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Mainstream, VOL LX No 36 New Delhi, August 27, 2022

Marx’s mathematical manuscripts - an overview | Sankar Ray

Friday 26 August 2022, by Sankar Ray

“Mathematics itself is always an empty calculus, pure syntax. To use it in social theory, we must give its terms meanings. We need semantics. We must ‘interpret’ the calculus”, wrote Roger Kopl, professor of finance at the Whitman School of Management of Syracuse. Marx scholar and polymath Pradip Baksi in his latest book, ‘Karl Marx and Mathematics, A Collection of Essays in Three Parts [1] refers to Bruno Bukhberger, professor of Computer Mathematics, Research Institute for Symbolic Computation, Johannes Kepler University: Mathematics is essentially a software. But is ‘mathematics purely abstract thought? If so, there seem to be an awful lot of drawing and visualisations involved. Graphs and co-graphs, number theory proofs with pattern blocks, and even simple proportions are used to find heights using shadows. All these are ways of seeing mathematics in action, and all heighten our mathematical intuitions’, states the editorial in The Journal of Humanistic Mathematics, penned by Mark Huber of Department of Mathematical Sciences, Claremont McKenna College and Gizem Karaali Department of Mathematics, Pomona College in California, USA.

Nearly three decades ago, Baksi’s seminal work in print, the English translation of ‘Karl Marx-Mathematical Manuscripts (he did a Bengali translation too) saw the light of the day and had a benignly seismic reaction among many Marx scholars but none in India had written about or reviewed it. “Social Scientist’, ‘The Marxist’ not even Economic Political Weekly carried any review article. Three years ago, its second edition (an Aakar Books title) was brought out. I read the book thrice but found it very difficult even to write an introductory text of it. I never thought of reviewing the book, nor am I venturing even now. I modestly endeavour to introduce it to readers. Now after the publication of Karl Marx and Mathematics, A Collection of Essays in Three Parts (1919). Mathematicians may read both books together. They could at least do so critically.

In the first edition, Baksi stated in the introduction, “While writing the Capital, Marx became specially interested in mathematics. In this connection, on 11 January 1858, he wrote to Engels: " In elaborating the principles of economics I have been so damnably held up by errors in calculation that in despair 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," quoted Baksi elsewhere.

Emil Julius Gumbel, the eminent German mathematician, was the first editor of MMM. Scanning 865 pages of MMM, he wrote a short and crisp preface in the mid-1920s. ‘They contain a free exposition of what Marx studied, combined with numerous chronological dates, and philosophical reflections on what he read’. Marx, Gumbel felt, ‘did not think of getting them published; some of these works were meant for Frederick Engels. Engels went through them and made his own additions. He corresponded with Samuel Moore’. However, David Borisovich Riazanov, arguably the greatest Marx scholar of the 20th century and the first director of the Institute of Marx and Engels (picked up by Lenin) commissioned Gumbel to decipher MMM in order to get it published in German. But Iosif Stalin’s emergence put a stop to the project. Gumbel left Moscow but Riazanov remained to be executed on false charges by Stalin. Albert Einstein was aware of MMM. Gumbel in a letter to Einstein on 30 April 1926 wrote: ‘’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.’ (machine translation from original in German).

Georg Wilhelm Friedrich Hegel too devoted himself to the study of mathematics in depth Unfortunately, Hegel’s mathematical notes are unavailable although both Marx and Engels believed – must have studied when the related text was available- that Hegel’s (‘my master’, as Marx always looked up to him until his last breath) grasp of mathematics was deeper than theirs... Unlike Hegel, Marx sought a philosophical explanation of the existing mathematics and mooted an innovative approach towards mathematics sans metaphysics, idealism, mysticism and obfuscation although both Marx and Engels viewed Hegel’s Mathematical perception was deeper. Cyril too noted that Marx’s ‘interest turned increasingly to the study of infinitesimal calculus, not just as a mathematical technique, but in relation to its philosophical basis’. But Hegel, alike Marx and Engels, was ‘deeply dissatisfied with the vagueness of the mathematicians about differentiation. Are the differentials dy, dx finite quantities, which can be divided into each other? Or are they zero?’. Nonetheless, dy or dx are not ’quanta’: ’a pan from their relation they are pure nullities’.

Engels wrote to F A Lange on 29 March 1865,” I cannot leave unmentioned a remark about old Hegel, to whom you deny a deeper mathematical scientific education. Hegel knew so much mathematics that none of his students were capable of editing the numerous mathematical manuscripts that he left. To my knowledge, the only man who understands enough mathematics and philosophy to do this is Marx" But Marx needed a new way of studying and doing mathematics, insulated from metaphysics, idealism, mysticism and obscurity. Without this, his project of critiquing political economy in entirety and the state of affairs therewith might not have been feasible. . Marx and Engels studied mathematics too in order to radically reconstruct the whole of human wisdom..

