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p Visualisation is a problem of considerable philosophical significance in modern physical theory, especially in quantum theory. Suffice it to recall that the absence of a mental picture in quantum mechanics, as it seemed (unlike classical theory), used to be employed by individual physicists as a proof of sorts simply of the tentative nature of those of its general ideas that gave it a revolutionary character. The problem could not help occupying a prominent place in the work of the founders of quantum theory.

p Dirac, for instance, said that nature’s ’fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies’. Quantum theory, moreover, according to him, is built up ’from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all’.^^1^^ We shall return to these ideas of Dirac’s below.

p Let us also note Niels Bohr’s analysis of issues relating in one way or another to this problem. One must agree, in particular, with his statement that ’however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms’.^^2^^

p At the same time, according to him, ’an adequate tool for a complementary way of description is offered precisely 79 by the quantum-mechanical formalism which represents a purely symbolic scheme permitting only predictions, on lines of the correspondence principle, as to results obtainable under conditions specified by means of classical concepts’.³ This apparatus, Bohr says, is an appropriate means of describing ’complementary phenomena’, i.e. phenomena observed in mutually exclusive experimental set-ups permitting the particle and wave properties of atomic objects to be discovered.

p The idea of complementarity thus served Bohr as the key to the visualised interpretation of atomic processes. That this is the case is strikingly clear in his essay ’Quantum Physics and Philosophy’, in which he wrote in particular: ’the limited commutability of the symbols by which such variables are represented in the quantum formalism li.e. the quantities that characterise the state of a physical system in classical mechanics—M. 0.] corresponds to the mutual exclusion of the experimental arrangements required for their unambiguous definition.’^^4^^

p The problem of forming a mental picture is directly related to the trends in philosophy. The concept of visualisation, which arose on the soil of everyday experience and is linked with common sense, is not accepted by idealism, which disclaims its significance in man’s cognitive activity. Materialism, on the contrary, accepts it and develops it in depth (which also applies to dialectical materialism).

p An example of a point of view on visualisation in quantum physics that is close to the views of objective idealism is provided by Heisenberg’s considerations on the complementarity of the mathematical symbolics relating to the atomic world, and its description in terms of the concepts of classical physics, which were discussed in Chapter II.

p Positivists, if they are consistent, do not think in the least how to connect the content of the mathematical concepts of quantum theory with those of the natural language. The mathematical apparatus of quantum mechanics makes it possible to give order to the observed results, e.g. to predict the possible results of some observations from those of other observations, and that quite suits positivists.^^6^^

p The line of materialism on the problem of a mental picture of a physical theory implies recognition of the dialectical unity of sensuous knowledge and abstract thinking reflecting objective reality. The combination in a single 80 whole of the mathematical formalism of the physical theory and the experimental data and results relating to this theory and expressed in the concepts of classical physics corresponds to the point of view of dialectical materialism on the problem of visualisation. One must bear in mind that the nature of this combination differs from how it is presented from the standpoint of non-dialectical materialism, as will become quite clear from the exposition that follows.

Born, incidentally, when discussing this problem and analysing various philosophical approaches in that connection, did not expound the point of view of Marxist philosophy on this problem in an adequate way. He stated that, according to dialectical materialism, it was sufficient to limit oneself to ’the objective world of formulas without relation to sensual intuition’.^^6^^ As has already been said above, the point is quite different: this chapter is devoted to elucidating issues relating to the problem of visualisability in physics from the point of view of dialectical materialism.

p In our view, quantum mechanics reflects exactly, in precise concepts, the motion of atomic objects that resembles the motion of particles in some experimental conditions and in others the spread of waves and that differs radically from them both (with which classical theory is concerned). At the same time—and this has to be emphasised in every possible way—this motion is not picturable, i.e. it cannot be expressed in a visual picture like that in which the motion of a macroscopic body or the ’wave motion’ of a certain continuum is represented. In this sense it is said that quantum theory cannot be visualised. As the German physicist Gerhard Heber puts it: ’Although we describe the nature of atomic objects mathematically, we cannot understand it on the model level. It is usually said in this connection that the nature of quantum-mechanical objects is "not obvious". I would assume that our inability to construct a visual model of the microworld is not final, and that it will be possible in the future to build a visual model of atomic objects, because our power of sensual intuition is as capable of development as our power of abstraction.’^^7^^

