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.
Engels, in his introduction to the second edition of Anti-Dühring, revealed that among the manuscripts which he inherited from Marx were some of mathematical content, to which Engels attached great importance and intended to publish later. Photocopies of theses manuscripts (nearly 1000 sheets) are kept in the archives of the Marx-Lenin Institute of the Central Committee of the Communist Party of the Soviet Union. In 1933, fifty years after the death of Marx, parts of these manuscripts, including Marx’s reflections on the essentials of the differential calculus, which he had summarised for Engels in 1881 in two manuscripts accompanied by preparatory material, were published in Russian translation, the first in the journal Under the Banner of Marxism (1933, no. 1, pp.15-73) and the second in the collection Marxism and Science (1933, pp.5-61). However, even these parts of the mathematical manuscripts have not been published in the original languages until now.
In the present edition all of the mathematical manuscripts of Marx having a more or less finished character or containing his own observations on the concepts of the calculus or other mathematical questions are published in full.
Marx’s mathematical manuscripts are of several varieties; some of them represent his own work in the differential calculus, its nature and history, while others contain outlines and annotations of books which Marx used. This volume is divided, accordingly, into two parts. Marx’s original works appear in the first part, while in the second are found full expository outlines and passages of mathematical content.* Both Marx’s own writings and his observations located in the surveys are published in the original language and in Russian translation.
Although Marx’s own work, naturally, is separated from the outlines and long passages quoting the works of others, a full understanding of Marx’s thought requires frequent acquaintance with his surveys of the literature. Only from the entire book, therefore, can a true presentation of the contents of Marx’s mathematicla writings be made complete.
Marx developed his interest in mathematics in connection with his work on Capital. In his letter to Engels dated January 11, 1858, Marx writes:
‘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.’ (K.Marx to F.Engels, Works, Vol.29, Berlin, 1963, p.256.)
Traces of Marx’s first studies in mathematics are scattered in passages in his first notebooks on political economy. Some algebraic expositions had already appeared in notebooks, principally those dated 1846. It does not follow, however, that they could not have been done on loose notebook sheets at a much later time. Some sketches of elementary geometry and several algebraic expositions on series and logarithms can be found in notebooks containing preparatory material for Critique of Political Economy dating from April-June 1858.
In this period, however, the mathematical ideas of Marx proceeded only by fits and starts, mostly when he was not occupied with anything else. Thus on November 23, 1860 Marx wrote to Engels: ‘For me to write is almost “out of the question”. Mathematics is the single subject for which I still have the necessary “quietness of mind”.’ (Marx-Engels, Works, Vol.30, Berlin, 1964, p.113) In spite of this he invariably went on with his mathematical ideas, and already on July 6 1863 he wrote to Engels:
‘In my free time I do differential and integral calculus. A propos! I have a surplus of books and will send one to you if you want to study this topic. I deem it almost indispensable for your military studies. By the way, it is a much easier part of mathematics (involving mere technique) than the higher parts of algebra, for instance. Outside of knowledge of the usual algebra and trigonometry there is nothing else necessary to study, except for general familiarity with the conic sections.’ (Ibid., p.362)
Also, in the appendix to an unpreserved letter from the end of 1865 or beginning of 1866 Marx explained to Engels the essentials of the differential calculus in an example of the problem of the tangent to the parabola.
However, he was still concerned first of all with the basics of mathematics in their connection with political economy. Thus in 1869, in relation to his studies of questions of the circulation of capital and the role of promissory notes in inter-governmental calculations, Marx familiarised himself with the long course of commercial arithmetic, Feller and Odermann, which he outlined in detail (cf.mss.2388 and 2400). It was characteristic of Marx’s survey techniques that, coming across some question of which he did not already feel himselft in command, Marx was not content until he had mastered it completely, down to its foundations. Every time Feller and Odermann used some mathematical technique, Marx considered it necessary to re-commit it to memory, even if it was known to him. In his surveys of commercial arithmetic - these and also much later ones, cf.mss.3881,3888,3981 - are found insertions, moreover, of purely mathematical content in which Marx advanced even further into fields of higher mathematics.
