of interaction.

An especially radical change in the conception of interaction was introduced by the theory of the electromagnetic field, which proved that the interactions of charged bodies are transmitted with finite velocity, and also established the existence of a new kind of force, essentially dependent on velocity—the Lorentz force. As a result of these discoveries, a reconsideration of the foundations of Newtonian mechanics began.

The most profound analysis of classical mechanics was given by Friedrich Engels.

Engels generalized the development of nineteenth-century natural science from the standpoint of dialectical materialism. For the first time in the history of natural science he gave the correct interpretation of the law of the conservation and transformation of energy, emphasizing in it the transformability of qualitatively different forms of motion. In contrast to natural scientists who tried to reduce all forms of motion to the mechanical, Engels substantiated a new understanding of motion, which in the general case is not simple displacement, but change in general: “Motion is not only change of place,” says Engels, “in supra-mechanical domains it is also a change of quality” (203)²*).

The mechanical form of motion usually plays the role of a secondary form, accompanying more complex changes of matter. “Every—

*) Here and below, the number in parentheses following quotations denotes the page of the work indicated in the list of cited literature. In what follows, the underlinings in quotations belong to the cited authors, unless there is a special stipulation.

... motion contains within itself mechanical motion, the displacement of larger or smallest parts of matter... But this mechanical motion does not exhaust motion in general” (203)².

From these positions Engels also analyzed the concepts of mechanics—force and work. Engels rejects the notion, current at that time, of force as a special principle that gives rise to motion and is independent of motion. He writes: “The defect: 1) force is usually treated as something existing independently...” (229)². Motion is not generated by force out of nothing; on the contrary, force is a quantity that characterizes the transfer of motion, in the general case the transformation of motion from a non-mechanical form into a mechanical form; namely, force characterizes the rate of change of the state of mechanical motion taking place in connection with other processes (or capable of taking place under known conditions).

The value of the concept of force is due to the fact that, knowing the quantitative dependence of the change in the quantity of motion of a body on certain quantities (distances, velocities, times, properties of bodies), one can solve many practical problems of mechanics, even without knowing the mechanism of the transformation of non-mechanical forms of motion into mechanical ones. However, in solving more complex mechanical problems, and still more in considering the foundations of mechanics, one must not forget the real meaning of the concept of force.

Proceeding from the interpretation of the concept of force set forth above, Engels also indicates the limits of application of this concept. Thus, when we are dealing with the chemical bond of bodies (“the force of affinity”), the category of force can no longer be applied. He writes: “Some chemists also speak of chemical force as of such a force which causes the combination of substances and holds them together. However, here we do not really have a transition, but have the merging of the motions of different bodies into one whole, and the concept of ‘force’ thus proves to be at the limits of its use” (228)². Indeed, in modern physics the chemical bond is characterized by the magnitude of the bond energy, and by no means by force. In general, in those cases where the interrelation of bodies cannot be given as a function solely of coordinates (or of their differentials), the concept of force proves to be at the limits of its application. Consequently, to identify force with the interaction of bodies, or to regard it in all cases as the principal measure of interaction, as is often written in textbooks on mechanics, is incorrect.

Attention is drawn to the remarkable fact that, half a century before the discovery of exchange interactions, Engels regarded the chemical bond as “the merging of the motions of different bodies into one whole,” and not simply as a certain static force.

From the point of view of the interconnection of mechanical motion with non-mechanical motion, Engels also interprets the concept of work. He criticizes the limited definition of work as a quantity produced by force

(product of force and path) and therefore is not an independent characteristic of motion; in essence this definition of work is far from always applicable. Engels pointed out that “the basic condition of all physical work is qualitative change, change of form” (73). Modern physics has fully confirmed this thought of Engels. In problems of a more general character, work and energy play the role of the basic measure of motion, the basic quantity, while force is a derived quantity.

Thus Engels, proceeding from a dialectical-materialist understanding of nature, as early as 1880 outlined the only correct interpretation of mechanics. The basic meaning of Engels’s interpretation of the concepts of mechanics consists in the fact that these concepts are considered from the standpoint of the unity of the forms of motion of matter, of the inseparable connection of mechanical motion with nonmechanical processes. Such an approach enabled Engels to disclose the true physical meaning of the concepts of force and work and to show that the concept of force is by no means a measure of interaction in all cases.

Engels’s principal works on the dialectics of natural science remained inaccessible to his contemporaries. Physicists, however, remaining captive to a metaphysical worldview, were unable correctly to appreciate the full deep significance of the law of conservation and transformation of energy. They interpreted this law metaphysically, only as a quantitative law of the conservation of a certain quantity, and did not see in it the expression of the connection of qualitatively different forms of motion. On the contrary, in the first period after its discovery, there was a tendency to interpret the law of conservation of energy in the spirit of Helmholtz—as a law confirming the possibility of reducing all physical processes to the mechanical motions of a system of material points between which central forces of attraction and repulsion act. For this reason, for a long time physicists could not draw those conclusions regarding the interpretation of mechanics that Engels drew. Only toward the end of the nineteenth century, when the existence of the electromagnetic field became generally recognized and the conception of forces acting at a distance proved untenable, did the progressive scientists of that time enter upon the path of a critical analysis of Newtonian conceptions of interaction in general and, in particular, upon the path of a critical analysis of the basic concept of Newtonian mechanics—the concept of force.

We shall consider here the statements on the foundations of mechanics by one of the outstanding Russian physicists of the late nineteenth century—Nikolai Alekseevich Umov.

N. A. Umov subjected the basic concepts and laws of classical mechanics, and above all the concept of force, to a thorough critical analysis. In his work “The Present State of Physical Theories” (published in 1900) he writes: “Classical

mechanics is unsatisfactory also in the very essence of the definitions laid at its foundation: these definitions contain an arbitrarily admitted hypothesis of the action of forces at finite distances. This hypothesis, with regard to which Newton expressed such restraint in the doctrine of universal gravitation (and considered it contradictory to the mind of the natural philosopher), nevertheless runs like a red thread through all of Newton’s axioms” (173)^3.

A similar thought was expressed by Umov in another work, “The Significance of Descartes in the History of the Physical Sciences,” published in 1896. N. A. Umov first of all subjects to criticism the second law of Newtonian mechanics, containing the assertion of the independence of the action of a force from the state of motion of the body subjected to this action, and also of the independence of the actions of forces from one another. This assertion, Umov indicates, is connected with the Newtonians’ view of the source of force. “If force is an immaterial property of bodies, then it is clear that, for example, the force with which the earth acts on a stone does not depend on the motion of the stone. In reality, however,” says Umov, “such independence will be a particular case, which cannot be elevated into an axiom” (116)^3. In fact, this law “admits that motion itself cannot become a source of force; yet we know that the motion of a magnet depends essentially on whether there are conductors of electricity in the surrounding space or not. The admission of the independence of action contained in the second law will be valid, again, only on the assumption of forces acting through a void” (174)^3.

From the same positions Umov also analyzes Newton’s third law. He writes: “The third law, which says that the actions of two bodies upon each other are always equal and directed to opposite sides, is again based on the hypothesis of forces acting at a distance” (174)^3. However, “the action of an electric current on a magnetic needle does not obey the third law” (175)^3.

Finally, “the first law of motion, the law of inertia, considers a body torn away from all nature and moving in empty space. From the point of view of modern physics, forces cannot act in empty space, and therefore the question arises—what preserves the body under consideration?” (174)^3.

A profound analysis of the content of mechanics, a critique of the Newtonian understanding of force, prepared the ground for its further generalization; the following statement by Umov is remarkable: “We also cannot assert that Newton’s axiom will be preserved in the simple case of the fall of bodies, if the bodies move with velocities comparable to the velocity of light” (116)^3. Thus N. A. Umov, as early as 1896, anticipated one of the most important propositions of the theory of relativity. N. A. Umov’s statement is not accidental; it is the logical result of a critical analysis of the laws

of Newton’s mechanics on the basis of all the achievements of nineteenth-century physics; this analysis inevitably led to the conclusion that a further generalization of the laws of classical mechanics was necessary. Thus, if one has in mind its positive content, the predecessors of the theory of relativity should be considered to be the leading materialist physicists of the nineteenth century, and not Mach at all, as bourgeois history of science portrays it.