p That atomic objects are only described mathematically by modern quantum mechanics, and that there are not as 81 yet the appropriate words and images for them, is only partially, so to say, true. One has to bear in mind that quantum theory, like the other ’not visualisable’ theories of modern physics, is being confirmed by experiment and has grown up on that basis. And that means that quantum mechanics employs visualisable concepts (and others directly related to them) in one way or another, because its truth is verified by means of instruments that are macroscopic bodies, and their readings, from which inferences about atomic objects are drawn, are perceived by man. If this question is posed more broadly it cannot be otherwise: the atomic objects a?e material realities, and matter is not simply and only that which exists objectively independently of the human mind, but is the objective reality acting on the human sense organs and producing sensations in them. Man would have known nothing about the atomic world existing independently of his mind if this world, so to say, had not given signs of itself through macroscopic phenomena perceived by him which are related in a regular manner with atomic and microscopic phenomena in general.

p Thus, visual concepts are one way or another inevitable in quantum theory. The question is, however, how and in what form do they come into quantum theory. To answer that let us first consider the definition of the concept of a mental picture or visualisation.

p A theory is most frequently called easy to visualise if it employs habitual concepts. The concept has been defined in roughly this way by the Austrian physicist Arthur March.* Such a rather psychological definition can hardly, because of its extremely arbitrary nature, be accepted as satisfactory. Many authors add that a picturable theory is one that deals with phenomena that can be perceived directly.^^9^^

p This last criterion of visualisation or obviousness of a theory, though to some extent satisfactory in itself, was not, in practice, separated in physics from certain other requirements of principle that in fact confused the matter. Thus, when mechanistic views were dominant, it actually meant a requirement for all physical phenomena to be reduced to mechanical ones. In this case Maxwell’s electromagnetic theory, for example, proved not to be ‘visualisable’ as Boltzmann, in particular, suggested. In our day, when physicists have become accustomed to the faet that 82 electro-magnetic phenomena cannot be reduced to mechanical one§, Maxwell’s theory has come to be called obvious.^^10^^

p When we turn to quantum mechanics, it would seem that the same situation as with Maxwell’s theory has developed in it. It has become quite clear that macroscopic phenomena cannot be reduced to atomic ones, and vice versa. Quantum mechanics has also become a ‘normal’ theory, but it is regarded, as we know, as a theory about which we cannot form a mental picture. L. I. Mandelstam, in particular, stressed this; March, too, drew attention to it in his many discussions of the philosophical problems of science. He called a theory obvious or visualisable ( anschaulich) that employed ’only concepts ... borrowed from the world of everyday experience’, but quantum mechanics ’forbids the use of certain concepts, in which we are accustomed to think, as misleading’.^^11^^ Then why, in his opinion, is Maxwell’s electromagnetic theory visualisable?

p So, we have not yet advanced a single step in our reasoning about ’mental picturing’ (anschaulichkeif). What theory is easy to visualise? Let us return to this question.

p Man in his historical practice has had to deal millions upon millions of times with macroscopic phenomena that occur at relatively low velocities. This practice also led to the theories and concepts of classical physics, the first scientific generalisation of notions about the inanimate nature perceived by man (historically the first such theory was Newton’s mechanics), acquiring a visual form in his mind. It meant that it was possible to imagine, on the basis of the propositions and concepts of the theory of an object, the sensual impressions and perceptions produced in man by the object being studied.

p When, for instance, starting from the concept of a moving particle in classical mechanics (an object characterised simultaneously by its position and by its momentum) we state that the particle is a visual concept, we associate notions of a stone, a pellet, a bullet, a grain of sand with the concept ‘particle’. In fact, we picture, say, that a bullet flying from the muzzle of a pistol passes through a thin cardboard disk leaving a hole in it.

p On the other hand, we have every reason to believe that the pistol bullet is exactly a particle. Let the flying bullet pass through two disks rotating at a high angular velocity around a common axis, at a short distance from each other 83 (with, such a system one can measure the bullet’s velocity).^ In this example the holes in the disks represent coordinates (position); the bullet’s momentum can be determined from its mass, the distance between the disks, and the time it took to cover this distance. In general the classical notion of a moving particle is the physical generalisation of notions of the mechanical behaviour of a stone, a grain of sand, a bullet, and other similar objects that man has to do with in everyday life. This concept, when considered from the formally theoretical aspect, corresponds to the system of axioms (basic principles) of Newton’s classical mechanics.