In the 1870s, starting in 1878, Marx’s thoughts on mathematics acquired a more systematic character. Concerning this period Engels in the introduction to the second edition of Capital:
‘After 1870 came another pause caused mainly by the painful illnesses of Marx. By habit, he usually filled his time studying; agronomy, American and especially Russian land relationships, monetary markets and banks, and finally natural science: geology and physiology, and particularly his own mathematical work, all go to make up the contents of numerous notebooks from this period.’ (Marx-Engels, Works, Vol;24, Berlin 1963, p.11)
At the same time the problems of applying mathematics to political economy continued to interest Marx. Thus in a letter to Engels of May 31, 1873 Marx wrote:
‘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 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.’ (Marx-Engels, Works, Vol.33, Berlin, 1966, p.82).
Thus it is clear that Marx was consciously leading up to the possibility of applying mathematics to political economy. Given the full text of all Marx’s mathematical manuscripts in the second part of our book, it still does not fully answer the question of what impelled Marx to proceed to the differential calculus from the study of algebra and commercial arithmetic. Indeed the mathematical manuscripts of Marx begin precisely in this period when Marx was concerned with elementary mathematics only in connection with problems arising from his study of differential calculus. His studies of trigonometry and the conic sections are found exactly in this context, which he suggested to Engels to be indispensable.
In differential calculus, however, there were difficulties, especially in its fundamentals - the methodological basis on which it was built. Much light was thrown on this condition in Engels’s Anti-Dühring.
‘With the introduction of variable magnitudes and the extension of their variability to the infinitely small and infinitely large, mathematics, in other respects so strictly moral, fell from grace; it ate of the tree of knowledge, which opened up to it a career of most colossal achievements, but at the same time a path of error. The virgin state of absolute validity and irrefutable certainty of everything mathematical was gone forever; mathematics entered the realm of controversy, and we have reached the point where most people differentiate and integrate not only because they understand what they are doing but from pure faith, because up to now it has always come out right.’ (Anti-Dühring, p.107)
Naturally Marx was not reconciled to this. To use his own words, we may say that ‘here, as everywhere’ it was important for him ‘to tear off the veil of mystery in science’. (see p.109) This was of the more importance, since the procedure of going from elementary mathematics to the mathematics of a variable quantity must be of an essentially dialectical character, and Marx and Engels considered themselves obliged to show how to reconcile the materialist dialectic not only with the social sciences, but also with the natural sciences and mathematics. The examination by dialectical means of mathematics of variable quantities may be accomplished only by fully investigating that which constitutes ‘a veil surrounded already in our time by quantities, which are used for calculating the infinitely small - the differentials and infinitely small quantities of various orders’. (Marx-Engels, Works, Vol.20, Berlin, 1962, p.30) Marx placed before himself exactly this problem, the elucidation of the dialectic of symbolic calculation, operating on values of the differential.
Marx thought about mathematics independently. The only person to whom he turned was his friend Samuel Moore, whose understanding of mathematics was at times rather limited. Moore could not render any essential help to Marx. Moreover, as can be observed in remarks that Moore made concerning the 1881 manuscripts (which Marx sent Engels) containing Marx’s expository ideas on the derivation and meaning of the symbolic differential calculus, Moore simply did not understand these ideas. (cf. Marx’s letter to Engels, this volume p.xxx)
Marx studied textbooks of differential calculus. He oriented himself with books used at courses in Cambridge University, where in the 17th century Newton held a chair of higher mathematics, the traditions of which were kept by the English up to Marx’s day. Indeed, there was a sharp struggle in the 20s and 30s of the last century between young English scholars, grouped about the ‘Analytical Society’ of mathematicians, and the opposing established and obsolete traditions, converted into untouchable ‘clerical’ dogma, represented by Newton. The latter applied the synthetic methods of his Principia with the stipulation that each problem had to be solved from the beginning without converting it into a more general problem which could then be solved with the apparatus of calculus.
