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p Let us first consider the meaning of the expression ’ physically exact concept.’ Above we stressed, in one connection or another, that a physical concept in any physical theory is neither an instrument reading nor a mathematical abstraction; in a physical concept that reflects the objectively real, the two are fused together as it were. Exact physical concepts are exact for the reason that they correspond to the objectively real (in the final analysis, this correspondence is established by experiment).

p So-called abstract physical concepts cannot be counterposed to so-called visualisable physical concepts on the planes of their relation to the objectively real, or of their accuracy. Both the former and the latter reflect the objectively real and, if they correspond to it, they are exact concepts. No physical concept exists without a connection with experimental data, but abstract concepts are connected with such data by a more complex logical chain of reasoning (implying knowledge of the laws of nature) than visualisable concepts.

p Both types of concept thus make use of the concepts of everyday language in their definitions, but the degree of this use cannot be compared; in the definitions of the visualisable concepts it is relatively easy to find the roots of their origin in experiment, but abstract concepts are connected with the experimental data in a mediated, and frequently very complicated, way. In the axiomatic construction of a theory its concepts and the relations between them are defined quite exhaustively.

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p At the same time, exact concepts can rightly be called exact only within the limits of a certain closed system; in that sense they are relatively exact or approximate concepts if there is no single closed system.

p The concepts of a theory thus contain both an element of abstract thinking and an element of imagination. This applies to both the classical and non-classical theories of physics. In a classical theory, however, its concepts are a direct generalisation of the experimental data (the corresponding concepts of everyday language being raised in it, so to say, to the first degree of abstraction); in quantum theory, on the other hand, its concepts are not such a direct generalisation of the experimental data; the data are generalised in it in a mediated way, through the use of classical concepts.

p From this standpoint classical concepts are not at all a priori with respect to quantum theory, in the sense that quantum theory employs only classical concepts (with the corresponding limitations). Quantum mechanics employs its own basic concepts and principles; accordingly, its concepts that differ qualitatively from classical ones, do not differ from them in any way in the sense of their certainty, clarity, and exactness. The same has to be said of the concepts of other constructed non-classical theories.

p The tendency to establish exact concepts in science that arose from axiomatics does not by itself ensure adequate cognition of nature. Nature is inexhaustible as a whole and in any of its parts. Science and its theories and concepts that reflect nature consequently have to change and develop, reflecting it more deeply and completely; the old concepts (and theories) cease to be exact as regards the new sphere of ‘finer’ phenomena of nature, new exact concepts and theories being developed that correspond to the new sphere. Thus, when physics masters a new area of natural phenomena, the limits of applicability of its old concepts and theories are determined on the one hand, and on the other hand new concepts and theories are developed. These twoj processes, which appertain to concepts and theories, are, as a matter of fact, a single process of the development of science. Initially the inadequacy of the old concepts in regard to the new sphere of phenomena is established empirically, and difficulties and paradoxes arise in the existing theory (this is, so to say, the new 349 theory’s period of uterine development). Later the development of scientific cognition leads to precise determination of the applicability of the old concepts and theories, but this means, at the same time, the formulation of new concepts and their system: the new theory begins its existence.

p That was how things stood with the theory of relativity and quantum mechanics, which now represent closed systems of concepts (axiomatised theories). Such is the situation with the modern theory of elementary particles, in which the presence of difficulties and paradoxes speaks of a need for fundamentally new concepts, and system of ’crazy ideas’.

p As a result, the conclusion is inevitable that science neither can nor does manage in its development just with exact concepts. In certain conditions, when a new theory is being born, i.e. when it is a theory only ’in itself and has no developed system of concepts, science uses, and cannot avoid using, imprecise concepts without which it is impossible in practice to construct a rigorous, consistent, complete theory.

The tendency in the development of science that leads to the establishment of exact concepts in it is thus interwoven with, and merges with, an opposite tendency whose specific feature is to employ imprecise concepts in science. Imprecise concepts are inevitable with every advance of science. They disappear when a certain cycle of its development is completed, so as to emerge again at a new stage of its development.

p There is not simply something in common between the question of the physically exact concept and that of the process of formation of this concept itself, but, as we see, an inseparable connection between them. This point was not considered, as a matter of fact, in eighteenth and nineteenth century physics and in essence could not be. It then seemed that the sole system of axioms had been found that covered existing physics and should embrace all future physics; the physical equations and related concepts corresponding to it seemed absolutely exact and not restricted by any limits in this exactness. This system of axioms was embodied in the system of principles of mechanics formulated in Newton’s Principia, which could only be modified 350 but whose basis remained the same; the theoretical development of physics was regarded as consistent application of Newton’s mechanics to broader and broader spheres of natural phenomena, discovered in experiment.

p Since the theory of relativity, and especially quantum mechanics, became established, this understanding of axiomatics in physics, of course, has radically altered; with the new understanding of fundamental principles the question arose of the formation of a physical concept as a certain logical process.

p In Section 1 we mentioned six ’closed systems’ of appropriately ordered concepts, definitions, and axioms in physics, each of which describes a certain sphere of natural phenomena, and all of them are connected to some extent with one another. These ’closed systems’ undoubtedly reflect the existence of discontinuities or leaps in nature and correspond to the fact that the forms of motion of matter are connected through transitions and differ qualitatively from one another.