The disclosure of the limitations of the principles of Newtonian mechanics impelled the conscious search for more general laws of motion. At the same time, it made it possible for leading physicists to evaluate more correctly those generalized laws of motion that had been established for complex mechanical systems as early as the end of the eighteenth and the first third of the nineteenth centuries by Lagrange, Hamilton, Ostrogradsky, and Jacobi. We have in mind the so-called variational principles of mechanics and the generalized differential equations of Lagrange.

As is known, Newton’s equations of motion are not directly applicable to the motion of complex constrained systems of bodies; moreover, the difficulties are by no means exhausted by the impossibility of solving Newton’s differential equations for the aggregate of bodies forming the system. For example, in cases where the motion of the system cannot be decomposed into the displacements of individual “material points” between which prescribed constraints exist, it is, naturally, not even possible to set up Newton’s differential equations (the motion of any complex mechanism belongs to such a case). In these cases, as is known, one has to apply Lagrange’s equations, which contain generalized coordinates, variable coefficients of inertia, and generalized forces (whose dimension does not at all coincide with the dimensions of the corresponding quantities entering Newton’s equation). The basic concepts with which one has to operate in these equations are kinetic and potential energy, while force is a derived quantity.

Until the last quarter of the nineteenth century these principles were interpreted as purely formal mathematical generalizations of Newton’s equations. This limited and, in essence, incorrect interpretation of variational principles and Lagrangian equations was due to the then-dominant conception in physics of forces as independent principles determining the change in the motion of bodies. These principles and equations were considered merely another expression of the law of action of the same objects—forces.

Of course, there is a connection between Newton’s equations and more general equations, for example Lagrange’s; by means of mathematical transformations one can pass from the former to the latter. However, this does not mean that they encompass one and the same range of natural phenomena and that their content is identical. Equations

Newton are only a special case of Lagrange’s equations. Usually, mathematical transformations of particular forms of laws into more general forms are forced by the search for new methods of posing and solving more complex problems that go beyond the limits of the original circle of phenomena. Unfortunately, a formal understanding of the generalized equations of mechanics as purely mathematically transformed Newton’s equations is encountered even at the present time.

The discovery of the law of conservation and transformation of energy, which established the unity of the various processes of nature, deepened the understanding of variational principles not simply as a “convenient” form for solving complex mechanical problems, but as a new stage in the development of classical mechanics.

It is especially important that the generalized principles of mechanics express not narrowly mechanical laws, but contain a characterization of transformations of energy from one form into another, going far beyond the limits of mechanics. Umov points out: “Potential and kinetic energies can be expressed according to the characteristics of a phenomenon, without constituting a preliminary mechanical image; such characteristics include quantities that have nothing in common with the quantities considered in classical mechanics (the spacing is ours.—S., Sh.). These include, for example, electromotive force, the force of an electric current, the force of a magnetic field, etc.” (175)^3. An example of the extension of generalized principles beyond the limits of mechanics is Maxwell’s derivation of the equations of the electromagnetic field on their basis.

However, the dominance among physicists at the end of the 19th century of the mechanistic worldview fettered their thinking and led them to a narrower understanding of the general principles of mechanics. Mechanist physicists regarded all forms of energy as different forms of kinetic energy. Therefore attempts were made to find the general principles of mechanics in a purely Cartesian spirit. These principles were to express the modes of transmission of motion from some bodies to others through invisible masses, through “hidden” motions. Guided by this idea, H. Hertz constructed his system of mechanics. N. A. Umov also placed great hopes in Hertz’s mechanics. However, these hopes were not justified. Hertz’s mechanics did not play, and could not play, any significant role, since it was based on erroneous notions about the possibility of reducing nonmechanical processes to mechanical displacements, notions that came into contradiction with the entire course of the development of physics, and for this reason it offered no way out into practice.

But, while noting that the limited worldview of the materialist physicists of the end of the 19th century prevented them from giving a consistently correct interpretation of mechanics, we must also emphasize something else.

The very fact of their criticism testified that the development of physics urgently required a more profound understanding of the foundations of mechanics, and in essence confirmed the correctness of Engels’ conception.

What conclusion, then, follows from Engels’ analysis of classical mechanics?

The conclusion that the mechanical motion of bodies is only one moment of motion in general, of motion understood as the change of bodies; consequently, the concepts of mechanics must be interpreted on the basis of the inseparable connection of mechanical motion with other, non-mechanical forms of motion. This does not mean that already within the limits of mechanics it is possible and necessary to disclose the nature of the interaction of bodies. However, in the interpretation of the laws and concepts of mechanics the existence of such connections must be shown.

The further conclusion is that the laws and concepts of mechanics, while not losing their applicability in a definite domain, at the same time, in the process of the development of science and with the increasing complexity of the tasks confronting it, receive a generalization—and moreover not only in the relativistic or quantum sense, but also in the sense of a deeper understanding and grounding of them within the bounds of classical mechanics. The notions that the foundations of classical mechanics, as given by Newton, existed for two hundred and twenty years (up to the theory of relativity) without essential changes are completely untenable. Over the course of more than two centuries classical mechanics developed not only in breadth but also in depth; it not only perfected its mathematical apparatus, but substantially developed its general laws as well, despite the fact that its initial premises were preserved.

A critical analysis of the very foundations of classical mechanics was necessary for the preparation of its relativistic generalization. The necessity of such a further generalization was clear to the progressive physicists at the end of the nineteenth century, who were able critically to evaluate the content of classical mechanics on the basis of an analysis of the newest results of physics.

A diametrically opposite position on questions of mechanics was taken by the idealist physicists—Mach, Duhem, Poincaré. Although there are certain differences among their views, in the main they belong to one camp. Mach exerted the greatest influence on bourgeois natural scientists.

Mach attempted to impose upon physicists his own system of views on mechanics, which represented an expression of his general philosophical conception. As is known, Mach revived Berkeleyan-Humean philosophy; he asserted that our knowledge begins and ends with sensations, beyond whose limits man cannot go. Concepts, laws, and finally all of science are only “convenient”

techniques for linking sensations. The judgment that sensations, concepts, and laws reflect some object and its properties is, according to Mach, unscientific “metaphysics,” from which one must rid oneself in every possible way.

This subjective-idealist position lay at the basis of Mach’s critique of classical mechanics. From these positions he criticized Newton’s concepts of absolute space, time, and motion—as concepts which are not given in sensations, cannot be directly measured, and are metaphysical concepts. From these same positions Mach also approached the substantiation of mechanics, the interpretation of its basic concepts.

Since Mach denied the objectivity of the material world, there could be no question for him of the emergence of the mechanical form of motion from other forms. Mechanical motion and its properties, according to Mach, are only an immediately given fact in sensations. The circle of ideas connected with this fact begins with the perception of the simplest fact of mechanical displacement; in the ascertainment of this fact Mach sees the entire content of mechanics. The concepts of mechanics developed subsequently are merely auxiliary devices of thought, having, according to Mach, no objective significance.

What, then, do we perceive in sensations as the “simplest” fact of experience, if we are speaking of mechanics? The configuration of bodies and its change, velocity and its change, i.e. acceleration—such is Mach’s answer.

On these initial concepts—the configuration of bodies in acceleration—Mach attempts to build his “foundations” of mechanics. In opposition to Newton’s formulation of the laws of mechanics, Mach proposes his own formulation, in which all concepts in one way or another connected with the idea of matter, causality, and the objectivity of various forms of motion and their interconnection are carefully banished.

Mach subjects Newton’s concept of mass as a quantity determined by the amount of matter to the most severe criticism. Mach seeks to prove that Newton’s definition of mass is a logically vicious circle; the viciousness supposedly consists in the assertion that mass is equal to the product of density by volume, while density is defined as the mass of a unit volume. “The true definition of mass,” Mach declares, “can be derived only from the dynamical relations of bodies” (209)4. Mach considers the basic fact of mechanics to be that two bodies impart to one another oppositely directed accelerations, the magnitudes of which stand in a definite ratio. The reciprocal ratio of these accelerations (with a negative sign) is what is called the ratio of masses. Mach believes that by this relation

exhausts the entire content of the concept of mass. He writes: “In our concept of mass there is no theory whatsoever; ‘quantity of matter’ is entirely superfluous in it; it contains only the precise definition, designation, and name of an actual fact” (182)\(^4\).

