Mathematical Manuscripts of Karl Marx – a historical overview
Sankar Ray
Emil Julius Gumbel in a letter to Albert Einstein on 30 April 1926 stated, ‘Ich war jetzt sechs Monate in Moskau und habe im Marx – Engels Institut die sogen. Mathematischen Manuskripte von Marx druckreif gemacht. Es handelt sich dabei um Notizen zur Differenzialrechnung, die ein gewisses philosophisches Interesse besitzen und zeigen, dass Marx die Anfangsgründe des Differenzierens wohl beherrscht hat. Meine Arbeitsbedingungen waren außerordentlich günstig. Allerdings lebt die Mehrzahl der dortigen Gelehrten in großer Notlage’. Roughly translated, it reads as  'I have been in Moscow for six months now and have been called the Marx  Engels Institute. Marx's mathematical manuscripts made printable. These are notes on differential calculus, which have a certain philosophical interest and show that Marx was well in command of the initial causes of differentiation. My working conditions were extremely favourable. However, the majority of scholars there live in great distress.' The letter (in part) is quoted in Oscar Sheynin’s Gumbel, Einstein and Russia, Moscow. Sputnik, 2003  http://www.sheynin.de/download/humb.pdf
Known for development of extreme value theory in the arena of Mathematical Probability (Gumbel Distribution of Statistical Extremes) having worked with legendary Statisticians, Leonard Tippett and Ronald Fisher, legendary pioneers of Statistical theory, Gumbel was the first editor of Marx's mathematical manuscripts It is evident from the abovementioned letter that Einstein was had a behindthescene role in sending Gumbel to the Marx Engels Institute, Moscow, for preparing the printcopy of MMM. Along with German biologist Julius Schaxel, he wrote to the chairperson of Soviet AllUnion Society for Cultural Relations with Foreign Countries (VOKS) Olga Davidovna Kameneva, on 2 July 1925 conveying Gumbel’s desire to work at the newly set up MEI whose director was David Borisovich Riazanov (Goldendach), arguably the best Marxscholar of the 20th century Riazanov replied on 17 July 1925: “ On my part, I agree to entrust Gumbel with the work on the manuscripts of Marx (concerning Mathematics) and to employ him generally for several months with other scientific work at our institute.” Kameneva was one of younger sisters of Leon Trotsky and wife of Lev Kamenev, a polit bureau member of Bolshevik Party – Russian Social Democratic Labour Party (Bolshevik).
It was Riazanov who discovered the MMM at the archives of the Social Democratic Party of Germany (SPD) before the beginning of World War I. He was toying with the idea of publishing it but it didn’t click at that time. The obstacle was overcome after the MEI was born. Pradip Baksi briefly narrated this in his seminal work, ‘Karl Marx and Mathematics’, Aakar Books, Delhi,2019).[ Prior to this, Baksi published ‘Karl Marx Mathematical Manuscripts in 1994 ( revised edition, brought out by the same publisher in 2018, bicentenary of Marx] . Riazanov noticed that a part of MMM was missing from the SPD archives and ‘located them at the residence of an important leader of SDP, Edward Bernstein’ whom is branded by the official Marxist parties as an enemy as the chief revisionist. Riazanov turned to Austrian social democrat Frederick Adler requesting to publish the MMM bur in vain. Things changed after MEI was set up with Riazanov at the helm (Baksi: A Note on the history of collecting, deciphering, editing MMM,1994’, Karl Marx and Mathematics, p 10). MEI had taken photographs of all mathematical manuscripts which were stored there when Gumbel arrived in Moscow (winter 1925).
However, at the instance of Riazanov, Gumbel prepared an 865page manuscript for publication by 1927. He classified it into four categories: A. Calculations; B. Extracts; C. Drafts; D. Independent works, but ‘they are deprived of any philosophical thought which could belong to Marx’, wrote Gumbel Marx could have just copied the original texts as they stood in the sources but sometimes Marx did not follow the development to be found in the source, and eventually many extracts are compound of short remarks the sources of which are impossible to discover’, he added .
In 1927 Gumbel published a summary of the MMM in the Soviet magazine Letopisi Marxisma (“Marxist chronicles”), which was the first public advance presentation of the contents of the Marxian manuscripts.4 However, the arrest in 1931 of Riazanov, who fell victim to the Stalinist purges and was then shot in 1938, blocked the project already ready for publication.
