Wednesday, 18 September 2013

Political Science: The lost culture of mathematics?

I have been thinking a lot about how political mathematicians should be. This had been at the back of my mind for a while but it came to the fore when I read Pindyck’s paper that motivated Insincerity in Climate Science. Today, mathematicians are supposed to be ‘pure scientists’ standing aloof of political discussion, but in the history of my own domain of mathematical probability there is a rich tradition of mathematicians being involved in political acts. This is intended to be the first of two posts, the second will consider the (negative) impact of the de-politicisation of mathematics.
Poincaré’s impact on mathematics was profound, and as influential as any of his contemporaries. But more than being an ‘ivory tower’ researcher, he was a teacher, a conscientious administrator, to the detriment of his own research, and took an active part in human affairs. For example, Mawhin reports an episode when the French papers linked the wet weather to the passage of a comet, which Poincaré debunked with “humour”.
However not all his activities were motivated by a sense of fun. In 1894, Alfred Dreyfus, a French army officer from a Jewish family in the Alsace, on the border with Germany, was arrested and then court-martialled for spying for the Germans. There was a public outcry that the trial had been a fraud and in 1899 there was a retrial. Part of the evidence against Dreyfus was supplied by Alphonse Bertillon, a biometrician who worked as a police officer and in the 1880s developed anthropometry, a technique for uniquely identifying people on the basis of eleven physical measurements, such as height, length of foot and length or ear and was the forerunner of fingerprinting. Sherlock Holmes is thought to have been born within a year of Bertillion and started his career as a consulting detective in 1881. Holmes refers to Bertillion in The Hound of the Baskervilles and in The Naval Treaty. Bertillion’s evidence at the second trial was emphatic
In the collection of observations and agreements that constitute my demonstration, there is no place for doubt; and this is with not only theoretical but material certitude, that with the feeling of responsibility following from such an absolute certitude, in all honesty, I affirm, today as in 1894, under oath, that the memorandum is the work of the defendant. [9]
Mawhin describes Poincaré’s involvement
Such a statement was philosophically unbearable to Poincaré. In a letter written at the request of Painlevé [another prominent French mathematician who was active in politics] and read to the court, he strongly reacted against the use of probability theory in Bertillon’s conclusions:

Nothing in it has any scientific character. I do not know if the defendant will be sentenced, but if he is, it must be on other evidence. It is impossible that such an argument makes any impression on free-minded people who have received a solid scientific education. [9]
Poincaré was first elected to the Académie des Sciences in 1887 becoming President in 1906 and on the basis of the breadth of his research he became the only person elected to every one of the five sections of the Académie, geometry, mechanics, physics, geography and navigation. In 1908 he was elected to the pantheon of French intellectuals, the Académie Française, following the footsteps of d’Alembert, Condorcet, Laplace and Fourier, being elected director just before his sudden death in 1912. Mawhin, a mathematician himself, notes that there has been no mathematician in the Académie Française since 1941, reflecting the fact that recently, mathematicians have not added any ‘style’ (perhaps élan is a better term) to their work [9].
Émile Borel was born in Montauban in 1871, his father a Protestant minister and his mother coming from a family of wealthy wool merchants. He was a brilliant student and came top of both the entrance examinations to the École Polytechnique and the École Normale Supérieure. His family were keen for Émile to go to the École Polytechnique as it was a better route into business, but Emile chose the École Normale, since it would lead to a career in science.
By the time Borel had completed his thesis in 1894 he had already published six papers on other topics, and was sent to the University of Lille for three years ‘apprenticeship’, returning to the École Normale in 1897 with a further twenty-two papers to his name. His rise was meteoric, becoming Deputy-Director of the École Normale in 1910. He has named after him at east ten mathematical notions, including the fundamental Borel sigma-algebra, and a crater on the moon.
Borel had become interested in probability around 1905, and in 1921 he was appointed to be the professor of Probability and Mathematical Physics at the University of Paris. Borel’s attitude to probability was noteable for such an outstanding mathematician. As Eberhard. Knobloch puts it
[Borel] took for the most part an opposed view [to other mathematicians at the time] because of his realistic attitude toward mathematics. He stressed the important and practical value of probability theory. He emphasised the applications to the different sociological, biological, physical, and mathematical sciences. He preferred to elucidate these applications instead of looking for an axiomatisation of probability theory. Its essential peculiarities were for him unpredictability, indeterminism, and discontinuity. [5]
Around 1921 he began to consider situations where winning a game did not depend solely on chance, but also on the skill of the players [6, p 33], such as the game of baccarrat which had been studied by Joseph Bertrand in 1899, and, unknown to Borel, Her that was solved in 1713 by the first Earl Waldegrave, an illegitimate grandson of James II/VII, while in exile in France using ‘mixed’ strategies over two hundred years before von Neumann-Morgenstern. Borel published a series of papers on the general theme of ‘Games that Involve Chance and the Skill of the Players’ in 1924.