In this regard, the facts are sufficiently clear that Marx began studying differential calculus with the work of the French abbot Sauri, Cours complet de mathématiques (1778), based on the methods of Leibnitz and written in his notation, and that he turned next to the De analyse per aequationes numero terminorum infinitas of. Newton (cf.ms.2763). Marx was so taken with Sauri’s use of the Leibnitzian algorithmic methods of differentiation that he sent an explanation of it (with application to the problem of the tangent to the parabola) in a special appendix to one of his letters to Engels.
Marx, however, did not limit himself to Sauri’s Cours. The next text to which he turned was the English, translation of a modern (1827) French textbook, J.-L. Boucharlat’s Eléments de calcul différentiel et du calcul intégral. Written in an ecletic spirit, in combined the ideas of d’Alembert and Lagrange. It went through eight editions in France alone and was translated into foreign languages (including Russian); the textbook, however, did not satisfy Marx, and he next turned to a series of monographs and survey-course books. Besides the classic works of Euler and MacLaurin (who popularised Newton) there were the university textbooks of Lacroix, Hind, Hemming, and others. Marx made scattered outlines and notations from all these books.
In these volumes Marx was interested primarily in the viewpoint of Lagrange, who attempted to cope with the characteristic difficulties of differential calculus and ways of converting calculus into an ‘algebraic’ form, i.e., without starting from the extremely vague Newtonian concepts of ‘infinitely small’ and ‘limit’. A detailed acquaintance with the ideas of Lagrange convinced Marx, however, that theses methods of solving the difficulties connected with the symbolic apparatus of differential calculus were insufficient. Marx then began to work out his own methods of explaining the nature of the calculus.
Possibly the arrangement of Marx’s mathematical writings as is done in the second half of the volume permits a clarification of the way in which Marx came onto these methods. We see, for example, beginning with the attempt to correct Lagrange’s outlook how Marx again turned to algebra with a complete understanding of the algebraic roots of the differential calculus. Naturally, his primary interest here was in the theorem of the multiple roots of an algebraic equation, the finding of which was closely connected with the successive differentiations of equations. This question was especially treated by Marx in the series of manuscripts 3932, 3933, appearing here under the titles ‘Algebra I’ and ‘Algebra II’. Marx paid special attention to the important theorems of Taylor and MacLaurin. Thus arrived his manuscripts 3933, 4000, and 4001, which are impossible to regard simply as outlines and the texts of which are, therefore, given in full.
Generally speaking in the outlines Marx began more and more to use his own notation. In a number of places he used special notation for the concept of function and in places dy/dx for 0/0. These symbols are met passim a number of other manuscripts (cf. 2763, 3888, 3932, 4302).
Convinced that the ‘pure algebraic’ method of Lagrange did not solve the difficulties of the foundations of the differential calculus and already having his own ideas on the nature and methods of the calculus, Marx once again began to collect textual material on the various ways of differentiating (cf. Mss. 4038 and 4040). Only after reading the expositions suggesting (for certain classes of functions) the methods of ‘algebraically’ differentiating, only after constructing sketches of the basic ideas did he express his point of view. These are exhibited here in the manuscripts and variants published in the first part of this volume. We now proceed to the contents of these manuscripts.
In the 1870s, from which date the overwhelming majority of Marx’s mathematical works, contemporary classical analysis and characteristic theories of the real numbers and limits were established on the European continent (principally in the works of Weierstrass, Dedeking and Cantor).
This more precise work was unknown in the English universities at that time. Not without reason did the well-known English mathematician Hardy comment in his Course of Pure Mathematics, written significantly later (1917): ‘It [this book] was written when analysis was neglected in Cambridge, and with an emphasis and enthusiasm which seem rather ridiculous now. If I were to rewrite it now I should not write (to use Prof. Littlewood’s simile) like a “missionary talking to cannibals”,’ (preface to the 1937 edition). Hardy had to note as a special achievement the fact that in monographs in analysis ‘even in England there is now [i.e., in 1937] no lack’.
It is not surprising therefore that Marx in his mathematical manuscripts may have been cut off from the more contemporary problems in mathematical analysis which were created at that time on the Continent. Nonetheless his ideas on the nature of symbolic differential calculus afford interest even now.
