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p Nature is one in its diversity and is matter in motion This idea of dialectical materialism has become a common view of modern physics that is reflected in its methodology and logic as well as in its content. The two principles of the development and the unity of nature are also linked by modern physics in its explanation of and search for new phenomena and laws; one of the main tasks of this chapter is to demonstrate this.

p Strictly speaking, the definite general view of nature (the world-outlook problem) held by physics at one period or another of its historical development has always been internally connected with the logic of research characteristic of it at the same period (the methodological problem). That was the situation before classical physics, when physical knowledge based on the everyday observation and lacking (with certain exceptions  [320•* ) systematic methods of research, corresponded wholly with the very general and indefinite views of the philosophers of the period and their sometimes (if one thinks of ancient philosophy) inspired natural- philosophic guesses. That was also how it was in classical physics when the method of research proclaimed by Newton, and later called the method of principles, which was a sort of modification of Euclid’s axiomatics, had a certain correspodence with the atomistic approach to nature (which was held by Newton himself).

p The unity of nature is reflected in the unity of cognition. 321 This unity found its original form in axiomatics; geometrical knowledge was the first part of the knowledge of its time to become a science, in being constructed axiomatically by Euclid. Euclid’s geometry is a logical system of geometrical concepts where statements follow from one another so precisely and consistently that from the point of view of the thinking mind none of them raises any doubt.

p At the same time, Euclid’s geometry was not created by reason out of itself and was by no means an a priori construction. The word ‘geometry’ (’land measuring’) itself witnesses that it developed out of practical needs, namely, from the requirements of measuring plots of land (in the states of antiquity) and from astronomical observations (in ancient Egypt and Babylon), as a generalisation of the corresponding observed data. Before Euclid, the mathematicians of the day were occupied in solving many mathematical problems of everyday life, the connection between which they did not always grasp, and with the properties of individual geometrical figures (the triangle, circle, etc.); they knew individual theorems but they could not deduce them from a single logical principle. This empirical approach to geometry (and in general to mathematics) was historically inevitable in its first stage of development. After Euclid (it has long been a commonplace) such an approach was no longer necessary.

p Does this circumstance mean that geometry (and, therefore, its development) has not drawn anything from experiment since Euclidean times? To answer that, let us compare the axiomatic construction in geometry (at least its general features) with the axiomatic approach (or axiomatic method of research in the broad sense of the term) in physics, which has been established in this science since Newton, and analysis of which is one of the main tasks of this chapter.

p The complete or closed system of one physical theory or another (classical mechanics was the first to take this path) consists of basic concepts and principles (called axioms in geometrical language) which link these concepts through certain relations, and of corollaries that are deduced from the axioms by logical deduction. It is these corollaries that should correspond to the experimental data (be checked in experiment). A physical theory cannot be physical without this, or rather experiment and only experiment 322 is the criterion of truth of physical theories, i.e. only experiment finally certifies that a theory reflects objective reality and, therefore, certifies that the mathematical apparatus (formalism) of the theory is appropriate.

p The axiomatic approach in physics thus enables its theories to master the truth through logical thinking. Modern physics distinguishes six closed systems of concepts, connected by axioms each of which describes a certain sphere of phenomena of nature. The first system is Newtonian or classical mechanics, which includes statics and dynamics. The second system formulated for the purposes of the theory of heat is connected with (but by no means is ‘reduced’ to) classical mechanics through the statistical approach. The third system was deduced from the study of electricity and magnetism (and given shape by Maxwell). The fourth system is the (special) theory of relativity, a kind of combination of classical mechanics and Maxwell’s electromagnetic theory that was given final form in the work of Einstein and Minkowski. The fifth system embraces primarily quantum mechanics, and through it the theory of atomic spectra and chemistry. Finally, the sixth closed system of concepts is the general theory of relativity, which was given this name by its author, Einstein, and has not yet found its final form as a physical theory (it mainly consists of a developed mathematical apparatus). In addition, the possibility of the existence of a seventh closed system of concepts must be mentioned which may have to be formulated in connection with the construction of a modern theory of elementary particles, and which would link quantum mechanics and the theory of relativity in a deep synthesis. Each system of concepts in physics has a corresponding mathematical apparatus (formalism) inherent to it, which describes a definite domain of physical phenomena, evidence about which is provided by experiment; the limits of the applicability of a system’s concepts are also established by experiment (as regards their correspondence to nature). We shall discuss the relation between these axiomatic systems in the sections that follow.

