Mandelstam’s Operationalism

Abstract

In this chapter, we discuss Mandelstam’s philosophical position, which, according to the present author, may be characterized as operationalism. In the world literature, the position of the American physicist, Nobel Prize winner for his work in high-pressure physics, Bridgman Percy (1882–1961) is called operationalism. In 1927, Bridgman published a book “The Logic of Modern Physics”, which contains the thesis, which became the banner of operationalism: “the concept is synonymous with a corresponding set of operations”. For example, “the concept of length is fixed when the operations by which length is measured is fixed, that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined”.

14.1 In Which Way the Term “Operationalism” Was Applied to Mandelstam’s World View

In this chapter, we discuss Mandelstam’s philosophical position, which, according to the present author, may be characterized as operationalism.

In the world literature, the position of the American physicist, Nobel Prize winner for his work in high-pressure physics, Bridgman Percy (1882–1961) is called operationalism. In 1927, Bridgman published a book “The Logic of Modern Physics”, which contains the thesis, which became the banner of operationalism: “the concept is synonymous with a corresponding set of operations”. For example, “the concept of length is fixed when the operations by which length is measured is fixed, that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined” [67, p. 5].

Percy Bridgman stressed that his operationalism is genetically associated with Einstein’s methodology of the theory of relativity. For Einstein, the spatial and temporal coordinates are what is measured by rulers and clocks, and the simultaneity loses its absolute status and requires clock synchronization.

The term “operationalism” was applied to L.I. Mandelstam’s philosophy of science by the famous mathematician Anatolii Danilovich Aleksandrov in January 1952 (Aleksandrov became Academician in 1964, in 1952, he was Corresponding Member of the USSR Academy of sciences). He delivered a lecture at the “United Institute Colloquium” of the Physics Institute of the Academy of Sciences (FIAN). This lecture was entitled “On ideological mistakes in some textbooks on physics”. Aleksandrov pointed to the source of the mistakes. This was the fifth volume of L.I. Mandelstam’s Complete Works, containing his lecture on the theory of relativity and quantum mechanics. Aleksandrov said: “This is a prescriptional view of the definition of scientific concepts, view which was developed by idealists Percy Bridgman and Philip Frank. This trend to reduce concepts to operations is a trend of subjective idealism, which tends to eliminate the objective reality and to reduce everything to immediate data” (cited in [324, p. 181]).

Aleksandrov’s lecture was criticized by I.E. Tamm, S.E. Chaikin, and some other colleagues of the late L.I. Mandelstam. They gave the quotations as evidence in favor of Mandelstam’s materialism. However, they agreed that Mandelstam’s philosophy of science was not consistent and he was not a proponent of dialectical materialism.¹

Now that the ideological battles of the early 1950s are history and not the closest history, it is hardly worth trying “to restore justice”. It is hardly necessary to prove specifically that the terms “operationalism” and “idealism” are not more offensive than the “social democrat” and “anti-globalists”. It is clear that Aleksandrov spoke from a position of orthodoxy, which should deal with all kinds of heresy. It is also clear that those who opposed him tried to be closer to orthodoxy. However, we have a claim that L.I. Mandelstam was operationalist and idealist. Aleksandrov, having philosophical parallels, came to the conclusion to which it was not hard to come: Mandelstam, like many physicists of his time, was in the orbit of ideas, dating back to Einstein’s first article on the theory of relativity (1905), to Heisenberg’s articles on quantum mechanics and eventually to the works of Ernst Mach. Aleksandrov described Mandelstam’s position as operationalism, bearing in mind its proximity to the concept of P. Bridgman. But Mandelstam did not refer to Bridgman. Calling the philosophical credo of Mandelstam operationalism, we are planning to analyze the relationship of this credo, formulated authentically, to the paradigmatic operationalism of Bridgman.

14.2 How Did Mandelstam Formulate His Operationalism?

Mandelstam’s operationalism is scattered among the notes of his lectures and seminars, which constitute the fourth and fifth volumes of his Complete Works. The most comprehensive and clearest of the formulations can be found in his 1933–1934 Lectures on the Theory of Relativity and 1939 Lectures on Quantum Mechanics. Mandelstam formulated the principles of his operationalism in his discussion of the nature of physical concepts and physical theories.

In Lectures on the Theory of Relativity , where he tended to follow directly Albert Einstein [1, Vol. 4, pp. 90–305, 3, pp. 83–285], Mandelstam spoke mainly about the nature of concepts. He clearly was aware of the philosophical character of the problem which he posed, although he had never used the word “philosophy”. Having outlined the problem of how to reconcile the principle of relativity which had been formed in classical physics and the principle of the constant velocity of light, the problem which, according to him, was Einstein’s point of departure, Mandelstam stated [1, Vol. 4, pp. 177–178]:

In order to do this, another discussion should be launched, the discussion of the structure of physical concepts. I can not speak about it in details, for (1) I am not a specialist in this field and do not know these matters enough and (2) these matters would take us far away from our problems. However, some important peculiarities, without which a physicist cannot work, we shall see. We shall see that we speak a lot of words which have no content, and confusions result from this. Let us look at some simple facts.