p One should not, however, confuse the ’mental picture’ of concepts and theories in classical physics with the ’mental picture’, say, of the concepts of common sense that had been developed by man in his everyday experience even before there was any science, or to confuse it with the imaginability of the objects being studied. One cannot, for instance, imagine a motion with a velocity of 300,000 kilometres per second as one can imagine motion with a velocity of five kilometres an hour (the motion of a pedestrian) or with a velocity of 100 kilometres an hour (the motion of a motor car). Ultraviolet radiation (a traditional example) cannot be imagined either.

p And at the same time the concepts with which classical physics operates are easy to visualise. Thus, the concept ’motion with a velocity of 300,000 kilometres per second’ is associated with the idea of a certain solid (a scale) that is laid a certain number of times along a straight line, and the idea of the time during which the hand of a watch passes through a certain interval on a certain line. The same can be said mutatis mutandis about the ’mental picture’ of many other concepts and statements of classical physics. The concepts ‘force’ or ‘mass’, for example, do not coincide with the corresponding ideas from everyday life; the law of inertia is by no means a commonsense statement.

p What we have said should illustrate the idea that ’ visualisation’ is not identical with the ‘representability’ or ‘imaginability’. Classical physics takes from ’sensuous representation’ (living contemplation) the appropriate material to be worked on by thought. Non-classical physics does the same thing but in its own manner. Classical theories, however, do not move as far from ‘representation’ as nonclassical ones do. The problem is to determine how the 84 concepts and abstract notions of classical and non-classical physics are related.

p In dealing with this problem Lenin’s following note concerning Hegel’s arguments about the relation of idea to thought is of decisive philosophical significance: ’"Come before consciousness without mutual contact" (the object)— that is the essence of anti-dialectics. It is only here that Hegel has, as it were, allowed the ass’s ears of idealism to show themselves—by referring time and space (in connection with sensuous representation) to something lower compared with thought. Incidentally, in a certain sense, sensuous representation is, of course, lower. The crux lies in the fact that thought must apprehend the whole “ representation” in its movement, but for that thought must be dialectical. Is sensuous representation closer to reality than thought? Both yes and no. Sensuous representation cannot apprehend movement as a whole, it cannot, for example, apprehend movement with a speed of 300,000 km per second, but thought does and must apprehend it. Thought, taken from sensuous representation, also reflects reality; time is a form of being of objective reality. Here, in the concept of time (and not in the relation of sensuous representation to thought) is the idealism of Hegel.’^^13^^

p In this note of Lenin’s the statements that are especially important for our topic are those from which it follows that sensuous representation and thought are interrelated, that thought, growing from sensuous representation and reflecting reality more deeply and completely than sensuous representation, embraces all sensuous representation in its movement as though making it an element of itself. From this point of view the abstractions and the relations between them that are contained in a physical theory in the form of a mathematical apparatus must necessarily be connected with the directly perceptible material provided by experiment. In classical physics, for instance, the values of its variables and functions, and in quantum mechanics the eigenvalues of its operators, correspond to the values of the corresponding physical quantities observed in the experiment.

In general, no theory in physics, if it is to and does grasp objective reality, can avoid establishing a connection between its mathematical concepts, on the one hand, and the perceptible readings of the experimental set-ups that inform about 85 this reality, on the other. It is only this connection that makes a theory in physics a truly physical theory. In this sense, no non-classical theory can be substantiated without classical physical theories, since it is impossible to describe the experimental results without employing classical concepts. The philosophical roots of all this are that nature, with which physics deals, represents moving matter, and that cognition of matter (of any of its forms and structures) is impossible without its (direct or indirect) effect on the human sense organs. In this case it is appropriate to quote the words ’the sensuous, physical (excellent equating!)’ that Lenin uttered in connection with certain ideas of Feuerbach’s.^^14^^

p How are sensuous representation and thought related and connected in physical theory?