Borel, driven by the ‘hunch’ that in exploring these problems, new mathematics would emerge [3, p 84], approached these problems first by considering a game of two players, both of whom could adopt three similar strategies, and working out whether the ‘best’ strategy existed, or if it did not exist, what was the best set of ‘mixed’ strategies. The main difference between Borel’s work and that of Waldegrave is described by the joint biographer of Norbert Weiner and John von Neumann, Steve Heims
The first step in a proper mathematical theory of games is to provide a suitable description of games in mathematical language. Such a description must contain all necessary information concerning any game but should contain no irrelevant information. Irrelevant information would impede insight into the mathematical problem. But one can devise such a suitable description only after one is very clear about the mathematical problems that one wishes to pose in relation to games. This was first done by ...Borel in 1921.[3, p 83]
Borel showed how the simple three strategy game could be extended to the case where there was a continuum, an infinite number, of strategies and commented that
The problems of probability and analysis that one might raise concerning the art of war or of economic and financial speculation are not without analogy to the problems concerning games, but they generally have a much higher degree of complexity [2, p 20]
Despite this clear precedent, von Neumann was clear in stating that when he was working on games, he was unaware of Borel’s efforts [6, p 45].
Borel had volunteered for military service in 1914 and initially commanded an artillery battery, but then worked in research for most of the war and was awarded the Croix de Guerre. The war had had a profound effect on Borel, half his students and his adopted son had been killed, and in 1920 he resigned from the Deputy Directorship of the École Normale and he became active in leftist politics, being elected to the Chamber of Deputies in 1924 and becoming the Minister of the Navy in 1925 under the mathematician Prime Minister Paul Painlevé’s government which lasted six months. (As a side note one of Borel’s political opponents was Raymond Poincaré, Henri’s cousin). During the Second World War, in his seventies, he was an active member of the French Resistance and wrote a number of books on the practical usefulness of probability. He died at the ripe old age of eighty-five.
In 1897 a young German mathematician. Felix Hausdorff wrote a paper Das Risico bei Zufallsspielen (‘The risk in random games’) that proposed measuring risk by the expected square of the shortfall, developing the expected shortfall proposed by Teitens. Hausdorff is perhaps the most tragic of the German mathematicians of his age. Born in 1868 in what was then Prussian Breslau, but is now Polish Wroclaw, into a family of wealthy Jewish textile merchants, he graduated from the University of Leipzig in 1891 with a doctorate in mathematics applied to astronomy. However, Hausdorff was more interested in Nietzsche’s modernist philosophy and contemporary literature than mathematics, publishing, the same year as his book on parlour games, Sant’Ilario. Gedanken aus der Landschaft Zarathustras (‘Sant’ Ilario: Thoughts from the Landscape of Zarathustra’), a collection of aphorisms related to the German philosopher, Friedrich Nietzsche’s work, ‘Thus spoke Zarathustra’.
Hausdorff’s involvement with Nietzschean philosophy was not political but cultural and had a profound effect on mathematics. At the time Nietzsche dominated the intellectual circles that Hausdorff was mixing with. At the same time as Poincaré was thinking about mathematical recurrence, Nietzsche discussed the philosophical concept of ‘eternal recurrence’ that featured in both Hindu and Ancient Egyptian religions, in Die fröhliche Wissenschaft (1882) (‘The Gay Science’). According to the historian of mathematics Moritz Epple, when Hausdorff tried to reconcile Nietzsche’s philosophy with Poincaré’s mathematics he was sucked into Cantor’s world of transfinite numbers and point-set topology. By 1904 Hausdorff stopped writing literature and in 1910 became a professor at the University of Bonn, publishing the Grundzüge der Mengenlehre (‘Foundations of Set Theory’) in 1914, which would become the standard text on the subject, introducing new concepts of space and measurement that would go on to twentieth century analysis.
