Mathematics in the Soviet Union: A Political Activity
Mathematicians of today are considered as academicians who are isolated in their offices working on problems no one will ever to try understand and on subjects no one knows of. Although at first sight this seems to apply to most of the natural sciences and in some cases even to social sciences, undoubtedly mathematics is considered as “the” discipline of working on unnecessary subjects that have nothing to do with real life.
This approach to mathematics and mathematicians is a result of the historical context we live in. An immediate example of a contrary approach would be that of ancient times where mathematics was the essential science for practicalities such as trading, architecture, farming, and military. Even though the connections remained tight, there is almost a consensus both in today’s society and in the academies that the role of mathematics in practical life diminished during professionalisation and specialisation in 19th and 20th centuries.
However, 'historical context’ is not limited to time but is indeed a function of space as well. In this essay, mathematics in the Soviet Union and its relationship with the political atmosphere of the time will be discussed. While the main focus will be on the period 1917-1941, supporting information from different periods will be included as well. As “the importance of Russian and Soviet mathematics is poorly reflected in English-language sources”; some of the examples and quotations will be on science in general and not on mathematics proper. Those examples are included for the purpose of illustrating the general approach to science and scientists in that time.
First, the contrast between the definitions and explanations of mathematics made in Encyclopaedia Britannica and Large Soviet Encyclopaedia, both printed around same years, will be pointed out. Next, a brief summary of the history of tsarist period in Russia and the Soviet Union will be given, mainly concentrating on the approach to science. Then, the politicisation of mathematics will be discussed. This section consists of two parts: First, the environment in which mathematical research was done; and second, the controversies on what mathematics is and what it should be. Lastly, an analysis of the development of mathematics, focusing on the underlying historical influences to it, will be made.
The main argument of this essay is that mathematics is a political activity, and that today’s approach to mathematics as an apolitical activity itself is a highly political stance.
1. What is Mathematics?Two important encyclopaedias were prepared in 1930s; the Encyclopaedia Britannica (hereafter EB) was printed in 1941, and the Bol’shaia Sovetskaia Entsiklopediaa (Large Soviet Encyclopaedia; hereafter BSE) in 1938. The main contrasts between EB and BSE concerning mathematics are “on the most essential questions of mathematics: What are the origins of mathematics? and What is the relationship between mathematics and the real world?” These questions, mainly dealing with the problem of how mathematical developments occur, are answered in completely different manners in the afore mentioned sources.
1.1. Bol’shaia Sovetskaia Entsiklopediaa
The entry “Mathematics” in BSE was written by A.N.Kolmogorov, one of the leading mathematicians of the Moscow school of probability who was also a student of N.N.Luzin. In his article, he “quoted Engel's views on how mathematics was a reflection of material relationships and answered practical needs in its early history, later grew to be a highly abstract field, but never lost its organic tie to material reality.” He asserted that mathematics arose out of “the most elementary needs of economic life.” He also noted that whilst mathematics became more and more abstract, its origins in real life were forgotten by mathematicians; but he reminded them that
“the abstractness of mathematics does not mean its divorce from material reality. In direct connection with the demands of technology and science the fund of knowledge of quantitative relations and spatial forms studied by mathematics constantly grows.”
Kolmogorov's definition is consistent with A.D.Aleksandrov's approach to the relationship between mathematics and dialectics, which will be discussed in section 3.2.2. .
1.2. Encyclopaedia Britannica
On the other hand, in EB, there is a very different approach to mathematics in the articles “Mathematics, Foundations of” by F.P.Ramsey and “Mathematics, Nature of” by A.N.Whitehead. They never referred to practical needs of the humanity, but treated mathematics as a discipline independent of physical reality. Indeed, Ramsey maintained that “as a branch mathematics, geometry has no essential reference to physical space” According to his analysis, the mathematician
“regards geometry as simply tracing the consequences of certain axioms dealing with undefined terms, which are really variables in the ordinary mathematical sense, like x and y. And he demands of his axioms, not that they should be true on some particular physical interpretation of the undefined terms, but merely that should be consistent with one another.”