p If we take a physical theory in its developed formnot simply in its aspect of formalism, but as a physical theory—it treats its subject-matter simultaneously as it were in its aspects of sensuous representation and thought. The experimental set-ups provide sensually perceptible data about phenomena being investigated by the physical theory, while the mathematical apparatus, which represents a system of abstract concepts, makes it possible to raise these data to the level of theoretical generalisation and so to reflect the laws of the phenomena concerned. In accordance with what we have said a physical concept appears as a result of a dual sensuous and abstract thought process of understanding objective reality. The transition from the perceptible readings of an instrument (which inform about the phenomenon concerned) to mathematical concepts, and the reverse transition from mathematical concepts to instrument readings are effected by certain rules and imply the existence of laws of the phenomena being studied. These rules of the connection or relationship between mathematical concepts and perceptible instrument readings should (and in reality do) reflect these laws, and find expression depending on the specific nature of the regularities of the field of the phenomena being investigated.

p Quantum mechanics broke with the mechanistic prejudice that the laws of macroscopic phenomena also operate in the microworld. This circumstance, however, only explicitly 86 affected the mathematical apparatus of the theory; as for the rules of the connection between mathematical concepts and instrument readings, this matter is frequently either presented in the literature of quantum mechanics in an uncertain way (without answering whether or not the corresponding rules in classical and quantum theory should coincide) or is posed in such a form that fundamental concepts of classical theory are only ‘limited’. Heisenberg, for instance, says that ’the first language that emerges from the process of scientific clarification is in theoretical physics usually a mathematical language, the mathematical scheme, which allows one to predict the results of experiments’.^^18^^

p The converse of this statement, as we know, is Heisenberg’s own statements that the classical fundamental concepts are in some sense a priori with respect to Einstein’s theory and quantum theory, i.e. that no new primary physical (and not just mathematical) concepts have allegedly been introduced by the theory of relativity and quantum theory. In order to follow through a point of view consistently that holds that the laws of macroscopic phenomena are qualitatively different from those of the microworld, it is necessary to follow this same point of view in matters relating to the combination of experimental facts and theories, to the combination of the sensually perceptible and mentally abstract in the understanding of physical phenomena, and to the connection between physical notions and mathematical concepts in the physical theory.

p Let us consider these questions in greater detail.

p A child who plays with a cat for the first time combines individual sensations (through the action of the first signalling system) (it sees the cat and hears its mewing) into an impression of a definite cat. At the same time a kind of ‘generalisation’ of the acquired conditional reflex occurs: the child reacts in the same way to similar cats with whom he happens to play. The general concept ‘cat’, however, only develops in the child’s mind when he learns about cats from grown-ups’ stories, reading of the appropriate books (in transfer of the results of mankind’s centuries-long practice the second signalling system plays the main role), and now every individual cat for him is a member of the genus ‘cat’. In that way, even first acquaintance with the objects surrounding man contains embryos of the connection 87 between and unity of the sensually perceptible and mentally abstract. The unity of sensual and abstract cognition ( underlying which is man’s practical relation to nature) reflects the dialectics of the objectively real world.

p This example, which is unrelated to physics, can help us in considering the problem posed above, since the process of forming a physical concept does not differ essentially, from the logical aspect, from the process of forming the concepts of everyday life.

p In classical theories physical concepts for the most part represent a direct generalisation of notions that are employed by so-called common sense. The physical concept of length, for instance, represents a generalisation of the fact that perceived things possess various extensions. The comparisons of dimensions made billions of times by man in practice before systematic scientific investigation of nature led to the development of scientific concepts of a constant scale and units of length, and through the latter to rules of the correspondence between the lengths of perceived things and certain numbers. The length of every perceived thing could thus now be measured precisely, i.e. generally speaking, the concepts developed in everyday experience and mathematical abstractions could now be unified in that profound synthesis of sensual and abstract cognition without which physics as a science does not exist.

p It would be the purest pedantry, of course, to demand that all the physical concepts figuring in classical and nonclassical theories should arise in exactly this way, i.e. in a way by which a physicist always proceeds from the perceptible readings of an instrument to mathematical abstraction. That way is typical of the concepts of classical mechanics, for the reason that the latter arose directly from everyday experience and took shape before the other theories of classical physics, serving for some time as their model of scientific cognition.

