Marx's Mathematical Manuscripts 1881

Written: August, 1881;

Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;

First published: in Russian translation, in *Pod znamenem marksizma*, 1933.

The creation of the scientific theory of the revolutionary struggle of the international proletariat to overthrow the capitalist system and to construct socialism made it necessary, as Marx himself indicated, to examine social conditions from the point of view of materialism and dialectics. These must be deduced from the entire complex of real phenomena and verified by the manifold totality, both of the facts of history and of the reality of nature, society and human thought. Thus, one of the necessary prerequisites for the creation of scientific communism was the mastery of the sciences which study the governing laws of the development of nature, the mastery of their results and methods. At the same time the study of the natural sciences, and mathematics as well, from the point of view of their history and interaction with the economic development of society, was necessary for the practical activity of the proletariat as a class coming to power in order consciously to transform society.

With respect to mathematics, dialectical materialism had to solve two closely interrelated problems. On the one hand, it was necessary to generalize the results of mathematics philosophically, and to incorporate them in the scientific world view, the world view of dialectical materialism. On the other hand, the method of materialist dialectics should be used to illuminate the crucial problems of mathematics, thereby enriching the dialectic method. In large measure this work (?)tell to the share of F[riedrich] Engels, since Marx was almost completely occupied with the validation of the laws of the economic development of capitalism and with the practical guidance of the international workers movement. In spite of this Marx persistently kept track of the progress of natural sciences and the technical achievements of his times, and for almost thirty years, from the late [18]50s right up to his death, was occupied with mathematics a great deal.

These studies were reflected in a number of observations scattered throughout the works of Marx, both on the influence of mathematics on philosophy and on the philosophic elucidation of specific problems of mathematics. In addition, they were expressed in his wide-ranging correspondence, particularly with Engels. Then they were used by Marx in the preparation of his most important work, *Capital*. Finally, the results of his studies were preserved in the extensive manuscripts left behind on Marx’s death. These papers were devoted to various problems of mathematics and its history, primarily the problem of the logical and philosophic basis of the differential calculus.

Marx had two motives for his mathematical studies: political economy and philosophy.

Although Marx repeatedly emphasized the specific nature and extraordinary complexity of economic phenomena and the impossibility of comparing them to biological, still less physical, phenomena, nonetheless he considered the application of mathematics not only possible but indeed necessary for the investigation of the general laws of economics. In *Capital* Marx employed a mathematical form in writing down economics laws, by no means solely for illustration. The analysis of the form of value and money, the composition of capital, the rate of surplus value, the rate of profit, the process of transformation of capital, its circulation and turnover, its reproduction, its accumulation, loan capital and credit, differential rents: - Marx accomplished all of this by employing mathematics. Proceeding by means of the simplest algebraic transformations from one formula to another, he next analyzed them, interpreted them economically, and formulated new laws. By just such means, for example, Marx derived the dependence of the rate of profit

P = M/(C +V)

(where C is constant capital, V is variable capital, and M is surplus value)^{*2} on the organic composition of capital

O = C/V

so that

P = A/(1 + O)

where A = M/V is the rate of production of surplus value), and established the law of the tendency of the average rate of profit to fall. (?)By the very same means he established the inter-relation between the two sectors of capitalist reproduction: the first sector is the production (?)of the means of production:

C_{1} + V_{1} + M_{1} = T_{1}

where T_{1} is the total value of the producers’ goods sector), and the second sector is the production of the means of consumption,

C_{2} + V_{2} + M_{2} = T_{2} ,

so that, for simple reproduction,^{*3}

C_{2} = V_{1} - M_{1} .

(?)He discovered thereby the general law of the formation of the costs of production and the economic ‘mechanism’ inevitably leading, under conditions of premonopolistic capitalism, to strongly periodic economic crises.^{†}

The still unpublished preparatory works to the third volume of *Capital* contain Marx’s detailed calculations of the quantity (A⋅O)/(1 + O), the difference of the rate of surplus value A and the rate of profit P, where Marx represented its variations in the form of a variety of curves. Since the third volume of *Capital*, which is devoted to the process of capitalist production taken as a whole, is a synthesis of the first volume - the immediate process of the production of capital - and the second volume - the process of transformation of capital - Marx tried in his rough drafts to supplement the complete and comprehensive qualitative picture provided in his previous work with a quantitative picture.

