Marx’s brilliant study of mathematics‘The Mathematical Manuscripts of Karl Marx’, New Park Publications, 280pp. £25 hardback, £15 paperback. Reviewed by Andy Blunden; ONE hundred years after the death of Karl Marx, New Park Publications this month publish for the first time in English the Mathematical Manuscripts of Karl Marx. The mathematical work of the greatest revolutionary thinker of our time had remained virtually closed book until 1968 when NAUK published the German edition which has provided the core of material for the present book. The. first extracts from the manuscripts were published in 1933 in Pod Znamenem Marksizma, the journal founded by the Bolsheviks, on the 50th anniversary of Marx’s death. This followed Lenin’s directive of 1921 to acquire copies of all Marx and Engels’ manuscripts available abroad. After that, 20 or 30 years elapsed before work was apparently resumed. Ernst Kolman and Sonia Yanovskaya, who contribute the articles, ‘Karl Marx and Mathematics’ (1968) — and the ‘Preface’ (1968) respectively, and a joint article ‘Hegel and Mathematics’ (1931), to the present volume, worked on the manuscripts both in the 1930s and in the more recent period. Kolman brought them to world attention at a congress in London in 193 1, and it is said that Yanovskaya made them her life’s Work, dying before the publication of the NAUK edition. No explanation is given, however, as to why nothing was published relating to the manuscripts between 1933 and 1968. Indeed, even in the 15 years since the authoritative edition of 1968 was published in the Soviet Union, no further translations or analyses. have emerged until New Park were able to compile. this volume, Even the otherwise complete Collected Works of Marx and Engels currently being published by Lawrence and Wishart is not plannned to include them. Is this because the Mathematical Manuscripts of Karl Marx are of little political, philosophical or historical interest? Are they an historical curiosity? Certainly not! And the excellent additional and explanatory material included in the New Park edition prove that this is not so and that the Soviet scholars who devoted many years of their lives to work on the manuscripts did not believe it so. As Ernst Kolman says in concluding his contribution: ‘As a result of all this work lasting many years 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 significance, ideas which, despite the headspinning pace of the development of mathematics in the ‘80s of the last century — among which and in particular including its logicalphilosophical 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. ...’ (p.234) And let us add — not only of interest to mathematicians and philosophers — but also an invaluable guide and inspiration for all revolutionary fighters, grappling with the advanced theoretical problems posed by the breakup, of world capitalism and the urgent need for the preparation of the working class for the ‘reversal of roles’ (see p. 50) — for the seizure of power. Many readers may be deterred by their lack of knowledge of mathematics and putoff by the array of algebraic formulae a flick through the pages of this book will show. It should be remembered that Marx wrote these manuscripts for the purpose of clarifying himself and Engels, who remarked in his letter of August 10, .18 8 1. (see p. XX VII): ‘Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and 1 was pleased to find that I did not need them. 1 compliment you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it.’ And mathematical beginners will feel for Marx when he wrote to Engels (January 11, 1858) during work on Capital: ‘I am so damnedly held up by mistakes in calculation in the working out of the economic principles that out of despair I intend to master algebra promptly. Arithmetic remains foreign to me. But I am again shooting my way rapidly along the algebraic route’. (p. VII) Study of the Mathematical Manuscripts is, of course, no light exercise, but the New Park volume contains an excellent collection of additional material which make the manuscripts themselves more accessible, develop the ideas contained Within them, and also stand independently as important contributions to the development of dialectical materialism. In particular, Kolman and Yanovskaya’s article of 1931, Hegel and Mathematics, contains a brief but monumental critique of Hegel’s philosophy, a critical analysis of the range of bourgeois philosophy, from Kant to Mach, via a study of the nature of mathematical, knowledge, and an important contrast between the work of Marx and Hegel in natural science generally. Yanovskaya’s preface to the 1968 edition outlines how Marx came to his study of mathematics sketches briefly the most. important aspects of his work on differential calculus which makes up. the bulk of the manuscripts, and, tells of the problems involved in their preparation for publication. The three letters — two from and one from Marx — bring out in sharp relief the unity of these two men their work on the sciences; several aspects are further elaborated, and, in particular, Engels’ first letter casts in characteristically bold terms the