Mach himself emphasizes, as a special merit of his concept of mass, that there is no theory whatsoever in it; this means that Mach, on principle, refuses to clarify the nature of mass, its origin, i.e. to explain the inertia of a body.

All these arguments of Mach’s about mass do not withstand criticism. First of all, it is incorrect that Newton’s definition of mass contains a vicious logical circle. Newton was an atomist: by density he understood not the mass of a unit volume, but the number of particles in a unit volume. He sought, above all, to give an explanation of the inertia of a body, and not only a method for measuring it. Newton proceeded from the experimental fact that two identical bodies possess twice as much mass as one body, i.e. that mass is an additive quantity; at the same time, in Newton’s time it was already well known that if a double volume of gas is compressed into one volume, its mass does not change. From these facts Newton drew the conclusion that mass is determined by the number of particles in a body. Of course, such an understanding of the nature of mass is limited; it is necessarily based on the assumption that all bodies consist of particles identical in their nature, differing only in magnitude (and form); moreover, the mass of each particle is taken to be proportional to its volume.

Thus, in Newton’s definition there is no vicious logical circle, as Mach asserts, but only a limitation of understanding characteristic of that time, which was revealed in the subsequent development of physics. As is known, it was later proved that the notion of the identity of all particles of matter is untenable, and that the mass of a particle turned out to be connected with its energy.

Mach strove to show that the question of the nature of mass is meaningless, since, allegedly, we always deal with only one and the same fact—the acceleration of bodies in their interaction, i.e. with the sole form of manifestation of mass. But this assertion of Mach’s is also incorrect. Newton had already shown the possibility of measuring mass by weight. Mach very easily, but without justification, “does away” with this method of measuring mass, reducing it to the first. He attempts to justify this reduction by the fact that the accelerations of counterbalanced bodies are destroyed by their interaction, and since the accelerations of all bodies in the gravitational field are the same, the masses of counterbalanced bodies are also equal. But Newton’s merit consisted precisely in the fact that, following Galileo, he confirmed experimentally (by swinging pendulums) that the accelerations of all bodies in the gravitational field

are identical; in other words, he proved that heavy mass is equal to inertial mass, and that therefore the weight of a body can serve as a measure of its mass (for one and the same place). Thus, besides the dynamical method of measuring masses, another method—weighing—has long been known. Finally, the connection between mass and energy, discovered later, gives the fundamental possibility of measuring the mass of a particle, for example in nuclear processes, by yet another method.

Thus, the measurement of mass in general is possible by several methods, and by no means solely by the dynamical one.

Mach confidently declared that his understanding of mass, as taken from “pure experience,” would never be shaken. He wrote: “Thus, once we, following the indications of experience, have considered the existence of a special defining property of bodies, our task is exhausted by the recognition and unambiguous designation of this fact. We cannot go further than the recognition of this fact, and every attempt to go further from this point leads only to obscurities. Every awkwardness disappears as soon as we have made clear to ourselves that the concept of mass contains no theory, but only experience” (185)4.

Physics very quickly refuted this “prediction” of Mach’s, as well as his other “predictions.” It was compelled to raise the question of the origin of mass already in the classical theory of electrons, and since then the task of creating a theory of mass, of explaining its nature, has invariably been at the center of physicists’ attention and has become especially urgent in the theory of “elementary” particles.

Thus, the viciousness of Mach’s reasoning about mass consists in the fact that he, on principle, renounces the possibility of any explanation whatsoever of the inertia of bodies, seeing in mass only a coefficient in a known equation; this viciousness consists, further, in the fact that he ignores the manifold manifestation of mass, its connections with other physical quantities characterizing other properties of bodies, connections that condition the possibility of various methods of measuring mass. The materialist physicist, defining mass as a measure of the inertia of a body, cannot confine himself to this definition and must show the deepening of the concept of mass in the subsequent development of science.

Mach continues the same line of emasculating the physical content of concepts with respect to force as well. “What we at present call in mechanics force,” says Mach, “is not something hidden in the processes, but a measurable, factual condition of motion, the product of mass by acceleration” (213)4.

Consequently, force, according to Mach, is not a definite characteristic of the connection of arising (or disappearing) mechanical motion with nonmechanical processes; according to Mach, it is not a concept having real physical meaning, but is merely a designation for the product of mass by

acceleration. According to Mach, physical conditions do not create motion, but only accompany it. He is especially opposed to the idea of a real cause of acceleration and praises Newton highly for not wishing to penetrate into the physical content of forces: “Newton’s repeated definite assurances that what is important to him is not speculations about hidden causes of phenomena, but the investigation and ascertainment of what is given in the facts..., characterize him as a philosopher of outstanding significance” (161)\(^4\).

The formal interpretation of force is refuted already by the fact that, in the absence of conditions for the occurrence of acceleration, other actions take place in bodies, other changes in the state of bodies—for example, deformations, electrical actions, and so on. It is curious that Mach himself, in passing, acknowledges the existence of this circumstance, which does not prevent him from completely ignoring the latter when interpreting the concept of force.

Mach is characterized by a complete disregard for the achievements of theoretical physics when considering the basic concepts of physical science. Thus, Mach’s entire conception in mechanics is connected with the denial of the reality of fields, which already in his time had begun to play a major role in physical theories.

Mach’s treatment of space and time as an “ordered system of series of sensations” was subjected to devastating criticism by V. I. Lenin in his work Materialism and Empirio-Criticism. Lenin writes: “This is obvious idealist nonsense, which inevitably follows from the doctrine that bodies are complexes of sensations. It is not man with his sensations that exists in space and time, but space and time exist in man, depend on man, are generated by man—that is what Mach arrives at. He feels that he is sliding into idealism and ‘resists,’ making a heap of reservations, drowning the question, like Dühring, in lengthy disquisitions (see especially Knowledge and Error) on the mutability of our concepts of space and time, on their relativity, and so forth. But this does not save him and cannot save him, for the idealist position on this question can really be overcome only by recognizing the objective reality of space and time. And this Mach does not want to do at any price. He constructs an epistemological theory of time and space on the principle of relativism—and that is all” (165)\(^1\).

As is known, the principle of relativism led Mach to recognize the equal status of the Ptolemaic and Copernican systems. The physical inconsistency and reactionary essence of this conclusion have already been exposed in a number of works published in our press.

Mach’s treatment of the foundations of mechanics found its clearest expression in that “system of construction of mechanics” which he proposes in place of Newton’s. Mach asserts that at the basis

this system of mechanics rests on the ascertainment of only a single fact: “Strictly speaking, only one fact has been established. Different pairs of bodies determine, independently of one another and in themselves, pairs of accelerations whose terms represent ratios that are invariant and characteristic for each pair of bodies” (212)^4. The ascertainment of this fact replaces, in Mach’s system, all the principles of mechanics. “Only the practical need of teaching can justify its partial expression (of the experimental fact.—S., Sh.) with the aid of many principles (the number of which is determined, properly speaking, only by scientific taste)” (212–213)^4.

What idea, then, did Mach put into his formulation of mechanics? The expulsion of the idea of the objectivity of motion. In the physical relativity of motion Mach sees a pretext for smuggling in philosophical relativism.

This is a classical illustration of the different approach of materialists and idealists to questions of physics. The dialectical materialist Engels clearly saw that “the motion of a single body does not exist—[of it one can speak] only in a relative sense—falling” (199)^2, but at the same time he everywhere emphasized the objective character of motion. The idealist Mach confuses the question of the physical connections of bodies manifested in motion (the physical relativity of motion) with the question of the objectivity of motion.

It is no accident that, having subjectivized motion, Mach strives to avoid an explicit formulation of the principles of inertia and of the equality of action and reaction. He reduces the idea of the conservation of motion to the position of a particular and inessential consequence of the indicated “fact.” This fully accords with Mach’s general view of the law of conservation of energy, in which he sees not a fundamental law of nature, but only a rule establishing an inessential form of connection between certain quantities.