Marx did not bring this work, which even in the case of simple reproduction demands rather complicated, although elementary, computations, to completion. The work, however, correctly posed the problem of the distribution of surplus value (in the costs of production) under conditions of large-scale reproduction in both sectors in order to obtain maximum profits and also derived the law of periodic crises. These are problems which can only be solved by means of contemporary methods of linear programming. The mechanism of economic crises, however, can also be studied empirically, a method concerning which Marx wrote to Engels on May 31, 1873:

‘I have just sent Moore a history which *privatim* had to be smuggled in. But he thinks that the question is unsolvable or at least *pro tempore* unsolvable in view of the many parts in which facts are still to be discovered relating to this question. The matter is as follows: you know tables in which prices, calculated by percent etc. etc. are represented in their growth in the course of a year etc. showing the increases, and decreases by zig-zag lines. I have repeatedly attempted, for the analysis of crises, to compute these “ups and downs” as fictional curves, and I thought (and even now I still think this possible with sufficient empirical material) to infer mathematically from this an important law of crises. Moore, as I already said, considers the problem rather impractical, and I have decided for the time being to give it up.’^{*4}

The mathematician Samuel Moore, who was Marx’s adviser in mathematics, was unfortunately not sufficiently well versed; he was obviously unacquainted with Fourier analysis, that branch of applied mathematics which deals with the detection of latent periodicities in complex oscillatory processes, the foundations of which were already contained in J. Fourier’s 1822 work, *Analytic Theory of Heat*.

Since Marx believed, according to Paul Lafargue,^{†2} that ‘a science is not really developed until it has learned to make use of mathematics’, he advanced the thesis of the possibility, indeed the necessity, of the application of the mathematical method to research in the social sciences, in political economy in particular. At the same time this did not mean the replacement of political economy and its general laws and methods by mathematics along the lines of the so-called ‘mathematical school’ of vulgar political economy, headed in England by W. Jevons and in Italy by V. Pareto and others, which had sprung up in the [18]80s in opposition to the bankrupt ‘historical school’ but which, like the latter, also argued for a ‘harmony of interests’ of all (?)classes of capitalist society. Marx made the following observation, in a (?)letter to Engels on March 6, 1868, regarding one of the representatives of this school, Macleod: ‘... a puffed-up ass, who 1) puts every banal (?)tautology into algebraic form and 2) represents it geometrically.’^{*5}

Thus, according to Marx, as in any other specialized science so in political economy, mathematics can be a powerful tool for research only within the limits of the validity of the theory of that specialized science. Therefore, as his acquaintance the Russian jurist and (?)publicist M.M. Kovalevskii wrote,^{†3} Marx devoted himself to the study of mathematics in order to obtain the ability to apply the mathematical method as well as to examine profoundly the works of the mathematical school.

Marx’s considered judgment on one of the most important problems of the foundations of geometry, which he expressed in ‘The Theory of Surplus Value’, the unfinished 4^{th} volume of *Capital*, in connection with a polemic with [Samuel] Bailey, who had incorrectly used the geometric analogy, may serve as an example of his philosophical conclusion on the questions of mathematics. Marx wrote:

‘If a thing is distant from another, the distance is in fact a relation between the one thing and the other; but at the same time this distance is something different from this relation between the two things. It is a dimension of space, it is a certain length which may as well express the distance of two other things besides those compared. But this is not all. When we speak of the distance as a relation between two things, we presuppose something “intrinsic”, some “property” of the things themselves, which enables them to be distant from each other. What is the distance between the syllable A and the table? The question would be nonsensical. In speaking of the distance of two things, we speak of the difference in space. Thus we suppose both of them to be contained in space, to be points of space. Thus we equalize them as being both existences of space, and only after having them equalized *sub specie spatii* we distinguish them as different points of space. To belong to space is their unity.’^{‡}

Here Marx, while analyzing the process of abstraction by means of which the geometric concept of ‘distance’ or ‘length’ originates, focuses attention not only on the materialistic origin of this concept, the basis of which lies in the ‘characteristic’ of two comparable objects, but also on its relative character, on its indissoluble connection with space, understood as a material, really existing entity. And all this was in 1861-1863, during the unbroken predominance in science of the Newtonian world view, some forty years before the appearance of the theory of relativity, in which Einstein boldly took to its logical conclusion the idea that ‘length’ is not simply a superficial abstract measure of a physical body but an integral characteristic of the spatial relationship of two bodies.