Hausdorff, like all other Jewish mathematicians in Germany at the time, was expelled from the University of Bonn in 1934 and tried, unsuccessfully, to emigrate. The University tried to protect him, but in January 1941 he received orders that he would be deported to a concentration camp, and on 25 January 1942 he wrote to a friend
By the time you receive these lines, we three will have solved the problem in another way - in the way which you have continually attempted to dissuade us. ...What has been done against the Jews in recent months arouses well-founded anxiety that we will no longer be allowed to experience a bearable situation. ...Forgive us, that we still cause you trouble beyond death; I am convinced that you will do what you are able to do (and which perhaps is not very much). Forgive us also our desertion! We wish you and all our friends will experience better times.
and the following day, along with his wife and sister-in-law, committed suicide.[10]
Andrei Andreyevich Markov was born in 1856 in the provincial city of Ryazan, but in the 1860s his father became the estate manager for a Russian princess and the family moved to St. Petersburg. Andrei falls into the class of mathematicians who were sickly as children but showed remarkable aptitude for mathematics (His younger brother Vladimir was similarly gifted in mathematics, but died of tuberculosis when he was 25.), and in 1874 entered St Petersburg University to study mathematics and physics. He submitted his Masters thesis in 1880, which caught the attention of Chebyshev, Russia’s greatest mathematician of the nineteenth century, and then studied for his doctorate while teaching at the university.
Markov was working on problems in probability in the midst of the Russian Revolution of 1905 (Bloody Sunday and the revolt on the Battleship Potemkin) and Markov was an active supporter of the revolutionaries. He had been elected to the Russian Academy of Sciences, but the Tsar forced his removal and when the Romanov’s celebrated the tercentenary of their rule over Russia in 1913, Markov ostentatiously celebrated the bicentenary of the Law of Large Numbers. After the February Revolution of 1917, Markov requested to be sent to the provinces, where he taught for the next four years and when he returned to St Petersburg he was seriously ill, and died in 1922. [10, Markov]
Andrei Nikolaevich Kolmogorov was born at the end of April 1903 in Tambov, about half-way between the Crimea and the Kolmogorov family home near Yaroslav, on the Volga River some 150 miles north-west of Moscow. Andrei’s mother was described as ‘independent’, which might explain why this unmarried woman was giving birth some 300 miles from home as she travelled north from the Crimea. Maryia did not survive her child’s birth and the baby Andrei was raised by one of his mother’s sisters, Vera. Despite the bad start to his life, Andrei was bought up in a loving home, made comfortable by the fact that his grandfather was a local, though minor, noble. Not much is known about Kolmogorov’s father, other than after training as an agriculturalist and becoming involved in revolutionary politics, he had been exiled to Yaroslav were he met the Kologorovs, who were also involved in revolutionary activity. After the Russian Revolution he was appointed tot the Ministry of Agriculture and was killed fighting the White Russian general, Denikin in 1919. ([4], [13])
Vera and Andrei moved to Moscow in 1910 and Andrei went to a progressive private school, where his favourite subjects were biology and history. On account of the difficult situation in Moscow in the aftermath of the October Revolution of 1917, Kolmogorov left the city and worked on the construction of a railway between 1918 and 1920. He returned to Moscow and was admitted, without any examination, to the University of Moscow to study mathematics and physics. Kolmogorov did not restrict his studies to mathematics, but also took courses in metallurgy and history. His first research paper was in fact on landholding in late-medieval Novgorod, and Kolmogorov often told a story that his his teacher said to him “You have supplied one proof of your thesis, and in the mathematics that you study this would perhaps suffice, but we historians prefer to have at least ten proofs.”.[13]
In the midst of the momentous events of the Revolution and the restrictions of War Communism, Kolmogorov questioned the relevance of mathematics, but the attraction of mathematics would not let him go. In 1922, whilst still still only 19, Kolmogorov came up with an important technical result he became an international sensation. He became a postgraduate student in 1925, the same year as his first paper on probability, and received his doctorate in 1929, having published 18 papers on mathematics. For the summer of 1929, Kolmogorov and another young mathematician, Pavel Alexandrov, who had recently returned from post-graduate work at Princeton, travelled from Yaroslav to the Caucuses and then on to France and Germany. In 1931 Kolmogorov was back at Moscow, as a Professor of Mathematics. [4]
Whilst in the Caucuses, Kolmogorov started to think more seriously about probability, and seems to have discussed his ideas in France and Germany. His trip to western Europe must had clarified everything for Kolmogorov, because when he returned to Moscow he was able to put together the most important work in probability ever, his Grundbegriffe der Wahrscheinlichkeitsrechnung (‘Foundations of Probability’), which was published, by the German company Springer, in 1933.