Differential calculus is characterised by its symbols and terminology, such notions as ‘differential’ and ‘infinitely small’ of different orders, such symbols as dx, dy, d²y, d³y ... dy/dx³, d²y/dx², d³y/dx³ and others. In the middle of the last century many of the instructional books used by Marx associated these concepts and symbols with special methods of constructing quantities different from the usual mathematical numbers and functions. Indeed, mathematical analysis was obliged to operate with these special quantities. This is not true at the present time: there are no special symbols in contemporary analysis; yet the symbols and terminology have been preserved, and even appear to be quite suitable. How? How can this happen, if the corresponding concepts have no meaning? The mathematical manuscripts of Karl Marx provide the best answer to this question. Indeed, such an answer which permits the understanding of the essence of all symbolic calculus, whose general theory was only recently constructed in contemporary mathematical logic.
The heart of the matter is the operational role of symbols in the calculus. For example, if one particular method of calculation is to be employed repeatedly for the solution of a range of problems then the special symbol appropriately chosen for this method briefly designates its generation, or as Marx calls it, its ‘strategy of action’. That symbol, which comes to stand for the process itself, as distinct from the symbolic designation introduced for the process, Marx called ‘real’.
Why then introduce an appropriately chosen new symbol for this? Marx’s answer consists in that this gives us the opportunity not to execute the entire process anew each time, but rather, using the fact of previously having executed it in several cases, to reduce the procedure in more complicated cases to the procedure of the more simple ones. For this it is only necessary, once the regularities of the particular method are well-known, to represent several general rules of operation with new symbols selected to accomplish this reduction. And with this step we obtain a calculus, operating with the new symbols, on its, as Marx called it, ‘own ground’. And Marx thoroughly clarifies, by means of the dialectic of the ‘inverted method’, this transition to the symbolic calculus. The rules of calculus allow us on the other hand not to cross over from the ‘real’ process to the symbolic one but to look for the ‘real’ process corresponding to the symbol, to make of the symbol an operator - the above-mentioned ‘strategy of action’.
Marx did all this in his two fundamental works written in 1881 and sent to Engels: ‘On the concept of the derived function’ (see p.3) and ‘On the differential’ (p.15). In the first work Marx considers the ‘real’ method, for several types of functions, to find the derived functions and differentials, and introduces appropriate symbols for this method (he calls it ‘algebraic’ differentiation). In the second work he obtains the ‘inverted method’ and transfers to the ‘own ground’ of differential calculus, employing for this aim first of all the theorem on the derivative of a product which permits the derivative of a product to be expressed as the sum of the derivatives of its factors. Employing his own words, ‘thus the symbolic differential coefficient becomes the autonomous starting point whose real equivalent is first to be found... Thereby, however,the differential calculus appears as a specific type of calculus which already operates independently on its own ground (Boden). For its starting points du/dx, dz/dx, belong only to it and are mathematical quantities characteristic of it.’ (pp.20-21). For this they ‘are suddenly transformed into operational symbols (Operationssymbole), into symbols of the process which must be carried out... to find their “derivatives”. Originally having arisen as the symbolic expression of the “derivative” and thus already finished, the symbolic differential coefficient now plays the role of the symbol of that operation of differentiation which is yet to be completed.’ (pp.20-21).
In the teachings of Marx there were not yet the rigorous definitions of the fundamental concepts of mathematical analysis characteristic of contemporary mathematics. At first glance the content of his manuscripts appear therefore to be archaic, not up to the requirements, say, of Lagrange, at the end of the 18th century. In actuality, the fundamental principle characteristic of the manuscripts of Marx has essential significance even in the present day. Marx was not acquainted with contemporary rigorous definitional concepts of real number, limit and continuity. But he obviously would not have been satisfied with the definitions, even if he had known them. The fact is Marx uses the ‘real’ method of the search for the derivative function, that is the algorithm, first, to answer the question whether there exists a derivative for a given function, and second, to find it, if it exists. As is well known, the concept of limit is not an algorithmic concept, and therefore such problems are only solvable for certain classes of functions. One class of functions, the class of algebraic functions, that is, functions composed of variables raised to any power, is represented by Marx as the object of ‘algebraic’ differentiation. In fact, Marx only deals with this sort of function. Nowadays the class of functions for which it is possible to answer both questions posed above has been significantly broadened, and operations may be performed on all those which satisfy the contemporary standards of rigour and precision. From the Marxian point of view, then, it is essential that transformations of limits were regarded in the light of their effective operation, or in other words, that mathematical analysis has been built on the basis of the theory of algorithms, which we have described here.