p The mathematical apparatus of a physical theory (which is interpreted here as a theory at the stage of formation), which represents a certain system of abstractions, possesses relative independence and has its own logic of development; by virtue of that certain concepts appear initially in certain conditions in a physical theory, which is becoming established, as a mathematical abstraction; only later is the physical meaning of the mathematical concepts revealed, 88 i.e.  they find, as one says, their physical or empirical interpretation. Discovery of the physical meaning of mathematical abstractions is a most important necessary aspect of the development of a physical theory. Without it the theory is, after all, a mathematical scheme and not a physical theory. Only this aspect gives mathematical abstractions physical flesh; consequently, only by taking it into account is it possible to formulate the laws of those physical phenomena that must be reflected by the theory; which means to give the physical theory a really developed form.

p In modern physics, which deals with phenomena that are not directly perceptible, the second way of forming physical concepts is typical, i.e. when the physicist proceeds from mathematical abstractions to perceptible instrument readings (which inform of the things being studied). This is shown convincingly by quantum mechanics.

p How were quantum concepts developed?

p Planck’s hypothesis about the discontinuity of the possible values of an oscillator’s energy, which diverged in principle from classical notions, made it possible to explain the laws of thermal radiation. Even at the first stage of quantum theory the development of this hypothesis led to outstanding discoveries: e.g. Einstein’s discovery of photons; creation of the theory of the heat capacity of solids; Bohr’s atomic model; and the explanation of Ritz’s empirical rule followed by the spectral lines of atoms, etc.

p One must note that even at the initial stage of quantum theory the physical content of the assumptions made remained unclear: the heuristic role of the mathematical form was pushed to the foreground (an example of that is scientists’ numerous attempts to comprehend physically Planck’s formula s = hv at the dawn of quantum theory).

p The further development of quantum theory consisted, above all, in finding the mathematical apparatus to express the statement of the discontinuity of energy. Only then did it become possible to bring out the physical sense of all the quantities involved in the mathematical apparatus, and consequently to carry solution of the problem of creating atomic mechanics through to completion. All this equally applies to the matrix mechanics of Heisenberg and Born and to the wave mechanics of de Broglie and Schrodinger, the two roots, as Born put it, of quantum mechanics, 89 which was given contemporary form by Dirac. Schrodinger, for instance, compared the hypothesis about the discontinuity of energy states in micro-objects and the mathematical equation he had formulated, which he had obtained by employing the mathematical apparatus of the classical theory of vibrations assuming that the quantities describing the behaviour of micro-objects were associated with relationships of this apparatus that had been altered in a certain way. He supposed, further, that some operator of a certain class corresponded to energy, and its eigenvalues to the energy values observed in the experiment.

p This assumption together with the established equation already made it possible to obtain fruitful results, e.g. to substantiate the Balmer series and explain the Stark effect.

p An important role has been played in the development of quantum mechanics by finding the physical meaning of the wave function which figures in the Schrodinger equation. This wave function characterises the state of a micro-object in certain macroscopic conditions. It is it that makes it possible to effect the transition from operators to the values of quantum quantities observed in experiment.

p If the wave function is the eigenfunction of an operator of a physical quantity (say, of position), then, according to the basic postulates of quantum mechanics, the operator’s eigenvalue corresponding to this function is a possible value of the quantity (in an experiment with electrons it corresponds to the position of the spot observed on the screen).

p If the wave function is not an eigenfunction of the ( physical) quantity’s operator (assuming, again, a quantum position) this quantity (in the state characterised by this wave function) has no definite value (in the experiment with electrons it corresponds to the distribution density of the observed spots on the screen). Max Born suggested interpreting the square of the modulus of the wave function | \\t (x) |² as the probability of an electron’s hitting a point with a coordinate x. In accordance with this interpretation of the wave function it is stated that a quantity in a state characterised by a wave function that is not an eigenfunction of its operator has only an average value. The value can be computed from the mathematical apparatus of quantum mechanics if the wave function characterising the state of the micro-object is known.

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p Generally speaking, only some of the quantities appertaining to a micro-object in a given state have definite values in quantum mechanics; all other quantities (in the same state) have only average values and not definite ones. This is closely related to the specific feature of the wave function which consists in its coinciding with the eigenfunctions of some operators and not with those of others.

p This last feature of the wave function follows from the central point of quantum mechanics, viz. the so-called commutation relation. From this relation it can be deduced that the state of a micro-object is a common eigenstate of any two quantum quantities only if the operators corresponding to these quantities are commutative. Thus, from the commutation relation for the operators of momentum
and position P_XX—XP_X—4 one can derive that momentum and position cannot have definite values simultaneously in quantum mechanics.