In the course of his work on Capital, Marx protractedly endeavoured to overcome his lack of knowledge in this field in order to apply algebraic methods to quantitative aspects of political economy. But, from 1863, his interest turned increasingly to the study of infinitesimal calculus, not just as a mathematical technique, but in relation to its philosophical basis. By 1881, he had prepared some material on this question, and this forms the greater part of this volume. It is clear that these manuscripts were not intended for publication, being aimed at the clarification of Engels and himself.

Nonetheless, Marx had been in quest of an innovative way to study mathematics, its existing reality – for him a necessity to scan the classical political economy in writing Capital-A Critique of Political Economy Marx and his friends were not contented with mere philosophical interpretations of the world and its sciences,since to them the significance of theoretic investigations were trans-philosophical in view of their objective of . They attempted a radical reconstruction of the entire structure of human knowledge. his interest turned increasingly to the study of infmitesimal calculus, not just as a mathematical technique, but in relation to its philosophical basis. By 1881, he had prepared some material on this question, and this forms the greater part of this volume. It is clear that these manuscripts were not intended for publication, being aimed at the clarification of Engels and himself

Marx found a relationship between mathematics and materialistic dialectics, which, thinks Brazilian mathematician Agamenon R. E. Oliveira, was for Marx’ an imperative to ‘philosophically generalise the results of mathematics to incorporate them into the conceptual framework of science ..and illuminate and point out solutions to the problems and difficulties of mathematics, in turn enriching the dialectical method itself. These results, Marx believed, could be used in the preparation of his magnum opus : Capital.’.

Ten years ago, Thomas Weston in a paper, ‘Marx on the Dialectics of Elliptical Motion’, published in Historical Materialism, wrote that at the instance of Engels, Marx explained how to calculate the tangent to a given curve through differential calculus and wrote that’ the calculus had originally arisen from the problem of drawing tangents to curves, including ellipses

Baksi devoted himself wholly to translating the text of MMM from Russian – a monumental work by Safya Aleksandrovna Yanovskaya ( who did not live to see this in print) - into English ( did it into Bengali too) to ensure that a lay reader such as this writer finds no scope to feel that it is translated. Even the 39-page note by Yavnovskaya, published by the Institute of Marxism-Leninism that helps one read MMM for clarity is a highly commendable work of translation. We understand Marx’s notes on differential calculus in the realm of political economy. Baksi has added a two-and-a half page note on ‘Constructivist mathematics that, he notes, ‘investigates the formal and quantitative relations of objective reality; but there are some differences that characterize them.’ At the International Congress of Mathematicians in Moscow in 1966, E Bishop made a presentation defending Constructivism’, noted Baksi.

Baksi studied MMM and MMM-related materials extensively and meticulously while translating the text alongside Marx’s letters. But the MMM, published so far, is not the whole of it, Baksi pointed out. Until MEGA2 1/28 and IV/30 are published, the full text will remain unknown ( Introduction in Baksi’s Karl Marx and Mathematics.. In the 1840s, 1850s, and 1860, Marx studied differential calculus, conic sections, trigonometry and commercial arithmetic etc in the 1860s and 1870s, putting marginal comments which are particularly interesting and debatable as well. In May 1865, Marx in a letter to Engels wrote of his studying of differential calculus.Around the same time he explained how to calculate the tangent to a given curve. Calculus, Marx wrote, had its origin in the problem of drawing tangents to curves, including ellipses. He elaborated this in the appendix to a letter to Engels (although the letter remains untraced) in 1866 –‘Problem of Tangent to the Parabola’ (pp 102-04). His source inter alia was Abbe Sauri’s ’Cours complet de mathematiques’.

Marx studied the history of differential calculus and termed Isaac Newton’s discovery of Binomial Theorem as ‘revolutionary’ and ‘greatest discovery of algebra proper’. Yet he was critical of both Newton and Gottfried Wilhelm Leibniz, inventors of the basic ideas of calculus in the 1660s, for their ‘mystical’ and ‘metaphysical’ approach. On Newton, he states “With the help of secret or evident metaphysical presuppositions, which in their turn led to metaphysical and non-mathematical consequences ; what happens is a forcible destruction of certain magnitudes, blocking the path of deduction”... (‘Second Draft on the differential Calculus) .. Newton wrote a paper on this in 1669 refused to publish it. He did the same thing on two additional papers on Calculus in 1671 and 1676 on calculus, written, but unpublished them in his lifetime independently invented calculus somewhere in the middle of the 1670s. He said that he conceived of the ideas in about 1674, and then published the ideas in 1684, ten years later. His paper on calculus was called “A New Method for Maxima and Minima, as Well Tangents, Which is not obstructed by Fractional or Irrational Quantities.” It was d’Alemburt, Marx noted, ‘tore off the shroud of mystery from the differential calculus and thereby took a great leap forward.