Marx’s statement on the statistical nature of economic mechanisms as mechanisms of large-scale processes has an exceptionally great methodological significance for mathematical statistics. These mechanisms express the interactions of individual processes in the laws of probability; they dominate over any variations from the mean Marx repeatedly returned to this problem. For example, in the *Grundrisse* of 1857-1858 he wrote, in the chapter on money:

‘The value of commodities as determined by labor time is only their *average value*. This average appears as an external abstraction if it is calculated out as an average figure of an epoch, e.g. a pound of coffee is one shilling if the average price of coffee is taken over, let us say, 25 years; but it is very real if it is at the same time recognized as the driving force and the moving principle of the oscillations which commodity prices run through during a given epoch. This reality is not merely of theoretical importance: it forms the basis of mercantile speculation, whose calculus of probabilities depends both on the median price averages which figure as the center of oscillation, and on the average peaks and average troughs of oscillation above or below this center.^{*6}

Despite the misconception, current for a long time among the majority of Marxists working in the field of economic statistics, that Marx’s statements on stochastic processes apply only to capitalist economics, a misconception based on the non-dialectical representation of the accidental and the necessary as two mutually (?)exclusive antitheses, these statements of Marx - to be sure, in a new interpretation - have enormous significance for a planned socialist economy, in which, since it is a commodity economy, the law of large numbers never ceases to operate.

Hegel’s *Science of Logic*, especially the second section to the first book, ‘Quantity’, was undoubtedly a philosophical stimulus for Marx’s mathematical studies. The article ‘ Hegel and Mathematics’, written by the present author together with S.A. Yanovskaya,^{*7} cites (?) in this connection the following words of Engels:

‘I cannot fail to comment on your remarks on the subject of Old Man Hegel, to whom you do not attribute a profound mathematical and scientific education. Hegel knew so much mathematics that not one of his students was capable of publishing the numerous mathematical manuscripts left behind after his death. The only person, so far as I know, sufficiently knowledgeable of mathematics and philosophy to perform such a task - is Marx.’^{†4}

In the ‘Philosophical Notebooks’ V.I. Lenin criticized^{‡2} the statements of Hegel on the calculus of infinitesimally small quantities contained in the chapter ‘Quantity’, specifically, that ‘... the justification [for neglecting higher-order infinitesimals - E.K.] has consisted *only* in the *correctness of the results* (“demonstrated on other grounds”) ... and not in the clearness of the subject ...’, that “... a certain inexactitude (conscious) is ignored, nevertheless the result obtained is not approximate but *absolutely* exact,’ that ‘notwithstanding this, to demand *Rechtfertigung* [justification - *Trans*.] here is “not as superfluous” “as to ask in the case of the nose for a demonstration of the right to use it”.’^{**} V.I. Lenin made the following remarks: ‘Hegel’s answer is complicated, *abstrus*, etc. etc. It is a question of *higher* mathematics ...’ ‘A most detailed consideration of the differential and integral calculus, with quotations - Newton, Lagrange, Carnot, Euler, Leibnitz etc., etc, - showing how interesting Hegel found this “vanishing” of infinitely small magnitudes, this “intermediate between Being and non-Being”. Without studying higher mathematics all this is incomprehensible. Characteristic is the title *Carnot*: “Refléxions sur la Métaphysique du calcul infinitésimal”!!!’

It is undoubtedly true that Marx, who had written in 1873:

‘The mystification which the dialectic suffered at the hands of Hegel does not obscure the fact that Hegel first gave a comprehensive and conscious representation of its general forms of motion. It is necessary to stand it on its feet, in order to reveal the rational kernel beneath the shell of mystification.’^{*8}

having already applied his dialectical materialist method which, in his own words, was not only fundamentally ‘different from the Hegelian, but is its direct antithesis’, since for Marx ‘the ideal is nothing other than the material, perceived in a human head and transformed within it’,^{†5} was extremely tempted to try to discover the secret which seemed to lie at the basis of differential calculus.

Marx’s studies of mathematics were known from his correspondence with Engels, particularly the letters from Marx to Engels of January 11, 1858, May 20 1865, July 6, 1863, and August 25, 1879, the letters from Engels to Marx of August 18, 1881 and November 21, 1882, and Marx’s answer of November 22, 1882. They may also be evaluated from references in Engels’s preface to the second volume of *Capital*, comments in Engels’s *Anti-Dühring*, and in his unfinished manuscript, *The Dialectics of Nature*, published for the first time in 1925 in Moscow in the second book of the [Russian-language] *Archives of Marx and Engels*. The Karl Marx-Friedrich Engels Institute, which was founded in 1920, in carrying out the instructions of V.I. Lenin in his letter of February 2, 1921^{‡3} to purchase the manuscripts of Marx and Engels located abroad (or photocopies of them), acquired a great many, including photocopies of Marx’s mathematical manuscripts preserved in the archive of the German Social-Democratic Party - 863 closely-written quarter-sheets, apparently incomplete; the missing pages were later added, however, so that the entire collection came to a thousand sheets. To work on them the Institute commissioned the German mathematician E. Gumbel, whom R. Mateika and R.S. Bogdan helped to decipher the extremely difficult text.