When we are speaking about some scientific laws, for example, Newton’s laws, we mean formulas containing x, y, z. To test these formulas we need to substitute certain numbers for x, y, z. However to do so, we must be able to measure length.

What does it mean to measure length for a physicist? At first he must have a unit of length. What is the unit? This is a distance between two marks on the rod which is kept in Paris…

This is not all. Once a physicist has a unit he also needs a technique for measuring. He needs a real process that gives him a number which is, by definition, the length of the object. A physicist must have a prescription for how to measure length.

Mandelstam emphasized Einstein’s philosophical contribution to physics. He said [1, Vol. 4, pp. 180, 196]:

A number of concepts is not experienced but accepted by definition in the course of cognition of the real world. Einstein shows that it is the point that has been overlooked and this is his great contribution to science. Einstein performed his great service when he elucidated that the concept of simultaneity is a concept like the concepts of length and the time of an event.

Discussing the physical concepts in his Lectures on the Quantum Mechanics [1, Vol. 4, pp. 347–415], Mandelstam seemed to follow Heisenberg’s celebrated 1927 paper where the uncertainty relations had been formulated (he had never referred explicitly to this article). In turn, with respect to his methodological part, Heisenberg apparently followed Einstein’s 1905 article.

(The real history was more complicated. As is well known, in the 1960s, Heisenberg reminiscently recalled how he spoke with Einstein in 1926, who told him the famous phrase that “it is the theory that decides what we can observe”. However, these recollections do not contradict the observation that both Einstein’s 1905 article in its kinematical part and Heisenberg’s 1927 uncertainty article were written in the same operationalist tenor. It is not surprising that, as Heisenberg pointed out, he recalled his conversation with Einstein just before writing his uncertainty paper. “Operational definitions of fundamental concepts subject to quantum mechanics and the uncertainty relations quickly followed. The theory did indeed decide what could or could not be observed and remembered” [74, p. 239]).

Mandelstam said [1, Vol. 4, p. 354]:

Quantum mechanics rightly abandons prejudice that laws of macro-world are valid in micro-world. But only the mathematical part of the theory completely proceeds from this point of view. In the text-books it does not take sufficiently into account that prescriptions for the transaction [from the math- ematical technique to the real objects] must differ from those in classics.

If in classics I state that x is the position of a material point than I mean a clear prescription: if I set properly a rigid rod graduated according to a definite prescription, then x numbers that marking with which the point coincides.

As far as we speak about the molecular issues this prescription is not performable… Thus having called x the position I only pretend that I establish the relation of it to the nature. With such a definition theory is in the air.

Mandelstam continued [1, Vol. 4, p. 358]:

The uncertainty relations trouble us, since calling x and p position and momentum respectively we are thinking about the corresponding classical magnitudes… Why do we called p momentum? This is self-delusion again… Until we have no new measuring prescriptions it would be better not to use old terms.

He explained the uncertainty relations as follows:

The very definition of quantities, with which the theory works, presupposes the theoretical impossibility of simultaneous exact values of x and p. The situation is the same as in classics. The question “What is the oscillation frequency of a pendulum at a particular instant of time?” is absurd. So, the thing is in the very definition of the concept.

Mandelstam gave the usual operational (for Mandelstam, “prescriptional” ) definitions of the position and the momentum of a particle: the position of the dot on a photographic plate resulting from the incidence of a particle on the plate and the curvature of the track of a particle in a cloud chamber, respectively. However, he pointed out the inadequacy of such an approach: The momentum of an uncharged particle (say, neutron) cannot be measured by measuring the curvature of the track of the particle in a cloud chamber. Mandelstam stressed that the direct measurements are exceptional and outlined the theory of indirect measurement, which was not articulated by Heisenberg and the other founders of quantum mechanics. This was an important move.

I.E. Tamm commented on this move in his essay of Mandelstam’s biography as follows [333] ([8, pp. 135–136] is cited; for an English translation, see [340, p. 275]):

As far as I know, Leonid Isaakovich was the first to include in lectures the very important distinction between direct and indirect measurements in quantum systems. The last stage in any measurement of a quantum system necessarily has a macroscopic character. L.I. calls measurement direct when the first measurement step is macroscopic. Example: An electron incident on a photographic film leaves a blackened spot. The macroscopic coordinate of the spot, by definition, is the coordinate of the electron upon its impact on the film. It is important to note that the direct measurements are possible only for free or nearly free particles in free fields. For example, it is impossible to determine the coordinate of an electron in a hydrogen atom by placing a photographic film inside the atom.

In addition to direct measurements, indirect ones are also possible. In these we force the quantum system on which we want to make measurements to interact with another micro-system on which direct measurement are possible. The date of these direct measurements we use for theoretical calculations of the values of the quantities relevant to the first system. Example: By measuring the angular distribution of electrons scattered by an atom, we can find the distribution of bound electrons in this atom.

Thus, Mandelstam extended the concept of operational definition by including “indirect operational definition” suggesting theoretical calculations. With this extended operationalism, he examined the foundations of quantum mechanics.