In essence, from Mach’s formulation of the foundations of mechanics the concept of force too has been thrown out, since a direct dependence of acceleration on spatial quantities is established. Mechanics is completely “kinematized.” This means that the phenomena of the motion of bodies are “described” in complete detachment from the physical processes that determine them. Mach’s ideal of “pure description” is thus attained, but at the cost of emasculating the real content of mechanics.

Mach asserted that it was precisely his program of the kinematization of mechanics that was implemented by Hertz, whose principles of mechanics we mentioned above. But Mach’s attempts to present Hertz as his consistent follower are untenable. The latter strives to construct a kinetic mechanics, i.e., to reduce all interactions of bodies to the influences of hidden motions or hidden masses. We have already said that this attempt was limited, that it expressed the mechanistic worldview of its author. However, nothing in common

with Mach’s kinematization of mechanics Hertz’s principles had nothing in common, since Hertz strove to explain the laws of interaction of bodies, and did not confine himself to a “pure description” of facts.

Thus, all the basic content of mechanics, in Mach’s opinion, is expressed in the fact stated above. According to Mach, the development of mechanics since Newton’s time has not enriched its content; the new regularities of mechanics established after Newton added nothing essential to Newton’s laws (which Mach nevertheless found it necessary “to correct a little” in order to fit them to his philosophical conception). “Newton’s principles are sufficient,” Mach asserted, “in order, without invoking any new principle, to consider every practically possible case of mechanics, whether in the domain of statics or of dynamics. If difficulties arise in doing so, they are always only difficulties of a mathematical (formal), but never of a fundamental, character” (237)^4. Consequently, both the law of conservation of energy (like the very concepts of work and energy) and the variational principles of mechanics, in Mach’s opinion, introduce nothing new into the content of mechanics: they are merely mathematical expressions convenient for computation. “It is possible, without doubt,” Mach declares, “to devise many other integrals whose variations yield the ordinary equations of motion, but which for that reason need not have any special physical significance” (319–320)^4. With their help one can only solve somewhat more simply problems which, generally speaking, can be solved by using Newton’s laws alone. To prove his assertion, Mach considers several simple problems of statics and dynamics and shows that they can be solved both by variational methods and with the aid of elementary laws. In statics, for example, the conditions of equilibrium of loads suspended from the ends of a rod can be found both by means of the principle of possible displacements and directly from the rule of the lever.

It is clear that such a demonstration of the supposedly equal power of elementary and generalized methods can be carried out only on elementary examples. The exceptional significance of generalized principles is manifested precisely in complex, and not in elementary, cases.

The most curious thing in Mach’s reasoning is that he himself gives examples refuting his assertions. Thus, for example, concerning the principle of possible displacements, he sets forth the well-known problem of the conditions of equilibrium of a hidden mechanism. “If some new machine were hidden in some box in such a way that only two levers for the application of the forces \(P\) and \(P'\) protruded, and we found that the simultaneous displacements of the latter are equal to \(h\) and \(h'\), we would immediately know that, in the case of equilibrium, \(Ph = P'h'\), whatever this machine might be in all other respects;”

machines” (58).⁴ Obviously, it is impossible to solve this problem with the aid of a simple parallelogram of forces, for we do not know all the connections between the two levers. This problem is solved in principle only with the aid of the principle of possible displacements; yet Mach sees in it no physical meaning, does not regard it as an expression of new physical regularities. This does not prevent Mach from interpreting the named principle in favor of his philosophy, according to which the aim of science consists in economy of thought: “...the principle has, consequently, a certain economic value”!

Thus, contrary to the obvious, the whole value of the principle of possible displacements is reduced to “economy of thought,” i.e. to economy of calculations. Such conclusions are met with everywhere in Mach. We shall not give further examples showing that the discovery of new regularities and concepts of mechanics was connected with the necessity of solving problems that proved beyond the power of the old methods not only because of mathematical complexity, but owing to difficulties of a principled character, as is the case in the example cited above.

Mach’s Mechanics, called by him a “historico-critical essay on its development,” presents the real history of science in a deliberately distorted form. It is enough to compare the assessment of variational principles given by Umov with Mach’s assessment, to recall what role these principles play in the investigation of the motion of such objects to which Newton’s laws are not directly applicable, in order to understand how Mach falsified the history of mechanics.

Let us sum up. Proceeding from his idealistic philosophy of “pure description,” which denies the existence of the objective world, Mach completely emasculated the content of mechanics, threw overboard the idea of the conservation of motion, the idea of the inseparable connection of mechanical motion with non-mechanical processes; reduced mechanics to pure kinematics—to a description of accelerations formally dependent on the configuration of bodies and on certain numerical parameters; degraded the general laws of mechanical processes, discovered in solving problems considerably more complex than the problem of the interaction of two material points, to the level of simple empirical rules. Thereby the relativist Mach absolutized mechanics as a science, given once and for all, having no prospects of development, since, allegedly, from Newton’s time it has dealt and in the future will deal with only one fact—the description of accelerations.

An analysis of the physical content of mechanics, of those problems which were posed and solved by it, shows how empty and anti-scientific Mach’s reduction of mechanics to a description of the dependence of the accelerations of bodies on their configuration is. Mechanics as a whole

never had as its main task the description of motions. Not only in contemporary mechanics, but also at earlier stages of its development, the central place in it was occupied by problems of the interaction of bodies (problems of strength, resistance of media, stability of motion, resonance, and many others).

Mechanical motion is only a moment of more complex processes that occur in real bodies. In the concepts of mechanics, the connection of mechanical motion with other, nonmechanical processes is reflected in a distinctive form. Therefore physicists are fundamentally wrong who seek to justify the kinematization of mechanics by considerations that mechanics is an abstract science devoted to the study only of mechanical motion; that therefore its concepts can and must reflect the properties only of mechanical motion “in itself”; and that, in view of this, mechanics has not set and has no right to set itself the task of revealing the connection of mechanical motion with nonmechanical processes. Of course, mechanics by itself cannot reveal the essence of this connection, the entire content of those objective properties of things which find their manifestation and reflection in the concepts of mechanics. But it does not follow from this fact that the existence of such a connection should not be reflected in the concepts of mechanics.

It is impossible to substantiate mechanics within the closed framework of mechanics itself when it is completely divorced from physical phenomena. Confinement within the limits of “pure mechanics,” if carried through straightforwardly, inevitably degenerates into the conception of “pure description.” Self-limitation within the framework of pure mechanics does not make it possible to determine the limits of applicability of the established laws and concepts of mechanics. This was shown with complete clarity by F. Engels and confirmed by N. A. Umov and other materialist physicists at the end of the last century, as was stated above. Only in the light of physical theory as a whole can one understand within what limits the established laws of mechanics are valid, and thereby arrive at new, more general laws.

It is enough to compare the two conceptions of mechanics—the conception of the materialists, on the one hand, and that of Mach and his like-minded followers, on the other—to see the entire reactionary character of the views of the latter. Mach’s conception in mechanics not only did not help to generalize new facts, not only did not lead science forward, but even came into direct contradiction with its further development.

Naturally, Mach’s ideas did not exert, and cannot exert, any influence on the real development of mechanics. Under the pressure of facts, and contrary to Mach’s forecast, the fundamental laws of classical mechanics were generalized; the materialists proved right, those who demanded a generalization of mechanics in connection with the discovery of new forms of motion.

It would seem that Mach’s ideas, completely refuted by the development of science, should have been discredited in the eyes of physicists. This, however, did not happen; if in the last century his conception had few adherents, of which he himself complained, then in the twentieth century the number of his supporters began to grow. This was a particular manifestation of the general crisis of physics, the causes of which were exhaustively revealed by V. I. Lenin in his classical work Materialism and Empirio-Criticism.

Mach’s reactionary views have been taken up by his present ideological heirs—the “logical empiricists.” Present-day Machism still proceeds from the position of a “pure description” of observed facts, according to which science merely registers facts, but does not explain them. Neo-Machism sees in the concepts of physics not a reflection of the properties of real objects and processes, but only a designation of certain measuring operations, i.e. it gives physics an “operationalist” character. In the laws established by physics it sees merely a simple statement of connections between the results of measurements, and nothing more.*)

How, then, do the neo-Machists interpret the basic concepts and laws of mechanics?