In 1927 Gumbel published a report in *Letopisi Marksizma* on the manuscripts,^{*9} giving a short description of them. He classified the manuscripts into categories: calculations without any text at all; extracts from works read by Marx; outlines of his own works; and finally, finished original works.

Gumbel correctly noted that Marx’s choice of sources seemed to be influenced by Hegel, and he presented a (far from complete) list of mathematical works which Marx had summarized: 13 authors and 18 titles. Of these works, the oldest in time was the *Philosophiae Naturalis Principia Mathematica* of Newton, 1687, and the most recent, the textbooks of T.J. Hall and J.W. Hemmings, 1852. They also included the classical works of d’Alembert, Landen, Lagrange, MacLaurin, Taylor and two other works of Newton, *De Analysi per Aequationes Numero Terminorum Infinitas* and *Analysis per Quantitatum Series, Fluxiones et Differentias*.

The contents of the manuscripts, Gumbel indicated, dealt with arithmetic (for example, the effect of a discount on the rate of exchange, the paying off of a bill of exchange, discounts and rebates, raising to a power and extracting the root of an equation, exercises in taking the logarithm, and so forth), geometry (trigonometry, analytic geometry, conic sections), algebra (the elementary theory of equations, infinite series, the concept of function, Cardan’s Rule, progressions, the method of indeterminate coefficients), and differential calculus (differentiation, maxima and minima, the Taylor theorem). He reported that the original works which Marx had completed would be published in the 16^{th} volume of [the Russian edition of] the *Complete Works* of Marx and Engels.

In 1931, with the appointment of the well-known activist of the Bolshevik Party V.V. Adoratskii to be director of the Institute, work on the manuscripts was given a new direction. As head of the Marx Study Center at the time, I was acquainted with the transcribed portion of the manuscripts and with the preparatory work toward their publication, and I was convinced that E. Gumbel was unable to appreciate completely either the importance of their publication or their philosophical and historical-mathematical significance. At my suggestion the board of directors of the Institute enlisted for the work on the manuscripts S.A. Yanovskaya, leading a team which was joined by the mathematicians D.A. Raikov and A.I. Nakhimovskaya.

In London in 1931 the Second International Congress of the History of Science and Technology took place, at which a Soviet delegation took part whose members included the author of these lines. The papers of our delegation came out as a separate book with the title *Science at the Crossroads*.^{*10} Among the papers included was my own, entitled: ‘A Brief Report on the Unpublished Works of Karl Marx pertaining to Mathematics, the Natural Sciences, Technology and Their Histories.’ This report discussed: first, the passages from 27 works of natural science which Marx copied and to which he supplied commentaries: on mechanics, physics, chemistry, geology, biology, as well as on electrical technology, metallurgy, agricultural chemistry, and others; second, his works on technology (primarily dating to 1863), treating the history of mills, the history of looms, the problem of automated production in mechanized factories, the development from tools to machines and from machines to mechanized factories, the effect of the mechanisation and rationalisation of production on the development of the textile industry in England and on the situation of the proletariat in the period 1815-1863, the changes in the social system of production at various stages of technological development, the interaction between labor and science, between city and countryside, and so on; and third - Marx’s mathematical manuscripts.

In Zurich in 1932 there convened an International Congress of Mathematicians in which a Soviet delegation took part. At the ‘Philosophy and History’ section of the congress I made the report, ‘A New Foundation of the Differential Calculus by Karl Marx’,^{†6} which discussed one of the works contained in Marx’s manuscripts. It was of great interest, both for the history of mathematics and for those dealing with the philosophical problems of the scientific worker, since it contains a sketch of the historical development of the concept of the differential and a statement of Marx’s viewpoint on the foundation of analysis. This work is of the third category of the manuscripts, and consists of five chapters: 1. The Derivative and the Differential Coefficient [the at that time so-called ratio, *dy/dx*, 2. The Differential and Differential Calculus, 3. The Historical Development of Differential Calculus, 4. The Theorem of Taylor and MacLaurin, 5. A Critique of Newton’s Method of Quadratures.

