p An ‘observable’ (about which we spoke in the first section) is a concept that has experimental meaning, or one based on experiment, or something that a statement about can be verified experimentally. One can often read in the literature that ’only thoroughly empirically based concepts’ should be employed to describe physical phenomena.^^16^^

p This last requirement seems quite reasonable from the standpoint of physics, but it is in fact ambiguous and therefore nearly useless in that form. The symbols which form the mathematical apparatus of any physical theory (without them there is no theory) do not as yet constitute a physical theory. For a theory in physics really to be a physical theory, concrete measurement formulae have to be provided for the symbols of its mathematical apparatus that relate these symbols to physical objects. In quantum mechanics, for example, the wave function in itself does not yet have a physical meaning, and is therefore not an observable quantity until a concrete formula is given that relates it to something physical. From that point of view Born was right when he said that the wave function was not an observable quantity.^^17^^

p The concept of observability (or non-observability) should not, in fact, be applied to the wave function as such: the latter is a mathematical quantity, while the observability concept applies only to the physical. Wave functions, or the vectors in Hilbert space, are mathematically quantum states (just as the operators that act on these functions are 101 mathematically quantum physical quantities, or ‘observables’). At one time, when quantum mechanics was acquiring the developed form we now know, the job was to determine the physical agent of the wave function (and if it was a question of an operator, to find its physical realisation). This was done by a probability interpretation of quantum mechanics: for given values of the variables that are the arguments of the wave function the square of its modulus equals the probability density at which the variables obtain their selected values during measurement.

p It was in this way that the physically realised wave function is an observable quantity in quantum mechanics, representing the most important physical characteristic of a system. Similarly, the properties of a quantum quantity are brought into correspondence with an operator in such a way that the possible values of a certain quantum physical quantity correspond to the eigenvalues of the operator representing this quantity.

p Thus, when it is a matter of the observability of a quantity, reference to the experiment, although necessary, is far from sufficient to solve the problem of this observability.

p Let us take some other, different examples. The concept of absolute simultaneity employed in classical mechanics agrees beautifully with a certain group of experimental mechanical data; sophisticated experiments relating to electromagnetic phenomena in moving bodies, however, do not correspond to this concept and form the basis for the concept of relative simultaneity of Einstein’s theory.

p Or take the concept of the atom. Strictly speaking this concept had not been substantiated experimentally before Perrin’s experiments. It had a hypothetical character; the atomic hypothesis, however, as we know, played an outstanding role in the development of physics.

p The appeal to experiment as the sole substantiation, in the final analysis, of physical knowledge thus still leaves too many uncertainties for unambiguous answers to be given relating to the description of physical phenomena and knowledge of the laws of nature. In matters of this sort physical theory also plays an important role, and therefore the requirement for a corresponding (logical) analysis of the statements of natural science. Even when quantum mechanics was taking its first triumphal steps, Max Planck drew attention to this circumstance.

p In Planck’s opinion the fact that quantum mechanics dealt with quantities observable in principle and problems having physical meaning was not its special advantage over other theories. The problem of observability in principle, he stressed, can never be solved a priori; it has only to be solved from the angle of a certain theory. ’The distinction between the different theories,’ he said, ’consists precisely in the fact that according to one theory a certain magnitude can in principle be observed, and a certain question have a meaning as applied to physics; while according to the other theory this is not the case. For example, according to the theories of Fresnel and Lorentz, with their assumption of a stationary ether, the absolute velocity of the Earth can in principle be observed; but according to the Theory of Relativity it cannot.... The choice between these two opposed theories depends not upon the nature of the theories in themselves, but upon experience.’^^18^^

p It is worth adding to Planck’s example that the negative result of the Michelson-Morley experiment substantiated to some extent (as far as absolute velocity was concerned) not only the theory of relativity but classical mechanics as well. This issue was resolved in favour of the theory of relativity by postulating the principle of the independence of the velocity of light of the motion of the source (a principle that accorded with Lorentz’s theory of motionless ether). The principle was not, however, postulated by itself but in a peculiar combination with the principle of the relativity of uniform, rectilinear motion, which contradicts it.

p By this addition, we would like to stress not only that the experimental basis of the theory of relativity is incomparably broader than the experimental basis of classical theories but also that the spread (generalisation) of established principles and basic concepts to a new field of phenomena means, in certain conditions, their alteration at certain points. Such generalisations are closely related to our theme.