The commutation relation expresses the unity of the opposite particle-wave properties of micro-objects in the form of mathematical abstractions. The establishing of this relation and the finding of its connection with the particle and wave pictures formed by micro-objects and observed in experiment are an excellent example of the dialectics of nature and its cognition by man.

p We can now bring together our discussion of the problem of the visualisation of classical and quantum concepts, including the question of the role of classical concepts in quantum theory.

p There is an element of imaginability in both classical and quantum physical concepts: if there is no connection between the perceptible readings of instruments and the values of variables (in the case of classical theory) and the eigenvalues of operators (in the case of quantum theory), there would be no physical concepts in either classical or quantum theory. In classical theory, however, concepts are a direct generalisation of the observation data; the concepts in quantum theory are not such a direct generalisation, but instead generalise the observation material in a mediated way through classical concepts.

p The eigenvalues of quantum operators correspond to 91 observation data (the readings of instruments) in exactly the same way as the values of classical variables correspond to the data of observation; for example, the position of a spot on the screen in an electron diffraction experiment and the position of the hole in a disk pierced by a bullet are measured by a constant scale with appropriate divisions. In other words, it is never possible in measuring quantum quantities to do without classical concepts.

p At the same time the dual particle-wave nature of microobjects is reflected in the mathematical apparatus of quantum mechanics and this puts its stamp on the classical concepts used in quantum theory. Thus one can infer from
the commutation relation P_XX—XP_X = - that eigenvalues of the momentum and position operators do not exist for one and the same state. It follows from this that the classical concept of a moving particle cannot be employed in quantum mechanics in exactly the same way as in classical theory (as we noted above the term ’moving particle’ is applied in classical mechanics to an object that simultaneously has position and momentum). In other words the particle and wave characteristics, when applied to atomic phenomena, lose their ‘classical’ independence and become connected as it were, implying one another.

p This means that the so-called relativity of the state of a micro-object (i.e. the fact that the state of an object is not determined in quantum mechanics regardless of the experimental set-up by which the object is being studied but only in connection with this set-up, in connection with the conditions fixed by it) is a manifestation of the dual particle-wave nature of micro-objects.

p Micro-objects do not behave in a single experiment in the same way as waves or particles of the macroworld. At the same time, since only classical concepts can be employed to describe the observed phenomena, the classical particle and wave concepts should be treated in experiments with micro-objects as mutually exclusive, and the experimental conditions under which corpuscular phenomena are observed as incompatible with the conditions in which one observes wave phenomena, and vice versa. The contradiction is resolved by introducing new physical quantum concepts that have features similar to those of the corresponding classical concepts but differ radically from them.^^16^^

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p In conclusion we shall try to answer the following question. In which precise language can one speak about the micro-objects themselves? Is the refined and developed natural language appropriate for this or is such a language unsuitable for this purpose?

p According to positivists, this question has no meaning; many scientists, however, including those whose philosophical sympathies are far from materialism, do pose it.^^17^^

p From the foregoing discussion we find that such a precise language does exist and that it has developed from the language of classical theories; the job of constructing such a language was solved by Bohr’s complementarity principle, which was developed further by his successors. The corresponding classical concepts (e.g. the concept of a moving particle), which proved to be imprecise when applied to phenomena on an atomic scale, were radically transformed and defined as concepts relating to the system of concepts and principles of quantum mechanics, which system differs radically from the system of concepts and principles underlying classical theories. This found expression in a new formalism and correspondingly new rules of the relation of mathematical concepts and the observed results in quantum mechanics (described in the language of classical notions), compared with the formalism and rules of connection in classical theories.

p An element of visualisation is present in quantum physical concepts and quantum theory but the concepts themselves and the theory are not easily visualised. Here a very important role was played by the development of the concept of relativity to the means of observation, without which it would have been impossible to comprehend quantum mechanics as a physical theory, and which is a further generalisation and development of the concept of relativity with respect to frames of reference in the theory of relativity and classical mechanics.