We are certainly well acquainted with Engels’s statement in the Dialectics of Nature that ‘the turning point in mathematics was Descartes’ introduction of variable quantities. Thanks to this movement came into mathematics and with it the dialectic and thanks to this rapidly became necessary differential and integral calculus, which arose simultaneously and which generally and on the whole were completed and not invented by Newton and Leibnitz’ (Dialectics of Nature p.258).
But what is this ‘variable quantity’? What is a ‘variable’ in mathematics in general? The eminent English philosopher Bertrand Russel says on this point, This, naturally, is one of the most difficult concepts to understand,’ and the mathematician Karl Menger counts up to six completely different meanings of this concept. To elucidate the concept of variables - in other words, of functions - and that of variables in mathematics in general, the mathematical manuscripts of Marx now represent objects of essential importance. Marx directly posed to himself the question of the various meanings of the concepts of function: the functions ‘of x’ and functions ‘in x’ - and he especially dwelt on how to represent the mathematical operation of change of variables, in what consists this change. On this question of the means of representation of the change of variables Marx placed special emphasis, so much so that one talks characteristically of the ‘algebraic’ method of differentiation, which he introduced.
The fact is, Marx strenuously objected to the representation of any change in the value of the variable as the increase (or decrease) of previously prepared values of the increment (its absolute value). It seems a sufficient idealisation of the real change of the value of some quantity or other, to make the assertion that we can precisely ascertain all the values which this quantity receives in the course of the change. Since in actuality all such values can be found only approximately, those assumptions on which the differential calculus is based must be such that one does not need information about the entirety of values of any such variable for the complete expression of the derivative function f’(x) from the given f(x), but that it be sufficient to have the expression f(x). For this it is only required to know that the value of the variable x changes actually in such a way that in a selected (no matter how small) neighbourhood of each value of the variable x (within the given range of its value) there exists a value x1, different from x, but no more than that. ‘x1 therefore remains just exactly as indefinite as x is.’ (p.88)
It stands to reason from this, that when x is change into x1, thereby generating the difference x1 - x, designated as Δx, then the resulting x1 becomes equal to x + Δx. Marx emphasised at this point that this occurs only as a result of the change of the value x into the value x1 and does not precede this change, and that to represent this x1 as known as the fixed expression x + Δx carries with it a distorted assumption about the representation of movement (and of all sorts of change in general). Distorted because in this case here, ‘although in x + Δx, Δx is equally as indeterminate in quantity as the undetermined variable x itself; Δx Is determined separetely from x, a distinct quantity, like the fruit of the mother’s womb, with which she is pregnant.’ (p.87)
In connection with this Marx now begins his determination of the derived function f’(x) from the function f(x) with the change of x into x1. As a result of this f(x) is changed into f(x1), and there arise both differences x1 - x and f(x1) - f(x), the first of which is obviously different from zero as long as x1 ≠ x.
‘Here the increased x, is distinguished as x1, from itself, before it grows, namely from x, but x1 does not appear as an x increased by Δx, so x1 therefore remains just exactly as indefinite as x is.’ (p.88)
The real mystery of differential calculus, according to Marx, consists in that in order to evaluate the derived function at the point x(at which the derivative exists) it is not only necessary to go into the neighbourhood of the point, to the point x1 different from x, and to form the ratio of the differences f(x1) - f(x) and x1 - x that is, the expression (f(x1) - f(x))/(x1 - x), but also to return again to the point x; and to return not without a detour, with special features relating to the concrete evaluation of the function f(x), since simply setting x1 = x in the expression (f(x1) - f(x))/(x1 - x) turns it into (f(x) - f(x))/(x - x), that is, into 0/0, or in other words into meaninglessness.