p The experiment, of course, has no direct connection with the closed systems of geometry, but the needs of the experimental sciences (i. e. all the sciences about nature) frequently present mathematics with certain tasks (the physical sciences usually do it through their formalisms), which 323 the latter either fulfills, or will later. In this sense (in a mediated way) the mathematical disciplines are also connected with experiment. Even in the direct sense, although not altogether conventionally, geometry may be an experimental science. One can read this in Newton’s works  [323•* , and in Einstein it is formulated even more definitely. ’If ...,’ he wrote, ’one regards Euclidean geometry as the science of the possible mutual relations of practically rigid bodies in space, that is to say, treats it as a physical science, without abstracting from its original empirical content, the logical homogeneity of geometry and theoretical physics becomes complete’.^^1^^

p The whole strength of mathematics is that it not only can but has to abstract from ’its original empirical content’ if it wants to obtain new scientific results. In certain conditions, however, especially when the matters of the relation between mathematics and objective reality, or the objective meaning of its concepts and theses are being studied, it returns to its ’empirical content’, which provides new stimuli for its development. This was the case, for example, when the differential and integral calculuses were formulated, or when Gauss failed to confirm the ideas of nonEuclidean geometry through measurements; and it was done in its own way, and at a higher level of development of mathematics and physics, by Einstein’s theory of gravitation.

Thus, it follows even from this preliminary sketch of the axiomatic and empirical approaches to geometry and physics, that these approaches do not contradict each other and do not totally exclude one another. Such a counterposing was quite frequent, nevertheless, in the rising science and philosophy of the ancient world, and in the Middle Ages; according to the thinkers of Ancient Greece, experiments were an improper occupation, while the medieval schoolmen who, of course, respected only the authority of Holy Scripture, succeeded only in developing formal logic. From the time of Renaissance, which corresponds to the starting point of the modern science (in the broad sense of 324 the term) and the new philosophy, counterposing of the axiomatic and empirical forms of approach as the absolute opposites slowly but consistently disappeared from basic research and the struggle of ideological trends. These opposites became relative, while the axiomatic and empirical approaches proved to be aspects of a single general method of research in modern science, though even now one runs across relics of the old counterposing of these approaches in the relevant literature.

p Let us consider certain features of the axiomatic method of research more closely.

p This method has changed and been enriched with new possibilities of explaining and predicting the phenomena studied since Euclid’s times. Whereas it could be spoken of in its initial, as one might say Euclidean, form as ’• informal’^^1^^ or ’material axiomatics’^^2^^ (as Kleene puts it), now, since Hilbert’s famous work and studies in mathematical logic, axiomatics appears as both ‘formal’ and ‘formalised’. These two notions differ from the first in the concepts and their relations appearing in them in their pure form, as it were, free of empirical content, and in formalised axiomatics the language of symbols (formalism) is employed instead of verbal language, while in the material axiomatics deduction is not in fact isolated from empiricism and visualisation.

p This also applies mutatis mutandis to axiomatic constructions in physics. In the axioms or principles (sometimes called fundamental laws) of Newton’s mechanics it is a matter of inertial mass and force, acceleration, space and time, and the relations between these concepts. They (i. e. the relations and concepts) are the original ones in the context of Newton’s mechanics and by themselves are idealised expressions of experimental facts. First expounded in Newton’s Principia, they can serve as a model of informal axiomatics in classical physics.

p The development of the axiomatic method in physics for the most part repeats its development in geometry. In modern physics, with its very complex and ramified mathematical apparatus, one has every right to speak of the existence of formal, and especially of formalised, axiomatics (which in a certain sense is the pinnacle of 325 development of the axiomatic method). This has become absolutely clear since the establishment and construction of the theories of non-classical physics. A rigorous, and to some degree exhaustive, analysis of the related questions would take us far beyond the framework of this chapter; we shall try to give just a general idea of what the issue is.