He studied numerous works on differential calculus such as ‘Differential Calculus’ of Leonhard Euler who attempted freeing of mathematics from the ‘treatment of differentials as actually infinitesimal magnitudes, , Colin Maclaurin’s and those of Sylvestre François Lacroix,Jean Louis Boucharlat , John Hind and Th.G Hall . He made critical comments on the Euler-Maclaurin theorem too.

Marx had a high opinion about Italian mathematician Joseph-Louis Lagrange who banked on algebraic foundation for initial equations in differential calculus. . Moreover, his great merit lay ‘not only in laying foundations of Taylor’s Theorem and differential calculus through purely algebraic analysis but in particular also in the introduction of the very concept of derived functions’ and hence logically-structured..

In May 1865, Marx wrote to Engels that he was studying differential calculus. In a letter to Engels in late 1865 or early 1866, Marx responded to Engels’s request for information on differential calculus by explaining how to calculate the tangent to a given curve. He wrote that the calculus had originally arisen from the problem of drawing tangents to curves, including ellipses.

In the course of his work on Capital, he continually strove to overcome his lack of knowledge in this field, so that he could apply algebraic methods to quantitative aspects of political economy. But, from 1863, his interest turned increasingly to the study of infinitesimal calculus, not just as a mathematical technique, but in relation to its philosophical basis. By 1881, he had prepared some material on this question, and this forms the greater part of this volume. It is clear that these manuscripts were not intended for publication, being aimed at the clarification of Engels and himself.

Marx mostly wrote on differential calculus, having studied it historically and critically. In ‘Notes and Extracts from John Heinrich Moritz Poppe’s Book on History of Mathematics and Mechanics’, jotted down by Marx, he noted Egyptians having ‘determined the height of pyramids in terms of the length of their shadows itself’ but pointed out the incompleteness of the method. Aside from the Greeks – ‘our teachers in mathematics’- flourishing of mathematics in Arab countries in the 7th century, Romans, Phoenicians, the Chinese, and the Tatars etc, he reminds us of Scottish John Napier who introduced logarithmic tables, geometry in and around 500 B.C, in Greece or Greek scholar Diophantus’s invention of Geometry, followed by Arabs in the 10th century and so on.

‘Karl Marx and Mathematics- A collection of Texts in Three Parts’, published a year after the second edition of MMM was published, was another outstanding work by Baksi, published by Aakar Books, following which its western edition was brought out by Routledge.It helps one understand MMM, let alone essays extreme obstacles faced by Riazanov in preparing the text of MMM. More importantly, it contains startling revelations. For instance, Yanovskaya made no mention of Gumbel and Ernst Kolman –both attached to the Marx- Engels Institute that undertook the project of publishing Mathematical Manuscripts. This book contains a paper, ‘Emil Julius Gumbel(1891-1966), the first editor of the mathematical manuscripts of Karl Marx by Annette Vogt, a senior research scholar at the Max-Planck-Institut für Wissenschaftsgeschichte and honorary professor, Humboldt-Universität zu Berlin translated from German by Baksi) . Kolman, the paper states (footnotes) presented a paper, Eine neue Grundlegung der Differentialrechnung durch Karl Marx (‘ A new foundation of differential calculus by Karl Marx’) at the International Mathematical Congress in Zurich in 1932.

This 229+xxx book helps us understand the MMM much more broadly while unspunning the little-known quasi-conspiratorial role of Stalin era in suppressing not only MMM but the grand endeavour of Riazanov and his team to publish Marx-Engels-Gesamtausgabe or complete works of Karl Marx and Friedrich Engels (MEGA) . But this writer is of the view that it is better discussed and reviewed separately. Baksi’s introduction and two notes on the MMM apart, Alan Alcouffe’s paper (translated from original in French), “The differential calculus, mathematicians and economists in the nineteenth century: K.Marx and H. Laurent, readers of J L Bouchariat helps one understand Marx’s perception of differential calculus while Annette Vogt’s essay on Gumbel as the first editor of MMM reveals the battle for publishing MMM against heavy odds during Stalin era. Two separate chapters, Investigation and Plural Mathematics – contain select papers that were translated by Baksi very lucidly and carefully. This is why this book ought to be introduced and reviewed exclusively.

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