p Let us consider the problems arising here. The concept of the observable in principle is not simply compatible with a certain theory. It is necessarily connected by a chain of corresponding conclusions with the theory’s basic concepts and principles. The definition of the observable in principle provides a method that allows us, on the basis 103 of experiments, to say whether the observable in principle corresponds to the objectively real. Examples of the observable in principle were given above; they provide an opportunity of clarifying that the observable in principle coincides in essence with the operationally definable.

p Operational definitions have long been in use in physics, especially in connection with the employment of mathematical concepts and methods in it and, accordingly, with the appearance of abstractions of an ever higher order. They were also dealt with in classical physics, but, remembering its history, they were not used explicitly in it. Their systematic application and development in explicit form one finds in non-classical theories.

p Operational definitions have advantages—in certain circumstances—over verbal ones. When the point concerns physical quantities and, in general, physical characteristics relating to idealised objects in the broad sense of the term (on the plane of logic of science they include, for instance, so-called constructs), and it is necessary to solve problems about these objects (finding, say, their physical characteristics) from instrument readings, operational definitions may be the only useful ones. From that point of view it is clear that the method of an operational definition has nothing to do with definitions of matter, a law of thought, and other philosophical concepts or, as we remarked above, either mathematical or biological concepts. For them there are other methods of definition.

p It is difficult to agree with Born who, while justly opposing the epistemological line of operationalism, suggested, however, that the domain of operational definitions was solely classical physics.^^19^^ Of course, one cannot, as we noted above, define a wave function or operator by an empirical ‘operation’, but the operational definition is not ’ responsible’, so to say, for the ‘philosophy’ of operationalism and its metaphysical, a priori ideas. The operational definition of a physical quantity thus means that it is defined by describing the operations needed to measure it within the limits of a certain theory. What we have stressed, however, is exactly what is ignored in the corresponding arguments by operationalism (and, in general, by positivism), which regards operations not as a means of reflecting the objectively real in the human brain but as the real itself. That is why operationalism even does not pose the question of 104 the conditions and limits of the applicability of operational definitions or of their variability.

p The applicability of any operational definition is in fact limited not simply in the trivial sense by the boundaries of the object being defined but also by the limits of the theory in which the definition occurs. In classical theory, for example (as became clear with the development of physics), the simultaneity of events at two points A and B removed from one another is defined by taking a clock from A to B, when it is affirmed that it goes synchronously at A and at B (after being brought there). In Einstein’s theory, however, simultaneity at two different points is defined as follows: (1) points A and B are connected, according to a certain rule, by a light signal; (2) the frame of reference to which the simultaneity argument applies is indicated. When formulating his theory of relativity Einstein thus altered the definition of simultaneity accepted in classical theory.

p A similar picture exists in quantum mechanics as well, when it is compared with classical theory. Not only is the mathematical apparatus of quantum mechanics different from that of classical theory, but also the rules for linking its concepts with the instrument readings or experimental data (without such a connection the concepts of its mathematical apparatus have no physical meaning), in other words, there are operational definitions of quantum quantities that do not coincide with operational definitions of analogous classical ones.  [104•*  In general, if a field of new fundamental laws has been discovered and a theory covering it established, the operational definitions of the corresponding objects should also be new. In short, in contradiction to the operationalist point of view, there is no universal criterion, the same for all theories, of when an assertion should be regarded as having (or not having) empirical meaning. Nature is infinitely richer than any of its domains and any of its aspects reflected in experience and the theories grown on it. As physical knowledge develops, penetrating 105 phenomena and processes of nature that have not been studied by earlier experience, peculiar situations can arise in which we have somehow or other to use concepts of the old theory that are losing their meaning, and at the same time to construct a new theory, selecting concepts corresponding to it.

p From the standpoint of a certain theory it would be useful to compare the observable in principle with the experimentally observable and hypothetical for a more definite identification of their specific nature.