This character of the evaluation of the derivative, in which is formed the non-zero difference x1 - x and the subsequent - after the construction of the ratio (f(x1) - f(x))/(x1 - x) - dialectical ‘removal’ of this difference, is still preserved in the present-day evaluation of the derivative where the removal of the difference x1 - x takes place with the help of the limit transition from x1 to x.
In his work ‘Appendix to the manuscript “On the history of the differential calculus”, Analysis of the Method of d’Alembert’ Marx also spoke of the ‘derivative’ essentially as the limit of the value of the ratio (f(x1) - f(x))/(x1 - x), although he denoted it with other terms. In fact the confusion surrounding the terms ‘limit’ and ‘limit value’, concerning which Marx observed, ‘the concept of value at the limit is easily misunderstood and is constantly misunderstood’, prompted him to replace the term ‘limit’ with ‘the absolute minimal expression’ in the determination of the derivative. But he did not insist on this replacement, however, foreseeing that the more precise definition of the concept of limit, with which he familiarised himself in Lacroix’s long Traité du calcul différentiel et du calcul intégral - a text which satisfied Marx significantly more than others - could result further on in the introduction of unnecessary new terms. In fact Marx wrote of the concept of limit, ‘this category which Lacroix in particular analytically broadened, only becomes important as a substitute for the category “minimal expression”.’ (see p.68).
Thus Marx clarified the essentials of the dialectic connected with the evaluation of the derivative even in contemporary mathematical analysis. This dialectic, not a formal contradiction, makes as will be shown below, the differential calculus of Newton and Leibnitz appear ‘mystical’. To see this it is only necessary to recall that Marx by no means totally denied the representation of any change in the value of the variable as the addition of some ‘increment’ already having a value. On the contrary, when one speaks of the evaluation of the result of the already introduced change, one is induced to speak equally of the increase of the value of the variable (for example, of the dependence of the increase of the function on the increase in the independent variable), and ‘the point of view of the sum’ x1 = x + Δx or x1 = x + h, as Marx calls it, becomes fully justified. To this transition from the ‘algebraic’ method to the ‘differential’ one Marx specially devoted himself in his last work ‘Taylor’s Theorem’, which unfortunately remains unfinished and is therefore only partially reproduced in the first part of the present book. (A very detailed description of this manuscript of Marx, with almost all of the text, appears in the second part of the book, [Yanovskaya, 1968 pp.498-562]).
Here Marx emphasises that, while in the ‘algebraic’ method x1 - x consists solely for us as the form of a difference, and not as some x1 - x = h and therefore not as the sum x1 = x + h, in the transition to the ‘differential’ method we may view h ‘as an increment (positive or negative) of x. This we have a right to do, since x1 - x = Δx and this same Δx can serve, after our way, as simply the symbol or sign of the differences of the x’s, that is of x1 - x, and also equally well as the quantity of the difference x1 - x, as indeterminate as x1 - x and changed with their changing.
‘Thus x1 - x = Δx or = the indeterminate quantity h. From this it follows that x1 = x + h and f(x1) or y1 is transformed into f(x + h).’ (Yanovskaya, 1968 p.522)
In this way it would be unfair to represent the viewpoint of Marx as requiring the rejection of all other methods employed in differential calculus. If these methods are successful Marx sets himself the task of clarifying the secret of their success. And after this is shown to him, that is, after the examined method has demonstrated its validity and the conditions for its use are fulfilled, Marx considers a transition to this method not only fully justified but even appropriate.
Following his 1881 manuscript containing the fundamental results of his thoughts on the essence of differential calculus, Marx chose to send Engels a third work, concerned with the history of the method of differential calculus. At first, he wanted to depict this history with concrete examples of the various methods of showing the theorems on the derivation of the derivative, but then he relinquished this resolve and passed on to the general characteristics of the fundamental periods in the history of the methods of differential calculus.
This third work was not fully put into shape by Marx. There remain only the indications that he had decided to write about it and sketches of the manuscript, form which we know how Marx constructed and undertook the plan of his historical essay on this theme. This rough copy is published in full in the first part of this book (see pp.73-106). All of Marx’s indications that there should be introduced into the text this or that page from other manuscripts are here followed in full. The manuscript gives us the possibility to explicate Marx’s viewpoint on the history of the fundamental methods of differential 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.