p Consider the equation

p „ d(mv)

p dt •

p It expresses Newton’s second law, which implies that the mass of a body is constant. This equation, however, can be also considered in the context of the special theory of relativity; in this case m denotes

p where m_u is the mass of a motionless body, (’the rest mass’), v is the body’s velocity, and c is the velocity of light. This equation thus expresses the law of relativistic mechanics which implies that the mass of a body changes with its velocity. It may also express the law of motion in quantum mechanics; it is well known that in quantum mechanics, and in classical mechanics, quantities are connected (as we know) by the same equations, but in quantum mechanics they contain operators, i. e. concepts of a different mathematical nature than those of classical mechanics.

p The reader may rightly ask what the basis for such ’ substitutions’ in the equations is (i. e. for replacing numbers by operators, and m by a more complex expression), and what is in general their logical meaning. To answer this means to talk about the very content of classical, relativistic, and quantum mechanics, the transition from the special theory and its concepts to the deeper, more general theory, whose concepts are more meaningful than those of the special theory, i. e. from the angle of the above_y it is equivalent to speaking about the understanding of mass in relativistic mechanics and of the way this understanding was reached, and of the fact that, in quantum mechanics, operators mathematically depict physical cases that are never met by classical theory, to talking about the very logic of the rise of the special theory of relativity and quantum mechanics.

p We would like, thereby, to emphasise that the formal 326 and formalised axiomatic constructions of a physical science embrace the development of its content, promoting deeper and deeper comprehension of nature. It must be noted here that the interpretation of its formalisms has special significance in physics compared with the analogous problem in mathematics; we shall discuss this below.

p In what way is the axiomatic method essential for physics? In its logical and methodological aspects, the significance of this method in physics (both in the form of material axiomatics and in its higher forms, the formal and formalised) is not simply great but, as we shall try to show, cannot be overestimated. When it is compared with other methods of research, one cannot but agree with Hilbert who said about the axiomatic method in mathematics, that ’notwithstanding the great pedagogical and heuristic value of the genetic method, the axiomatic method is preferable for the final representation and full logical substantiation of the content of our knowledge’.^^3^^

p It seems to us, we repeat, that what Hilbert said about the axiomatic method in mathematics applies as well to physical axiomatics. In this case, of course, as always, one must not succumb to the extreme view and exaggerate Hilbert’s profound thought.

p Let us begin with the genetic method mentioned by Hilbert. We shall touch on its content, but in a way rather different from Hilbert’s.^^4^^ We wish to speak of the role of the genetic method in cognition and at the same time, unlike Hilbert, to stress that it is, in its own manner, ‘included’ in the axiomatic one.

p How is the concept of number introduced? Assuming the existence of zero and starting from the statement that when a number is increased by one the next number emerges, we obtain a natural series of numbers and develop the laws of counting with them. If a natural number a is considered and one is added to it b times, we obtain a number a -f- b and thereby define (introduce) the operation of addition of natural numbers (and with it a result called the sum).

p Let us now add a numbers b times; we thus define ( introduce) the operation of multiplication of natural numbers and will call the result of the operation the product of a and b, denoting it by ab. In a similar manner, omitting the corresponding argument, we define the operation of raising to a power.

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p Consider the so-called inverse operations of addition, multiplication, and raising to a power. Assume that we have numbers a and b; how can one find the number x that satisfies the equations a -f x = b, ax = b, x^a = fc? If a -f- x = = b, then x is determined by substraction; x = b—a (the result of which is called the difference). The operations of division, extracting the root, and computing the logarithm (the last two operations are inverse in relation to raising to a power) are introduced in a similar manner.

p With these definitions as the basis, it is possible to construct the axiomatics of natural numbers. The relevant axioms can be grouped: a) axioms of conjunction; b) computational axioms; c) axioms of order; d) axioms of continuity. We shall not dwell on their analysis (Hilbert’s book has everything necessary about this, the difference between his presentation and ours being, however, that in his work these axioms function as those of a real number; this point will be clarified below).

p We have come to the main point of our argument. As follows from the practice of looking for solutions of equations in which these numbers appear (they were interpreted as natural numbers, but from the standpoint of arithmetical axiomatics they are usually considered as the positive integral parts of a rational number; it seems to us that this is only correct when they are approached retrospectively), the inverse operations (subtraction, division, root extraction) cannot be performed in every case. We shall not cite the relevant facts; they are well known now to any schoolchild, but will assume that the inverse operations are realised in all cases. As a matter of fact, this assumption came true in arithmetic during its historical development, and in the end as a kind of logical resume of this development, whole numbers of integers and fractions, positive and negative numbers, rational and irrational numbers entered it.

