By **DALETE FERNANDES***

*The German thinker and the history of his method and his mathematical writings can help in the popularization of mathematics and in the understanding of society*

Karl Marx and Friedrich Engels are well known for their philosophical and economic studies, but their research has branched out into different fields of investigation and among these fields is mathematics.[I]. For as demonstrated by Professor Wilson do Nascimento Barbosa[ii], the curiosity that leads to the search for knowledge will also lead to the search for new tools that make it possible to organize information. In this way, the use of mathematics, statistics, logic and calculation will also emerge as indispensable tools for Marx and Engels, helping in research and analysis.

Mathematics will also appear in Marx's biography as an intellectual respite, "especially in days of great spiritual or physical pain", exerting a "calming influence"[iii]. Marx will study and entertain himself with mathematics, making it an object of study and a hobby, taking this interest to Engels as well.

Marx shared his mathematical reflections with Engels, allowing him to follow every part of the development of his research. What made Engels feel responsible for collecting and publishing “the most important mathematical manuscripts left behind by Marx”?[iv]. In the preface to volume II of *The capital*, Engels writes: “After 1870 there was a new pause, determined mainly by illness. As usual, Marx filled this time studying […] original mathematical works form the contents of the numerous notebooks of the period”.[v]

However, only in 1933 was a partial version of Marx's mathematical manuscripts (MMM) presented in Moscow, and only in 1968 was an extended version published in German and Russian.

The mathematician and anthropologist Paulus Gerdes, who has his works on Marx's mathematical writings as a reference, believes that despite having developed many articles on Marx's relations with mathematics, the subject remains little known, as the publication is very recent.[vi]

In Brazil, economist Sylvio Massa de Campos[vii] published in 2006 some notes on Marx's mathematical manuscripts. Realizing the absence of studies on the subject, it aimed to arouse interest in mathematicians and economists. Physicist Fernando Bunchaft, on the other hand,[viii] who researched Marx's method of derivation, had part of his research published in 2013.

According to some researchers, mathematics “is thought moving within the scope of complete abstraction”,[ix] and Marx to “achieve a rigorous formulation, was willing to think abstractly as much as necessary”[X], so it is possible to understand the refuge and the search for rigor that mathematics provided to Marx.

Marx's studies took place in parallel with the development of calculus, since from the XNUMXth century onwards, mathematicians began to focus on the laws of variation, that is, on the functions[xi]. So Marx often worked with concepts that were still flourishing, therefore, had not been proved mathematically. In this way Marx began to seek greater rigor for the demonstrations and concepts he became aware of, also seeking to include part of the studies in his historical, social and economic analyses.

For Engels: “Hegel's mathematical knowledge was of such magnitude that none of his disciples was able to edit the numerous mathematical manuscripts found among his papers. As far as I know, the only man who knows enough mathematics and philosophy to do that is Marx.” [xii].

Karl Marx is considered a great mathematician not only by his friend Friedrich Engels[xiii], who claimed that even in mathematics, [Marx] made autonomous discoveries; but also by other intellectuals such as the mathematician Paulus Gerdes[xiv], which talks about the possibility of popularizing mathematics through the study of Marx's mathematical writings; and economist Andrea Ricci[xv], who considers Marx a forerunner of modern computational mathematics.

Some researchers who tried to analyze the MMM considered it superficial or even outdated; stating that Marx sought the general and abstract through simple examples. However, other researchers claim that the MMM are part of the revolutionary work[xvi] of Marx and the Marxist knowledge of eliminating uncertainties[xvii].

In the early 1850s, letters appear with information about some of Marx's mathematical reflections, but the first demonstrations that Marx had begun his mathematical research are from the late 1850s and early 1860s.

In a letter dated February 3, 1851[xviii], Marx reports to Engels the development of his theories and talks about crisis, imports of goods and possible balance sheets of the Bank of England. As early as January 11, 1858[xx] Marx writes that, in the elaboration of economic principles, he was so held back by errors in calculation that in despair he began to go through algebra again.* *

In a letter to Engels of July 6, 1863[xx] Marx writes, “In my spare time I study differential and integral calculus. By the way! I have many writings on this subject and I wanted to send them to you”.