The characteristic features of the methods of Newton and Leibnitz revealed, according to Marx, the fact that their creators did not see the ‘algebraic’ kernel of differential calculus: they began immediately with their operational formulae, the origins and the meaning of which remained therefore misunderstood and even mysterious, so that the calculus stood out as ‘a characteristic manner of calculation different from the usual algebra’ (p.84), as a discovery, a completely special discipline of mathematics as ‘different from the usual algebra as Heaven is wide’ (p.113).
To the question, ‘By what means... was the starting point chosen for the differential symbols as operational formulae’ Marx answers, ‘either through covertly or through overtly metaphysical assumptions, which themselves lead once more to metaphysical, unmathematical consequences, and so it is at that point that the violent suppression is made certain, the derivation is made to start its way, and indeed quantities made to proceed from themselves.’ (p.64)
Elsewhere Marx writes concerning the methods of Newton and Leibnitz: ‘x1 = + Δx from the beginning changes into x1 = x + dx ... where dx is assumed by a metaphysical explanation. First, it exists, then it is explained.’ ‘From the arbitrary assumption the consequence follows that ... terms ... must be juggled away, in order to obtain the correct result.’ (p.91)
In other words, so long as the meaning of introduction into mathematics of the differential symbols remains unexplained - more than that, generally false, since the differentials dx, dy are identified simply with the increments Δx, Δy - then the means of their removal appear unjustified, obtained by a ‘forcible’, ‘juggling’ suppression. We have to devise certain metaphysical, actually infinitely small quantities, which are to be treated simultaneously both as the usual different-from-zero (nowadays called ‘Archimedean’) quantities and as quantities which ‘vanish’ (transmute into zero) in comparison with the finite or infinitely small quantities of a lower order (that is, as ‘non-Archimedean’ quantities); or, simply put, as both zero and non-zero at the same time. ‘Therefore nothing more remains,’ writes Marx in this connection, ‘than to imagine the increments h of the variable to be infinitely small increments and to give them as such independent existence, in the symbols x., y. etc, or dx, dy [etc] for example. But infinitely small quantities are quantities, just like those which are infinitely large (the word infinitely [small] only means in fact indefinitely small); the dy, dx ... therefore also take part in the calculation just like ordinary algebraic quantities, and in the equation (y + k) - y or k = 2xdx + dxdx the dxdx has the same right to existence as 2xdx does.’ .. ‘the reasoning is therefore most peculiar by which it is forcibly suppressed’. (p.83)
The presence of these actually infinitely small, that is, formally contradictory, items which are not introduced by means of operations of mathematically grounded consistency but are hypothesised on the basis of metaphysical ‘explanations’ and are removed by means of ‘tricks’ gives the calculus of Newton and Leibnitz, according to Marx, a ‘mystical’ quality, despite the many advantages they bring to it, thanks to which it begins immediately with operating formulae.
At the same time Marx rated very highly the historical significance of the methods of Newton and Leibnitz. ‘Therefore,’ he writes, ‘mathematicians 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) result 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.’ (p.94)
The next stage in the development of the methods of differential calculus, according to Marx, was the ‘rational differential calculus’ of d’Alembert and Euler. The mathematically incorrect methods of Newton and Leibnitz are here corrected, but the starting point remains the same. ‘d’Alembert starts directly from the point de départ of Newton and Leibnitz, x1 = x + dx. But he immediately makes the fundamental correction: x1 = x + Δx, that is x and an undefined, but prima facie finite increment*2 which he calls h. The transformation of this h or Δx into dx ... is the final result of the development, or at the least just before the gate swings shut, while in the mystics and the initiators of the calculus as its starting point.’ (p.94) And Marx emphasised that with this the removal of the differential symbols from the final result proceeds then ‘by means of correct mathematical operation. They are thus now discarded without sleight of hand.’ (p.96)
Marx therefore rated highly the historical significance of d’Alembert’s method. ‘d’Alembert stripped the mystical veil from the differential calculus, and took an enormous step forward,’ he writes (p.97).