p In accordance with this understanding of physical axiomatics the equations of physics, and the quantities figuring in them, are never absolutely exact; or rather they are absolutely exact only within their applicability, and beyond it the question no longer arises. A more general and deeper theory (for instance, the theory of relativity or quantum mechanics) determines the field of applicability at certain points of the special and simpler theory from which it developed (in our example of classical mechanics, it is the limiting case with the tendency of the velocity of light to approach infinity and Planck’s constant—zero); in this case the more special and simpler theory is an approximation of a theory that is deeper and more general, while the corresponding quantities of the simple theory become approximate ones (for example, absolute simultaneity is an approximate quantity, i.e. a quantity such as preserves its meaning only within certain limits established by the theory of relativity).

p We must stress that approximate quantities are no ‘worse’ (or ‘better’) than exact ones (which relate to a more general theory) in the sense of their being adequate to objective reality in the same way as the laws of Newton’s mechanics are valid within their sphere of applicability, and cannot be ‘improved’, whereas the laws of mechanics of the theory 351 of relativity are valid in a broader sphere of phenomena than that reflected by classical mechanics, without discarding Newton’s laws. The more general theory, moreover, employs approximate quantities in the appropriate conditions without which its quantities would have no physical meaning (suffice it to note that the practical procedures of measurement in the theory of relativity and quantum mechanics implied use of approximate quantities that were, in the first place, quantities of classical physics).

p Thus, the approximate nature (i.e. the limits of significance) of a quantity in such-and-such a theory is determined by the more general, deeper theory; the quantity then sheds the illegitimate universalism that is inevitable until a certain time.

p The approximate nature of a quantity in a certain theory is discovered through the more general, deeper theory developing from it with its new underlying principles and basic (fundamental) concepts. But the process of physical cognition can also go in the opposite direction, when a theory is transformed into a more particular one with the formation of concepts that did not exist in the original theory. Fock, who brought out the fundamental significance of approximate methods in theoretical physics, reviewed this process of physical cognition (which is inseparably connected with so-called approximate methods in physics).^^28^^

p We shall not go into details of the philosophical problems associated with the analysis of approximate methods in physics, except to make the following comment. Modern physics rejects the metaphysical prejudice that the cognitive value of a special theory is less than that of a more general theory; a special theory covers a narrower sphere of phenomena—in that sense alone does the general theory provide more complete and therefore more adequate knowledge of objective reality; but in their own spheres of applicability these theories are equivalent as regards coverage of their corresponding spheres of phenomena; from the standpoint of cognition it is all the same whether we move in one direction along a genetic series of axiomatic systems or in the opposite direction. For modern physics the absolute is by no means, therefore, ‘better’ (or ‘worse’) than the relative as regards cognition; the same holds for the relation between the exact and the approximate. To 352 illustrate this, let us assume that in certain phenomena in which atomic objects are involved the wave nature of matter is not essential (particles moving in a cloud chamber); then, in these conditions, we would be justified in abstracting from the uncertainty relation, which limits the concept of a particle; classical physics and its concept of the trajectory of a particle, i.e. concepts that are impossible in quantum mechanics, come to the fore.

p When one reasons abstractly and takes into account the view now widely held, one can say that every physical theory and every physical concept are in principle approximate.

p Why do we make the stipulation: ’takes into account the view...’ etc? The point is that physicists have now become so accustomed to the idea of the variability, relativity, uncertainty, it would seem, of everything, that acceptance of the relativity of the fundamental statements of science does not cause much perplexity. What appeared to be heresy in eighteenth and nineteenth century science, when its theoretical foundation rested on Newton’s indisputable mechanics, is regarded in twentieth century science as almost hackneyed, in view of the idea of the variability of fundamental physical propositions. On the other hand, the idea that physics can arrive at something constant and final in the sense of its principles now appears strange, although it was considered quite normal in the days of classical physics. Meanwhile the idea that physics can be ‘completed’ in the sense of construction of its principles is now being voiced by individual scientists^^29^^; because of that our stipulation above was necessary.

p Furthermore, by employing the expression ’when one reasons abstractly’, etc., we thereby stress the fact that a physical theory (physical concept) contains a number of elements of physical neglect about which nothing is known at a given stage of development of physics but it is assumed that something will be known in the future. In the examples above certain physical theories (certain physical quantities) figured as really approximate theories ( quantities), and not just approximate in principle, and their approximate nature was demonstrated. Here, on the other hand, we mean theory in general and the fact that it contains neglected elements in principle.

p This idea is based on semi-empirical/semi-general considerations; since the rise of non-classical physics, 353 fundamental physical theories have been replaced by more general, deeper ones, and physics is now apparently on the eve of a new, similar change, in connection with the difficulties of constructing a theory of elementary particles and formulating a new cosmological theory; so it was and is now, and so it will be in future.