Through the letters, some of Engels' reactions can be analyzed, which in addition to correcting some errors - as he demonstrated in a letter to Marx of May 30, 1864[xxx], in which he claims to be immersed in a book on arithmetic from which he thinks Marx would have distanced himself, because of regrettable numerical errors – he also praises the development of studies and Marx’s didactics, in a letter of August 18, 1881[xxiii]:

Yesterday I decided to study your mathematical manuscripts and even without the help of textbooks; I happily verified that I didn't need them, which I congratulate you on. […] When we say that in y = f(x), x and y are variables, it means that as long as it is not modified […] x and y remain, provisionally, constant. Only when they are actually modified, that is, within the function, do they effectively become variables. […] This story has obsessed me to the point that I think about it all day long.

Marx was interested in the foundation of calculus and found some of the concepts and foundations of the books he consulted on the subject to be unsatisfactory. So Marx sought to research and study the different definitions he found, seeking to develop his own views.

Engels, although he was unable to publish the MMM, despite having studied and analyzed parts of them, left in his writings several references to studies and discussions on the subject. In *Dialectic of Nature *states that mathematical methods, mainly, were established in what was essential and presents some of the mathematicians studied by them. In *Anti-Duhring*, already in the preface, Engels states that “a conception of history, at the same time dialectical and materialist, requires knowledge of mathematics and the natural sciences”[xxiii]*.*

Marx would have observed that there were different opinions about some basic mathematical principles and definitions, which caused a lot of confusion. There was no consensus on differential calculus – which is dedicated to the rates of change of magnitudes – with divergent interpretations occurring. Marx is fascinated by these differences and seeks a non-mystical definition of the problem.*.*

But after Engels' death in 1895, MMM went unmentioned for several decades. According to Ricci, after some of Marx's mathematical mistakes were pointed out, a view was popularized that Marx had little aptitude for mathematics. Historian Eric Hobsbawm[xxv] states that in this period the Marxist discussion took the form of speculation and increased the debate on the need to revise the theses of Marx and Engels.

Julius Gumbel[xxiv], one of the first mathematicians to work on the organization of MMM, writes, in a summary of the manuscripts published in 1927, that the mathematical research of Marx and Engels is evident through correspondence and exchange of information, bringing descriptions about the study of calculus in 1865 and the intensification of such research in the early 1880s.

Gumbel divides the contents of the manuscripts into arithmetic, geometry, algebra, differential calculus, Taylor's Theorem, drafts, notes, and independent mathematical works; making a brief description of each of these divisions.

After Gumbel, mathematician Sofya Yanovskaya[xxv] takes the lead in organizing Marx's mathematical material. In the preface to the 1968 edition he describes how the first publications were made.

In the first publication of 1933 part of the almost 1000 pages of photocopies of these manuscripts was published in Russian. In the second publication of 1968 it was possible to publish Marx's comments on some of his mathematical reflections and on the history of differential calculus. Summaries and notes on the sources used by Marx were also presented.

Yanovskaya still presents reports on the preparation of the material and talks about the enormous work required for organization. Because the papers needed to be deciphered and dated. The statements, summaries and notes made by Marx were separated and organized. There would still have been many pages that were not transferred to the photocopy version, kept in disorganized groups. The sources used and mentioned by Marx were verified, all of Marx's comments were organized and comparisons of the notes with the original sources were made. The work involved libraries from different countries and researchers from different areas.

According to the organizers, the main language used in these manuscripts was German, but pages in French and English were also found. Some languages and dates were difficult for researchers to identify. As part of the manuscripts were personal studies, there was no concern by Marx with dating and detailed descriptions.