p The physical concepts in any physical theory, either classical or non-classical, are thus not the instrument readings and not mathematical abstractions; in physical concepts reflecting objective reality the two are combined, and it is they that are precise physical concepts; they are precise because they correspond to objective reality.^^18^^

p There are other conceptions in the literature of the matter under discussion. We shall only mention the attempt to 93 formulate an exact language corresponding to the mathematical formalism of quantum mechanics but having nothing to do with ‘visualisation’ and suggesting a change in the laws of conventional formal logic.

p So-called quantum logic (Reichenbach, Weizsacker) ascribes not two values of truth to statements (‘truth’ and ’ falsity’), as conventional logic does, but three—‘truth’, ’ falsity’ and ‘uncertainty’. This ‘uncertainty’ is not equivalent to ‘ignorance’; rather it characterises a special type of situation. The principle of the excluded third (’a statement is either true or false, tertium non datur’) does not operate in quantum logic. The following example is quite demonstrative in this respect: if one says that an electron passing through a screen with two apertures ’has not passed through a certain aperture’, it still does not follow that ’it certainly passed through some other aperture’; there is a third possibility: ’the electron’s passage through the aperture is uncertain’ (this possibility is by no means equivalent to our ignorance of which hole the electron has passed through).

p From the point of view of ’quantum logic’ in Weizsacker’s presentation of it, it follows that any visualisation is excluded from quantum mechanics, and that there is a logical and epistemological gap between it and classical theory. This follows if only because ordinary logic is the logic of the everyday, refined natural language that is the language of the concepts of classical physics.

p The question is whether deviations from everyday language and ordinary logic like Weizsacker’s quantum logic are still needed if one has in mind explanation of the phenomena on an atomic scale that quantum theory deals with.

p The content of what we have said provides an answer to this question. Here we would only add that Bohr, Pauli, and other physicists have disagreed with employing a multivalued logic in order to get a more ‘precise’ representation of the situation that has built up in quantum mechanics.^^19^^

p REFERENCES

p  ^^1^^ P. A. M. Dirac. The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1958), p vii.

p  ^^2^^ Niels Bohr. Atomic Physics and Human Knowledge (John Wiley & Sons, New York, Chapman & Hall, London, 1958), p 39.

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p s tbid., p 46.

p  ^^4^^ Niels Bohr. Essays 1958-1962 on Atomic Physics and Human Knowledge (N. Y., London, Interscience Publ., 1963), p 5.

p ! One can be convinced concretely of this by looking, for instance, at Philosophy of Science by the well-known positivist Philipp Frank (Prentice-Hall, Englewood Cliffs, 1957).

p  ^^6^^ Max Born. Bemerkungen zur statistischen Deutung der Quantenmechanik. In: Werner Heisenberg und die Physik unserer Zeit (Vieweg & Sohn, Brunswick, 1961), p 107.

p  ^^7^^ Gerhard Heber. Uber einige philosophisch wichtige Aspekte der Quantentheorie. In: Naturwissenschaft und Philosophic. Edited by Gerhard Harig and Josef Sehleifstein (Akademie Verlag, Berlin, 1960), p 30.

p  ^^8^^ Arthur March. Die physikalische Erkenntnis und ihre Grenzen (Vieweg & Sohn, Brunswick, 1955), p 12.

p  ^^8^^ L. I. Mandelstam. Lectures on the Fundamentals of Quantum Mechanics. Polnoye sobranie sochinenii, Vol. 5 (Leningrad, 1950), p 404.

p  ^^10^^ Arthur March. Op. cit., p 12.

p  ^^11^^ Ibid.

p  ^^12^^ A description of this device (a chronograph) can be found in: R. W. Pohl’s Mechanik, Akustik und Wdrmelehre (Springer Verlag, Berlin and Gottingen, 1947), p 12.

p  ^^13^^ V. I. Lenin. Philosophical Notebooks. Collected Works, Vol. 38 (Progress Publishers, Moscow), p 228.

p “ Ibid., p 75.

p  ^^15^^ Werner Heisenberg. Physics and Philosophy (George Allen & Unwin, London, 1959), p 145.

p  ^^16^^ The matter posed here about the dialectical contradiction in quantum mechanics is considered in greater detail in Chapter V.

p  ^^17^^ See, for instance, Werner Heisenberg. Op. cit., pp 154-155.

p  ^^18^^ These points are developed further in Chapter X.

 ^^19^^ Niels Bohr. Essays 1958-1962 on Atomic Physics and Human Knowledge, p 5.