This kind of dividing of a natural number into these two opposite elements, and the relation between them led to the concept of relative number, of number as a ratio, and of a real number; the last consequently developed from the simple concept of a natural number through successive generalisations. The concept of a real number is being developed further in modern arithmetic, but what we have said is sufficient for our purposes.

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p To go deeper in points considered above would mean to sink ourselves completely in the content of scientific disciplines appertaining to mathematics that were created to analyse them. We are interested only in what follows.

p In assuming that inverse operations are realisable in all cases, for instance, if one has subtraction in mind, that it is possible to subtract a bigger number from a smaller one (i.e. to solve the equation a—b = x where b >a), we thereby already go beyond the context of the theory of natural numbers. Assumptions of this kind imply looking for and introducing (denning) new concepts of number, broader and more meaningful in their totality than the concept of a natpral number. To do that axioms formulated during introduction of natural numbers are employed, which are regarded as embracing new numbers which make it possible to give appropriate definitions of the latter. One can demonstrate in an example of subtraction that as a result of applying axioms we can obtain, say, (14—6)—10 = = (10-2) -10 or 4-6 = 0-2.

p Similarly one can define (introduce) the numbers 0—1 or—1, 0—2 or—2 (which we have done), 0—3 or—3, 0—4 or—4, 0—5 or—5, etc. By comparing negative numbers with natural numbers, we can easily see the opposition between them; therefore, if the natural numbers are denoted as positive integers, the new numbers should be called negative integers.

p Introducing natural numbers we deduced the axioms applying to them; now these axioms define a new class of numbers which appear as positive and negative integers (i.e. as relative numbers). The formal aspect of the axioms themselves remains the same, but their content becomes richer: it is already impossible to define the addition of negative numbers, and the addition of negative and positive numbers as a consecutive increase of a number a by one b times, although all forms of addition unknown to the theory of natural numbers (as well as the addition of natural numbers) are covered by the system of axioms formulated as the axiomatics of natural numbers. Only now the symbols in the axiom equations signify new numbers found through these axioms, rather than natural numbers.

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p Fractions and integers, irrational and rational numbers are introduced in a similar manner. From this angle a series of natural numbers is a set of positive integers and rational numbers that oppose negative integers, fractions and irrational numbers that do not belong in this series.

p Feynmaii called application of the axiomatic method to determine new, broader classes of numbers (in our example, natural numbers—>• relative numbers—>• numbers as ratios —*• real numbers), an application in which the heuristic function of axiomatics, so to say, becomes explicit, an ’ abstraction and generalisation’.^^5^^ This method was, as a matter of fact, used by Marx in his Mathematical Manuscripts, when he developed the dialectics of the transition from algebra to differential calculus.^^6^^ A dialectical analysis of certain aspects of questions arising can be found in the work of I. A. Akchurin et al (1968).^^7^^

p In this application of axiomatic method it is stressed in fact that axiomatics by no means excludes acceptance of the variability of basic concepts and logically closed theories; on the contrary, it implies the necessity for new basic concepts and principles to arise. Everything that makes the axiomatic method so valuable for the logical shaping and the full logical substantiation of scientific theories gets its true (and not in the formal-logical sense) completion and expression adequate to reality through this kind of application of axiomatics.

p Bourbaki expressed this beautifully about mathematics: ’The unity which it [the axiomatic method—Ed.] gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to "substitute ideas for calculations".’^^8^^

p The situation in physics is much the same. The principle of relativity, for instance, which is a consequence of the principles of Newton’s mechanics, i.e. the principle of relativity in Galileo’s form of it, did not hold in the case of the propagation of light. The phenomenon was governed by the principles of electromagnetic theory. It thus became a matter of expanding the sphere of applicability of 330 mechanical principles by including electromagnetic phenomena in it. This meant, however, that the principles of Newton’s mechanics should form a single integral system with the principles of electromagnetic theory. Their combining led to the birth of new concepts that were broader and more meaningful than those of classical mechanics. The concepts of space and time were the first to undergo change; the concepts of absolute space and absolute time disappeared; they were replaced by the concepts of relative space and relative time, which proved to be aspects of a single four-dimensional space-time continuum. Galileo’s transformation (which connects inertial reference frames in Newtonian mechanics and implies absolute space and time) was accordingly replaced by the Lorentz transformation (which connects inertial reference frames and implies relative space and time). The principle of relativity had already appeared in its generalised Einsteinian form, and relativistic mechanics emerged.