The Grundbegriffe axiomatises probability theory firmly within mathematical analysis, synthesising Poincaré with the work of Borel and Hausdorf. Maistrov notes
The axiomatisation resulted in abstracting the notion of probability from its frequency interpretation, but at the same time made it possible to always pass over from a formal system to real-world processes [8, p 264]
In axiomatising probability Kolmogorov achieves two things, he frees probability from being tied to frequencies, a link first made by de Moivre and embedded by Laplace: a probability can be any measure of a set, concrete or abstract. This, on its own, would be useless, but Kolmogorov also ensures that the abstract mathematics is tied to the real-world. Poincaré had highlighted the importance of probability to science as the means for establishing inferences, in fact the importance of probability is much deeper, it links the physical world of events, to the formal, abstract, hyper—real world of mathematics. The importance of probability to science is much more fundamental than being a simple tool.
The purpose of the axiomatisation was to lay the foundations of probability, and the rest of the Grundbegriffe builds up a coherent, mathematical, theory of probability. One aspect was in clarifying the idea of conditional expectation, that the expectation is ‘conditional’ on what is known. This observation may not seem too revolutionary, but at the time it was. This is captured by the Bulletin of the American Mathematical Society review of the Grundbegriffe which appeared in 1934
Moreover, the theorem of Bayes, concerning whose validity there have been many controversies, is also an almost immediate consequence of the system of postulates, but the reviewer does not think this derivation of the theorem of Bayes settles the old contention relative to the validity of inferring the characteristics of a statistical population from a sample by means of the theorem of Bayes. [11]
In proving Bayes’s Theorem, by establishing the mathematical definition of conditional expectation, a proof that had rested on accepting that probability measures where abstract measure on sets and not defined in terms of relative frequencies, Kolmogorov made de Finetti’s subjectivist approach to probability, acceptable.
This, in Stalin’s Soviet Union, was not a politically neutral achievement. At exactly the same time as Kolmogorov was formulating the foundations of probability, 1929—1933, Stalin was, literally, executing his ‘Second Revolution’, the economic and social revolution that involved the extermination of the kulak, landowning peasants, and the forced collectivisation of the Soviet Union. At the time, the Marxist Dirk Struik, who was Dutch but in 1926 he was invited to both Moscow and MIT, he chose to move to the US to work with Norbert Weiner, was arguing that no subjectivist interpretation of probability was compatible with Marxist ideology, and that probability should be
a physical theory, and not a subjective theory, and a theory in which one investigates the relationship between causal and random events.[12, Quoting Struik, (1934) On the foundation of the theory of probability (in Russian).]
Dialectical materialism was, like logical positivism, opposed to metaphysical explanations of phenomena, effects had their causes that the materialistic scientist should discover. To get an idea of the politicisation of mathematics in the 1930s, Struik accused the logical-positivist Mises of taking a ‘metaphysical’ position in developing the frequentist approach to probability [12, Note 8 on page 351].
The risks Kolmogorov was taking were not abstract. In 1936 his PhD supervisor, Nikolai Luzin, became a victim of the second Revolution. Luzin’s mathematics had been critisiced in 1930 for being too abstract, and his PhD supervisor, Dmitri Egorov, was convicted for anti-Soviet activities because he opposed religious persecution. Luzin survived this purge but was criticised in 1936 for publishing abroad and of plagiarism in a campaign that was in part orchestrated by Pavel Aleksandrov under the guidance of Ernst Kolman. George Lorentz tells the story of this politically motivated attack
Famous mathematicians formed the interrogating commission at the Academy’s trial. Of these, Lyusternik, Shnirelman, and Gel’fond already belonged to the “initiating group” responsible for Egorov’s downfall. They were joined by Sobol’ev. Luzin’s former students were represented by Aleksandrov, Kolmogorov, and Khinchin. This revealed a split among Luzin’s students: Lavrentiev and P. S.Novikov were present, but did not say a word against Luzin, a sign of civil courage, while Menshov and Nina Bari (one of the best Soviet female mathematicians) were missing altogether. Actually, Kolmogorov said very little. [7]
Luzin was re-habillitated in 2012 but there is still some mystery as to why Aleksandrov, Luzin’s student, was a part of the group leading the prosecution. A widespread theory is that Aleksandrov was gay and the secret service had incriminating evidence on him, the deal was deliver a bigger fish and you can go free. There are views that Kolmogorov was drawn into the plot against his will because he had had a homosexual relationship with Aleksandrov. Or, Luzin was gay and corrupted Aleksandrov and possibly Kolmogorov; Kolmogorov famously and inexplicably slapped Luzin in public in 1946. Others argue that the homosexual context was invented after Stalin’s death to justify Luzin’s prosecution, which was motivated by Aleksandrov’s desire to advance in the bureaucracy.