p Let us return again to real theories. In order to demonstrate that such-and-such a theory is approximate, we have to prove that it does not cover certain phenomena that are covered by a more general theory. The methodology of this question is, in fact, the methodology of the quest for and construction of a new theory, and here (the point concerns fundamental theories), non-classical physics has its own theoretical methods (unknown to the old physics), which achieve their end. These methods (the principle of observability, mathematical hypothesis, etc.) have the following inextricably connected premises in common: (1) by cognising something unknown, i.e. by going beyond the limits of the cognised, we extend established concepts, principles, and theory to this something; (2) this extension does not exclude but implies, on the contrary, that one may have to alter (revise) some of the theory’s established basic concepts and principles qualitatively and, therefore, in the final result, to construct new basic concepts and principles, i.e. a new theory. These two premises, in spite of their opposite nature, are essentially one, but depending on the conditions, which also include the cognised something, one or other of them comes to the fore.

p As regards the first, it can be thought that it would not be justified to extend principles and concepts that reflect the circle of known phenomena to the unknown things, for it would be wrong to extend the concepts of trajectory and particle to atomic phenomena—that could be demonstrated after long discussions and various theoretical misadventures when, as it seemed, one had to proceed directly from the appropriate thesis, and the truth would be found more quickly!

p The point, however, is that a new theory cannot be constructed, as it is, in general, impossible to cognise, of nothing and, therefore, one cannot, in cognising the unknown, do without established knowledge. An established theory (or one or other of its bits), when applied to 354 unknown phenomena so as to cognise them, functions in respect to them only as a hypothesis, with all the propositions and conclusions following from that fact. Without hypothesis discoveries in science are impossible, of course, and a hypothesis, understandably, in order to fulfil its task, must satisfy certain requirements.

p A hypothesis is internally connected with fantasy, but fantasy cannot be unrestrained and unchecked in science. Which theoretical structure can be more ideal in this respect than an established scientific theory, verified by experiment, which operates as a hypothesis!

p It is therefore logically justified that physicists, after a really unexpected discovery, do not immediately put forward staggering ideas and theories in order, so to say, to catch the unexpected phenomenon in the net of cognition, but study the discovered phenomenon very thoroughly, with, it would seem, unnecessary sluggishness, by means of the old theories and established principles. The discovery of radium did not immediately destroy the notion of the atom’s invariance; the Michelson-Morley experiment was analysed many times on the basis of the theories of classical physics; the same must be said of the phenomena with discovery of which quantum theory began its development.

p When we extend everything we already know to unknown phenomena, or to new spheres of natural processes, moreover, it is only thus that we open the way to scientific progress. What kind of science would it be, if it enabled cognitive problems to be solved (and solved them) only from the sphere of the known! The boundary between the two premises of physical cognition discussed above passes exactly through this point.

p As for the second of these premises, the most essential thing relating to it, in our view, has been analysed to one degree or another in Marxist literature on the methodology of modern physics, and we refer the reader to it.^^30^^

p All these problems gravitate to the idea of the dialectical unity of absolute and relative truths. For physicists who do not consciously accept dialectical materialism, this idea is frequently a stumbling block. A vivid example of this, on the plane of the issues discussed above, are the statements of Richard Feynman, a distinguished physicist who unconsciously, as we have often seen, applies the principles of dialectics to resolve the problems of his science.

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p According to him, ’there is always the possibility of proving any definite theory wrong; but ... we can never prove it right.... Newton ... guessed the law of gravitation, calculated all kinds of consequences for the (Solar) system and so on, compared them with experiment—and it took several hundred years before the slight error of the motion of Mercury was observed. During all that time the theory had not been proved wrong and could be taken temporarily to be right. But it could never be proved right, because tomorrow’s experiment might succeed in proving wrong what you thought was right.... However, it is rather remarkable how we can have some ideas which will last so long’^^1^^ (my italics— M.O.).^^31^^

One finds such passages quite often in Feynman’s book. From the standpoint of the unity of the exact and the approximate previously discussed, it is not very difficult to disprove Feynman’s seemingly factual considerations: the deviation of Mercury from the motion predicted by Newton’s theory can be explained by Einstein’s theory of gravitation, which is correct in a broader sphere of application than Newton’s; the latter, on the other hand, is a limiting case of Einstein’s gravitational theory. In his argument Feynman touched on statement that experiment is the criterion of a theory’s truth, but he, it must be assumed, is not familiar with the dialectical idea of the relative nature of this criterion.^^32^^ In Feynman’s opinion, it would seem, a physical theory, if it is correct (true), should be universal and final; in his view physical science is moving in essence to the latter; at least, we find the following concluding lines in his book: ’Ultimately, if it turns out that all is known, or it gets very dull, the vigorous philosophy and the careful attention to all these things that I have been talking about will gradually disappear.’^^33^^

p It is held, and rightly so, that the presence of a system of axioms in a theory is an indication of its logical completeness (closed state), but in the history of knowledge and science the logical completeness of a theory has usually been regarded as a synonym of sorts of its universality and invariance. This was justified historically, we may say, by 2000 years’ reign (to the middle of the nineteenth century) of Euclid’s geometry as the sole geometrical 356 system, or the 200 years supremacy (to the twentieth century) of Newton’s mechanics as the final, indisputable theoretical system of physics. We have tried to show the illusory nature of this notion when axiomatic ideas are considered on the logical plane. The logical completeness of a theory does not preclude its development but, on the contrary, implies it; we propose to examine this idea more definitely in the concluding part of this section.