The first part of the third chapter, which forms the nucleus of the entire work, contains a brief account of the methods of Newton, Leibnitz, d’Alembert and Lagrange. The second part, which summarizes the first, consists of three sections with the following contents: 1. Mystical Differential Calculus, 2. Rational Differential Calculus, 3. Purely Algebraic Differential Calculus. In another fragment Marx contrasts his own differential method to the methods of d’Alembert and Lagrange. His method differs from the method of Lagrange because Marx really differentiates, thanks to which differential *symbols* appear, while Lagrange applies differentiation to the algebraic binomial expansion.

It is clear from both fragments that Marx, like Hegel, considered all efforts to provide a purely formal-logical foundation for analysis hopeless, just as the attempts to give, beginning with the graphic method, a purely intuitive-visual foundation to it had been naive. He set himself the task of providing a foundation for analysis dialectically, relying on the unity of the historical and logical aspects.

Marx demonstrated both that the new differential and integral calculus came into existence from elementary mathematics, on its own ground, ‘as a specific type of calculation which already operates independently on its own ground,’ and that ‘the algebraic method therefore inverts itself into its exact opposite, the differential method’. (See p.21 in this edition.) Marx valued highly the work of Lagrange, but he did not consider him - as he was usually considered and as Hegel considered him - a formalist and conventionalist who introduced the basic concepts of analysis into mathematics in a purely superficial and derivative manner. Marx appreciated just the opposite in Lagrange, namely, that he revealed the connection between algebra and analysis, that he showed how analysis develops out of algebra. ‘The real and therefore the simplest connection of the new with the old is discovered as soon as this new reaches its completed form, and one may say that differential calculus gained this relation through the theorems of Taylor and MacLaurin.’ (See p.113)

At the same time, however, Marx reproached Lagrange for not perceiving the dialectical character of this development, for sticking for too long to the domain of algebra, and for insufficiently appreciating the general laws and methods proper to analysis, so that ‘in this regard he should only be used as a starting point’. (See Yanovskaya, 1968, p.417) Thus Marx, like a genuine dialectician, rejected both the purely analytic reduction of the new to the old characteristic of the methodology of the mechanistic materialism of the 18^{th} century, and the purely synthetic introduction of the new from outside so characteristic of Hegel.

Reports and articles concerning Marx’s mathematical manuscripts also appeared in 1932 in the journals *Za Marksistsko-Leninskoe Estestvoznanie, Vestnik Kommunisticheskoe Akademii*, and *Front Nauki I Tekhniki*.^{*11} There was a great deal of interest in the manuscripts among the Soviet, as well as the foreign, learned public. Only in 1933,^{†7} however, did it become possible, as a result of the work of the team of scholars mentioned above, to publish the first extracts from the manuscripts, in the journal *Pod Znamenem Marksizma* and simultaneously in the collection Marksizm I Estestvzoznanie, issued on the 50^{th} anniversary of Marx’s death by the Marx-Engels Institute. In both publications, the extracts from the manuscripts were accompanied by the article ‘On the Mathematical Manuscripts of K. Marx’ ^{‡4} by the team leader S.A. Yanovskaya. The published extracts are three works of Marx dating from the [18]70s and the beginning of the [18]80s. Marx completely finished and prepared to send to Engels the first two - ‘The Derivative and the Symbolic Differential Coefficient’ and ‘The Differential and Differential Calculus’. The third work, ‘A Historical Sketch’, is an unfinished draft. From the latter, which includes the sections: 1. Mystical Differential Calculus (that is, Newton and Leibnitz), 2. Rational Differential Calculus (that is, d’Alembert) and 3. Purely Algebraic Differential Calculus (that is, Lagrange); we introduce here in the team’s translation, section I, in order to acquaint the reader with Marx’s exposition. (pp.91-92)

‘*1. Mystical Differential Calculus. x _{1} = x + Δx* from the beginning changes into

‘Then, however, *y _{1} = y + dy* or

‘If the increment of *x = Δx* and the increment of the variable dependent on it *= Δy*, then it is self-evident (*versteht sich von selbst*) that *Δy/Δx* represents the ratio of the increments of *x* and *y*. This implies, however, that *Δx* figures in the denominator, that is the increase of the independent variable is in the denominator instead of the numerator, not the reverse; while the final result of the development of the differential form, namely the *differential*, is also given in the very beginning by the assumed differentials.*(?)