At the end of the 1930s Kolmogorov turned his attention to biology, which was a courageous act in the Soviet Union, which was undergoing Stalin’s first purge that involved the disappearance of around 20 million [1, p 543]. At the time an agriculturalist, Trofim Lysenko, was advocating an idea, Lamarkism, that ‘acquired’ characteristics could be inherited, for example the muscles of a blacksmith would be inherited by their children (Darwin acknowledged this process as a possibility, which he called pangenesis, in Variation in Plants and Animals under Domestication). This was completely at odds with Medel’s mathematical genetics but fitted nicely with a Stalinist view that bad behaviour would disappear as people acquired socialist habits which would be inherited and reinforced. As a result, in the early 1930s Lysenko began to dominate Soviet biology, and by the end of the decade geneticists, such as Nikolai Vavilov, were being purged and sent to their death in prison-camps. In 1938 Kolmogorov, inspired by R. A. Fisher’s 1930 book The Genetical Theory of Natural Selection that integrated Mendelian probability with Darwinian biometrics, derived a differential equation that described how the ‘concentration’ of a species changed in time. In 1940 he went further and published a paper with the challenging title On a new confirmation of Mendel’s laws. ([13, p 899], [12, p 342])
During the war, like Wiener, Kolmogorov became involved in fire control and the best way to distribute anti-aircraft balloons around Moscow. However he worked on many other topics as well, including further work related to probability and analysis in general and the question of turbulence in fluids, an area in which his impact was as great as that in probability [4, Batchelor, p 47]. In the 1950s, Kolmogorov turned his attention to information theory, dynamical systems and complexity [4, Moffatt and Lorentz], including solving Hilbert’s 13th problem.
In celebrating the bicentenary of Newton’s death, Kolmogorov’s assessment, as the Russian science historian A. P. Youschkevitch writes, contrasted the “sound brightness of Newton’s mentality” with the “mathematical mysticism of Leibnitz and Euler”. In Kolmogorov’s words
Newton not only made fundamental discoveries by applying mathematics to the natural sciences ...Newton was also the first to conduct a unified mathematical study of all mechanical, physical, and astronomical phenomena. Speaking about the time of Newton, one may also discuss the subordination of isolated fragments of the natural sciences to mathematics. Of course, Leibniz’s ideas about the possibility of the mathematization of all human knowledge were even more universal. But, precisely because of their absolute generality and abstraction they proved to be fruitless.[14]
Having focused on the applications of mathematics in the 1950s, in the 1960s he laid the theoretical foundations of information theory, in the context of algorithms, developing Turing’s ideas. In doing this work, Kolmogorov highlighted the usefulness of some of Richard von Mises ideas in probability. [4], [13] In 1960 Kolmogorov was appointed to be the first Director of the Laboratory of Statistical Methods that introduced modern statistical techniques into Russia, in particular those developed by Jerzy Neyman in the 1930s. As well as applying statistical methods in the physical sciences Kolmogorov turned his attention to the ‘Applications of mathematical probability and statistics to poetics’, an area that he continued to publish in up to 1985 [4, p 40].
In 1963, as the country rebuilt itself following the death of Stalin in 1953, the Education Ministry opened four specialist schools for mathematics and physics in Leningrad (St Petersburg), Moscow, Kiev and Novosibirsk. Kolmogorov became so involved with the Moscow school that it became known as ‘Kolmogorov’s school’ and for fifteen years, until he was seventy-five, he not only gave lectures but introduced students to the arts and broader extra-curricular activities.
Kolmogorov was asked how youngsters should be introduced to science, he observed that ‘celebrated scientists’ had been nurtured by teachers, lecturers and doctoral supervisors, but in addition had been surrounded by supportive friends.