In addition, one should note that Whitehead asserted that the “act of counting” is “irrelevant to the idea of number.”
Taking these two sources into consideration, it should be observed that, as mentioned in the introduction, the notion of mathematics today as a logical system independent of real life is not intrinsic to the mathematics itself but is a result of the social context in which mathematics is defined.
2. Historical Background
Before opening a discussion on the development of mathematics in the revolutionary period of the Soviet Union, it is essential to recall the history of mathematics and science in that region. Even though it would be useful to consider the early periods of Russian history; namely the Kievan Rus' period from the ninth century A.D. to approximately 1240, the period of Mongol role from 1240 to 1480, and the time of Muscotive primacy from 1480 to 1700; due to space limitations, the focus will be restricted to the Imperial period of Russia, the revolutions in 1917, and the early years of the socialist state.
2.1. The Imperial period of Russia
The Imperial period of Russia is marked by the modernisation efforts started by Peter the Great. In his reign, from 1689 to 1725; in addition to establishing a navigation school, a Naval Academy, an artillery school, an engineering school and a medical school; most important for our study, he initiated the establishment of the Academy of Sciences in St.Petersburg, the foundation of which was finished in 1725, after his death. The founders and early mathematicians of the Academy include Daniel and Nicholas Bernoulli, Jacob Hermann and Leonhard Euler.
The reforms continued during the reigns of Catherine the Great (1762-1796), Alexander I (1801-1825) and Alexander II (1855-1881), while there were repressive periods as well. The Moscow University was founded in 1755. This was followed by the foundations of universities in Dorpat (1802), Vilna (1802), Kazan (1804), Kharkov (1804) and St.Petersburg (1819) by Alexander I. It is interesting to note that “the founding dates of professional societies such as the Russian Physical-Chemical Society  were comparable to the birth dates of similar societies in the United States.” The professional societies specialised in mathematical sciences that are established at the Imperial period of Russia were the Society of Mathematicians (1811), the Moscow Mathematical Society (1864), the Kharkov Mathematical Society (1879), The Kazan Mathematical Society (1890). the Physics and Mathematics Society of Kiev University (1890), and the St.Petersburg Mathematical Society (1890).
Even though St.Petersburg remained as the centre for mathematical research; by the end of the nineteenth century, Moscow University became an important institution attractive for many scholars.
2.2. A Marxist revolutionBefore 1917 Russian science lagged behind the leading countries of the West, but it had achieved an inertia of its own. Even in the repressive period of Ivan Delianov's ministry of education from 1882 to 1897, mathematics in Russia continued to flourish. Russian scholars like Lobachevskii, Chebyshev and Markov occupied solid positions in the history of mathematics.
“While most scientists, engineers and physicians in the Russian Empire greeted the February Revolution as an event that held much promise for political freedom and scientific research, their reaction to the advent of the Bolsheviks in October was one of suspicion and hostility.”
The technical intelligentsia felt that its social status was in danger. However, Lenin “often personally received the representatives of the Academy, Vice-President V.A.Steklov and Permanent Secretary S.F.Ol'denburg, and talked with them about the work and needs of the Academy.” And with the introduction of the New Economic Policy in 1921, which was a strategical step backwards from the socialist ideals, the psychological distance between the technical intelligentsia and the Socialist State diminished.
On one hand, no member of the Academy of Sciences was a member of the Communist Party for many years; but on the other, “the number of scientific workers in institutes of the Academy grew from 154 in 1917 to 413 in 1925.”
2.3. The Soviet Union
In the Stalinist period, the attitude of the Socialist State towards technical intelligentsia changed dramatically. The Cultural Revolution, and the “Great Break” in general, in 1928-31, created a repressive atmosphere for independent research. While these actions are mainly explained with Stalin's personality and ideological stance; it is important to note that they were the realisation of the reconstruction of Soviet state as a proletarian state that was considered, by revolutionary groups, to be delayed. During the Stalinist period, the Soviet Union created a new generation of technical intelligentsia that is said to accord with the socialistic ideals.
