Chapter Two of the projected *Itogi izuchenii matematicheskoi Rukopisi K. Marksa*

consists of the translation of Morell & Meo

Antonio Drago [Institute of Theoretical Physics, University of Naples]

*The Theoretical Implications of the Mathematical Manuscripts of Karl Marx*

During the last years of his life, in the period 1880 to his death in 1883 especially, Marx devoted himself to the study of mathematics. In the preface to the second [1885] edition of *Anti-Dühring *Engels described the manuscripts of these studies as of crucial importance, but they remained unpublished, resting unnoticed in the Institute of Social History in Amsterdam, until 1931 with the discovery of a photocopy taken and carefully hidden within the holdings of the Marx-Engels Institute in Moscow; a preliminary extract was published in 1933 in Russian, but it took until 1968 for the complete edition to appear.

*Miguel Cabron dige: The Soviet Union obtained copies of the mathematical manuscripts from Berlin in the early 1920s through the efforts of David Riazanov, director of the Marx-Engels Institute in Moscow. Unfortunately, Riazanov was purged by Stalin in the early 1930s and apparently became a non-person in the Soviet Union, for he is mentioned neither by Yanovskaya nor by Kolman.*

It was only in 1972 that an extract saw the light in Italian translation; finally in 1975 Dedalo editions has made available the longer manuscripts. Even so, the latter edition was one of the first to appear in a Western country, but it suffers, unfortunately, from a translation which leaves much to be desired: so much so that it often obscures or covers up the meaning of the text and makes the reading of it tiring. Even more important, moreover, this edition also repeats the order of the manuscripts reproduced in the Russian edition, an order which is not a chronological one and which could have been motivated either by the belief that the manuscripts being placed first were the ones written by Marx — and this is something yet to be debated or verified — or by the desire to place in the first rank the only manuscripts which open a door to an interpretation consistent with the official [Soviet] doctrine of dialectical materialism, since in a paranthetical note it talks about “negation of the negation”.

(Actually, this particular manuscript was dedicated to Engels for his birthday and one can therefore conclude that the parenthetical statement was meant as homage to the ideas of his friend more than as a statement of a research method used by Marx.)

It is a further misfortune upon the reader that the manuscripts thereby placed at the end are the ones which are the most discursive and most illuminating, the most enlightening ones.

**2. **Commentaries on the mathematical works of Karl Marx are few, and they would lead one to believe that Marx wrote notes and exercises as an autodidact, with perhaps an occasional thought put in. No one can reconstruct with any certainty Marx’s research program. Perhaps the best comment, even though a rather hasty one, was made by Lombardo Radice, who recognizes in Marx a precise intent within the framework of the philosophy of science. The others merely try to connect (without any real proof, it seems to me) the manuscripts to dialectical materialism, so that they present the more readily what Engels thought about mathematics than what was written by Marx.

Readers of the manuscripts, besides dealing with the difficulties arising from problems in the Italian translation and those inherent in works not meant for publication, also require a working knowledge of mathematics right up to the study of infinitesimal calculus, such as is required at the first-year university level: a substantial difficulty in Marx’s day but a level which at least one person in a hundred nowadays has attained by means of his education.

At first glance even a mathematical beginner would see that Marx is not just a student who engages in purely scholastic exercises, even to the point where he several times warns his readers about points glossed over in modern-day presentations — Marx was eminently justified in retaining his attitude of profound skepticism toward all kinds of abstract formulation. Nor are his ideas occasional or disjointed: Marx proposes nothing less than the reconstruction of the debate over the philosophic foundations of differential calculus and re resultant progressive unveiling of “the ‘mysteries’ of differential calculus”(Marx uses this expression as a demonstration of his excllusion of all metaphysical explanation). Finally, it is remarkable that Marx does not tackle these problems either by means of an abstract discussion or by the use of a discussion which is restricted to a particular formalized method. Rather, has statements desplay both a deep insight into the purely mathematical aspect of the difficulty and quite frequesnt reference to the historic, philosophic, and critical notations of previous mathematicians. In other words the fundamental premises of this study share little in common with those of Engels in *Anti-Dühring*, and even less with the *Dialectics of Nature*.

**3. **The central issue in the reconstruction of the contents of the manuscripts is the dating of the manuscripts. The notice to the reader os [the 1975 Dedalo edition under review] relates the conclusions of research done in Russia. The two manuscripts which are given first were in fact among the last written. All the others can be dated in general after 1880, whereas the one which appears to have been the first to be composed is the manuscript entitled “Taylor’s Theorem.”

In this manuscript Marx rejects, based on the consideration of Taylor’s theorem (a result of capital importance in differential calculus), the idealism of definitions and symbols not constructed on the basis of algebraic operations, the fundamental of which is Newton’s binomial theorem, which for Marx is of “revolutionary” significance.

The validity of this rejection is then demonstrated by reconstructing the historic debate which centered on the philosophic foundations of the differential calculus, a reconstruction which views the debate not only as a progressive advance in the precision and power of the instrument of calculus, but also as a progressive unveiling of the effective basis upon which it rests. Marx brings out a series of difficulties in the traditional interpretation of differential calculus and focuses on the operative processes which have to take place.

He therefore aligns himself in the same orientation as the work of Lagrange, which in fact attempted to reduce all of differential calculus back to algebraic calculus. Fron theis Marx derives a program for the reformulation of the philosophic foundations of differential calculus, a program which aims at going beyond the work of d’Alembert and of Lgrange himself. This manuscript indicates clearly that Marx had his own program of research which was based on the critical reformulation of what had beren done before him on the subject by the greatest mathematicians.