Stalinist milieu forced Gumbel to go back to the University of Heidelberg and resume working as a professor of statistics at. The work on the MMM was entrusted to a team of Soviet mathematicians coordinated by Sofia Yanovskaya of the University of Moscow. One of the members of the working group was Russian mathematician Kolman. Both Yanovskaya and Kolman owed allegiance to Stalin and resorted to belittle Gumbel academically. But that’s a different topic. Kolman wrote a synoptic discourse of MMM in 1931. The expanded (a little more than one hundred pages than Gumbel’s, after more pages were found out) version was brought out during the 150th birth anniversary of Marx, edited by Yanovskaya (although she passed away in 1966). But interest in Marx's mathematical reflections began to cross the borders of the Soviet bloc. However, partial translations were published in English, French, Italian, Spanish, Portuguese, Chinese, Bengali, and Japanese. The Bengali translation was by Baksi, published in 1994.
Once it was mistakenly thought that the 1968 edition was MMM in entirety. In 1987, with the blossoming of perestroika, the publication of the complete and definitive edition of the MMM was announced. But the disintegration of the USSR in 1991 set many things at naught. However, the scheme resurfaced with the birth of the new historicalcritical edition of the works of Marx and Engels (the second MarxEngelsGesamtausgabe, or MEGA2). The publication of the complete version of MMM was again announced  the two part volumes of MEGA2: MEGA I/28 in two parts and MEGA IV/30.
Marx’s interest in mathematics was necessitated in writing of his magnum opus, Das Kapital. He wrote to Frederick Engels on 11 January 1858: “In elaborating the principles of economics I have been so damnably held up by errors in calculation that in despair I have applied myself to a rapid revision of algebra. I have never felt at home with arithmetic. But by making a detour via algebra, I shall quickly get back into the way of things.” A few years later he wrote his lifelong friend again on 6 July 1863, “My spare time is now devoted to differential and integral calculus”.
In Das Kapital, Marx explained probabilistic tendencies (statistical averages) in economic laws, evident in the interaction of innumerable individual behaviours adopted by a myriad of subjects. This is due to Belgian Adolphe Quetelet, one of the greatest statisticians of the 19th century, the first to apply quantitative methods to the study of social and human phenomena, and cited several times by Marx. On the level of the conceptual foundations of pure mathematics, by contrast, MMM is mostly devoted to differential calculus, evident in his correspondence of last years of his life, even during the last months of his life.
Marx studied differential calculus in an original way, reflected in his extensive notes. He had learnt mathematics logically, conceptually and critically and this helped him write the most comprehensive critique of political economy which he defined as a ‘bourgeois science’. His investigative method b was of a historicalgenetic type, starting off with critical analysis of the theoretical development of differential calculus. It helped him identify internal logical contradictions and the attempts to resolve them. Marx identified three different methods in the development of the theory of differential calculus. The first was the “mystical” method, developed independently by Isaac Newton who was more interested in mechanical and dynamical aspects while Gottfried Wilhelm (von) Leibniz’s was more rigorous in the definition of formal and symbolical aspects. From the beginning the mathematical procedure, adopted by the two envisioned the use of infinitely small quantities, defined in the form of differentials (dy and dx) by Leibniz. During the calculations, these quantities were sometimes considered as positive numbers and at other times arbitrarily suppressed and treated as numerical zeros, thus contravening elementary algebraic rules. Because of this strange behavior, differentials appeared to orthodox mathematicians of the time as mysterious metaphysical entities. The amazing thing was that, through what Marx calls “jugglery”, correct results were obtained “by a positively incorrect mathematical procedure”, in which the solution known from the beginning was the hidden assumption of the whole argument.
The second method is the “rational” one formulated by D'Alembert and Euler replacing the infinitely small quantities with finite increments, defined with the symbols Δy and Δx, to which the ordinary algebraic rules are applied, since they are ordinary numbers. Only at the end of the procedure, by setting as equal to zero the increments in the difference quotient Δy/Δx, are the differentials dy and dx introduced. They arrived at the same inference as Newton and Leibniz but sans “jugglery” also free from formal errors. But this method lacks logical consistency. Indeed, differentials behave differently within the same equation because in the righthand side they disappear as numerical zeros, while in the lefthand side they remain in the form of a ratio dy/dx, which replaces the expression 0/0. The ratio 0/0 is indeterminate ( any number, multiplied by zero being zero) and the value of the derivative represents an arbitrary imposition, justified only by the need to obtain the correct derivation result, already known at the outset. While representing an undeniable formal step forward, the rational method did not solve the contradictions of the differential calculus, whose conceptual foundations remained shrouded in mystery, susceptible of being understood only intuitively through a deliberate act of thought and not by means of a logically coherent development.