Now, when our country is in need of many capable and well-educated researchers in the most diverse branches of science and technology it becomes imperative to establish a wide system of institutional measures with extracurricular lessons with the senior school children: specialized schools, various types of non-school activities, wide familiarization of the young with the specific nature of work in the universities and technical colleges of the new technology, proper organization of entrance examinations, and wide involvement in research of students in colleges, where the teaching of future researchers is subsidiary only.[13, p 927]
Kolmogorov advocated a holistic approach to education, School children should be taught beyond the classroom, being exposed to the work of research intensive universities and applied technical colleges and students of the universities should mix with those in the technical colleges. In 1970, as part of this programme, he created a magazine, Quantum, on mathematics and physics for school-students.
Pedagogy dominated Kolmogorov’s work in the 1970s-1980s and alongside his mathematical legacy, Kolmogorov has left a more tangible legacy in the many Russian mathematicians who occupy academic posts across the world and were taught by a system he moulded.[13]
Kolmogorov died in October 1987 of a lung condition and having been suffering from Parkinson’s disease for a few years. He had had been highly decorated by the Soviet State and his obituary was signed by Russia’s leaders. While Kolmogorov was a truly great mathematician, he does not fit the stereotype of an academic living in an ivory tower manipulating symbols in an abstract game. He was, like Poincaré, a mathematician guided by his intuition [4, Hyland, p 63], rather than Hilbert’s formalism, and motivated by practical issues, from topics as diverse as biology, geology, fluid dynamics and poetry. He attacked the false statistics of Lysenko, risking his life, just as Poincaré had risked his reputation in attacking Bertillion. While he did not leave an explicit philosophy of science, he did leave a legacy in how science should be taught.
Outside of mathematics and eastern Europe, Kolmogorov’s name is not well known, certainly it is not as familiar as Newton, Leibnitz, Gauss, Riemann or Poincaré. Kolmogorov’s reputation has been partly hamstrung by the fact that he worked beyond the Iron Curtain, and The West was unlikely to lionise a Soviet scientist, and partly by the attitude of many prominent mathematicians to the Grundbegriffe, which challenged the status quo surrounding the basis of probability.


[1] A. Bullock. Hitler and Stalin: Parallel lives. Fontana, 1993.
[2] R. W. Dimand and M. A. Dimand. The early history of the theory of strategic games from Waldegrave to Borel. In E. R. Weintraub, editor, Toward a History of Game Theory, pages 15—28. Duke University Press, 1992.
[3] S. J. Heims. John von Neumann and Norbert Weiner: From Mathematicians to the Technologies of Life and Death. MIT Press, 1980.
[4] D. G. Kendall, G. K. Batchelor, N. H. Bingham, W. K. Hayman, J. M. E. Hyland, G. G. Lorentz, H. K. Moffatt, W. Parry, A. A. Razborov, C. A. Robinson, , and P. Whittle. Andrei Nikolaevich Kolmogorov (1903—1987). Bulletin of the London Mathematical Society, 22(1):31—100, 1990.
[5] E. Knobloch. Émile Borel as a probabilist. In L. Kruger, L. J. Daston, and M. Heidelberger, editors, The Probabilistic Revolution: Volume 1: Ideas in History. MIT Press, 1987.
[6] R. J. Leonard. Creating a context for game theory. In E. R. Weintraub, editor, Toward a History of Game Theory, pages 29—76. Duke University Press, 1992.
[7] G.G. Lorentz. Who discovered analytic sets? The Mathematical Intelligencer, 23(4):28—32, 2001.
[8] L. E. Maistrov. Probability Theory: A Historical Sketch, Translated by S. Kotz. Academic Press, 1974.
[9] J. Mawhin. Henri Poincaré. a life in the service of science. Notices of the American Mathematical Society, 52(9):1036—1044, 2005.
[10] J. O’Connor and E. F. Robertson. MacTutor History of Mathematics archive. University of St Andrews, 2010.
[11] H.L. Reitz. Review of Grundbegriffe der Wahrscheinlichkeitsrechnung. Bulletin of the American Mathematical Society, 40(7):522—523, 1934.
[12] E. Seneta. Mathematics, religion, and Marxism in the Soviet Union in the 1930s. Historia Mathematica, 31(3):337—367, 2004.
[13] A. N. Shiryaev. Kolmogorov: Life and Creative Activities. The Annals of Probability, 17(3):866—944, 1989.

[14] A. P. Youschkevitch. A. N. Kolmogorov: Historian and philosopher. Historia Mathematica, 10:383—395, 1983. 

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