p The first blow against the ideal of the classical understanding of axiomatic construction in physics was struck by Maxwell’s electromagnetic theory. In fact, however, the essence of this understanding of axiomatics did not change; during the heyday of the electromagnetic picture of the world many physicists replaced the bodies of mechanics and Newton’s axioms by an electromagnetic Held and Maxwell’s equations (an i Newton’s mechanics itself seemed refuted, with no relation to the foundations of the Universe). At that time the point of view of dialectical materialism on this issue was expressed by Lenin. When the electromagnetic picture of the world was being built up he pointed out the inconsistency of the opinion that materialism asserted ’a “mechanical”, and not an electromagnetic, or some other, immeasurably more complex, picture of the world of moving matter’.^ And, as the development of physics since Maxwell’s electromagnetic theory has demonstrated, Lenin was right.

p The final blow to the classical understanding of axiomatics in physics was dealt by the theory of relativity, and especially by the development of quantum mechanics when it took on its contemporary shape.

p It became clear (this fact was mentioned above in connection with other matters), that Newtonian mechanics had limits to the realm of phenomena that it was expected to explain and predict, i.e. limits to its applicability, and that electromagnetic phenomena, on moving bodies, and also atomic phenomena, could not be described and explained by the concepts and principles of Newtonian mechanics. Experimental studies of the relevant phenomena, plus analysis of the theoretical situations arising in classical physics, led on the one hand to the theory of relativity, and on the other hand to quantum mechanics. Now, of course, the physicist has become accustomed to the idea that no closed physical theory is absolute, that there are limits 357 of its applicability, and that, in this sense, it is approximate. But how is one to find the limit of applicability of a theory? And what is this limit? Let us begin with the second question. There are phenomena that cannot be described in the language of the concepts of a certain theory or, if they can be so described, cannot be explained by it; such a theory leaves out the sphere of these phenomena, i.e. the sphere of applicability of such-and-such a theory; this is that realm of phenomena that is, or can be, explained by it*_B’ In other words, another theory, already different in principle, operates (i.e. describes, explains, and therefore predicts) beyond the limits of applicability of such- andsuch a theory.

p We shall not analyse the matter of a theory’s limit of applicability in detail. One aspect, however, deserves attention. The expression ’the limit of a theory’s development’ is frequently used and, apparently, quite logically. What does it mean? And how is it related to the expression ’the limit of a theory’s applicability’ just discussed?

p This question only seems artificial. The point is that ordinarily it is said to be meaningless to speak of the development of an axiomatic system. Indeed, all the theorems of an axiomatic system can be interpreted as being implicitly contained in its axioms and rules of inference. Only the activity of [a mathematician (or a corresponding device) can make any theorem contained in it explicit (and there is an infinite number of such theorems of various degrees of ordering in an axiomatic system). At the same time everyone knows that it is by no means easy to infer (or deduce) theorems from axioms; and the obtaining of, say, a geometrical (or mechanical) fact and statement from the corresponding system of axioms is, as always with cognition, the solution of a problem of searching for an unknown from known data! Engels said that even formal logic was a method for finding new results.

p The deductive method (which includes the axiomatic method proper), like any method employing formal and dialectical logic, cannot do without imagination or fantasy. It is worth recalling once more that, according to Lenin, even the most elementary generalisation contains an element of fantasy.^^35^^ The role of imagination increases greatly, of course, when it is a matter of the ever broader and deeper generalisations with which science is concerned 358 and without which it ceases to be science;  [358•*  it is a gratifying task to study this role.

p Thus, in so far as the deductive method or, considering its higher form, the axiomatic method, leads from the known to the unknown and increases scientific knowledge, an axiomatic system should be regarded as a theoretical one that can and does develop in certain conditions. The development of an axiomatised theory is the obtaining of new, previously unknown facts and propositions within the limits of its applicability. As follows from its definition, this development of a theory occurs, so to say, within itself. The theory does not go beyond its limits in this development but remains the same from the standpoint of its principles (system of axioms).

p Let us turn to the question of how to find the limit of applicability or limit of development of an axiomatised theory.

p The answer, of course, cannot be reduced to demonstrating that one constructed theory contains another constructed theory, with the first determining the limits of applicability of the second in a way inherent to it and snowing that the latter is its limiting case. It is not a method of solving the problem; rather it implies the existence of such a solution. Can the limits of applicability of a theory, or the boundaries of the realm of phenomena explained by it, be found empirically?

p It depends on the circumstances. The result of MichelsonMorley experiments or the so-called ultraviolet catastrophe became in fact the limits of the applicability of classical theories: from these two ’little clouds’ in the clear sky of classical physics there developed the theory of relativity and quantum mechanics. However, the motion of Mercury’s perihelion which had been known for quite some time and was not covered by Newton’s theory of gravitation, had not by any means become the limiting point of the theory’s applicability. Einstein’s theory of gravitation, which 359 determined the limits of applicability of Newton’s, was not found along the methodological path on which the theory of relativity and quantum mechanics arose. The decisive role in the creation of Einstein’s theory of gravitation was played by the principle of equivalence, which implies the identity of inertia and gravitation, i.e. an experimental fact played an essential role, namely that the acceleration of all bodies falling in vacuo is the same; this fact was known to Newton who did not include it in the theoretical content of his theory of gravitation but accepted it simply empirically.