Let us consider, by way of example, how one of them, namely Ph. Frank, analyzes the concept of force. Is force, for Frank, a “physical reality”? Frank answers this question in the affirmative: yes, “it is possible to give an operational meaning to the assertion that forces are physical realities” (19).⁷ At first glance one might think that Frank is here entering into contradiction with the Machian understanding of force, for which it is only the product of mass and acceleration. In fact, in physics one usually understands by “physical reality” something existing objectively, independently of human consciousness. But the operationalist Frank understands by “physical reality” not at all an objectively existing object or an objective property of moving matter, but something quite different. For him physical reality is only the result of measurements, and nothing more. Therefore he is satisfied with the totality of measurements and does not find it necessary to reveal the nature of this reality. Thus operationalism identifies physical reality with the description of the methods of measurement; at the same time it identifies the physical concept with the physical quantity.

*) Of course, the process of measurement is a necessary moment of physical cognition; quantitative determinations of the action of an object, of its manifestations, are always necessary. But, by limiting the tasks of science to measuring operations, neo-Machism castrates the real content of concepts and deprives them of objective meaning. The object is always richer than any one of its manifestations, expressed quantitatively in a particular measuring operation.

Force, according to Frank, is a physical reality because in certain cases it is measured by independent methods; such, for example, are gravitational and electromagnetic forces. However, this does not mean that Frank sees in the concept of force an objective characteristic of interaction. He does not set himself the task of revealing its physical content. From the point of view of scientific materialism, a physical quantity that reflects an objective property of moving matter can be measured in various ways because this property manifests itself in manifold ways under different physical conditions; for example, some nonmechanical form of motion may be transformed either into mechanical motion, or may lead to changes in other states of bodies; accordingly, force may be measured either through acceleration, or through deformation, or through the weight of a body, and so forth. One may cite an example from another domain of physics. A change in the mean energy of the chaotic motion of molecules, depending on conditions, may manifest itself in the expansion of bodies, in changes in the character of the radiation of a body, in the formation of an electric current, and so on; accordingly, temperature, which characterizes the intensity of chaotic molecular motion, may be measured either with the aid of a thermometer, or by means of a bolometer, or, finally, with a thermoelement. But this does not mean that temperature is a physical quantity because it is measurable in various ways, and that the physical content of the concept of temperature consists in these measurements. Thus, the diversity of manifestations of an objective property of matter makes possible diverse methods of measuring the quantity that characterizes this property. For Frank, on the contrary, the measurability of a quantity by independent methods serves as the criterion of physical reality, into the concept of which he introduces no objective content.

Newton’s second law itself is, according to Frank, only a definition of force, since only mass and acceleration can be measured. Besides this, for both of the above-mentioned forces there also exist other methods of measurement: for the gravitational force—through measurement of masses and distances; for the electromagnetic force—through measurement of charges (currents) and distances. Experience, according to Frank, confirms only the totality of equations relating $ma$ to $f$ and $f$ to $\dfrac{mM}{r^2}$ (and correspondingly for electromagnetic forces)*). Strictly speaking, experience, according to Frank’s assertion, confirms only the connection of $ma$ with $\dfrac{mM}{r^2}$, i.e., the connection of a number of “observable” quantities, si—

*) Here $m$ and $M$ are the masses of the interacting bodies, $r$ is the distance between them, $f$ is the acting force, and $a$ is the acceleration.

—According to Frank, it is a quantity that is unobservable: it serves only as one of the ways of describing the connection between observable quantities.

Whether it makes sense to apply the concept of force in mechanics is determined not even by whether force is a physical reality in the sense of operationalism or not, but exclusively by the “simplicity” of its expression through “observable quantities.” Frank writes: “If this expression of the force (gravitational.—S. Sh.) were as complicated as the equation of the curves described by the planets, there would be no sense in replacing the geometrical description by a dynamical one” (16–17)7. These arguments of Frank’s reveal with the utmost clarity the subjectivist meaning that the neo-Machists attach to the concept of “physical reality”: in the final analysis only “simplicity and convenience” prove to be the decisive criterion for introducing into science one or another concept that determines “physical reality.”

We see from this that Frank’s conception is in essence no different from Mach’s conception. The question of the objective properties of matter that are expressed by the concept of “force” Frank evades in every possible way; he emphasizes that “the only logical, sound way of posing the problem of the ‘existence of force’ consists in avoiding here as well what Carnap calls the ‘material mode’ of expression, and in adhering to the ‘formal mode’” (18)7. And by the “material mode of expression” these gentlemen understand the characterization of those objective properties of things which are reflected by a physical concept.

Neo-Machism, like Mach, furiously opposes the healthy striving of science to explain the objective properties of things, to understand their necessity; it wages war against “metaphysics,” by which it understands materialism. These gentlemen mask their idealism with the demand for “strictness” in the definition of concepts; but we have seen that beneath the mask of this “strictness” is hidden the emasculation of the objective content of concepts. Operationalism has extended its influence to many bourgeois physicists, even to those who formerly stood on the position of spontaneous materialism*).

The exposure of this most malicious and most dangerous enemy of materialism is the direct duty of Soviet scientists.

) As an example one may point to the well-known German physicist A. Sommerfeld; in his course Mechanics*, recently translated into Russian, he writes: “Of the concept of force we may say the same as of all physical concepts and terms: verbal definitions are devoid of content; true definitions are given by indicating a method of measurement, which, generally speaking, may be only theoretical and not necessarily practical” (11)8. Thus Sommerfeld reduces the entire content of the concept to the description of a certain measuring operation, i.e., in essence, he adopts the position of operationalism. This operationalist assertion, which deprives all concepts of any objective content, the editor of the translation, D. V. Sivukhin, left without any critical comment.

There is no doubt as to the path that the Soviet physicist should follow in substantiating mechanics; it is the path indicated by F. Engels. From the Engelsian understanding of mechanics, as was already said above, it follows that the essence of the basic laws and concepts of mechanics can be revealed only by considering mechanical motion in inseparable connection with the other physical processes that condition it. Of course, within certain limits one can and must abstract from this connection; for example, one may study the flight of a bullet under the action only of the force of gravity, or also under a quantitatively specified law of resistance of the medium. However, such a formal posing of the question is inadmissible when it is a matter of generalizing experience as a whole, of revealing the essence of the basic laws and concepts of mechanics.

Many authors of textbooks on mechanics (in physics courses) do not set themselves the goal of revealing the physical essence of its laws and concepts. S. E. Khaikin set this goal before himself consciously. In the preface to the first edition of the course Mechanics, S. E. Khaikin writes: “The character of the exposition in the present course differs in many respects from the generally accepted methods of presenting the section of mechanics in a general course of physics. I have sought to explain as fully as possible the physical content of those concepts that mechanics uses, and to draw as clearly as possible that physical picture which is concealed behind the schemes of reasoning usually employed in mechanics” (8)5. In his earlier book What Are Forces of Inertia (A Physical Introduction to Mechanics), S. E. Khaikin likewise emphasizes that he is making “an attempt to draw that physical picture which is contained in the basic ideas and laws of mechanics” (5)6.

But what interpretation, precisely, does he give to the laws and concepts of mechanics?

An analysis of both books shows that S. E. Khaikin not only does not follow the path most consistently pursued by F. Engels, but has chosen another path, which—whether S. E. Khaikin wants this or not—on essential points adjoins the line of the present-day Machists. In what follows we shall try to demonstrate this assertion; at the same time we shall be interested not so much in his individual formulations (which the author greatly changed in the second edition of Mechanics) as in the general understanding he has of the foundations of mechanics (which changed little in the second edition of Mechanics).

In what does S. E. Khaikin see the fundamental proposition of mechanics?

In the second edition of Mechanics we can read: “For a system of bodies, at any instant of time, the second derivatives with respect to time of the coordinates of the bodies are uniquely...

are determined by the coordinates of these bodies. This means that to determine the accelerations in a system of bodies at any moment of time it is necessary to know only the configuration of all the bodies of the system at that moment of time” (our italics.—S., Sh.) (90)5.

To strengthen the impression made by this formulation, S. E. Khaikin continues: “Figuratively speaking, to determine the accelerations of a system of bodies at some moment of time it is necessary to have only an instantaneous photograph of the system corresponding to that moment, and it is quite unnecessary to know whether the bodies are moving or at rest” (our italics.—S., Sh.) (90)5.