It is more difficult to trace Mandelstam’s outline of the structure of a physical theory to its philosophical sources. To some extent, this outline was close to that which had been given by some ideologists of modern physics at the turn of the century (H. Poincaré, P. Duhem, K. Pearson, et al.). In short, it could be expressed by the scheme: mathematics + experiment. Mandelstam, however, emphasized the prescriptional character of those rules which relate the mathematical technique of a theory to nature.

It is remarkable that Mandelstam developed in essence his discussion of a physical theory along the same line as the positivistically inclined philosophers of his time (Hans Reichenbach, Rudolf Carnap, Henry Margenau) and philosophers who came out later (Karl Hempel, Ernest Nagel) kept in their discussion of this matter. One can read in Mandelstam’s Lectures on Quantum Mechanics:

Every physical theory consists of two parts that supply each other. I shall start by indicating what the second part is. This is a set of equations of a theory (Maxwell’s equations, Newton’s equations, Schrödinger’s equation, etc.). Certain symbols are contained in these equations (x, y, z, vectors E and H, etc.). With this, the second part is completed.

The first part of a physical theory consists of the connections of these symbols (quantities) with the physical objects, connections, which proceed in accordance with the specific prescriptions (we must have the real objects as standards and a real measurement technique). [1, Vol. 4, p. 349]

The building of a physical theory can be divided into two stages.

First of all, one should introduce physical quantities that depend on the field to which this theory refers. Among them, we assume mathematical relations (e.g., in the form of differential equations).

The second stage consists of connecting the mathematical quantities with the physical objects. To achieve this, for every quantity, we must formulate a definite prescription for how to attach a numerical value to this quantity. [1, Vol. 4, p. 408]

Having reviewed the mathematical scheme of quantum mechanics (self-adjoint operators, the wave function, the Schrödinger equation), Mandelstam said [1, Vol. 4, p. 359]:

We need to coordinate the symbols, belonging to the Schrödinger equation, with the objects of nature. For a physicist to state such a relation means to give an actual prescription according to which numerical values of physical quantities could be extracted from the real objects.

Mandelstam carefully formulated his operationalism in his discussion of the non-classical physical theories; he was explicitly following Einstein and implicitly following Heisenberg. However, in his 1930–1932 Lectures on Oscillations (fourth volume) and in 1932–1933 Lectures on Selected Issues in Optics (fifth volume), Mandelstam treated some fundamental concepts of classic physics, concepts with which he was in touch within the main portion of his research and teaching, proceeding from the operationalist point of view. One can point to the operationalist essays in his big article representing his lecture delivered at the 1931 All-Union (National) Conference on Oscillations [1, Vol. 3, pp. 52–86] (this conference was taken into consideration in Chap. 9).

It is probable that Mandelstam accepted operationalism by studding fundamental articles of Einstein and Heisenberg and after that he applied it in his analysis of classical physical theories to which he basically contributed. It is also possible that Mandelstam came close to the operationalist point of view by solving the conceptual problems which arose in his research in classical physics and Einstein’s and Heisenberg’s writings only supported his operationalist inclinations and stimulated their elaboration. In any way, in his lectures and articles on classical physics, he was not an ostensible operationalist: He did not use the term “prescriptions” and did not discuss the definitions of concepts, but he emphasized that the problem of physical reality (in order to escape being a pseudo-problem) must be posed with respect to experiments and measurements which are able to fix things whose reality is under question.

In the next section, we shall consider two examples of Mandelstam’s operationalist treatment of the conceptual problems of classical physics.

14.3 Operationalism in Classical Physics: Two Examples

The two examples are concerned with the reality of components resulting from the Fourier analysis of physical phenomena. Mandelstam summarized his view of problems which arose in connection with the Fourier expansion by stating the following: “Every expansion is correct and reasonable in relation to the experimental device which is in usage” [1, Vol. 4, p. 173] and “The question of the reality of the expansion into a sinus series often arises. This question reaches meaning when it is put in connection with the apparatus that receives oscillations” [1, Vol. 4, p. 119].

As G.S. Gorelik recalled, at one of the seminars, Mandelstam explained his approach to physical reality as follows. There is a collection of balls, which are big and small, ferrous, and cupric. If we are sorting the balls with a sieve, the collection consists of big and small balls. If we are sorting them by magnet, the collection consists of ferrous and cupric ball (cited [138, p. 153]).

The examples, which we plan to discuss in this section, differ from each other in the physical contexts in which the problem of the reality of the Fourier components arose. The first is borrowed from radiophysics.

In 1930, in his paper in Nature, the outstanding English radio-engineer Ambrose Fleming struggled with “widely defused belief in a certain theory of wireless telephonic transmission that for securing good effects it was necessary to restrict or include operations within certain width of “wave band” [113, p. 92] (about Mandelstam’s criticism of Fleming see Chap. 3).

According to Fleming, “wave band” was merely a kind of mathematical fiction and does not correspond to any reality in Nature” [ibidem].