p It happens sometimes that an established theory does not explain certain known experimental facts; scientists become accustomed to that; but, as it turns out, their theoretical interpretation or explanation (justification) goes beyond the limits of the established theory, and sometimes only a person of genius can see this circumstance. That is how it was with the general theory of relativity, or Einstein’s theory of gravitation, which rested on the same experimental material (the same experimental base) as Newton’s theory at the time it was formulated, but added a set of new ideas to it that were alien to classical conceptions. The logical aspect of the rise of a theory in this way will be discussed below.

p So, how can one find the limits of the applicability of an axiomatised theory and of its principles and concepts, i.e. determine the realm of phenomena beyond which it is no longer valid and a new theory is required?

p A logically constructed theory or axiomatised theoretical system that functions correctly within the context of its applicability should be consistent and complete. Godel has shown that the consistency and completeness of a system itself cannot be proved by its theoretical means. It is usually accepted without proof (it was tacitly implied during the historical development of Euclidean geometry and Newtonian mechanics) that such-and-such a theory is consistent and complete if the specifically opposite is not required, in the same way as it is accepted without proof that a theory is universal if there are no facts contradicting it (as was noted above). The consistency and completeness of any theoretical system means that none of the statements which it contains implicitly and explicitly can be in contradiction with it and all should be explained by it, i.e. that all of 360 them are finally explained in it on the basis of its axioms and fundamental concepts.

p It follows from this that, if a phenomenon which is (say) to be explained within the context of a given theory not only cannot be explained but, on the contrary, contradictions (paradoxes) arise that cannot be resolved by this theory when explanation is attempted, we would be justified in considering their presence as an indication that the theory is nearing its limit.

p It is possible of course that after due reflection stimulated by the contradiction individual statements and concepts of the theory may be revised, and the contradiction resolved in terms of the given theory; in that case the contradiction and the way it is resolved serve only to improve the theory logically in terms of its principles. The same holds mutatis mutandis for the question of a theory’s completeness. At one time Einstein, Rosen, and Podolsky formulated propositions from which it seemingly followed that quantum mechanics in Bohr’s probabilistic interpretation was incomplete. It became clear, however (as Bohr showed), that Einstein was wrong: the initial proposition of his paradox in relation to the problems of quantum mechanics was ambiguous.³⁶ We are not interested in such cases: they appertain to the problem of logically perfecting a given theory in relation to its axiomatics, and not to that of the limits of its applicability.

p Let us now turn to the paradoxes that develop in a theory and are not resolved by its means; they are indications that the theory is nearing its limit, as was noted above. But that means (and we draw attention to it) that the necessity is arising to look for a new theory whose principles and fundamental concepts differ from those of the first, for a theory such as would resolve said paradoxes (or rather, in which they would not exist). The main task of all our further exposition is to analyse the corresponding problems.

p First of all we would stress that the logical path (and expression) of the historical movement from classical to modern physics was the birth of said paradoxes in a ( classical) theory and their resolution. To some extent this feature was also characteristic of Maxwell’s electromagnetic theory, the closest precursor of non-classical theories. Maxwell, who unified all the experimental data on electricity and magnetism found by Faraday, and expressed them in the language of mathematical concepts, saw a contradiction of sorts between 361 the resulting equations. In order to correct the situation, he added an expression to the equation without any experimental justification (it appeared later), and the theory of electromagnetism was born. Maxwell’s method of mathematical hypothesis also proved to be extremely fruitful in further research,  [361•*  it has frequently brought whole theories to physics.

p Einstein’s theory of relativity can serve as another example. It was created at the junction of classical mechanics and classical electrodynamics, as a result of resolving a paradox, a contradiction between Galileo’s principle of relativity and the principle of the velocity of light in vacua being independent of the motion of the radiating source when these principles were considered together. Podgoretsky and Smorodinsky have called such ‘junction’ paradoxes ’encounter contradictions’.^^37^^ The paradox above and its resolution are an excellent model of dialectical contradiction in relation to major problems of modern physics, on which one can find relevant studies in Marxist philosophical literature^^38^^. A most important role in resolving this paradox, i.e. in formulating the theory of relativity, was played by the method of fundamental observability.

p Quantum mechanics also developed in a certain sense as a result of resolving an ’encounter contradiction’, in this case, that of classical corpuscular mechanics (again Newton’s mechanics) and classical wave theory. The role of the wave theory, however, was played here not by the corresponding theory of matter but by the theory of electromagnetism; the ‘encounter’ was therefore by no means as ‘simple’ as with the (special) theory of relativity. Quantum mechanics developed as a result of resolving not only an ’encounter contradiction’ but also a number of other contradictions, some of which will be considered below. Here it is essential to point out that a rising new theory is, in the language of modern logic, a metatheory of sorts in relation to the original ones (this also applies to the theory of relativity).