After Stalin, periods of repression followed by periods of liberalisation constituted a fluctuating environment for the scientists. Unfortunately, this period of Soviet history will not be included in this essay.
3. Politicisation of Mathematics
History of mathematics is full of stories where ideologies heavily affect the activities of the mathematicians. Historically, the first well-known example is the story of Hippasus of Metapontum, who is said to be drowned by Pythagoras since he proved, contrary to the belief of Pythagoreans that all numbers can be expressed as a ratio of two natural numbers, the irrationality of the square root of 2.
While the examples that come into the mind are merely stories of mathematicians influenced by the political environment they work in, the Soviet Union is the unique example where the mathematical activities themselves are considered as political.
3.1. A Political Environment
Scientific research in the Soviet Union was under a threat of being labelled as 'bourgeois science' and hence 'against the ideology of the victorious revolution'.:
“When some Soviet intellectuals tried to discuss Freud and the development of psychoanalysis, the critics answered by connecting Freud's views to the bourgeois culture of middle Europe, with its guilt-ridden neurosis. ... When some economists raised doubts about the ability of the Soviet Union to achieve unheard-of industrial growth rates in face of limited resources, radical economists accused them of trying to slow the development of the socialist state. ... When certain Soviet authors tried to interpret the meaning of the new developments in relativity and quantum physics, showing that the old concepts of materialism and causality were no longer adequate, the Soviet critics often replied that bourgeois intellectuals were engaged in a reactionary effort to discredit scientific materialism.”
Soviet intellectuals were always on the alert due to concerns that their activities, both in content and in presentation, should accord with the ideology of the Socialist State.
3.1.1. N.N.Luzin’s Case
An interesting case is the affair of Nikolai Nikolaevich Luzin (1883-1950), the worldwide known mathematician who founded the Moscow school of theory of functions. In 1936, Luzin was the chairman of the presidium of the Mathematical Group of the Department of Physical and Mathematical Sciences, the head of the Mathematical Certification Commission, and the head of a department at the academy's Mathematical Institute. “He was a philosopher with a non-Marxist orientation. He was a religious believer who did not accept the new regime.” But also, “in his public life, Luzin was a conformist.” So, even though he was never openly attacked in Soviet mass media, he had more than what one needs to be attacked.
In July and August 1936, in several unsigned articles published in the leading newspaper Pravda, he was accused of “blatant plagiarism, openly biased reviewing of mathematical texts, overpraising of obviously weak works” and “publishing his best manuscripts abroad.” Luzin admitted his misbehaviour, but was dismissed from his administrative positions mentioned above. On August 6th, an editorial was issued in Pravda, declaring that the “Soviet scientific youth” had “finally gained control over higher mathematics” and “assisted in demasking the class enemy;”concluding the campaign against Luzin. “In 1941, he was given back his former office at the Mathematical Institute, and two years later he returned to the university.”
The operation against Luzin was a part of the ideological shift process away from internationalism towards a rhetorically masked nationalism. This can be seen in the following passage from one of the articles against Luzin:
“Perhaps they do not possess a feeling of national pride in even the tiniest dose? ... Perhaps, they do not experience Soviet patriotism at all? ... Such a situation must not be tolerated any further. The Soviet Union is not Mexico or some kind of Uruguay, it is a great socialist power.”
This ideological shift can be said to start in February 1934 and end in March 1939. Thus, 'the Luzin affair' was just a step in this campaign.
N.N.Luzin's case is a clear illustration of the influence of political state of affairs in the Soviet Union on academicians.
3.2. A Political Profession
“One of the remarkable aspects of the Russian Revolution was that it presented not only a prescription for a different political and economic order but also an alternative form of knowledge of the natural world; it called for a Marxist interpretation of nature consciously opposed to existing “bourgeois” descriptions. No other revolution in history contained a radical epistemological and cognitive system to the same degree.”
In the following, three examples of this approach will be given. First, a discussion on what probability theory should be will be mentioned. Secondly, A.D.Aleksandrov's opinions on mathematics will be summarised. Lastly, an approach to the history of science that is initiated by Boris Hessen, namely 'externalism', which complements historical materialist approach in that discipline, will be discussed.