Fleming referred to the “wave band theory” which was implied by the series expansion of the modulated signal, emitted by the transmitter. “When we sign or speak to affect the microphone at a broadcasting studio”, Fleming wrote, “the result is to cause the emitted vibrations, which are called the carrier waves, to fluctuate in amplitude but not to alter the number of waves sent out per sec”. Suppose the broadcasting station emits a carrier wave of frequency p. If q is the acoustics frequency, then the modulated vibration can be expressed by the function:

$$a = A\cos qt\sin pt.$$

(14.1)

However, this function can be expanded as follows:

$$a = \frac{A}{2}\left[ {\sin (p + q)t + \cos (p - q)t} \right]$$

(14.2)

In Fleming’s opinion, the modulated signal (14.1) corresponds to something in Reality, whereas the “wave bands”, presented in (14.2), are merely a kind of mathematical fiction. Fleming with his paper about the “wave band” theory launched a polemics in Nature of 1930.

Oliver Lodge, the outstanding English physicist, contributed to the polemics [221]. He greeted that Fleming, “in his admirably clear article”, raised the question “Whether a mathematical alternative does or does not invariably correspond with some physical reality”. In contrast to Fleming, he, based on physical properties of electromagnetic field, argued that the “wave bands” existed.

In reacting to the “wave band” discussion, Mandelstam did not mention O. Lodge’s contribution to it. As regards Fleming’s position, Mandelstam’s aim was to disavow it.

According to Mandelstam, any question about reality should be put against an apparatus or instrument by means of which an object, whose reality is a question that has no sense. This is not a way to put a question. A transition from formula (14.1) to formula (14.2) is mere trigonometry. No reception apparatus can detect whether there is one modulated wave or three (Mandelstam’s formulas slightly differed from Fleming’s—A.P.) nonmodulated waves from three transmitters. The question about the reality of wave bands is one of the kinds: Which is actually true, 10 = 2 + 8 or 10 = 5 + 5? [1, Vol. 4, p. 177].

If we are interested in applying a higher selective receiver, the representation (14.1) is not helpful. This receiver gives physical reality to that component of the sum (14.2) to which it tunes. However, a regular (not very selective) receiver gives physical reality to a single modulated wave.

Mandelstam also expressed this as follows: If we use as a receiver a tuning fork, it distinguishes between the components of the sum (14.2). But a human ear hears the single modulated signal.

The second example is in touch with the more complicated problem [1, Vol. 4, pp. 66–74]. In 1932, in his “Lectures on Selected Issues in Optics”, Mandelstam discussed a paradox which arose in physical applications of the Fourier integral. This paradox was physical rather than mathematical, and it was eventually connected with the same principles of the resonance theory as the above reality of the “wave bands”. However, here the problem was more complicated, insofar as a continuous spectrum is involved.

Let f(t) represent a wave packet, where

$$\begin{aligned} f(t) & = \sin nt,\quad \left| t \right| \prec \frac{T}{2}, \\ f(t) & = 0,\quad \left| t \right| \succ \frac{T}{2}. \\ \end{aligned}$$

We also admit that nT = 2πN, where N is an integer. Thereby we provide continuity of the function f (t) at t = ±T/2.

The expansion of f(t) into the Fourier integral is

$$f(t) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(u){\text{e}}^{iut} } {\text{d}}u.$$

Having calculated the Fourier factor, we arrive at

$$f(t) = \frac{n}{\pi }\int\limits_{ - \infty }^{ + \infty } {\frac{{\sin (u - n)\frac{T}{2}}}{{u^{2} - n^{2} }}{\text{e}}^{iuT} } {\text{d}}u$$

(14.3)

The last formula is a superposition of infinite sinusoids which extended from t = −∞ to t = +∞. The paradox is formulated as follows: Before t = −t/2, the function f(t) is equal to zero. In which way the “infinite sum” of sinusoids each of which is not equal to zero turns out to be equal to zero?

If the sinusoids had been real, we should see light before the light has been switched on. It is natural to say that these sinusoids, in contrast to their sum, are not real. However, this answer suggests a non-operationalist (according to Mandelstam, non-physical) notion of reality.

Mandelstam proposed another solution. Taking into account the equivalence between the LHS and the RHS of (14.3), one can only conclude that the actions of the infinite sinusoids, into which f(t) is expanded, are summed in such way that the result of their summation is zero. “We shall prove”, Mandelstam said “that this is the case even for a resonator of which damping is so small as we like” [1, Vol. 4, p. 69].

Mandelstam proved that even a higher selective receiver sets off a continuous bond of infinite sinusoids rather than a sinusoid, with the bond sinusoids taken together adding up to zero.

Nevertheless, an infinite sinusoid reaches physical reality too. Mandelstam considered two limit cases: (1) if $nT \gg \frac{{\omega_{o} }}{\delta }$ and (2) if ${\text{nT}} \ll \frac{{_{{}} \omega_{o} }}{\delta }$, where ω₀ is the proper (eigen) frequency of a resonator and δ is the constant of its damping.

In the first case (the damping is large), the resonator shows the spectrum which does not depend on the duration of the train represented by LHS and corresponds to the action of an infinite sinusoid to it. In the second case (the damping is very small), the resonator shows the shape of the train which is represented by f(t).

Reading the notes of Mandelstam’s lectures and seminars, we come to his treatments of reality of the Fourier components in connection with some other physical problems (“anomalous dispersion and the principle of relativity”, “the light beats”). However, from a methodological point of view, these Mandelstam’s comments do not add something essential to the above discussion.