‘If I assume the simplest possible (*allereinfachste*) ratio of the dependent variable y to the independent variable *x*, then *y = x*. Then I know that *dy = dx* or *y. = x.*. Since, however, I seek the derivative of the independent [variable] *x*, which here *= x.*, I therefore have to divide both sides by *x.* or *dx*; so that:

*dy/dx* or *y./x. = 1*

‘I therefore know once and for all that in the symbolic differential coefficient the increment [of the independent variable] must be placed in the denominator and not in the numerator.

‘Beginning, however, with functions of *x* in the second degree, the *derivative* is found immediately by means of the binomial theorem [which provides an expansion] where it appears ready-made (*fix und fertig*) in the second term combined with *dx* or *x.*; that is with the increment of the first degree + the terms to be juggled away. The *sleight of hand (Eskamotage)* however, is unwittingly mathematically correct, because it only juggles away errors of calculation arising from the original sleight of hand in the very beginning.

*x _{1} = x + Δx* is to be changed to

*x _{1}` = x + dx* or

whence this differential binomial may then be treated as are the usual binomial, which from the technical standpoint would be very convenient.

‘The only question which still could be raised: why the mysterious suppression of the terms standing in the way? That specifically assumes that one knows they stand in the way and do not truly belong to the derivative.

‘The *answer* is very simple: this is found purely by experiment. Not only have the true derivatives been known for a long time, both of many more complicated functions of *x* as well as of their analytic forms as equations of curves, etc., but they have also been discovered by means of the most decisive experiment possible, namely by the treatment of the simplest algebraic function of second degree, e.g.:

*y = x²*

*y + dy = (x + dx)² = x² + 2xdx + dx² ,*

*y + y. = (x + x.)² + x² + 2xx. + x.² .*

‘If we subtract the original function, x²(y = x²) from both sides, then:

*dy = 2xdx + dx²*

*y. = 2xx. + x.x. ;*

I suppress the last terms on both [right] sides; then:

*dy = 2xdx, y. = 2xx. ,*

and further

*dy/dx = 2x ,*

or

*y./x. = 2x .*

‘We know, however, that the first term out of *(x + a)²* is *x²*; the second *2xa*; if I divide this expression by *a*, as above *2xdx* by *dx* or *2xx.* By *x.*, we then obtain *2x* as the first derivative of *x²*, namely th increase in *x*, which the binomial has added to *x²*. Therefore the *dx²* or *x.x.* had to be suppressed in order to find the derivative; completely neglecting the fact that nothing could begin with *dx²* or *x.x.* in themselves.

‘In the experimental method, therefore, one comes - right at the second step - necessarily to the insight that *dx²* or *x.x.* has to be juggled away, not only to obtain the true result but any result at all.

‘Secondly, however, we had in

*2xdx + dx²* or *2xx. +x.x.*

the true mathematical expression (second and third terms) of the binomial *(x + dx)²* or (*x + x.)²*. That *this mathematically correct result* rests on *the mathematically basically false assumption* that *x _{1} - x = Δx* is from the beginning

‘In other words, instead of using sleight of hand, one obtained the same result by means of an algebraic operation of the simplest kind and presented it to the mathematical world.

‘Therefore, mathematicians (*man ... selbst*) really believed in the mysterious character of the newly-discovered means of calculation which led to the correct (and, particularly in the geometric application, surprising) results by means of a positively false mathematical procedure. In this manner they became themselves mystified, rated the new discovery all the more highly, enraged all the more greatly the crowd of old orthodox mathematicians, and elicited the shrieks of hostility which echoed even in the world of non-specialists and which were necessary for the blazing of this new path.’

In an analogous manner Marx critically analyzed both the method of d’Alembert as well as that of Lagrange and, as already mentioned, opposed all three methods with his own. It consists of first forming, for *y = f(x)*, the ‘preliminary derivative’,