p Our second example is provided by quantum mechanics. In this theory (which is discussed here with its logically closed form in mind) there is a basic postulate: namely that for each physical quantity (dynamic variable) in classical mechanics there is a certain linear operator in quantum mechanics which acts on the wave function; it is assumed that the relations between these linear operators are the same as between the corresponding quantities in classical mechanics.

p In quantum mechanics a postulate that connects an operator with the value of a quantity characterising the reading of the measuring instrument (by means of which knowledge of the micro-object is obtained) also has a basic role.

p Our two examples represent a kind of logical summary of the state of affairs that had developed in the theory of relativity and quantum mechanics when these theories were constructed. Like any summary, it does not depict the whole diversity of the logical and actual situations that had built up when these theories were being created, does not reproduce the details of the combination of thought and experiment which brought the principles of these leading theories of modern physics into being. In order to avoid possible misunderstandings in clarifying the method discussed here by which the new concepts were found (by means of axiomatics) we must draw special attention to the fact that 331 axioms, having been deduced in the defining of such- andsuch fundamental concepts become in turn the basis for deducing hew, broader, more meaningful fundamental concepts than the initial ones. The equations expressing the axioms now contain symbols without real meaning; and the essence of the matter is how to find these real meanings, i.e. to find new concepts (and that means to construct a new theory). This is done, as we know, by the methods of mathematical hypothesis, fundamental observability, and other theoretical techniques of modern physics.

When one takes such circumstances into account, it becomes clear that although (we shall take a well-known example) the structure of ,the axioms of relative numbers or real numbers is identical with the structure of natural number, it is impossible still to learn just from this isomorphy, how, say, one should add or multiply negative numbers. Similarly, the fact that the structure of the principles in classical, relativistic, and quantum mechanics is the same does not by itself guarantee knowledge of the main laws of relativistic and quantum mechanics (when the laws of classical mechanics are known). It is useful to recall Engels’ remark here on the law of the negation of the negation. Knowledge of the fact that this law of dialectics covers the development of grain and the calculus of infinitesimals, he said, ’does not enable me either to grow barley .successfully or to differentiate and integrate^^9^^’. As we have seen, the situation with axiomatics is similar. That, however, does not diminish the fruitful methodological role of the laws of dialectics, or of axiomatics, in any way; it is worth stressing once more that this methodological role is not the dogmatic finger of the Almighty even when this deity appears disguised as a scientist.

p The problems of axiomatics considered above necessarily include the problem of interpretation. We shall discuss it here in concluding this section.

p The concept ‘interpretation’, as it originated in mathematics and was adopted in modern logic, does not coincide with the normal usage of the concept (or of ‘comment’). Interpretation brings out the meaning of the symbols and formulas in scientific theories in which the axiomatic method plays a leading role (in the deductive sciences which, 332 according to Einstein, include physics as a fundamental science). Interpretation rests not on a visualisation but on certain logical foundations which we shall not analyse here. It establishes a system of objects that form a domain of values of the symbols used in a theory, and its job as a logical operation consists precisely in determining the objects in which the symbols and formulas of the theoretical system can be realised.

p The following difference in the meaning of the concept of interpretation in mathematics and physics, which individual authors sometimes find it difficult to see, is of great significance. The logical operation of interpretation can play a decisive role in mathematics in certain conditions in clarifying matters that are very important for mathematical knowledge (for instance, the problem there used to be of the consistency of non-Euclidean geometry). But for all that, the main point is not the connection of the symbols (terms) and formulas (relations) of its theories with the objectively real world. Mathematics is not liberated in principle from this connection, of course, regardless of attempts of idealistically-minded authors to do so (about which we have spoken above). There is no immanent necessity, however, in mathematics itself for the values of the symbols of its theories, and through them also the symbols, to be connected with the data of observation and experiment. This fundamental feature of mathematical theories is expressed quite clearly and definitely in the systems of formal axiomatics. In Hilbert’s Grundlagen der Geometric (Foundations of Geometry) for example, ‘points’, ’staight lines’ and ‘planes’ denote those things (objects) and their relations in respect of which only one assumption is made, namely, that they satisfy the axioms; in Hilbert’s geometry one abstracts oneself from the visualised points and straight lines of Euclid’s meaningful geometry (which represent idealisations of the normal solids, and the relations between them).