14.4 The Main Points of the Comparison with P. Bridgman’s Operationalism

Now, we are able to compare Mandelstam’s operationalism with the philosophy of science of the American physicist Percy William Bridgman, a philosophy which spread very wide, became well known, attracted severe criticism, and received the name which we apply to Mandelstam’s philosophical conception.

Mandelstam had much in common with Percy Bridgman. Both came to the philosophy of science as working physicists; both, in developing their philosophical views, were guided by Einstein’s methodology contained in his 1905 article on special relativity, both emphasized the importance of experiment and measurement for the clarification of physical concepts, both saw in their philosophical accounts a tool for criticism rather than a “doctrine”, both attacked “pseudo-problems” and struggled against “language stereotypes” (we deliberately use a term which neither Bridgman nor Mandelstam used and which hence is neutral with respect to them), and both Bridgman and Mandelstam explained their philosophical accounts by the same examples with “length”, “time”, and “simultaneity”.

We, however, concentrate here on the difference between Bridgman’s and Mandelstam’s approaches. This difference can be summarized in the following four points:

• 1. In contrast to the “essentially American philosophy”, as G. Holton worded it, of P.W. Bridgman [158, p. 132], Mandelstam emphasized the intersubjective character of his operationalism: According to him, the operations which supply scientific terms with meaning must be repeatable and reproducible. In his Lectures on the Theory of Relativity, he especially pointed out that operations should meet the request of “invariability and unambiguity” [1, Vol. 4, p. 182]. Bridgman also spoke about his operations as physical and hence reproducible for fellows and colleagues. “In principle the operations… should be uniquely specified” [65, p. 10]. However, his operationalism shows a solipsist tenor which becomes stronger in his later writings. As early as 1936, his Nature of Physical Theory demonstrated that he regarded operations as “private”, as “mine and naught else” [66, p. 14] (see also [67, p. 158], compared with [68, p. 8] where Bridgman writes about “objectivity” of operations).

• 2. Originally Bridgman restricted his “operational analysis” to the physical operations which could be actually performed. In such a way, he secured knowledge with respect to contradictions which could penetrate into it through the “mental operations”. In his later writings, he extended his operationalism by including the “paper and pen” operations, that is, theoretical calculations [66, p. 123, 68]. Mandelstam had never formulated his operationalism in such a rigid formula as Bridgman’s original conjunction “the concept is synonymous with the corresponding set of operations” and “if the concept is physical…, the operations are actual physical operations” [65, p. 5]. In his Lectures on Quantum Mechanics, he formulated his concept of operations by including theoretical calculations as an essential part of the operations: As was mentioned above, he developed the theory of indirect measurement in these Lectures. In contrast to Bridgman’s vague “paper and pen operations”, Mandelstam’s indirect measurements obeyed the definite rules which quantum mechanics suggested.

• 3. Describing the structure of physics, Bridgman tended to fix a sequence of the dichotomies: “theories and factual knowledge”, “mathematical equations and text”, “mathematical models and physical models” [65, pp. 1, 59, 62]. This was in the style of the approach common to the philosophy of science of his years and afterward. Due to his theory of indirect measurement, Mandelstam turned out to be close to a holistic approach to a physical theory, the approach which was expressed by W.V.O. Quine and K. Hempel in the 1950s. If (at least) two operational (“prescriptional”) definitions can be formulated for a physical concept, one of them can be treated as an empirical sentence testable against experiment and observations and the other as a proper definition which is a convention. In another situation, the former can replace the latter and the latter can replace the former. From the point of view of the theory of indirect measurement, all the quantum postulates are “rules” of measurement, and all of them provide the “prescriptions” for how to conduct indirect measurement. This means that this physical theory consists in operational definitions. However, these definitions can be testable in turn: One of them is considered to be an empirical sentence; the others provide a test as definitions, that is, conventions.

  Mandelstam did not use terms an “empirical sentence” and a “convention”. However, what he said can be treated with the aid of these terms and his position can be labeled as holistic: The theory as a whole, rather than its individual sentences, is testable against observations and experiment.

• 4. Bridgman did not devote much room to “physical reality” and “truth” in his writings. He was in touch with them only critically: his aim was the “operational criticism” of these philosophical concepts and the limitation of their applicability. Mandelstam was not concerned with these concepts either. However, his position can be called realistic. He permanently mentioned that “operations”, “prescriptions” related the mathematical symbols to “nature”, to the “real objects”, to the “reality” rather than “experience”, “observations”, “sense data”. Apart from this when he wrote and spoke about the confirmation of theories, he applied the qualifications “true” and “false” to theories.

True, his realism was of a kind of scientific realism: It was restricted by his holistic approach to a physical theory. Mandelstam rejected the “a priori concepts”, which were given “by themselves” (see [1, Vol. 5, pp. 183, 406]). Through his lectures, the picture of nature was open to discussion. Nevertheless, he retained some ontological parameters. Sometimes, Mandelstam, for example, was inclined to accept “ontological determinism”.