*φ(x _{1}, x) = (f(x_{1}) - f(x))/(x_{1} - x)*

which is assumed to be continuous at *x _{1} - x* and whose value at

*dy = f’(x)dx (*)*

is mathematically correct and is not reduced to the tautology

*0 = 0*

it therefore is an *operative* formula [emphasis in original - Trans.], applicable to complicated functions, making it possible to reduce an entire differentiation of its constituent functions. In this way, he points out, we obtain the dialectical *reversal* of the method: we now proceed not only from the real mathematical process of the formation of the derivative to its symbolic expression, but rather on the contrary, operating on the symbolic formula (*) and forming the ratio *dy/dx* we arrive at the expression of the derivative of the function. Consequently Marx, having not only discovered that the differential is the major linear portion of the increment but it also an operative symbol, proceeded along a path which we today would call *algorithmic*, in the sense that it consists of a search for an exact instruction for the solution, by means of a finite number of steps, of a certain class of problems. He was on a path which has been the fundamental path of the development of mathematics. Thanks to the dialectical materialist method which in his hands was a powerful, effective tool of research, Marx was able, without being a mathematician, to reveal the property of the differential used as an operational symbol, thus anticipating, as the Soviet mathematician V.I. Glivenko has shown, the idea of the eminent French mathematician G. Hadamard, enunciated in 1911 in connection with the application of this concept of functional analysis.^{*12}

Despite the philosophical and historical significance of the foundation of differential calculus provided by Marx, it did not enter into mathematics, which developed another path unknown to him. The sources which he studied (and their number was significantly greater than Gumbel reported in his article, which did not mention even those textbooks of analysis, such as those of J.-L. Boucharlat and J. Hind, which Marx outlined in detail) made no mention of the works of A Cauchy (*Cours d’analyse and Résumé des leçons sur le calcul infinitésimal*) in which in 1821-1823 he developed he theory of limits, a theory which, although it contained shortcomings which were later (1880) cleared up by K. Weierstrass, nonetheless incorporated a great deal of rigor and rendered the foundation proposed by Marx superfluous, although it did not diminish its historical and philosophical value. Marx did not know and could not have known of the work of the outstanding logician, mathematician and philosopher of Prague, B. Bolzano, who in 1816-1817 defined the concepts of limit, continuity, the convergence of series, and others - concepts which laid the basis of present-day analysis - since these works as well as others of 1830-1848 which contained the beginnings of set theory and the theory of real number s remained unknown for a long time. Only a hundred years later did they become the property of mathematicians(?). Naturally, Marx did not consider, therefore, the problems of continuity, the differentiability of functions, the axiomatisation of analysis, and so on.

The value of Marx’s mathematical manuscripts, however, is by no means restricted to his method providing a foundation for differential calculus and his critique of preceding methods. The complete significance of the manuscripts was only revealed when they were all deciphered and scientifically systematized. Beginning with 1932 and with the publication in 1933 of the three works mentioned from the deciphered manuscripts (which Gumbel had not given the attention they deserved), the Swedish mathematician Wildhaber first began working on behalf of the Marx-Engels Institute. Work on the manuscripts was resumed in the 1950s, and somewhat later (1960-1962) G.F.Rybkin became interested. All this work - deciphering, translation, research, and compilation of sources - was conducted under the leadership of S.A. Yanovskaya, who, despite an extraordinary load of teaching and preparing graduate students, despite a painful illness, gave the enterprise all of her energy and her enormous knowledge of the history of mathematics and its philosophical problems, transforming it into her life’s work. S.A. Yanovskaya’s commentaries on the manuscripts (both the one cited above and those contained in the volume prepared by the Institute of Marxism-Leninism of the Central Committee of the CPSU) by themselves constitute an important scientific work. One of her many students. K.A. Rybnikov, performed significant work in the preparation of the manuscripts for publication (in particular, the difficult research and collation of sources). The volume was prepared for publication by the historian O.K. Senekina, member of the Institute of Marxism-Leninism, and the mathematician A.Z. Rybkin, editor of the Nauka press.

As a result of all this work lasting many years (S.A. Yanovskaya labored on the manuscripts until her death in October 1966), a book has appeared which contains Marx’s ideas on a series of the most important problems in the history of mathematics as a whole and of its individual concepts, as well as on their epistemological [original: ‘gnoseological’ - *Trans*.] significance, ideas which, despite the head-spinning pace of the development of mathematics in the ‘80’s of the last century - among which and in particular including its logical-philosophical basis - have not lost their contemporaneity in the slightest. For historians of mathematics and for philosophers working with the philosophical problems of mathematics, Marx’s views will serve as a guide - not in the form of a quotation, every letter of which is followed as if counting out an emergency ration, but rather in the form of a matchless example of creative, concrete application of dialectical thinking.