most important, philosophical conclusions Marx and he were drawing. Six appendices supplied by the Soviet editors give excerpts from, and commentary on, textbooks and mathematical classics that Marx consulted, which are essential to understanding Marx’s own work and place it in g$ historical context. Kolman’s review, ‘Karl Marx and Mathematics: On the Mathematical Manuscripts of Marx’ (1968), is an excellent discussion of the relation of mathematics to the struggle for social ism, to political economy and to, philosophy,’ gives a detailed history of the manuscripts, and a concise summary of Marx’s conclusions relating to calculus and an explanation of Marx’s own preferred method of, differentiation, outlining the significance of later developments in these areas and drawing out the essence of, Marx’s contribution. Cyril Smith has contributed an essay, Hegel, Marx and the Calculus, written specially. for this volume, explaining the significance of Marx’s mathematical work as part of a living struggle. Smith shows the role that the problem of infinity has played since ancient times and especially in relation to Hegel and Marx, and its significance in all the philosophical crises in mathematics. He concludes with an analysis of the nature of mathematical knowledge. The New Park edition has reproduced, unabridged, the useful and diverse notes given in the Russian edition, plus an index to Marx’s Mathematical Manuscripts. The manuscripts themselves contain two finished articles, ‘On the Concept of the Derived Function’, and ‘On the Differential’, with drafts and supplements; a series of drafts and extracts on the history of differential calculus; an article on Taylor’s theorem and, three appendices on limit calculus and on D'Alembert’s method, of differentiation. Marx developed his interest in mathematics. in connection with his work on Capital (see Yanovskaya p. VIII), beginning in earnest around 1858 and continuing to the end of his life. He was driven to do so partly by the necessity of understanding and criticising contemporary commercial and economic literature but principally by the necessity of giving quantitative expression to the fundamental laws of value in capitalist society and developing the interconnection of the laws (see Kolman p. 218222). Marx clearly understood the necessity for political economy to learn to use mathematics if it was to reach its fullest development and indeed anticipated the possibility of statistical analysis of the laws of the capitalist crisis. M the same time Marx vigorously opposed the mathematical fetishism, characteristic of the bourgeois moneymentality, maintaining that the mathematical methods could never overreach the ‘boundary of the underlying, qualitative theory on which it was based. On p. 252253, Kolman and Yanovskaya explain: ‘The increasing difficulties offered to the mathematics of complicated forms of motion, piling up in an ascending series in leaps from mechanics to physics, from physics to chemistry, from there to biology and onwards to the social sciences, do not, in the dialectical materialist conception, entirely block its path, but allow it the prospect of even “determining the main laws of capitalist economic crisis” (Marx to Engels, May 31, 1873)’. Around 1863, Marx’s interest was taken by differential calculus and his interest in this area continued side by side with his study of commercial arithmetic, etc. until 1883. Marx studied the latest textbooks and studying problems as a pastime as well as in concentrated study. To anyone who has studied mathematics to the level of the calculus, it is clear why calculus attracted interest. For here, above all other areas, mathematics dialectical leap in a way which is so explicit, and so obstinate in its contradictoriness the face of all the attempts of mathematical gradualism to reform it. away. Hegel also paid great attention to the calculus. ‘Nevertheless, elementary mathematics , just like the formal logic, is not nonsense, it must reflect something in reality and therefore it must contain certain elements of dialectics.’ (Kolman and Yanovskaya p.250251). And: ‘The turning point in mathematics was Descartes’ variable magnitude, With that came motion and hence dialectics in mathematics and at once, too, of necessity the differential and integral calculus, ...’ (Dialectics of Nature, Engels, p.258) Thus, in the history of mathematics, Marx found a whole new source of dialectical knowledge, reaching its sharpest point in the birth of calculus out of elementary algebra itself, ‘driven to do so by the requirements of “application,” i., of practice, technique,, science’. (See Kolman and Yanovskaya, p.245) In his speech at Marx’s graveside, ‘Engels said: ‘But in every 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’. (see p.234). The legacy of Marx’s mathematical studies was 1,000 manuscript pages. References were made to the manuscripts in a number of letters between Marx and Engels and in Engels’s AntiDühring and Dialectics of Nature. Following Lenin’s directive of 1921, the Karl MarxFriedrich Engels Institute purchased photocopies of 863 closely written sheets from the German SocialDemocratic