Let us turn again to Mandelstam’s Lectures on Quantum Mechanics. Mandelstam joined von Neumann’s discussion of the completeness of quantum theory. This does not mean that Mandelstam shared von Neumann’s philosophy of causality. He tended to avoid the indeterminism which von Neumann proclaimed. Thus, von Neumann wrote that “es gibt gegenwaertig keinen Anlass und keine Entschuldigung dafuer, von der Kausalitaet in der Natur zu reden” [373, p. 167]. Mandelstam said, however, the following [1, Vol. 4, pp. 403, 414]:

They say that von Neumann demonstrated that the construction of the theory on the base of determinism is impossible. I think that such phrases say next to nothing.

If they sometimes say that von Neumann demonstrated that the causal theory of the atom phenomena is impossible then this is not the case.

This was not identical with von Neumann’s claim above.

To conclude, let us turn once more to Bridgman’s operationalism. Although Mandelstam’s philosophy of science can be summed up under heading “operationalism” indicating Bridgman’s teaching, it does not lack originality. With this, we arrive at the question by which the next section is entitled.

14.5 What Philosophical Tradition Was Lying Behind Mandelstam’s Operationalism?

Was Mandelstam influenced by Bridgman’s celebrated The Logic of Modern Physics? Materials at our disposal give no hint to answer “yes, he was” to this question. It is likely that Mandelstam arrived at his operationalism independently by studying Einstein and Heisenberg’s articles and meditating on the foundations of physics.

There is a reason to answer in such a way to the question about Bridgman’s influence on Mandelstam. As mentioned above, Mandelstam met Richard von Mises in his Strasbourg years and went on to communicate with him when they were stranded on opposite sides of the border in later years.

According to Papalexy’s recollections, Mandelstam and Richard von Mises discussed the philosophical foundations of physics.

Mandelstam and Richard von Mises exchanged letters in the 1920s and 1930s (Chaps. 5 and 6). Mandelstam visited Richard von Mises when he was in Germany in 1923 and he stayed at the Richard von Mises’ when he was visiting Germany in 1930. It is very probable that they continued their discussion on the philosophy of physics when they met together.

Mandelstam and his disciples enthusiastically greeted Richard von Mises’ book Wahrscheinlichkeit, Statistik, und Wahrheit (J. Springer, 1928) in which (in essence) the operational definition of probability was proposed and a Machist ideology of this definition was developed. (This book was a popular presentation of his ideas which were first published in 1919.) As was noted in Chaps. 2 and 6, Mandelstam contributed to the rapid publication of its Russian translation, and he and his disciples inspired the discussion of it at seminars and in journals in the USSR.

Richard von Mises was among the classics of neopositivism and a great propagandist of Machism. However, it is remarkable that although the German positivists made a note of The Logic of Modern Physics, Richard von Mises did not mention Bridgman’s operationalism in his 1939 Kleines Lehrbuch des Positivismus [370] and in previous philosophical writings. He mentioned it only in his 1951 English version of it entitled Positivism . “The physicist P.W. Bridgman”, Richard von Mises wrote, “devised in his operationalism a theory of knowledge that is closely related to, and in full agreement with, the main teachings of Mach” [371, p. 361].

As was mentioned above, Richard von Mises in essence developed the operational definition of probability in his book Wahrscheinlichkeit, Statistik, und Wahrheit. It should be noted that he did not use the term “operational definition” then and, in contrast to Bridgmans’, his definition referred to theoretical principles and included theoretical considerations.

Attempting to answer the question about the generic relation of Mandelstam to Bridgman, we are in the same difficult position in which Max Jammer seemed to be when he discussed Heisenberg’s operationalism as it was expressed in his 1927 article where the uncertainty relations had been formulated. On the one hand, Heisenberg’s article shows apparent operationalism. On the other hand, “it would be rash to classify Heisenberg as a pure operationalist” [166, p. 58]. To justify this, he referred to the facts that P. Bridgman did not approve of Heisenberg’s interpretation of the uncertainty relations and Heisenberg, in turn, did not accept Bridgman’s 1929 explanation of these relations [166, p. 472].

Thus, like in Mandelstam’s case, there is no indication that Heisenberg’s operationalist interpretation of his uncertainty relations and Bridgman’s operationalism are genetically connected. Heisenberg developed in his 1927 paper his own version of operationalism which can be traced back to the kinematical part of Einstein’s 1905 article and to E. Mach’s positivism. Indeed, although the main portion of Mach’s book and papers is written in a descriptivist tenor, his teaching admits interpretation in operationalist way. In fact, Mach was close to operationalism when he provided this definition of some physical concepts, for example, the definition of mass (see also Mach’s definition of the strength of illumination in [222]).

Mandelstam was prepared to elaborate operationalist philosophy of science by his education at Strasbourg University and his following scientific contacts. He was familiar with German positivist literature (see Chap. 2, Sect. 2.4; Chap. 5, Sect. 5.5; Chap. 6, Sect. 6.7). Apart from what followed his contacts with Richard von Mises, one can refer to his Borovoie diary in which he cited, for example, L. Wittgenstein (Chap. 12).

So, Mandelstam brought to Soviet science the methodology of the German community of physicists. Having interacted with the scientific and pedagogical problems which Mandelstam posed working in the Soviet Union, this methodology yielded his operationalist philosophy.