3.2.1. Politically Correct Probability
The intellectual attacks towards the use of probability theory and statistics to demonstrate that religion was supported by mathematical arguments were made mainly by Ernst Kolman (1892-1979) and Mikhail Khrisanovich Orlov (1897-1944). Below is an attack made by Orlov to the pre-Revolutionary Nekrasovites, i.e. the Moscow Mathematical School, as opposed to the St.Petersburg School lead by A.A.Markov (1856-1922), an atheist and a volatile fighter for scientific purity and for justice who was much acceptable to the incoming regime.
The reader should first recall Bayes' theorem:
If A1, A2, ... ,An is a set of mutually exclusive and exhaustive events, and B is any event; then
P(AiB) = P(BAi) P(Ai) / ∑ j P(BAj) P(Aj)
V.Y.Buniakovsky used Bayes' formula “in the context of the testimony of witnesses affecting the probabilities of unlikely events in the remote past, and thus using probability mathematics in support of religion.” Suppose A1 is an unlikely event (perhaps a miracle in remote past) with very small probability ε>0 and A2 is the complementary of that event. Now suppose that B is the event that m witnesses testify that the event has occurred, and each witness, independent of what others say, tells the truth by a probability 1-δ. Then Bayes' theorem gives that if (1-δ)>1/2, then P(A1B) approaches 1 as m becomes large, and the approach is rapid.
Orlov cites Nekrasov's praising Buniakovsky and states:
“Religion is now an inseparable part of the bourgeois apparatus for the repression of the masses ... Amongst other sciences, mathematics is also used by “representatives of the enlightened bourgeoisie” to strengthen their ideological positions ... This is why the exposure of such arch-reactionary attempts is of great importance in the general battle of the proletariat against capitalism.”
He then concludes that “in his 'Bible' ... Nekrasov [speaks] of posterior probabilities, but no mathematical manipulations give the least occasion to speak of 'god' on the basis of Bayes' Theorem” This discussion is an interesting example of the use of mathematics to support an ideological stance and the reactions to such an attempt.
3.2.2. Mathematics and Dialectics
Aleksandr Danilovich Aleksandrov (1912-1999), a distinguished mathematician educated in the Revolution's most idealistic period, was a sincere Marxist. He was the rector of Leningrad University from 1952 to 1964. Below is an analysis of an article he wrote on mathematics and dialectics due to the one hundredth anniversary of the birth of Lenin.
He cites Bertrand Russell's assertion that “mathematics is that doctrine in which we do not know whereof we speak nor whether what we say is true” and proposes that “mathematics creates its apparatus and it is absurd to speak of what is true and false: the apparatus either works or it does not, and if it works it either works productively or poorly.”
He emphasises that “the value and validity of mathematics consists in its applications”, not forgetting to remark that “exclusive preoccupation with certain applications is similar to the working of a machine only as a cutter or of industry only in production of objects of consumption.”
Aleksandrov considers the development of mathematics as a dialectical progress of the unity of opposites and notes that
“the development of mathematics is not confined to the establishment of new theorems, the invention of new methods and definitions of concepts in an arena already formulated. It includes also the development of essentially new concepts, the incorporation of new objects, and the establishment of fundamentally new theories.”
His opinions on mathematics are interesting because of two reasons: First, even though the Soviet State was generally considered to be unsuccessful in creating its own socialist intelligentsia, Aleksandrov was a prominent academician to have international academical recognition and strong ties with Marxism at the same time. Secondly, the consistency and compatibility of his ideas with Kolmogorov's definition of mathematics in BSE are striking.
3.2.3. Boris Hessen and History of ScienceIt is of great importance to mention Boris Hessen (1893-1936) and his approach to the history of science in this essay, because the story of Hessen and the Second International Congress of the History of Science in 1931 is an astonishing example to display almost all the features of the Soviet science mentioned in this essay: the relationship between science and the political economical context, the discussions about what science should be, and the highly political environment in which academicians work.