In addition, the mathematical manuscripts of Marx once again confirm the truth of the words Engels spoke at the graveside of his great friend. Speaking of Marx as the scientist who had discovered the law of the development of human history and the law of motion of capitalist production, Engels said: ‘Two such discoveries would be enough for one lifetime. Happy the man to whom it is granted to make even one such discovery. But in every single field which Marx investigated - and he investigated very many fields, none of them superficially - in every field, even in that of mathematics, he made independent discoveries.’^{*13}

^{*}
Translation of ‘K. Marks I Matematika (O ‘Matematicheskikh rukopisyakh’ K. Marksa)’, *Voprosy istorii estestviznaniya I tekhniki*, 1968, No.25, pp.101-112.

^{*2}
Constant capital is capital investment; variable capital is labour wages; surplus value is usually written S in English-language economic texts - *Trans*.

^{*3}
In simple reproduction all the value added to the producers goods is invested in the machinery to produce consumers goods - *Trans*.

^{†}
The significance of these schema for socialist economic planning is examined in the work of M. Ebeseldt (GDR), ‘Marx’s Schema of Reproduction and the interpretation of Ambiguous Variables’, (in Russian) *Ekonomika I matematicheskie metody*, 1968, Vol, IV, No.4, pp.531-535.

^{*4}
Karl Marx-Friedrich Engels *Werke* [German edition], Vol.33, Berlin, Dietz, 1966, p.82.

^{†2}
*Reminiscences of Marx and Engels*, Moscow [1956], p.75.

^{*5}
Karl Marx-Friedrich Engels *Sochineniya* [Russian edition], Moscow, Vol.32, p.33.

^{†3}
Reminiscences of Marx and Engels, p.325.

^{‡}
Karl Marx, *Theories of Surplus Value: Volume IV of Capital*, part III, Cohen and Ryazanskaya, trans., London, Lawrence & Wishart, 1972, p.143. Editors Ryazanskaya and Dixon note that ‘Marx wrote this paragraph in English’.

^{*6}
Karl Marx, *Grundrisse: Foundations of the Critique of Political Economy*, trans. M. Nicolaus, Penguin Books, London, p.137.

^{*7}
This edition p.235

^{†4}
Afterword to 2^{nd} German edition of *Capital*

^{‡2}
V. I. Lenin, *Collected Works*, Vol.38, Moscow, Foreign Languages Publishing House, 1963, pp.117-118.

^{**}
Note provided by editor of Lenin text: ‘An allusion to the couplet “The Question of Right” from Schiller’s satirical poem “The Philosophers”, which may be translated as follows:
‘Long have I used my nose for a sense of smell,’
‘Indeed, what right have I to this, pray tell?’

^{*8}
Karl Marx-Friedrich Engels, *Sochineniya*, Vol.23, p.22.

^{†5}
*Ibid*.

^{‡3}
Leninskii Sbornik, Moscow, 1942, Vol. 34, pp.401-402.

^{*9}
E. Gumbel, ‘On the Mathematical Manuscripts of K. Marx’, (in Russian) *Letopisi Marksizma*, Moscow, 1927, Vol.3, pp.56-60.

^{*10}
*Science at the Crossroads*: Papers presented to the International Congress of the History of Science and Technology held in London from June 29^{th} to July 3^{rd} , 1931, by the Delegation of the USSR, Kniga Ltd., Bush House, Aldwych, London WC2, 1931. Republished in 1971.

^{†6}
E. Kol’man, ‘A New Foundation of the Differential Calculus’ by Karl Marx’, [in German], *Verhandlungen des Internationalen Mathematiker-Kongresses*, Vol.2, Sektions-Verträge, Zurich, 1932, pp.349-351.

^{*11}
*Za Marksistsko-Leninskoe Estestvoznanie*, 1932, No.5-6 pp.163-168; *Vestnik Kommunisticheskoi Akademii*, 1932, No.9-10, pp.136-138; *Front Nauki I Tekhniki*, 1932, No.10, pp.65-69.

^{†7}
The original has 1932, an obvious misprint.

^{‡4}
*Pod znamenem marksizma*, 1933, No.1, pp.14-115; *Marksizm I estestvoznanie*, 1933, pp.136-180.

^{*12}
V.I. Glivenko, ‘The Concept of the Differential in Marx and Hadamard’ (in Russian *Pod Znamenem Marksizma*, 1934, No.5, pp.79-85.

^{*13}
Quoted from Marx-Engels *Selected Works*, Volume Two, p.153-154, Foreign Language Publishing House, Moscow. The speech was re-translated into English from the only written version, in the German-language *Sozialdemokrat*, Zurich, March 22 1883.