14.6 Andronov’s and Chaikin’s Principle of an Expedient Idealization

As an example of the development of Mandelstam’s operationalism in his scientific community, we consider the principle of the expedient idealization, which was first formulated in Chaikin’s 1935 preface to Russian translation of Baltasar Van der Pol’s review on the theory of non-linear oscillations [75], in the Introduction to Andronov, Vitt, and Chaikin’s 1937 book Theory of Oscillation, in Chaikin’s 1948 Mechanics [76] and then in his Physical Foundations of Mechanics. Andronov also formulated this principle (to avoid any associations with “bourgeoise philosophy”, he called it the “principle of a correct idealization”) in his 1944 article, written at length, “Mandelstam and the Theory of Non-Linear Oscillations” [18].

In Andronov, Vitt, and Chaikin’s book, we read [26, pp. xv–xvii]:

In every theoretical investigation of a real physical system we are always forced to simplify or idealize to a greater or smaller extent the properties of the system. The nature of idealization permissible in the analysis of a problem is determined by the problem in its entirety.

Thus one and the same idealization can be both “permissible” and “impermissible”, or better, expedient or inexpedient depending on questions to which we want to answer. An idealization of the properties of a real system, i.e. use of a mathematical model, enables us to obtain current answers to certain questions about the behavior of the system but does not, generally speaking, give us the possibility of answering other questions correctly about the behavior of the same system.

To trace Mandelstam’s root of this principle, let us turn again to Mandelstam’s lectures, where he gave the warning “No idealization can be extended to infinity. One needs idealize sensibly, keeping in mind the limits” [1, Vol. 4, p. 148]. By referring to the operations, Mandelstam not only struggled against “pseudo-problems” and took traditional concepts under examination but also explained how to develop and change idealizations.

A good example is provided in Mandelstam’s 1930–1932 lectures [1, Vol. 5, p. 72]. The standard idealization of a spring with a small bob on its end, which is hung vertically and whose mass is negligible in comparison with the mass of a bob, is a simple pendulum. One can come across this idealization in many elementary textbooks on mechanics and physics. However, this idealization hangs on our initial operations of a kind aimed to excite oscillations. If we pull down the bob and then allow it to move freely, then the standard idealization is good. However, if we pull the same string down from its halfway point and allow it to move freely, the standard idealization will fail to be helpful. We need to idealize the same set up not as a concentrated system but as a continuous medium system.

To feel better Mandelstam’s approach to idealizations, let us recall the example which illustrated his operationalist approach to physical reality (Sect. 14.5). This example fits his approach to idealizations too. If we work with a sieve, we take the idealization of big and small balls; if we work with a magnet, we take the idealization of ferrous and cupric balls.

Idealizations are not new to physicists. Many of them declared that physics presupposed idealizations. To access that development of Mandelstam’s philosophy of science which Andronov and Chaikin elaborated upon, one should recognize that for them, an expedient idealization became a working device. In I. Lakatos’ terminology (see [196]), one can say that for Mandelstam and his followers, it became a kind of “positive heuristic” (see, e.g., Andronov’s sketch of the history how he struggled with the idealization of Abraham–Bloch’s multivibrator and how Mandelstam and Papalexy helped him in Chap. 9).

In this connection, it is interesting to compare the Mandelstam–Andronov–Chaikin methods with those of the other (to some extent rival) Soviet school in non-linear science, the Nikolai Mitrofanovich Krylov–Nikolai Nikolaevich Bogoliubov school (see Chap. 10, Sect. 10.2).

By calling their subject “nonlinear mechanics”, Krylov and Bogoliubov emphasized that their approach differed from that of Mandelstam, Andronov, and Chaikin using the terminology of the theory of nonlinear oscillations as the name of their subject. Certainly, Krylov and Bogoliubov employed idealizations, too. However, this was not the main point of their methodology. This was approximations (first of all the asymptotic methods) in the theory of differential equations. The method of expedient idealization was used in Andronov, Vitt, and Chaikin’s book as the following two steps procedure: (1) idealization of the phenomenon under consideration, (2) the strict mathematical solution of the differential equation which resulted from this idealization (“the theoretical treatment of the idealized scheme should be carried out with full rigor” [27, p. xv, 26, pp. 18–19]). In turn, Krylov and Bogoliubov tended to take the following course of actions: (1) the strict description of the phenomenon under consideration (or the description of it with minimal idealizations), (2) the solution of the differential equations, which provides this strict description, by means of methods of approximations.

Although Mandelstam and his disciples, on the one hand, and Krylov and Bogoliubov, on the other hand, referred to each other, they did not really discuss works written by the competitive school. To compare their methods, it is worth to turn to Nikolas Minorsky’s book [236]. Thus, in his discussion, so-called relaxation oscillations, Minorsky referred to the idealized discontinuous treatment of relaxation oscillations in Mandelstam’s and Chaikin’s papers. He also mentioned the treatment of these oscillations given within the framework of the asymptotic methods “without involving any a priori idealization” [238, p. 600]. This treatment was an improvement of the method of the Dutch engineer Balthasar Van der Pol who strictly described the relaxation oscillations but solved the differential equation by the isocline method.