In this interpretation of the fundamental proposition of mechanics there is an essential defect: in such a form it can be formulated only on the basis of considering a certain “idealized” nature, in which the range of phenomena being generalized is artificially restricted and the inseparable connection of mechanics with physics is ignored.

S. E. Khaikin himself points out that, in order to express Newton’s fundamental proposition of mechanics in the form cited above, it is necessary to abstract from frictional forces, which plainly depend on velocity and therefore do not fit into this formulation (“Mechanics,” 2nd ed., p. 89). In the book What Are the Forces of Inertia?, in which the same interpretation of the fundamental proposition of mechanics is carried out, he makes yet another reservation: that the phenomena of mechanical hysteresis should not be taken into account (14)6.

But even this does not exhaust all the necessary restrictions. S. E. Khaikin is further compelled to exclude also the Lorentz force, which depends essentially on velocity (“Mechanics,” 2nd ed., p. 76).

Thus we see that S. E. Khaikin strives to construct mechanics on the basis of considering only forces “depending on configuration.” We shall return to the question of whether such a program is lawful; for the moment let us note that S. E. Khaikin, apparently, himself feels that his formulation of the fundamental law of mechanics is based on too narrow a foundation. But instead of giving a more generalized formulation of the fundamental laws of mechanics, one embracing all phenomena in which mechanical processes are manifested, or at least instead of indicating the necessity of subsequently giving such a generalization, S. E. Khaikin takes a path directly opposed and physically untenable: he asserts that the forces he has excluded from consideration are wholly reducible to microscopic forces, which in the final analysis depend on the configuration of molecular particles. In the book What Are the Forces of Inertia? he tried to substantiate this assertion at length. On frictional forces he wrote there: “The origin of frictional forces and their connection ...”

with the configuration of surface layers have not yet been fully clarified. However, it may be regarded as almost obvious that the emergence and change of frictional forces are caused by certain changes in the configuration of surface layers” (40)6. Despite acknowledging that complete clarity is lacking in these questions, S. E. Khaikin nevertheless, as arguments in favor of reducing frictional forces to forces dependent on configuration, points to a number of “theoretical” models of the forces of static friction, sliding friction, and rolling friction, and concludes: “Thus, in the case of friction of solid bodies against one another, we are dealing with forces directly dependent on configuration and only indirectly dependent on velocities. Therefore these frictional forces fit quite well into the conception of forces which we introduced at the very beginning” (42)6. He draws the same conclusion with regard to viscous forces, and concludes: “Thus, in mechanics we do not encounter elementary forces dependent on velocities” (43)6.

In the textbook Mechanics, S. E. Khaikin distinguishes frictional forces from elastic forces and universal gravitational forces: “The difference consists in the fact that frictional forces, to one degree or another, depend not only on the configuration of bodies, but also on their velocities.” However, he immediately makes reservations which again reduce frictional forces to forces dependent only on configuration: “Nevertheless, this distinction does not go as deep as one might have thought. Probably, in the final analysis, frictional forces do nevertheless depend only on the configuration of bodies, but this dependence does not appear as explicitly as in the case of universal gravitational and elastic forces. However, the question of the origin of frictional forces lies beyond the limits of mechanics” (119)5.

But whatever the situation may be with frictional forces, there still remains the Lorentz force, which essentially depends on velocity. Here S. E. Khaikin is compelled to retreat: even the method of “theoretical models” does not allow him to reduce all forces to forces dependent only on configuration. Yet he still thinks that, with the aid of this method, he has already conquered a broad domain without it. Having in mind gravitational, elastic, and frictional forces, he writes: “Remaining within the bounds of classical mechanics and considering only forces of mechanical origin (our spacing.—S., Sh.), we apparently may assume that in essence all forces depend only on the configuration of bodies” (43)6.

What content, then, is put into the words “forces of mechanical origin”? Is this term legitimate at all? After all, S. E. Khaikin himself acknowledges, as we saw above, that the question of the origin of frictional forces lies beyond the limits of mechanics, which is, of course, correct. If what is meant here is the mechanical action of forces, then one must ask why the mechanical actions of conductors traversed by current are excluded from consideration in mechanics. This contradiction in the treatment of forces remains

in the works of S. E. Khaikin unresolved. But the author’s tendencies, which led him to contradictions, are entirely clear: he wants to preserve the formulation of the fundamental law of mechanics in such a way that the accelerations arising in bodies would be connected only with the configurations of bodies, and at the same time he wants somehow to justify the excessive limitation of this formulation, the narrowness of the real physical basis of which it may be a generalization.

Thus, what can we establish from all of S. E. Khaikin’s arguments about the nature of forces?

Nothing can be done with Lorentz forces: they are irreducible to forces depending only on configuration. With respect to friction forces only hopes are expressed that they “apparently,” “probably,” in the final analysis depend only on configuration; but for the time being what is known about them is that they manifestly depend on velocity.

As a result of this, in order to obtain the fundamental law of classical mechanics in the desired form (the accelerations of bodies are uniquely determined only by the configuration of bodies), S. E. Khaikin has to refuse to consider several of the most important kinds of interaction. But, one asks, can a proposition be regarded as the fundamental law of mechanics if it does not encompass several kinds of interactions of bodies? Obviously, it cannot. If we express the fundamental law of some domain of phenomena, then it must necessarily encompass that entire domain; otherwise it loses its generality and cannot be considered a fundamental law.

It is not difficult to see that S. E. Khaikin’s conception fundamentally contradicts the views of F. Engels, for whom mechanical motion is only a moment of motion in the more general sense of the word; this clearly shows that, as a result of this, the concepts of mechanics (force, work, etc.) inevitably express—though in implicit form—the connections of mechanical motion with non-mechanical forms of motion. From Engels’ views it follows that it is illegitimate to derive the fundamental law of mechanics by idealizing nature, through the consideration of mechanical motion “in itself”; in view of the fact that such motion does not exist in nature, as Engels emphasized with particular force, a general law of mechanics formulated in this way will inevitably express only certain special cases, whereas the general cases must be reflected in laws which, in implicit form, express the real genetic connection of mechanical motion with non-mechanical forms.

But in that case, why did S. E. Khaikin need to commit such violence against the facts? To this question he himself replies: in order to obtain a “simple and clear picture.” In the second edition of Mechanics he writes: “If we exclude from consideration the forces of friction (for completeness one ought to have said:

as well as Lorentz forces and the phenomenon of mechanical hysteresis. — S., Sh.), then it may be considered that forces in mechanics depend only on the configuration of bodies. This will allow us subsequently to especially clearly imagine to ourselves the content of the basic laws of mechanics” (78)* (emphasis ours. — S., Sh.)

In the book What Are Forces of Inertia S. E. Khaikin gave the same answer: “If from the very outset we excluded friction and the resistance of the medium (let us add: as well as Lorentz forces and the phenomenon of mechanical hysteresis. — S., Sh.), then the basic content of Newton’s laws would not appear so opaque” (16)*.

Thus, violence against the facts is committed in order that the laws of mechanics might be presented “clearly” and “transparently.” But what is to be done if nature is not “transparent,” but diverse and complex, and cannot be fitted into an elementary formula? S. E. Khaikin is not troubled by this subjective criterion which he applies.

What, then, does “transparency,” according to S. E. Khaikin, consist in? It consists in the fact that the fundamental law of mechanics is presented as the ascertainment of a single fact, namely, that “by the configuration of the system the accelerations of all the bodies of the system are uniquely determined” (14)* (see also the second edition of Mechanics, p. 90). This means that S. E. Khaikin regards the interaction of bodies not as a physical process, reflected only very approximately by certain parameters (in particular, distance), but as a certain externally prescribed condition of acceleration. In this formulation the concept of force is in fact removed from mechanics, i.e. the mechanics of S. E. Khaikin, as with Mach, is kinematized. In the book What Are Forces of Inertia S. E. Khaikin writes about this with complete clarity: “In our formulation of the problem of mechanics the term ‘force’ is entirely absent. And indeed, one can pose and solve problems of mechanics without at all making use of the concept of force and reducing the problem to establishing connections directly between the configuration of bodies and the accelerations experienced by them” (17—18)*).