p The problem that could be called that of the stability of the structure of ordinary bodies, molecules, corpuscles (particles) or of the atoms that, from the standpoint of Newtonian mechanics, underlie matter and the motion of 362 which determines, in the end, all universal changes, was of the greatest significance for understanding how quantum mechanics arose. Newton ’found a way out’ by postulating the infinite hardness (of divine origin) of the primary atoms, etc.^^39^^ The same problem arose in all its direct visualability, so to say, when it became clear that the ‘primary’ atom was a system consisting of electrically charged particles (a positive nucleus and negative electrons), and the problem of its stability had to be solved from the standpoint of classical electromagnetic theory. ’Rutherford’s atom’, as we know, was unstable, but the problem was solved by a (then young) Danish physicist Niels Bohr who constructed an atomic model, applying Planck’s hypothesis of quanta to ’ Rutherford’s atom’. ’Bohr’s atom’ proved to be really stable, which was explained in terms of the laws of nature, i.e. the ancient atom finally acquired stability, and not because somebody tried to convince himself and others of this in his own name or that of God, but because it was necessitated by the quantum laws of the motion of matter. From there, too, development of the main stem of the quantum theory sprang, whose content absorbed the idea of the unity of corpuscular and wave properties of micro-objects and led in 1924-1926 to the creation of quantum mechanics.

p Nevertheless, when one thinks deeply about how the problem of the stability of the structure of the atomic particles of matter was solved, the idea that it could have been done differently and not as it was even seems strange. For in fact, the properties and motion of macro-objects can only be explained by the laws of motion and properties of the micro-objects composing them when the latter are not ascribed the properties and motion of macro-objects, if one does not want to fall into regresus ad infinitum. That is what was done by quantum mechanics, which brilliantly demonstrated that the laws governing micro-objects are quite different from those governing macro-objects. But then the hardness of macro-bodies, the constancy of standards of length and time, i.e. the physical characteristics of macroobjects without which measurements and, therefore, physical cognition, are impossible, must get their substantiation in quantum mechanics, as the mechanics of objects at atomic level.

p On the other hand, man (if we may be allowed to express it so) is a macroscopic being; he learns about the microworld 363 only when the micro-objects act on macro-objects that he links to his sense organs; these macro-objects (they become measuring instruments for him) enable man to learn about the microworld in a mediated way. Thus man, when cognising the micro-objects, cannot help but use classical concepts, since only in terms of them can he describe the readings of instruments, i.e. since as he measures he cannot do without using classical theories.

p Such is the relationship between quantum and classical mechanics, to put it briefly; it leads us to an understanding of the relationship between the basic principles of the theory of physics which, it seems to vis, is typical of twentieth century physics.

p Note first that the mechanics of the atomic world ( quantum mechanics) not only cannot be reduced to the mechanics of macro-bodies (classical mechanics) (the theory of electromagnetism also cannot be reduced to classical mechanics, and does not absorb the latter), but the relationship between them contains something more. Quantum mechanics, as was stated above, is the basis, in a certain sense, of classical mechanics; it justifies some of its fundamental concepts that reflect the properties of macro-objects, i.e. it deals with these concepts in the same way as classical mechanics, in which the derivative concepts are justified by axioms.

p It must also be added to this that the fundamental concepts, in their connections that form the basic equations of classical mechanics, were developed from notions taken from everyday experience (hardness, inertia, force) and the relations between them. That lends the axioms of mechanics the necessary physical meaning without which these equations would be converted into purely formal ones, and it would be impossible to call them physical. As for the main fundamental concepts and their connections, expressed by the basic equations of non-classical theories, the mathematical abstractions corresponding to these equations are connected with nature (i.e. have, so to say, become physicised) in each theory according to the rules inherent in it, using the concepts of classical physics.

p From the standpoint of what has been said, a theory’s axiomatic system contains basic concepts and their connections that are not logically justified in this system but are postulated on the basis of certain considerations, which are taken into account when the system is being 364 constructed. In this respect the theory is called incomplete (and open), but this incompleteness is different in principle from, say, that of quantum mechanics which Einstein had in mind in his discussion with Bohr mentioned earlier. The fundamental concepts and connections that form the axiomatic system of a theory can be substantiated by a deeper, broader theory than it, with new axiomatics, etc. On the logical plane the status of the ‘substantiation’ of fundamental concepts and their connections in the axiomatics of a theory is similar to that of an axiomatic system’s consistency and completeness, which, as Godel demonstrated, cannot be justified by the means of this system. Or, in more general form, the basic statements of a theoretical system cannot be obtained by its logical means, but they can be found by the logical means of a broader, deeper theory^^40^^. Using the same logical terminology one can say that quantum mechanics is a kind of metatheory of classical mechanics.

p Let us return to the example of Einstein’s theory of gravitation discussed above. Newton’s theory, and classical mechanics did not ‘brood’ over the proportionality or (with the appropriate choice of units) equality of the gravitational and the inertial mass of a body; it was just stated by classical mechanics. Toj find the justification of the equality between the gravitational and inertial masses of a body, or better justification of the statement that ’the gravitational and inertia masses of a body are equal’ would have meant to go beyond the limits of Newton’s gravitational theory and to construct one that would be a novel metatheory with respect to it. This was what Einstein did when he created a new theory of gravitation, or, as he called it, the general theory of relativity. We shall speak about this in Einstein’s own words, with citations from his works, limiting ourselves just to comments.