In the 1931 Congress, Hessen presented a report on Isaac Newton. “Most previous treatments of Newton had depicted him as a genius whose creativity transcended human understanding,” but Hessen announced that Newton and his work “could not be understood outside the context of the rise of mercantile capitalism in England.” He asserted that the new technologies the industrial revolution demanded for could only be obtained by applying his three laws of physics to ballistics, mechanics and hydrostatics. Studying Newton's personal life and his philosophical ideas, Hessen maintained that “Newton was the typical representative of the rising bourgeoisie, and in his philosophy he embodies the characteristic features of his class.” Thus, he formulated 'externalism' in the historiography of science, a discipline that stresses social, economic, and other non-scientific influences on the development of science.
However, there was an underlying argument in Hessen's paper. To see it, one should note that he was a defender of the compatibility of modern physical theories like relativity and quantum mechanics, and Marxism. Einstein's controversial ideas about how to interpret the results obtained from modern physics were heavily criticised by Soviet ideologists on the grounds that he related the results with the existence of some sort of a divine power. But Hessen argued that Einstein's personal philosophical stance can be separated from the validity of these scientific accomplishments. For that purpose, he demonstrated the parallel situation in Newton's case, where there was complete consensus about the validity. Hence, Hessen's report was a part of the ongoing discussions in the Soviet Union about which scientific theories are 'politically correct'.
Hessen's stance was strongly criticised by E.Kol'man. Kol'man tried to show how the wreckers in physics were trying to discredit materialism:
“The wreckers do not dare to say directly that they want to restore capitalism, they have to hide behind a convenient mask. And there is no more impenetrable mask to hide behind than a curtain of mathematical abstraction.”
Even though Hessen's ideas on the history of science were internationally accepted in the course of time, in a personal sense, he failed in the debate. “He died in prison in 1938, a victim of the purges, along with six members of the eight-man delegation to London in 1931.”
4. A Historical Materialist Analysis of the Development of Historical Materialist Mathematics in the Soviet Union
The examples mentioned in Section 3 show, besides from other interesting features of the Soviet science, how important the attitude of the Socialist state was for scientific studies. In this section, the evolution of the attitude of the state in the Soviet Union will be discussed.
4.1. Different Attitudes towards the Technical Intelligentsia
As mentioned in Section 1, the modernisation of Russia started during the reign of Peter the Great. His method was criticised due to the fact that he attempted to bring science and technology from the Western Europe to Russia by starting from the top, with an academy of sciences. Peter replied to these critics:
“I have to harvest large shock of grain, but I have no mill; and there is not enough water close by to build a water mill; but there is water enough at a distance; only I shall have no time to make a canal, for the length of my life is uncertain, and therefore I am building the mill first and have only given orders for the canal to be begun, which will force my successors to bring water to the completed mill.”
Considering the accomplishments of the Academy in the following century, it can be said that Peter correctly anticipated the development of science in Russia.
After the October Revolution, “when the Communist leaders inherited this extraordinary institution, they faced a decision -abolish it, as in the French Revolution; support it at the existing level while expanding research in other institutions such as the universities; or build a structure of scientific research in which it would be the central and critical element. They decided to adopt the last choice” due to the economic crisis and famine, and hence the need for technical advice in these matters. There was strong insistence by the radical revolutionaries, asserting that bourgeois intellectuals must be replaced by socialist scientists in order to obtain a purely proletarian science. Nevertheless, Lenin's opinion on the subject was somewhat different: “We must take all of culture which capitalism left to us and build socialism out of it. We must take all science, technology, all knowledge, art. Without them we cannot construct the life of a communist society. And science, technology and art are in the hands and heads of the specialists.” Thus, Lenin defended the 'transformation' of the technical intelligentsia.