Party archives and later the remaining sheets were found. The German mathematician Gumbel led a team to decipher them and published a report in 1927 listing the wide range of subjects dealt with, sometimes only in the form of copied extracts, but also including finished works — all very difficult to decipher. In 1931 there was a change and Kolman and Yanovskaya took over the work with a new team. In June 1931, Kolman attended the Second International Congress of the History of Science and Technology with a group of Soviet scientists led by Bukharin. The Bolshevik contributions at the congress had a profound effect on a generation of young British scientists. Apart from his report on Marx’s mathematical manuscripts, Kolman spoke on the crises of mathematics (see Science at the Crossroads, Kniga Ltd., 1971) By 1933, the Soviet workers were able to publish extensive excerpts to commemorate the 50th anniversary of Marx’s death, including the article on, ‘Hegel and Mathematics’ included in the New Park edition. Work was renewed in the 1950s and strengthened from 1960 onward and a team led by Sonia Yanovskaya meticulously deciphered, translated and edited the manuscripts and researched the sources Marx had used from all over. Europe., Yanovskaya died in October 1966, but the manuscripts were finally published in 1968. Kolman left the Soviet Union and died in Sweden in 1979. Publication of the manuscripts in English together with the unique collection of related material must provide an enormous stimulus now for sharpening the ideological struggle against bourgeois ideology — not only deepening the grasp of young cadres of the dialectical materialist method, but also winning over sections from the scientific community attracted to the great strength of Marxist theory at a time when science is being stultified, starved and degraded by the death agony of capitalism. Mathematics was correctly defined by Hegel as the science of quantity (see p. 239). Quantity is an objective concept derived from the movement of matter in all its spheres. Mathematical knowledge therefore contains real knowledge of the material world, although At a high level of abstraction. It is no, coincidence then that mathematical forms find ‘application’ in material processes since it was from the same material universe that they were abstracted in the first place. (see p.268 et seq., and AntiDühring p.5152) Mathematics thus provides stud dents of dialectical materialism with a source of new knowledge from two aspects. Firstly, by working over the .positive results of mathematics from both its historical and logical aspects we can deepen and concretise our grasp of the laws of dialectics. Secondly, at any given historical juncture, mathematics undergoes its particular crises, giving us the opportunity to sharpen our dialectical weapons in the struggle to resolve these problems. In contrasting Marx’s method to that of Hegel, Kolman and Yanovskaya state: ‘Whereas Hegel merely tries to substantiate what already exists, it is a matter here of a transformation, the conscious change, the reconstruction of science on the basis of the guiding role of practice.’ (p.253) Marx’s writing on differential calculus contained in this volume clearly shows that Marx set himself the task of critically reworking and resolving the problems of the foundations of the’ calculus and was by no means content to passively describe the dialectical moments of the calculus. And he has made a valuable contribution in this field. The actual historical development of mathematics over the last century and a half has moved on from the particular problems with which Marx was concerned in relation to calculus, but has confirmed the essence of Marx’s ideas on the subject. (See ‘Preface’ p. XIIIXVII) These problems, however, continually reemerge at a deeper level and are linked, to methodological. problems in all branches of science and close study of Marx’s method is essential — his synthesis of the logical and historical; his insistence on the highest standards of precision in science and contempt for all ‘sleight of hand’ etc.; his insistence on the sublation of the old into the new, established more concretely with every further development of the new. Only by his struggle to develop his algebraic method of differentiation could Marx bring out what was new in the calculus. As Marx remarks (p. 3): ‘First making the differentiation and then removing it therefore leads literally to nothing. The whole difficulty in understanding the differential operation (as in the negation of the negation generally) lies precisely in seeing how it differs from such a simple procedure and therefore leads to real results.’ What a far cry is this from the positivists of the school of Mach and the modernday revisionists who want to relegate philosophy to the job of a filingclerk for natural science and to revolutionary theory the role of a passive spectator. Kolman and Yanovskaya’s 1931 article ‘Hegel and Mathematics’ is a masterpiece which should be read and reread by all those struggling to develop Marxism. It is a continuation of Lenin’s theoretical work, notably ‘of Volumes 14 and 3 8 of the Collected Works, written on the eve of the Moscow trials. One wonders not so much that the authors ceased to publish, but that they survived to pick up their work again 30 years later. Independently of whether Kolman and Yanovskaya were conscious of L the political implications of their philosophical contribution, it is only the Trotskyists who today can carry forward Marx’s legacy as a guide — to revolutionary practice. Kolman and Yanovskaya (see p. 236238) credit Hegel’s work on mathematics with the following discoveries: That mathematics is the science of quantity — quantity as one moment of the notion, not as its ,essence (quantityfetishism), derived out of the dialectic of quality. That calculus ‘contains qualitative moments’ and thus cannot be reduced to elementary mathematics. That the essence of calculus is in its application, its origin in the requirements of practice, outside of mathematics. That without the aid of philosophy, mathematics cannot justify its own methods. The weaknesses of Hegel’s view they summarise as follows: Hegel denied the qualitative (dialectical) moments in elementary mathematics and insisted that calculus was alien to mathematics, ‘brought into mathematics in an external manner, through external reflection’, but ‘is forced to carry the true dialectics of the development of mathematics over to his philosophical system’, although in a ‘mystifying way’. They also say Hegel ‘denied the possibility of constructing a mathematics which would consciously apply the dialectical method’ and ‘consequently . . . is forced to jog along behind the mathematics of his day’ and frequently ‘creates an apparent solution where he should have sharply posed an unsolved problem’. A study of these points, which are given above in a much abbreviated form, will provide a rich source of material to deepen our understanding of how idealism imposes upon the external world images derived earlier in the process. In the struggle to subordinate the world to this idea, the idealist becomes trapped by his images, cut off from the dialectical source in external nature. By recognising the origin of the materialist ‘kernel’ of Hegel’s speculations, in the external world, Marx’s method builds up dialectics in a struggle to change the world, and derive new knowledge. ‘Marx, like Hegel, considered all efforts to provide a purely formallogical foundation for analysis hopeless ... (and) to give ... a purely intuitivevisual foundation to it ... naive’. (Kolman p.227) ‘Marx appreciated ... . (how Lagrange) ... revealed the connection between algebra and analysis, t he showed how analysis develops out of algebra ... ‘ (ibid) Marx critically analysed both the method of d'Alembert as well as that of Lagrange (and Leibnitz and Newton). ... and opposed all three methods with his own.’ (ibid p.232) On page 232, Kolman succinctly summarises Marx’s method Muting out that Marx recognised the differential equation as an ‘operative formula’ — ‘a strategy of action’ which, when it arises, constitutes a reversal of the differential process, since the ‘real’ algebraic processes then arise out of the symbolic operational equation, which originally itself arose out of a ‘real’ algebraic process. By means of Lagrange’s application of Taylor’s theorem Marx ‘discovered that the differential is the major linear portion of the increment’ which established the basis of the continuity with elementary algebra. The great merit of. his contribution is that under historical circumstances in which Marx could not have completed his task, he strove ‘for ah algorithmic method ('an exact instruction for the solution, by means of a finite number of steps, of a certain class of problems’) in which ‘he was on a path which has been the fundamental path of the development, of mathematics’ (Kolman p. 232). Developments in these areas unknown to Marx are outlined by Kolman (p. 232233) and Smith (p. 266268), including the way in which these problems were resolved in actuality. Marx’s method is described (Kolman p. 227) as the ‘unity of the historical and logical aspects’. So we could begin by outlining Marx’s periodisation of the history of the calculus: 1. ‘The mystical differential calculus’ of Newton and Leibnitz. 2. ‘The rational differential calculus’ of Euler and D'Alembert. 3. ‘The pure algebraic calculus’ of Lagrange. (Preface p. XX). Newton and Leibnitz, like the majority of the successors from the beginning performed operations on the ground of the differential calculus ... All of their intelligence was concentrated on that.’ (p. 7 8) Th e founders of calculus were not concerned with how and whence their discovery had arisen from the old mathematics. They were content to juggle away the terms that experiment showed were extraneous and operate with their new method of calculation, covering their tracks with mysticism. Newton attempted to use the proven applicability of the derivatives to the representation of material motion, as if this could substitute for mathematical proof. If y is a variable quantity dependent on another variable x , it is possible to calculate the increment of y resulting from a certain increment of x. The ratio between these increments is a measure of the degree of variation of y in relation to x over that interval. If y was distance and x time, the ratio of the finite differences would be the average velocity. The leap comes when the increment in x is taken as zero, (i.e. no increment at all) and the resulting change in y is zero also. But Newton and Leibnitz, by ‘juggling away’ terms which were ‘even more zero’, were able to determine a definite value for the ratio of zeros (c.f. instantaneous speed). The calculus arose in this mystifying way. In the next period of Man’s historical analysis, Rational Calculus, D'Alembert attempts to correct and demystify the calculus of Newton and Leibnitz by separating out the derivative by purely, algebraic means, before transforming the finite differences into infinitesimals. Contrary to prevailing opinion m the time, Marx credits Lagrange, *M initiation of the next period of development of the foundations 6f calculus. Lagrange utilised Taylor’s theorem, which allows the expression of any function in terms of a power series, of finite or infinite degree. Taylor’s theorem was a product of the calculus, but it enabled the comprehension of the differentiation of functions without degree in terms of the algebraic functions of definite degree (i.e. a finite number of terms) and the various derivatives appear ‘readymade’ in the coefficients of the various terms, which can be calculated by methods not involving the ‘mystical’ transformations of differential calculus. (seep. 91100) Philosophy had wrestled with the problem of comprehending motion for millennia before Newton and Leibnitz formulated calculus. Armed with dialectical logic. Marx critically dissected the different forms of expression for the change in the value of variables to which the mathematicians had hitherto paid no attention. In the historically first method, difference appears in its positive form as an increment independent of the variable to which it is added, and which then infinitely contracts so that it can be ‘juggled away’. In the second, D'Alembert’s ‘correction.’ of the ‘mystics’, the difference 1 appears directly in its negative for” but in the end returns to where the ‘mystics’ set out. In the ‘algebraic’ method, there are two independent values of the variable and the difference is developed so that a preliminary derivative is formed by separating the difference as a factor. The identity of the two different values then gives the definite derivative (see p. 127131). The development of the difference is well known for algebraic functions and effective methods exist fora wide class of other functions. Marx sought to generalise the, algorithmic feature of this method by use of Taylor’s theorem. Marx showed, with his example of the differentiation of a quotient from the product rule (see p.70), how the operational formulae reverse them selves — being derived as symbolic results of ‘real’ processes they then express themselves as a ‘strategy of action’ for ‘real’ functions — ‘as a specific type of calculation which already operates independently on its own ground’ (p. 2021). The founders of calculus introduced the differential ‘metaphysically’, without bothering to show how the differential was derived from the finite difference — it was first assumed to exist and then afterwards it was explained. Similarly, an increment was supposed which had an existence independent of the variable which was differentiating. Marx insisted on tracing the transition from the lower to the higher, although; contradictorily, this required reversing the historical process of the foundation of calculus. Marx was also interested in the equation in mathematics — with its opposite poles, with the development .and initiative shifting from one pole to another. The equation is the fundamental dialectical contradiction in mathematics, and it is characteristic that the mathematicians have paid no attention to its development, since for formal logic, if A=B, then B=A, and. that is that. The reversal of the transition from ‘real’ to symbolic expressions into the reverse transition is also a matter of indifference to the mathematicians, although it is the essence of the transition. Marx’s Mathematical Manuscripts must be seen as an outstanding model of dialectical practice, and read from the standpoint from which it was written — the struggle for dialectical theory as a guide to revolutionary social practice. Marx, for whom ‘arithmetic remain(ed) foreign’ in 1858 (aged 38), enriched the dialectical method not only by critical generalisation of the results of the sciences, but also by application of dialectical materialist theory to the resolution of the problems of the sciences, including mathematics. Engels’ Dialectics of Nature, and in particular, The Part Played by Labour in the Transition from Ape to Man, were part of the same revolutionary practice. The Mathematical Manuscripts of Karl Marx confirm that Marx and Engels worked as one. Attempts to separate off Engels’ work on the sciences are demolished by publication of this book, which could by compared to a literary timebomb with a hundredyearlong fuse.