p What is the situation in physics in this respect? In both physics and mathematics the formalisms of their theories can be given more than one interpretation. For instance, the axioms formulated by Ililbert can be interpreted in a way in which they are given in Euclid’s Elements, and also in such objects of theoretical arithmetic as real numbers (analytic interpretation). And in physics, if we take, for 333 example, the expression
it can be interpreted as a formula of total energy in classical mechanics, and also as the Schrodinger equation if one has in mind the operator form of this expression; it then becomes
where

p h d

p h a , =—T—, etc

p i t)_x ’

p Notwithstanding this, however, the symbols of formalisms in physics are necessarily connected with the readings of instruments and with observational and experimental data, i.e. the symbols in physical theories must be connected through interpretation with the objectively real world. Mathematics, as we have seen, operates differently. Without so-called empirical, or natural interpretation a physical theory is not a physical theory, in the same way as without formalism a physical theory is not a physical theory.

p From this standpoint, if every scientific theory  [333•*  in general is built up from logical (or, in certain special conditions, mathematical) apparatus (formalism) corresponding to it and its interpretation, a physical theory consists of a formalism (mathematical part) and an empirical interpretation (the visualised part). This does not, of course, mean that such-and-such a physical theory (or rather, its formalism) cannot be interpreted through the objects of another physical theory; it means only that no physical theory, if, we repeat, it is to be called a physical theory, can do without empirical interpretation. The transition from the data of observation or experiment to the formalism of theory, and from the formalism of theory to the data of observation or experiment (it is only as a result of such a transition that a real theory can be formed) is a very complex process, a leap that can only be analysed by dialectical logic.

p With construction of the theories of modern physics, the view became common that the laws of classical mechanics, 334 say, were not absolute universal laws of nature and were limited to a certain realm of physical phenomena or (in a rather modified and more general form) that it was impossible to regard the laws of the macroworld as valid in the microworld. This understanding of the laws of physics was consistently introduced in relation to the mathematical apparatus of the theories (in the developed formalism of the special theory of relativity four-dimensional vectors were employed, in quantum mechanics linear operators). But with respect to the rules of the relation of the concepts of the formalism (i.e. mathematical abstractions) to the experimental data (without which these concepts have no real physical meaning) such understanding of physical laws is not accepted by all authors, or rather they do not allow sufficiently for the fact that the rules of the connection between the formal concepts and the experimental data (or the receipts for the transition from mathematical quantities to physical ones) do not necessarily coincide in classical and non-classical theories. It is worth recalling here the interpretations given to the mathematical apparatus of the special theory of relativity and quantum mechanics by the opponents of Einstein’s and Bohr’s conceptions.

p One can consider the rules of connection of the concepts of a theory’s mathematical apparatus with the experimental data, or the concrete receipts that govern the connection of the mathematical quantities in the formalism with the physical ones, as expressions (definitions) of the corresponding physical concepts. This means, from this standpoint, that the construction of a new fundamental theory implies the exclusion of certain old fundamental concepts (which are retained in the old theory) and the introduction of new ones. The exclusion of the old fundamental physical concepts in the cause of creating a new theory is far from simple. So far the idea is still maintained in one form or another in the scientific literature that it is necessary and sufficient to be satisfied with the fundamental concepts of classical physics (space, time, motion, particle) in any physical theory. The receipts of the transition here from mathematical quantities to physical ones in the new theory do not so much make it possible to embrace the new objective reality at which the old theory came to a stop, as serve simply as a means of computation. Questions relating to this have been considered throughout our book.

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Translation of the language of formalism into the language of experiment, and vice versa, in one physical theory or another or the reflection of objective reality by a theory, or, in short, the physicist’s conversation with nature is a process of a kind in which one cannot take a step without dialectics.