However, the 'reconstruction' of it was realised in the Stalinist period. The 'Great Break' separated the sheep from the goat by eliminating the options into two: to support the Socialist State by heart, or to be an enemy of socialistic ideals. The Luzin Affair, the debates on probability theory, and the case of Boris Hessen are examples of this period. In an interview in 1934, Stalin voiced his opinion on how the engineers should work. He pointed out that “the engineer, the organizer of production, does not work as he would like to, but as he is ordered. ... It must not be thought that the technical intelligentsia can play an independent role.” Stalinist period marks a paranoid horror created amongst scientists.
The development of science cannot be separated from these attitudes of these leaders.
4.2. Soviet Mathematics and its DevelopmentThe number of articles published on mathematics increased from 40 in 1917 to 302 in 1929. At the same time, the number of authors who published these articles increased from 24 to 136. Also, the number of articles on mathematics published abroad rose from 13 to 102. In 1967, about a quarter of all scientific publications in the realm of mathematics consisted of works from the USSR. And “by the early 1980s, the Soviet Union had 10 to 30 percent more scientists and engineers than the United States, depending on the definition of degrees and fields.”
There are several reasons for this remarkable development. Three of them will be mentioned below.
First, it must be admitted that, whatever the political repression and manipulation there were, financial support for academical works from the state never stopped in the Soviet Union. Even in the periods of economic crisis, Communist leaders gave emphasis on supporting science and technology.
Second is the so-called 'blackboard rule', meaning that the Soviet scientists could be expected to excel on those topics where world-rank work could be done with tools no more complicated than blackboards and chalk. Although Soviet science flourished in those areas where central governing is essential (such as space research), blackboard rule had considerable accuracy.
Thirdly, for a student in the Soviet Union who is talented in mathematics, it was relatively easy to work independently in mathematics rather than in engineering and in most of the natural sciences, because those subject were too close to social and political issues.
Hence, there were good reasons for the development of mathematical sciences in the Soviet Union.
To summarise, mathematics in the Soviet Union was a highly political activity, both in the sense that the academicians were considered as a part of the political composition and that mathematics itself was considered a political discipline which has strong ties with the material life.
But is this approach to mathematics; namely, to consider it as a political concept; exclusive for Soviet intellectuals, or is it a general rule? Are today's mathematicians are immune to the political conditions they live in? If it is true that the history of mathematical sciences is an organic part of the development of humanity, can we say that we have a comprehensive understanding of studies in mathematics that are presently being done in the universities?
This essay was aimed at showing that today, there is a need of a thorough analysis of mathematics, which covers not only the internal discussions in mathematics but also the political economical context these discussions are made.
 It must be noted that the society and the academies today are two different agencies that exclude each other, which in itself provides evidence to the professionalisation and specialisation mentioned.
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.296
 Ibid., p.118
 In the comparison between EB and BSE, I heavily rely on Chapter 5: The role of dialectical materialism: The authentic phase in Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.118-119
 Bol'shaia Sovetskaia Entsiklopediia, vol.38; Moscow, 1938; col. 359. Quoted from ibid. p.118
 Bol'shaia Sovetskaia Entsiklopediia, vol.26; Moscow, 1954; p.464. Quoted from ibid. p.118
 Ramsey, F.P.; ”Mathematics, Foundations of,” Encyclopaedia Britannica, vol.15; 1941; p.83. Quoted from ibid., p.119
 Whitehead, A.N.; ”Mathematics, Nature of,” Encyclopaedia Britannica, vol.15; 1941; p.87-88. Quoted from ibid., p.119