p Having spoken about the proposition that ’the gravitational mass of a body is equal to its inertial mass’, Einstein said further that it ’had hitherto been recorded in mechanics, but it had not been interpreted’^^1^^ (in this case we employ the expression: classical mechanics did not substantiate, did not find grounds). And he concluded: ’A satisfactory interpretation can be obtained only if we recognise the following fact: The same quality of a body manifests itself according to circumstances as “inertia” or as “weight” (lit. “heaviness”).^^41^^ By having formulated this idea he thus gave 365 grounds for the equality of the gravitational and inertial mass, empirically stated in classical theory, and laid the basis for his theory of gravitation.

p The following excerpt (which we give without comment) from his paper What Is the Theory of Relativity? can serve to illustrate his basic idea: ’Imagine a coordinate system which is rotating uniformly with respect to an inertial system in the Newtonian manner. The centrifugal forces which manifest themselves in relation to this system must, according to Newton’s teaching, be regarded as effects of inertia. But these centrifugal forces are, exactly like the forces of gravity, proportional to the masses of the bodies. Ought it not to be possible in this case to regard the coordinate system as stationary and the centrifugal forces as gravitational forces? This seems the obvious view, but classical mechanics forbids it’.^^42^^

p If one draws together everything that has been said about a theory and its metatheory, the following conclusion suggests itself. The paradoxes arising in a theory that cannot be resolved by its logical means are an indication that the theory has reached its limits of applicability, and that its axiomatics (axiomatic construction) is its highest logical completion possible from the standpoint of its actual content and axiomatic form. Such paradoxes differ fundamentally from those that develop in a theory and are resolved by its logical means, i.e. from those that provide evidence of the theory’s logical imperfection (incorrectness in the reasoning or inaccuracy in the premises). The existence of paradoxes that are not resolvable by a theory’s logical means indicates the need to search for more general, deeper theories in terms of which they are resolved (the resolving usually coincides with the construction of the general theory being sought). The existence of this kind of paradox thus means, in fact, that the physical cognition of objects does not stay long at the level of such-and-such a theory but develops further, embracing new aspects of material reality, without discarding the knowledge already achieved by it; the existence of this type of paradox also means that the theory that contains paradoxes but does not resolve them in its own terms, potentially contains a theory that is more general and deeper than it. From this position every axiomatised theory necessarily contains knowledge that cannot be substantiated in its terms; otherwise cognition would become frozen at a 366 certain point, and the knowledge gained would be converted into a metaphysical absolute.

p The development of a theory of contemporary physics is ensured by a genetic series of theoretical systems representing axiomatic structures that are either closed or under logical construction and connected through certain relationships, the more general theoretical system in the genetic series of such structures growing out of the more special one. The single axiomatic system of the whole of physics, in the spirit of the mechanistic ideals of the eighteenth and nineteenth centuries, was buried by the development of physics. This system also proved to be logically impossible, as Godel’s theorems have shown; the logical development of a theory and of physical science as a whole is expressed by a genetic hierarchy of axiomatic systems combining a tendency to stability with one toward variability, which are inherent in individual axiomatic systems and in their aggregate.

p A single axiomatic system (structure) in the spirit of classical physics has been put an end to, but in the realm of ideas, more than in any other, the dead clings to the living. A single axiomatic system is being reborn in modern physics, too, though in a form seemingly far removed from its ’ classical’ model. In our day we can find the following conception about physical science in the literature: physics is constructed in principle as a rigorous, consistent, axiomatic system covering all its branches, in which the historically earlier theory (and its axiomatics) is the limiting special case of the historically later one (which proves to be broader than the first). In due course the same happens to the last theory, and so on. Feynman paints approximately such a picture in the axiomatics of modern physics. The following question then arises, however: does this ’and so on’ continue to infinity? We shall not go into its details and shall try to answer it.

p It is asked whether there really is an indisputable single axiomatics embracing all physics that allows for its present and possible future development.

p The question is answered in essence by the material above on the relationship of a ‘theory’ and a ‘metatheory’ in physics. All that remains here is to stress certain aspects of the problem.

p When a theory is generalised, i.e. when we pass from the 367 special to the general theory, the former by no means disappears completely in the latter, and the latter does not at all become the sole true theoretical system in physics, as would be the case if there were a single axiomatics in physical science. In reality, the special theory is preserved in the general one in a modified form (this also holds in respect of certain of its concepts); it remains in the general theory as an approximate one, and its concepts are also preserved as approximate. From this angle we can also speak of absolute simultaneity in Einstein’s theory of relativity. A theory is not discarded when it passes into a more general theory, but remains as relative truth, i.e. absolute truth within certain limits; this is the very ‘best’ for a theory from the standpoint of its relation to objective reality, since it is being found how far it is true.

p All this is associated with answering the following questions (some of them considered above). Why is it necessary to use Euclidean geometry in seeking the ’non-Euclidean nature’ of a certain spatial form? Why do we learn about the properties of the space-time continuum from separate measurements of space and time? Why are the concepts of classical mechanics employed to describe the experiments that constitute the experimental basis of quantum mechanics?

p We are convinced that dialectical contradiction, the source of every development of life, also operates in axiomatics.

p REFERENCES

p  ^^1^^ Albert Einstein. Ideas and Opinions (Alvin Redman, London, 1956), p 272.

p  ^^2^^ S. C. Kleene. Introduction to Metamathematics (Van Nostrand, New York, Toronto, 1952), p 28.