From the last assertion there inevitably follows the conclusion that, if force is introduced into mechanics, then it is not as an objective category, but as an auxiliary quantity. S. E. Khaikin himself

*) In a note to this assertion S. E. Khaikin refers to Hertz, who allegedly successfully constructed “the most consistent and harmonious mechanics,” in which the concept of force is absent. “However, even the most perfect mechanics of H. Hertz proved too cumbersome and practically of little use.” Above, in criticizing Mach for a similar reference to Hertz, we have already pointed to the fundamental difference between Hertz’s approach to mechanics and to the illegitimacy of such an analogy. It is also permissible to ask wherein lies the success of constructing mechanics without forces, if it proves practically of little use?

writes of this: “Using the concept of force, we can divide the problem of mechanics as formulated by us into two parts. Instead of establishing a direct connection between the configuration of a system and accelerations, we can now, on the one hand, establish a connection between the configuration and the forces, and on the other—between the forces and accelerations. In such a division of the problem into two parts it is again essential that the forces depend only on the configuration, i.e., on the coordinates of the system, and do not depend on the accelerations.” Having emphasized that only the independence of forces from accelerations makes it possible “to divide the problem of mechanics into two independent problems,” S. E. Khaikin continues: “Such a division of the fundamental problem into two independent problems very substantially facilitates the establishment of those relations that exist between configuration and accelerations. The relations between accelerations and forces, on the one hand, and between forces and configurations, on the other hand, prove to be far more transparent than the direct relation between accelerations and configurations” (24)6.

In other words, S. E. Khaikin sets forth the essence of the basic law of mechanics in the same way as Mach. It turned out that S. E. Khaikin needed the narrowing of the physical basis of the basic law of mechanics in order to present this law as “the statement of a simple fact.” This is the point of convergence with Mach’s conception of mechanics—the second, philosophical, defect in S. E. Khaikin’s exposition of mechanics.

Mach begins with the “statement of a simple fact” and builds upon it his entire conception of mechanics. All the concepts that he introduces into mechanics he either connects with this “simple fact” (for example, the concept of mass), or presents as an inessential form of connection of the initial concepts (for example, the concepts of force, work, etc.). The same features are also characteristic of S. E. Khaikin’s treatment. For him as well, conservation of motion turns out to be a particular consequence, and not the principal content and starting point of the laws of mechanics.

In S. E. Khaikin’s Mechanics the concept of force is not excluded, and it is even made to precede his formulation of the basic law of mechanics. It cannot be denied that this is a departure from the direct Machist formulations given by him in the book What Are the Forces of Inertia?

However, we cannot admit that S. E. Khaikin has thereby overcome the Machist conception of mechanics. This can be seen at least from the following two facts.

First, from the fact that the introduction and preservation of the concept of force do not prevent S. E. Khaikin, immediately thereafter, from formulating the basic law of mechanics as a statement of the fact of the dependence of the accelerations of bodies on the configuration of bodies (see Mechanics, 2nd ed., p. 90), and from developing a conception of mechanics in the spirit of the idealization of phenomena, of which we spoke above.

Secondly, this also follows from the very interpretation of force. In fact, S. E. Khaikin introduces the concept of force in the following way. First he states that “accelerations are always the result of the interaction of bodies” (75)\(^5\). Then he defines: “These actions of bodies upon one another, as a result of which the interacting bodies can impart accelerations to one another, we call forces” (ibid.). However, by this S. E. Khaikin does not reveal the objective meaning, the real content of the concept of force, as F. Engels does. S. E. Khaikin carries out no analysis whatever of the general physical concept of force. The above-quoted statement by S. E. Khaikin about force is only a formal definition of force.

The criterion of the reality of force, according to S. E. Khaikin, consists in the fact that force can be measured by at least two independent methods. One of them is provided by the second law of motion, in which force is measured through acceleration. But if there were no other independent method of measuring force (and mass), then the second law of motion would have to be regarded, according to S. E. Khaikin, not as an assertion but as a definition of force, in which force appears as the name for the product of the mass of a body by its acceleration. Only the possibility of another, independent measurement of force through the deformation of bodies produced by it gives, according to S. E. Khaikin, a physical meaning to force and thereby gives the character of an assertion (and not of a definition) to the second law of motion\(^*\).

It is enough to compare S. E. Khaikin’s criterion of the reality of force with the criterion of “physical reality” that F. Frank applies, in order to be convinced that, even while using the concept of force, S. E. Khaikin does not interpret it materialistically. We have seen that Frank, too, regards force as a physical reality only because it can be measured by independent methods; this did not prevent him from asserting that, at the same time, force is an intermediate quantity introduced for convenience of description.

The whole question of the reality of force is turned upside down by S. E. Khaikin, precisely as it is posed by the operationalists—

\(^*\) S. E. Khaikin writes: “Any relation has the character of an assertion only when the quantities entering into it can each be measured separately.... If, however, we have no method of measuring any of the quantities entering into a given relation, then this relation can at best be considered as the definition of a method of measuring certain quantities, but by no means as an assertion. We shall consider Newton’s second law as an assertion, as a physical law. For this we have first of all established a method of measuring force independent of Newton’s law” (italics ours.—S. Sh.) (28–29)\(^6\). S. E. Khaikin also applies this method in the 2nd edition of Mechanics.

stances, the untenability of whose views was examined above.

In the second edition of Mechanics, S. E. Khaikin omitted many openly Machist formulations concerning the concept of force, which were given in the book What Are Forces of Inertia. This does not mean that in the second edition of Mechanics he sets forth the correct conception of force. Here he retains the formulation of the general law of mechanics in which the concept of force is excluded (see above); as in the first book, he conducts the same reasoning about the second law of motion (an analysis of the cases when it is only a definition, and when it is also an assertion); as in Forces of Inertia, he does not find it necessary to reveal the objective meaning of the concept of force as a definite characteristic of the transformations of nonmechanical forms of motion into mechanical motion (in particular, he does not even introduce initial, if only purely qualitative, notions of physical fields). Meanwhile, to reveal the objective meaning of such a general concept as force is necessary already in mechanics, if we do not wish to turn the latter into a formal computational scheme, but seek to establish its genuine physical content.

We have already noted the peculiarity of the operationalist method of defining physical concepts, which leads to the fact that it excludes the possibility of clarifying the essence of those objective properties of motion that are characterized by these concepts. S. E. Khaikin criticizes, for example, Newton’s attempt to explain the inertia of a body, but he himself does not find it necessary even to point to the necessity of another explanation in the light of the results of modern physics.

It is enough to compare S. E. Khaikin’s exposition of the foundations of mechanics with the ideas of N. A. Umov set forth above, who, on the basis of the physics contemporary to him, succeeded in giving a materialist analysis of the foundations of Newtonian mechanics, in order to understand what an erroneous path S. E. Khaikin is following.*

The summit of formalism and operationalism in S. E. Khaikin is the exposition of the chapter “Mechanics of the Special Theory of Relativity” in the second edition of the course Mechanics. We shall not here analyze the substance of S. E. Khaikin’s treatment of the basic concepts of relativistic mechanics, since we believe that considering these basic concepts in isolation from electrodynamics, on the soil of which it developed, is fundamentally incorrect.

*) The absence of a concrete physical analysis of the equivalence of forces of inertia and forces of attraction led S. E. Khaikin to an incorrect formalist treatment of this equivalence, ignoring the physical conditions of its applicability and its local character; with such a treatment there inevitably follows, although not formulated by the author, the reactionary and anti-scientific conclusion, made in his time by Mach, about the equal rights of the systems of Copernicus and Ptolemy.

In Khaikin, however, relativistic mechanics is set forth in complete separation from electrodynamics; it has been reduced exclusively to an analysis of new methods for measuring mechanical quantities.

S. E. Khaikin writes: “The basic qualitative content of classical dynamics is preserved also in the dynamics of the theory of relativity. As in Newtonian mechanics, in the mechanics of the theory of relativity the accelerations of bodies (in inertial coordinate systems) are caused by forces, and forces represent the action of bodies upon one another” (550)5. This assertion is, in essence, incorrect. Here the question of the objective properties of the object of investigation has been omitted. If classical mechanics admits the conception of an ideally rigid body, then relativistic mechanics, which developed on the basis of the doctrine of the electromagnetic field, rejects the very possibility of the existence of ideally rigid bodies and of the instantaneous transmission of action—something already contained in implicit form in the very postulates of the theory of relativity. Without considering the new objective properties of bodies studied by relativistic mechanics—in particular, without considering physical fields—by reducing the content of the concepts of relativistic mechanics to methods of measurement, it is impossible to give a correct conception of its new content.