For Yanovskaya, the main reason for Marx's mathematical studies was the deepening of economic analysis. However, he claims that these studies went beyond economic analysis, as Marx wrote several purely mathematical commentaries.

In your famous book *The capital, *Marx presents several mathematical references; compares price with imaginary mathematical quantities[xxviii] and presents mathematical examples when defining the added value* [xxviii]*.

in the presentation of *The capital*[xxix], historian Jacob Gorender emphasizes the maturation of Marx's intellectual trajectory through the development of his methodology. About the Russian translation of *The capital,* Gorender claims that despite the Tsarist censorship declaring the book undoubtedly socialist, it also considered the book inaccessible to the majority because of the mathematical form of scientific demonstration. [xxx], but it was released with notable sales success.

For Ricci some letters from Marx – such as the one of May 30, 1873[xxxii], in which he describes his reflections on dialectics and the movement of bodies, or variations, and from May 31, 1873[xxxi], in which he describes attempts to calculate financial price movements, discounts and other fluctuations to analyze crises – seem to anticipate the birth of econometrics, applying mathematical tools to analyze possible economic crises.

Paul Lafargue, who observed part of Marx's mathematical research, in his memoirs in homage to Marx, talks about the MMM: “Mathematics interested him. Algebra was for him like a moral restorative and served him as a refuge in the most difficult and painful moments of his agitated existence [...] mathematicians who compelled it. Indeed, its publication in the Complete Works of Marx is being considered. In the field of higher mathematics, he recovered dialectical activity in its greatest logical simplicity. He was of the opinion that a science could not really develop unless it allowed the study of mathematics”.[xxxii]

When researching the mathematical foundation, through his materialist and dialectical method, Marx found definitions that he considered mystical. This is also why he decides to delve deeper into mathematical studies. Marx noticed beauty in mathematical demonstrations and did not limit himself to accepting hypotheses that seemed insufficient to him; he sought to substantiate every application he made.

Marx's effort to demystify mathematics is part of the history of mathematics and the effort of many researchers; an effort to demonstrate hypotheses and prove concepts.

Marx's study also dialogued with interdisciplinarity and the use of mathematical tools to assist in the observation of society and economic phenomena. Through functions, calculation and statistics, attempts were made to measure economic phenomena such as crises. Bringing mathematics into economics and social analysis, with great originality, Marx introduces concepts of econometrics, thus being one of its precursors.

Marx and the history of his mathematical method and writings can therefore help in the popularization of mathematics and in the understanding of society. Because it brings as objective a research that seeks answers*s not mystical *through the critical study of mathematical concepts and ideas.

***Dalete Fernandes*** Mathematics and majoring in History at USP.*

**Notes**

[I] This article developed at USP under the guidance of Professor Lincoln Secco.

[ii] BARBOSA, Wilson do Nascimento. *The Importance of Statistics for the Historian*. São Paulo, 2016. Available at: Accessed on: 30/10/2021.

[iii] MEHRING, Franz. *Karl Marx: life and work.* Vol. II. Lisbon: Editora Presença, 1974; pg. 275.

[iv] Available in: Accessed on: 1877/2/31.

[v] Marx, Carl. *Capital: critique of political economy. *Vol. II. São Paulo: Abril Cultural, 1983; pg. 07.

[vi] Gerdes, Paulus. *Karl Marx's Philosophical-Mathematical Manuscripts on Differential Calculus: An Introduction*. Maputo/Mozambique: TLANU, Magazine of Mathematics Education, 2008; pg. 24.

[vii] Campos, Sylvio Massa de. *Notes on Karl Marx's "Mathematical Manuscripts"*. Rio de Janeiro: Editora Europa, 2006.

[viii] Freire Jr, Olival; Carneiro, Saulo (org.). *Science, philosophy and politics: a tribute to Fernando Bunchaft. *Salvador: EDUFBA, 2013.

[ix] Whitehead, Alfred North*. Science and the Modern World. *São Paulo: Paulus, 2006; pg. 37.