The third method was ‘Purely algenraic differential calculus’ of Joseph LouisLagrange in his ‘Theory of analytical functions’ (1797 and 1813)’. Marx wrote in MMM, “d’alembert simply algebraicised (x + dx ) into (x +h). Lagrange imparted a purely algebraic character to the entire expression,having counterposed to it, as a general unexpanded expression(ital in original), the expamded series, which must be deduced from it’ (Baksi: MMM p 69)
“In every single field, wherever Marx has conducted investigations, even in mathematics, he has obtained independent results.” MMM and writing of Marx’s magnum opus were interrelated. Vasily Ivanovich Przhesmitsky, of the Department of Theoretical Mechanics, of the Moscow State University of Civil Engineering in a paper, 'On the operational logical apparatus working in Karl Marx’s Capital and Mathematical Manuscripts', published in: Voprosy dialekticheskoi logiki: Printsipy i formy myshleniya (materialy postoyanno deistvuyuschevo simpoziuma po dialekticheskoi logike) [Problems of dialectical logic: principles and forms of thought (materials of the permanently functioning symposium on dialectical logic)]. AN SSSR. Institut Filosofii [Academy of the Sciences of USSR, Institute of Philosophy] M., 1985. s. 7081.
‘Baksi deserves thanks for lucid translation of Przhesmitsky’s and other relevant papers by Valery Ivanovich Glivenko and Vladimit Nikolayevich relating to MMM from Russian into English. He asserts that “the logic of ‘Capital’ is distinct from ordinary logic’ proceeding from the fact ‘concrete truths happen to be both nonantinomical and antinomial’. And there lies the distinctive logic of ‘Capital’ a common experience we have in reading ‘Capital’. This distinctiveness lends support to David McLellan’s statement that Marx was the last and best of classical economics (David McLellan, ‘Marx’ 1977).
Baksi translated all the five papers at the AllUnion Symposium on “The Regularities and Modern Tendencies of the Development of Mathematics”, held in September 1985, at Obninsk, comprising the third chapter, ‘Plural Mathematics’ in ‘Karl Marx and Mathematics’. Looking like written originally, the two books, specially MMM, are unlikely to be reviewed anywhere as there is a deplorable scarcity of competent scholars who know both Marx and mathematics (including ethnomathematics and history of mathematics) well. The first edition of MMM , the first English translation of MMM’s 1968 Russian edition was published in 1994 remains unreviewed (revised second edition, published by Aakar Books in 2018)., not even in the Economic and Political Weekly and Social Scientist, although it’s mentioned by western scholars like Alain Alcouffe , Julian Wells and Andrea Ricci. True, reviewing the MM is a very tough proposition, but not something insurmountable. Dr Sudeb Mitra who is a faculty in mathematics City University of New York and a Marx scholar in a mail to Prof Paresh Chattopadhyay, India’s bestknown Marx scholar, expressed his pleasant surprise to find Marx's command over Logarithms “It's truly remarkable how versatile and exceptionally erudite this man was  he cites the paper ‘Meditatio juridicomathematica de interusurio simplice" by Leibnitz! He writes, ‘The theory of the calculation of interest owes its first improvements to the GREAT LEIBNITZ . . . " But Mitra also noted Marx’s gaps in mathematical knowledge. “I guess Marx did not have a chance to see Cauchy's works  so, his approach to calculus missed the rigorous foundation that Cauchy gave”.
Be it noted, Marx is a forerunner of mathematical economics. At the 50th annual general meeting in 1938, Wassily Leontief , later a Nobel laureate in economic sciences, in a paper , The Significance of Marxian Economics for Presentday Economic Theory (published in the American Economic Journal)
“toward the end of his life Marx actually anticipated the statistical, mathematical approach to the business cycle analysis…. The significance of Marx for modern economic theory is that of an inexhaustible source of direct observation. Much of the presentday theorising is purely derivative, secondhand theorising. We often theorise not about business enterprises, wages, or business cycles but about other people's theories of profits, other people's theories of wages, and other people's theories of business cycles.”
There is no denying that Marx had a distinctive mathematical talent, reflected in the MMM although official Marxist ideologues remained mysteriously apathetic towards it.
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Jul 15, 2020
Sankar Ray sankar.2010@hotmail.com
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