**4. **If we take this manuscript as a *point de départ *we can arrange the others intoa tentative chronological order:

*Miguel Cabron dige: For Anglophone readers the locations in Aronson-Meo are indicated in square brackets.*

*Mathematical Manuscripts of Karl Marx*

Theoretical Chronology of Composition

Title (pp. in Dedalo edition) [pp. in Aronson-Meo]

Taylor’s Theorem (153-160) [109-115]

More on Taylor’s Theorem (161-164) [116-119]

The Course of Historical

Development (35-44) [91-100]

First Outlines or Sketches (121-134) [76-91]

Continuation of Outlines (145-150) [101-106]

Contrast between the method of

d’Alembert and the algebraic method (171-174) [127-131]

Analysis of d’Alembert’s method (175-181) [132-139]

On the ambiguity of the terms ‘limit’

and ‘limit value’ (167-170) [123-126]

On the concept of the derived function (45-56) [3-14]

On the differential (57-75) [15-33]

Addenda to ‘On the differential’ (79-115) [35-68]

On emust acknowledge that Marx does not succeed in carrying his research program to completion: misled by Lagrange, he fails to evaluate properly the concept of limit, and this failure prevents him from transcending the very difficulty he identified at the very beginning — the difficulty of introducing purely symbolic relationships without immediately introducing the corresponding operatives.

But Marx’s research program, only now being considered by scholars after a lapse of almost a hundred years, is surprisingly close to that of the radical critics of the philosophical foundations of mathematics, a number of whom, independently of one another, has sought to reconstruct its fundamentals. Among others one recalls the mathematical-logical school of Intuitionism, which succeeded in producing a mathematical method and a theory independent of and incompatible with classical mathematics; the constructivists; and Harvard physicist Percy W. Bridgman, the founder of operationalism.

This indicates tha Marx anticipated by almost 20 years an alternative to traditional mathematics, an alternative which developed only after the turn of this century, when the Platonist foundations of traditional mathematics had become more and more evident. Marx antidipated this development in the sense that he correctly expressed a valid historical judgement on traditional mathematics and he aprticipated in the criticism and partly in the program of mathematicians who were searching for an alternative both before and after his time (those who succeeded Marx, incidentally, addressed themselves not to the calculus alone but to the very definition of number).

**5. **Motivation for Marx to study mathematics came not only from a purely intellectual desire and as a result of this program to criticize all aspects of bourgeois society, but also from the requirement to respond to the change in methodology of contemporary economists, who relied to a great extent on the differential calculus in developing their new marginalize economics He was anxious to improve his drafts of *Das Kapital*. A passage [recently translated into Italian] from the unpublished chapter six of book one of *Das Kapital* emphasizes the close connection between economics and mathematics; the argument there shows the the increase of the value of the independent variable was seen by classical economics as a model of the accretion of capital. This increase was seen by classical economists as the result of parthenogenesis; by Marx on the other hand as the process of exploitation of what classical economics systematically ignores, that is, workers’ labor. And since classical economics necessarily mystifies the real process of social life, so as well in the traditional philosophic interpretation of the differential calculus a mystical interpretation which conceals the true nature of the operations which it claims to explain. Therefor Marx aims to uncover these real operations. He wants to eliminate mysticism and to connect mathematics to reality.

In the study of mathematics Marx proceeds just as he did in economics. Based on the logical consequences of his recognition of the necessary existence of the proletariat, Marx reconstructs the historical debate and formulates his own research program: he critically extrapolates the scientific program of his chosen [scientific] predecessor — Ricardo in economics and Lagrange in the philosophic foundations of the differential calculus — until he succeeds in inverting their premises.

The above observations are congruent with the recent interpretation of Marx’s method advanced by A. Paci. While there is no doubt that the debate over Marx’s method will continue ling into the future, it is at least true that it will have from now on to consider the fact that Marx criticized two sciences with the same method, two sciences, significantly enough, a the extreme poles of bourgeois science: economics in the one of most practical application and mathematics in the most abstract.

Parenthetically, it is evident the the present-day polemic between “internalist” and “externalist” historians of science is transcended. Marx is an internalist insofar as he examines the history of the fundamental conceptions of the science, but alternatively he is an externalist as far as he bases his work on social factors and aims to bring even the most abstract symbols to social concreteness.

**6. ** As regards philosphy, the manuscripts which have so far been translated into Italian contain a parallel of impressive concision: Lagrange’s work places an algebraic foundation for Taylor’s Theorem

but it does not go at all back to the effective bases of Taylor, . . . and even less did he go further back, asking himself the question of why Newton’s binomial theorem . . . appears as a general operating formula . . . . Thus Fichte followed Kant, Schelling followed Fichte, Hegel followed Schelling, without Fichte nor Schelling nor Hegel having discussed the general basis of Kant, which is idealism; otherwise they could onto have continued developing it.

(pp. 163-164) [pp. 118-119]

It is notable that in the consideration of both historical processes, Marx places himself (or the proletariat) in the position of a refounder of a long historical debate developed atop the same undiscussed premises. Marx places no links between the two historical processes but considers them in parallel.

One therefor sees that philosophically as well, Marx’s research program in the study of differential calculus differs totally from that of Engels, for whom infinitesimals were the overcoming of mathematics founded on rigid concepts and the origin of ‘dialectical’ mathematics in which the concepts could not be made clearly precise of clearly defined.

*Miguel Cabron dige: In a recent (when the Drago article was translated, that is) assessment of nonstandard analysis, we find: “Engels was about 100 years behind the times on mathematics. For example, he said preposterous things about imaginary numbers. As far as he was concerned they existed as outright contradictions in mathematics itself. . . Again I want to say that Engels’ error and his joke about the square root of -1 are not very funny.” — Martin Davis, Courant Institute of New York University, writing in *Science and Nature: the Annual of Marxist Philosophy for Natural Scientists, No. 7/8, 1986, page 41.