[X] MORISHIMA, Michio; CATEPHORES, George. *Value, Exploration and Growth. *Rio de Janeiro: Zahar, 1980; page 24.

[xi] ROQUE, Tatiana. *History of mathematics: a critical view, undoing myths and legends. *Rio de Janeiro: Zahar, 2012. Page 344.

[xii] MARX, Carl. ENGELS, Friedrich. *Letters on the Sciences of Nature and Mathematics*. Barcelona: Editorial Anagrama, 1975; pg. 36-37.

[xiii] ENGELS, Friederich. *Speech at the Tomb of Karl Marx.* Available: . Accessed: 1883/03/22.

[xiv] Gerdes, Paulus. *Karl Marx's Philosophical-Mathematical Manuscripts on Differential Calculus: An Introduction*. Maputo/Mozambique: TLANU, Magazine of Mathematics Education, 2008; pg. 101-102.

[xv] RICCI, Andrea. *The Mathematics of Marx. *Lett. Mat. Pristem 106, 20–25 (2018). Available: Accessed: 10.1007/10031/018 & *The Mathematics of Marx*. Lett Mat Int 6, 221–225 (2018). Available: Accessed: 10.1007/40329/018.

[xvi] Gerdes, Paulus. *Karl Marx's Philosophical-Mathematical Manuscripts on Differential Calculus: An Introduction*. Maputo/Mozambique: TLANU, Magazine of Mathematics Education, 2008; pg. 101.

[xvii] Campos, Sylvio Massa de. *Notes on Karl Marx's 'Mathematical Manuscripts'*. Rio de Janeiro: Editora Europa, 2006; pg. 43.

[xviii] MARX, Karl; ENGELS, Friedrich. *Letters on capital. *São Paulo: Popular Expression, 2020; pg. 71-76.

[xx] Gerdes, Paulus. *Karl Marx's Philosophical-Mathematical Manuscripts on Differential Calculus: An Introduction*. Maputo/Mozambique: TLANU, Magazine of Mathematics Education, 2008. Pág. 16.

[xx] MARX, Carl. ENGELS, Friedrich. *Letters on the Sciences of Nature and Mathematics*. Barcelona: Editorial Anagrama, 1975; pg. 28 & Gerdes, Paulus. *Karl Marx's Philosophical-Mathematical Manuscripts on Differential Calculus: An Introduction*. Maputo/Mozambique: TLANU, Magazine of Mathematics Education, 2008; pg. 17.

[xxx] MARX, Carl. ENGELS, Friedrich. *Letters on the Sciences of Nature and Mathematics*. Barcelona: Editorial Anagrama, 1975, p. 30-31.

[xxiii] Ibid.; ag. 97-99.

[xxiii] ENGELS, Friedrich. *Anti-Dühring.* Rio de Janeiro: Peace and Land, 1979; pg. 10/11.

[xxv] HOBSBAWN, Eric. *History of Marxism: Marxism in the Time of Marx. *Rio de Janeiro: Peace and Land, 1983; pg. 423-443.

[xxiv] MARX, Carl. *Mathematical Manuscripts*. Delhi: AAkar Books, 2018.

[xxv] Ibid.

[xxviii] MARX, Carl. *Capital: critique of political economy*. Volume I – Volume 1. São Paulo: Abril Cultural, 1983; pg. 92-93.

[xxviii] Ibid.; pg. 174.

[xxix] Ibid.; pg. VII-LXXII.

[xxx] Ibid.; pg. XXI.

[xxxii] MARX, Carl. ENGELS, Friedrich*. Letters on the Sciences of Nature and Mathematics*. Barcelona: Editorial Anagrama, 1975; pg. 78-80.

[xxxi] MARX, Karl; ENGELS, Friedrich. *Letters on capital. *São Paulo: Popular Expression, 2020; pg. 309-310.

[xxxii] LAFARGUE, Paul. Memories of the Intimate Life of Carlos Marx. September/1890. Available: .