The whole paragraph on mass and force in the mechanics of the theory of relativity is devoted only to the methods of measuring them. It is precisely in certain changes in the methods of measuring mass that S. E. Khaikin sees the new element that determines the difference between the laws of relativistic mechanics and the laws of classical mechanics. He writes: “The law of motion of the mechanics of the theory of relativity outwardly has the same form as Newton’s second law in classical mechanics... However, the method of measuring one of the quantities entering into this law, namely mass, adopted in the mechanics of the theory of relativity, differs from the method of measuring mass in classical mechanics. Therefore, in essence, the law of motion of the mechanics of the theory of relativity represents a new law, fundamentally different from Newton’s second law” (spaced emphasis ours.—S., Sh.) (563)5.

Thus, the essence of the law of mechanics becomes new, “fundamentally different,” only by virtue of a change in the method of measuring one of the quantities entering into this law! The author’s operationalist approach is expressed in these words with the utmost “transparency.”

The connection between mass and energy in the theory of relativity is given by S. E. Khaikin merely as a collateral conclusion, in no way connected with the concept of mass, which he introduces into the content of the basic law of relativistic mechanics.

The formalistic treatment of relativistic mechanics in S. E. Khaikin’s textbook shows with particular clarity how much operationalism harms science. Instead of revealing the objective meaning of concepts and laws, showing precisely what objective—

... properties of moving matter are expressed in them, operationalism obscures the genuine content of science and does not make it possible to clarify what is new that is revealed by science in the process of its development.

It is characteristic that S. E. Khaikin considered it possible in his course to set forth the foundations of relativistic mechanics, which is very complex in its mathematical apparatus. However, he did not find it necessary to reveal, even by means of individual examples, that generalization within the limits of classical mechanics which is contained in the variational principles. Moreover, even work is treated by him only as the simple product of force by path, but it is not shown that it is the basic measure of the mutual transition of mechanical and non-mechanical forms of motion.

Thus, S. E. Khaikin does not show classical mechanics as a developing science *).

Critics of S. E. Khaikin directed their main attention to the erroneousness of certain general propositions expressed by him—about a law as an assertion, about idealization, and so on. Without doubt, S. E. Khaikin’s operationalist views were fully manifested in these formulations. S. E. Khaikin conceives the foundations of science as a set of empirical rules, as a set of simple statements of empirical facts, from which a number of consequences can be logically derived. Experience, in his view, appears in the form of a definite measurement. This coincides with the concept of experience among the neo-Machists. The foundations of science thus appear as the sum of assertions establishing the facts of the dependence of some quantities (the results of measurements) on others.

In contrast to assertions, definitions, according to S. E. Khaikin, are simple names or designations for certain physical quantities. He sees his task only in a successful classification of “assertions” and “definitions,” so that empirical facts may be presented in an easily surveyable sequence. This sequence may be one or another. In one sequence a law appears as a definition that does not require experimental verification; in another sequence a law appears already as an “assertion” requiring experimental verification.

All these ideas of S. E. Khaikin about science have nothing in common with the theory of knowledge of dialectical materialism. In reality, in S. E. Khaikin science appears not as a process

*) Some believe that it would be methodologically incorrect to explain to students that some concepts or others have a limited domain of application, beyond the limits of which they are generalized, since this allegedly introduces an element of vagueness into the exposition of the foundations of science.

We believe that these judgments are incorrect. Revealing the domain of application of one or another physical concept not only does not make it vague, but, on the contrary, helps to clarify its objective physical meaning and teaches the Soviet student to think correctly and scientifically.

an ever deeper and more faithful reflection of the objective nature of things, not as a movement through truths relative to absolute truth, as the Leninist theory of knowledge requires; science appears to him as an antihistorical scheme, as a sum of statements of facts concerning the existence of connections among the results of measurements.

Closely connected with such a view of science are S. E. Khaikin’s statements about idealization as a necessary moment of cognition. “Idealization” in S. E. Khaikin by no means coincides with the concept of abstraction in dialectical materialism. In idealization he sees only a way of arbitrarily restricting a problem. As we saw above, he applies such idealization in formulating the fundamental law of mechanics: certain kinds of forces are simply not taken into account, are arbitrarily excluded from consideration. Abstraction, however, as understood by dialectical materialism, is a way of revealing the most general and essential; in passing from one abstraction to another, science cognizes ever more deeply the essence of nature, its fundamental laws.

Under the pressure of criticism, S. E. Khaikin acknowledged some formulations as ambiguous and tried to correct them with additions in which he emphasizes his recognition of the objectivity of the existence of things and their properties, without thereby changing his views on the process of cognition.

However, in order to be a dialectical materialist, it is not enough to recognize the objectivity of the world. It is also necessary to accept the theory of knowledge of dialectical materialism and to be able to carry it through consistently in one’s special science, expelling from it an idealist theory of knowledge. Without this, no assurances by a scientist that he recognizes the objective world will make his works materialist. For S. E. Khaikin this means that he must renounce operationalist views on science and radically reconstruct the foundations of mechanics.

Let us sum up.

Only a consistent materialist interpretation of the foundations of mechanics, the path indicated by Engels, makes it possible to reveal the real content of the laws and concepts of mechanics, the connection of mechanics with the other physical sciences.

The positivist, Machist interpretation of mechanics reduces it to a system of equations connecting the results of measurements. It does not make it possible to determine the domain of application of Newton’s equations themselves and closes the path to understanding the new elements created at the subsequent stages of the development of classical mechanics after Newton. The positivist is incapable of evaluating the relative significance of the various stages in the development of mechanics. In fact, by presenting it as an aggregate of various systems of equa-

... he sees in the transition from one stage to another not a new, deeper understanding of the foundations of mechanics, but only formal mathematical transformations.

The development of science is not reducible to a simple extension of knowledge to ever new objects; in the light of new knowledge, the cognition of objects that would seem to have long since been studied and known is also deepened. Cognition is a process not only extensive, but also intensive. Each new stage in the development of physics compels us to reexamine even those branches of it that are considered classical, “established.” In the light of the new physics, the basic concepts and laws of “classical” physics find a deeper interpretation; their mutual relations change; the system of dividing laws and concepts into “basic” and “derived” is refined and sometimes radically reconstructed; the physical necessity of propositions that were previously treated as axioms is recognized. A theory never attains full “completeness,” as W. Heisenberg asserts in one of his recent articles.^9

Everything that has been said leads to the conclusion that philosophy is linked with natural science far more closely and organically than this seems to some physicists. It manifests itself in the whole “picture of the world,” in the entire aggregate of the natural scientist’s concrete views on the object and the process of cognition. Philosophy is not an “appendage” to science; it permeates the entire understanding of the foundations of science.

The task of developing the foundations of science in the spirit of dialectical materialism is a complex matter: it requires much work by Soviet physicists and philosophers. Our Party points out to us the necessity and urgency of this task.

CITED LITERATURE

1. V. I. Lenin, Works, vol. 14.
2. F. Engels, Dialectics of Nature, Gospolitizdat, 1946.
3. N. A. Umov, Collected Works, vol. 3, 1916.
4. E. Mach, Mechanics, translated from the 6th German ed., St. Petersburg, 1909.
5. C. E. Huygens, Mechanics, Gostekhizdat, 1947, 2nd ed.
6. C. E. Huygens, What Are the Forces of Inertia? (A Physical Introduction to Mechanics), Moscow, 1940.
7. Philip Frank, Foundations of Physics, International Encyclopedia of Unified Science, vol. I, Number 7, University of Chicago Press, 1946.
8. A. Sommerfeld, Mechanics, translated from German by T. E. Tamm, edited by Sivukhin, State Publishing House for Foreign Literature, Moscow, 1947.
9. W. Heisenberg, “Der Begriff «Abgeschlossene Theorie» in der modernen Naturwissenschaft,” Dialectica, 7/8 (v. 2, 3/4), 1948.