So, the method of expedient idealization is worth to be mentioned not only due to its philosophical connotations. This method is not trivial as a tool at a physicists’ hand.

14.7 A.D. Aleksandrov Versus L.I. Mandelstam

Let us recall the first section of the present chapter. This brings up the question: Why A.D. Aleksandrov, an outstanding mathematician, a topologist, the teacher (perhaps, one of the teachers) of the great mathematician Grigorii Perelman, took a floor to criticize the lectures which had been delivered by the prominent physicist, Member of the Soviet Academy of Sciences, L.I. Mandelstam, the lectures published posthumously, five years later L.I. Mandelstam’s death? We cannot find the only answer to this question.

First of all, we should take the “mathematicians—physicists controversy” into account. Operationalism is the physicist’s world view. It was put forward by the physicist Percy Bridgman who took the methodological considerations which the other physicists provided into account. As he himself wrote, he proceeded from Einstein’s definition of simultaneous events, the definition which refers to the synchronization of clocks distant one from other.

To illustrate his position Bridgman put the question: What does mean the length? The length means the operations which allow us to measure the length of any object, the operations with a ruler or any other instrument to measure the length.

Operationalism has never been popular among mathematicians. Mathematicians look for the foundations of mathematics by considering the set theory and logic. True, there is constructivism as the philosophy of mathematics. There are constructivists’ theories which appeal to the concept of algorithm. However, this concept is mathematical. It has a mathematical definition, and it is included in the context of mathematical theory.

If a mathematician is asked what the length is, he would refer to the concept of metric. Metric is a mathematical formalization of the distance. Length can be treated as a special case of the distance. However, the distance has a mathematical definition.

This example helps us to understand the idiosyncrasy of A.D. Aleksandrov as a mathematician with respect to the methodology of operationalism.

However, as was mentioned, there are other reasons for such idiosyncrasy. Operationalism was inconsistent with the symbols of the Soviet citizen’s belief. It was inconsistent with the concept of the “matter”, “objective reality”, “dialectics”, since it refers to the human being’s action and in the long run to the consciousness. By criticizing Mandelstam’s lectures, A.D. Aleksandrov met the Soviet citizens’ common sense. Here, by speaking of the Soviet citizens, we mean engineers, researchers, journalists, officials, students, schoolteachers, etc., in other words, we mean the educated Soviet citizens. We do not take into account the intellectuals educated in the Russian prerevolutionary traditions. Their amount was small, and they were not influential.

A.D. Aleksandrov wrote in the stile of the Soviet ideology in 1958, too (The All-Union meeting on the philosophy of contemporary science). However, this was another period in the development of the Communist ideology. Heavy battles against the bourgeois philosophy which took place at the end of the 1940s became a thing of the past. Stalin was dead, the head of Stalin’s security service Beria had been executed… The problem to criticize the bourgeois ideology became one of the problems which the Communist Party put before the Soviet philosophy. The authorities became to proclaim the creative development of Marxism–Leninism. In his Introductory Address President of the Academy of Sciences Nesmeianov told about the “objective laws” of the development of science. Philosophers should take these laws into account.

Nevertheless, A.D. Aleksandrov wrote the following:

The theory of relativity is a physical theory of space and time and the fundamental concepts “motion”, “mass”; “energy”, etc. are connected with this theory…

Space and time are the forms of being of matter. This means that the space-time relations don’t exist in themselves, as they are; they are determined by the material interconnections of things and of phenomena. Correspondingly, the laws of these relations (the properties of space and time) are the laws of the general structure of the material interconnections of things and phenomena…” [9, p. 93]

It is remarkable that Aleksandrov’s opponents who came out at the 1952 all-institute colloquium took the concepts of matter, of reality non-critically, too. They tended to explain that L.I. Mandelstam actually was a materialist. They claimed that one really could point to some small deviations from materialistic orthodoxy in his lectures which were published as the fifth volume of his “Complete Works”.

There is a comparison of the communist ideology with religion [189]. This comparison is rather productive. Marxist–Leninist excursions to the scientific concepts did not lead to any productive shifts in science. These excursions could not be falsified, operationally improved, productively criticized, in a word they have not led to any positive steps in the development of science.

Above, we have been concerned with two possible reasons of A.D. Aleksandrov’s attack of Mandelstam’s fifth volume.

The third possible point is career. The Soviet science has been constructed hierarchically. The great amount of research workers makes the “ground floor”, the Corresponding fellows of the Academy of Sciences make the “second floor”, and the Academicians make the “upper floor”. There were academicians of the Academies of the Union Republics. They approximately correspond the Corresponding members of the USSR Academy of Sciences. Among research workers, there was a hierarchy too. There were scientists with the Doctor of Science degree, with Candidate of Science degree, and the researchers without any degree.

Aleksandrov was Corresponding Member of the USSR Academy of Sciences. Naturally, he wanted to become an Academician. In his lecture, A.D. Aleksandrov referred that by analyzing Mandelstam’s writings he carried out the request of the “one organization”. He referred to the “one organization” in his closing speech too. It is very probable that the “one organization” was the Central Committee of the USSR Communist Party. This organization could be helpful for a person who wanted to make a step in his career.

This section is based on material containing in [324, 325].