p When, given the appropriate necessary and adequate conditions the observable in principle is still not observed in the experiment, this often has far-reaching consequences for the theory. That is how the hypothetical classical ether became obsolete in modern physics, and the corresponding theory (in one version or another) was preserved only as a historical relic if one disregards some of its ‘revivals’ through ad hoc hypotheses. The observable in principle may prove (in certain conditions) to be observable in experiment, or experimentally observable. Formally that means confirmation of the theory at a definite (and sometimes decisive) point. From this seemingly trivial point of view, Hertz’s discovery of electromagnetic waves was a most important confirmation of the validity of Maxwell’s theory of electromagnetic field; or J. G. Galle’s discovery of the planet Neptune after it had been ’discovered by pen’ (i.e. predicted) by J. C. Adams, and independently of him by U.J.J. Leverrier, became proof of the validity of Copernicus’ system, which had, strictly speaking, before that, to be considered a hypothesis. Frederick Engels wrote about this in his Ludwig Feuerbach and the End of Classical German Philosophy.*^^0^^

p On the other hand if the experimentally ‘observables’ figure in a certain system of concepts (which by itself is not a closed physical theory), i.e. are only experimentally ‘observable’ in the given system, they are simply the scaffolding for a possible closed theory. The mechanical characteristics of electron motion, for example, such as the electron’s position in an orbit or its period of revolution, were ‘expelled’ from Heisenberg’s matrix mechanics, and matrices put in their place. Matrix mechanics yielded fruitful results confirmable by experiment; the question of what it meant to use matrices instead of position and momentum 106 in this theory remained, however, outside its field of view. The mechanics of the atomic world had to deal with problems of this kind when it decided to be a really physical theory and not just empirical magic. That happened with discovery of the uncertainty relation and the formulation of Bohr’s complementarity principle, as is now well known.

p While the observable in principle and the experimentally observable thus may exist separately in certain systems of concepts, they tend towards each other, as it were, as these systems develop, and after certain theoretical ‘adventures’ they combine to form ‘normal’ physical concepts in a closed physical theory. The ’observables’ in quantum mechanics, represented mathematically by corresponding operators, can serve as an example of such ‘combinations’.

p To conclude this section, let us take a remark of Arnold Sommerfeld’s so as to stress the need to distinguish between the observable in principle, the observable in experiment, and the hypothetical, although these concepts, as follows from what has been said, do have undoubted points of contact. Sommerfeld, who made a great contribution to the development of quantum theory, wrote: ’The declared intention of Heisenberg’s first work on quantum mechanics (i.e. on matrix mechanics—M. 0.) was to develop a method that would be based exclusively on the connections between quantities observable in principle.’  [106•*  Such concepts as ’the position of an electron’, ’period of rotation’, ’the shape of the orbit’ were to be excluded from consideration. ’This restriction to the directly observable is based, in the last analysis, on Mach’s philosophy.’

p Sommerfeld noted further that Wilhelm Ostwald’s energetics that Mach and his supporters had propagandised also stemmed from a striving to limit himself to the directly observable. But, Sommerfeld concluded: ’energetics could be counterposed to the very fruitful kinetic theory of gases in which the positions and velocities of gas molecules, though not observable in detail, could not be left out as entropies of the theory. In the same way we can counterpose to Heisenberg’s point of view the wave mechanics in which eigenfunctions can just as little be checked in detail through experiment as the earlier electron orbits.’^^21^^

p In this comment of Sommerfeld’s correct ideas are mixed with statements with which we cannot agree in such a way that the latter set the tone. First of all, it is wrong to say that the observable in principle in physics rests on Mach’s philosophy. That is not only made clear by the whole content of our book, but also by Heisenberg himself who, having paid tribute to positivism in the twenties, pointed out in his Physics and Philosophy that positivism and the principle of observability differed from one another.^^22^^

p Furthermore, Sommerfeld apparently does not draw a sufficiently clear distinction between the ’observable in principle’ and the ’observable in experiment’. Quantum mechanics, as we know, has its own observables in principle. Suffice it to recall—and Sommerfeld is wrong here, too— that the wave function in its probability interpretation is a physical characteristic. Heisenberg adopted it as a concept needed by quantum mechanics. These facts, incidentally, disprove the view that there is a philosophical similarity between Heisenberg’s standpoint, which rejects electron orbits, and that of Mach who did not recognise atoms.

Finally, the position and velocity of an individual atom, treated by Sommerfeld as unobservables, should be rather classed as hypothetical; they may be included, moreover, among the observable in principle from the standpoint of classical mechanics, since the kinetic theory of gases is very closely associated with the latter.