 Grattan-Guiness,I. and Cooke,Roger; ”Russia and the Soviet Union” in Companion Encyclopaedia of the History and Philosophy of the Mathematical Sciences; Routledge, London, 1994; pp.1477-1483
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.80
 Lapko,A.F. And Lyusternik,L.A.; From the history of Soviet mathematics; Russian Mathematical Surveys, 22(6):11-136; 1967. p.15
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.80
 Ibid., p.82
 Lapko,A.F. And Lyusternik,L.A.; From the history of Soviet mathematics; Russian Mathematical Surveys, 22(6):11-136; 1967. p.21
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.82
 Lapko,A.F. And Lyusternik,L.A.; From the history of Soviet mathematics; Russian Mathematical Surveys, 22(6):11-136; 1967. p.22
 For more information on this period: Loren R.Graham, Science and Philosophy in the Soviet Union; Knopf, New York, 1972
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.92
 Levin, Aleksey E.; Anatomy of a Public Campaign:”Academician Luzin's Case” in Soviet Political History; Slavic Review, Vol.49 No.1 (Spring 1990), pp90-103. p.91
 Demidov, Sergei S. and Ford, Charles E.; N.N.Luzin and the affair of the ”National Fascist Center” in Joseph W.Dauben, Menso Folkerts, Eberhard Knobloch, and Hans Wussing, editors; History of Mathematics. States of the Art. Flores quadrivii – Studies in Honor of Christoph J.Scriba. Academic Press, San Diego etc., 1996; pp137-148. p.138
 Levin, Aleksey E.; Anatomy of a Public Campaign:”Academician Luzin's Case” in Soviet Political History; Slavic Review, Vol.49 No.1 (Spring 1990), pp90-103. p.91
 Ibid., p.92
 Pravda, 6 August 1936, p.3 ; quoted from ibid. p.94-95
 Levin, Aleksey E.; Anatomy of a Public Campaign:”Academician Luzin's Case” in Soviet Political History; Slavic Review, Vol.49 No.1 (Spring 1990), pp90-103. p.95
 Ibid., p.101
 Pravda, 9 July 1936, p.3 quoted from ibid. p.93
 Levin, Aleksey E.; Anatomy of a Public Campaign:”Academician Luzin's Case” in Soviet Political History; Slavic Review, Vol.49 No.1 (Spring 1990), pp90-103. p.103
 For another such case, see: Demidov, Sergei S. and Ford, Charles E.; N.N.Luzin and the affair of the ”National Fascist Center” in Joseph W.Dauben, Menso Folkerts, Eberhard Knobloch, and Hans Wussing, editors; History of Mathematics. States of the Art. Flores quadrivii – Studies in Honor of Christoph J.Scriba. Academic Press, San Diego etc., 1996; pp137-148
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.99
 In the following, I heavily rely on: Seneta, Eugene; Mathematics, religion, and Marxism in the Soviet Union in the 1930s; Historia Mathematica 31 (2004) pp.337-367
 Seneta, Eugene; Mathematics, religion, and Marxism in the Soviet Union in the 1930s; Historia Mathematica 31 (2004) pp.337-367 ; p.352
 Ibid., p.352-353
 Orlov, M.; Matematyka i Religiia [Mathematics and Religion]; Proletar, Partvydav; section 1. quoted from ibid. p.351
 Ibid., section 11.
 Aleksandrov,A.D.; On the one hundredth anniversary of the birth of V.I.Lenin: Mathematics and dialectics; Siberian Mathematical Journal, 11(2), pp185-197, March 1970; p.189
 Ibid., p.189-190
 Ibid., p.186
 In the following, I heavily rely on Chapter 7: Soviet attitudes toward the social and historical study of science in Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.144
 Ibid., p.145
 Bukharin et.al.; Science at the Cross Roads; p.182; quoted from ibid. p.145
 It is interesting to note that E. Kolman was the major actor in all the cases mentioned in this essay. He is said to be the author of the unsigned articles against Luzin (cf. 3.1.1.). He was also part of the discussions on probability theory. (cf. 3.2.1.)
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.148
 Kol'man, E.: Vreditel'stvo v nauke: Bol'shevik, 2 (1931), p76 ; quoted from ibid. p.148
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.151
 Ibid., p.31
 Quoted from ibid. p.31
 Ibid., p.81
 Quoted from ibid., p.271
 Quoted from ibid., p.162
 Lapko,A.F. And Lyusternik,L.A.; From the history of Soviet mathematics; Russian Mathematical Surveys, 22(6):11-136; 1967. p.133
 Fifty Years of Soviet Mathematics; Russian Mathematical Surveys, editorial, p.2
 Graham, Loren R.; Science in Russia and the Soviet Union: A Short History; Cambridge University Press; London, 1993; p.261
 Ibid., p.207