p  ^^3^^ David Hilbert. Grundlagen der Geometrie (Verlag und Druck von B. G. Teubner, Leipzig, 1930), p 242.

p  ^^4^^ Ibid., p 241.

p  ^^5^^ R. P. Feynman, R. B. Leighton, M. Sands. The Feynman Lectures on Physics, Vol. 1 (Addison-Wesley Publishing Co., Reading, Mass, 19), pp 22-23.

p  ^^6^^ Karl Marx, Matematicheskie rukopisi (Mathematical Manuscripts) (Nauka Publishers, Moscow, 1968).

p  ^^7^^ I. A. Akchurin, M. F. Vedenov, and Yu. V. Sachkov. The Dialectical Contradictoriness of the Development of Modern Science. In: 368 M. E. Omelyanovsky (Ed.). Afateriatisticheskaya dialektika i metody estestvennykh" nauk (Nauka Publishers, Moscow, 1968), pp 43-69.

p  ^^8^^ N. Bourbaki. The Architecture of Mathematics. The American Mathematical Monthly, 1950, 57, 4: 231.

p ^a Frederick Engels. Anti-Diihring (Progress Publishers, Moscow, 1969), p 162.

p  ^^10^^ N. Bourbaki. Art. cit., p 223.

p  ^^11^^ Albert Einstein. Op. cit., p 274.

p  ^^12^^ Richard Feynman. The Character of Physical Law (BBC, London, 1965), p 54.

p  ^^13^^ Ibid., pp 172-173.

p  ^^14^^ Ibid., pp 55-56.

p  ^^15^^ Ibid., p 58.

p  ^^16^^ Albert Einstein, Op. cit., p 274.

p  ^^17^^ Ibid., p 271.

p  ^^18^^ Ibid., p 274.

p  ^^19^^ See also M. E. Omelyanovsky. Modern Philosophical Problems of Physics and Dialectical Materialism. In: M. V. Keldysh (Ed.). Lenin i sovremennaya nauka, Vol. I (Nauka Publishers, Moscow, 1970), p 230.

p  ^^20^^ Albert Einstein. Mein Weltbild (Ullstein Biicher, Berlin, 1934), p 109.

p  ^^21^^ V. I. Lenin. Philosophical Notebooks. Collected Works, Vol. 38 (Progress Publishers, Moscow), p 372.

p  ^^22^^ Albert Einstein. Op. cit., p 273.

p  ^^23^^ V. I. Lenin. Op. cit., p 372.

p  ^^24^^ Sir Isaac Newton. Op. cit., pp 397-400.

p  ^^25^^ Albert Einstein. Op. cit., p 271.

p  ^^26^^ Ibid., p 274.

p  ^^27^^ V. A. Fock discusses this definition in his paper Space, Time, and Gravitation. In: A. M. Nesmeyanov (Ed.). Glazami uchonogo (AN SSSR, Moscow, 1963), p 13.

p  ^^28^^ V. A. Fock. The Fundamental Significance of Approximate Methods in Theoretical Physics. Uspekhi fizicheskikh nauk, 1936, 16, 8.

p  ^^29^^ See, for instance, Richard Feynman. Op. cit., p 173.

p  ^^30^^ See, for instance, M. E. Omelyanovsky (Ed.). Materialisticheskaya dialektika i metody estestvennykh nauk (Materialist Dialectics and the Methods of the Natural Sciences) (Nauka Publishers, Moscow, 1968), which also contains a bibliography on the relevant matters.

p  ^^31^^ Richard Feynman. Op. cit., pp 157-158.

p  ^^32^^ V. I. Lenin. Materialism and Empiric-criticism. Collected Works, Vol. 14, pp 142-143.

p  ^^33^^ Richard Feynman. Op. cit., p 173.

369

p  ^^34^^ V. I. Lenin. Materialism and Empiric-criticism. Op. cit., p 280.

p  ^^35^^ V. I. Lenin. Philosophical Notebooks. Op. cit., p 372.

p  ^^38^^ A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? The Physical Review, 1935, 47, 10: 777. N. Bohr. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 1935, 48, 8: 696.

p  ^^37^^ M. I. Podgoretsky, Ya. A. Smorodinsky. On the Axiomatic Structure of Physical Theories. Voprosy teorii poznaniya, No. 1, (Moscow, 1969), p 74.

p  ^^38^^ For more details see M. E. Omelyanovsky. On Dialectical Coatradictoriness in Physics. Voprosy filosofii, 1970, No. 11.

p  ^^39^^ Sir Isaac Newton. Mathematical Principles of Natural Philosophy. Optics (Encyclopaedia Britannica, Chicago, 1952), p 541.

p  ^^40^^ Kurt von Godel. Ober formal unentscheidbare Sa’tze der Principia Mathematica und verwandter Systemei. Monatshefte fur Mathematik und Physik, 1931, 38: 173-198. On this point there is excellent material in V. A. Fock’s article cited above (Uspekhi fizicheskikh nauk, 1936, 16, 8). It reviews the historical development of theoretical physics to some extent reversing the eourse of this development.

p  ^^41^^ Albert Einstein. Relativity. The Special and General Theory ( Hartsdale House, New York, 1947), p 77.

 ^^42^^ Albert Einstein. Ideas and Opinions, p 231.