However, so long as d’Alembert’s starting point remains the representation of the variable x as the sum x + an existing element, independent of the variable x, the increment Δx - then d’Alembert has not yet discovered the true dialectic process of differentiation. And Marx makes the critical observation regarding d’Alembert: ‘d’Alembert begins with (x + dx) but corrects the expression to (x + Δx), alias (x + h); a development now becomes necessary in which Δx or h is transformed into dx, but all of that development really proceeds.’ (p.128)
As is well known, in order to obtain the result dy/dx from the ratio of finity differences Δy/Δx, d’Alembert resorted to the ‘limit process’. In the textbooks which Marx utilised, this passage to the limit foreshadowed the expansion of the expression f(x + h) into all the powers of h, in which revealed in the coefficient of h raised to the first power was the ‘already contained’ derivative f’(x).
The problem therefore became that of ‘liberating’ the derivative from the factor h and the other terms in the series. This was done naturally, so to speak, by simply defining the derivative as the coeficient of h raised to the first power in the expansion of f(x + h) into a series of powers of h.
Indeed, ‘in the first method 1), as well as the rational one 2), the real coefficient sought is fabricated ready-made by means of the binomial theorem; it is found at once in the second term of the series expansion, the term which therefore is necessarily combined with h¹. All the rest of the differential process then, whether in 1) or in 2), is a luxury. We therefore throw the needless ballast overboard.’ (p.98)
The same thing was done by Lagrange, the founder of the next stage in the development of the differential calculus: ‘pure algebraic’ calculus, in Marx’s periodisation.
At first Marx liked very much Lagrange’s method, ‘a theory of the derived function which gave a new foundation to the differential calculus’. Taylor’s theorem, with which was usually obtained the expansion of f(x + h) into a series of powers of h, and which historically arose as the crowning construction of the entire differential calculus, with this method was turned into the starting point of differential calculus, connecting it immediately with the mathematics preceding calculus (yet not employing its specific symbols). Marx noted with respect to this, ‘the real and therefore the simplest interconnection of the new with the old is discovered as soon as the new gains its final form, and one may say, the differential calculus gained this relation through the theorems of Talyor and MacLaurin.*3 Therefore the thought first occurred to Lagrange to return the differential calculus to a firm algebraic foundation.’ (p.113)
Marx found at once, however, that Lagrange did not make use of this insight. As is well known, Lagrange tried to show that ‘generally speaking’ - that is, with the exception of ‘several special cases’ in which differential calculus is ‘inapplicable’ - the expression f(x + h) is expandable into the series
f(x) + ph + qh² + rh³ + ...,
where p, q, r, ... the coefficients for the power of h, are new functions of x, independent of h, and ‘derivable’ from f(x).
But Lagrange’s proof of this theorem - in fact without much precise mathematic meaning - did not arise naturally. ‘This leap from ordinary algebra, and besides by means of ordinary functions representing movement and change in general is as a fait accompli, it is not proved and is prima facie in contradiction to all the laws of conventional algebra ...’ (p.177), writes Marx about this proof of Lagrange’s.
And Marx concludes with respect to the ‘initial equation’ of Lagrange, that not only is it not proved, but also that ‘the derivation of this equation from algebra therefore appears to rest on a deception’ (p.117).
In the concluding part of the manuscript the method of Lagrange appears as the completion of the method initiated by Newton and Leibnitz and corrected by d’Alembert; as the ‘algebraicisation’ based on Taylor by means of the method of formulae. ‘In just such a manner Fichte followed Kant, Schelling Fichte, Hegel Schelling, and neither Fichte nor Schelling nor Hegel investigated the general foundations of Kant, of idealism in general: for otherwise they would not have been able to develop it further.’ (p.119)
We can see that in a historical sketch Marx gives us a graphic example of what in his opinion should be the application of the method of dialectical materialism in such a science as the history of mathematics.
Completion of the present edition of Mathematical Manuscripts of Karl Marx required a great deal of preparation. The text of the manuscripts was translated in full; they were arranged chronologically; excerpts and summaries were separated from Marx’s own statements; on the basis of analysis of their mathematical content the manuscripts were collected into units which can be read as a whole (in fact, many of the manuscripts do not make up notebooks, but are rather of separate sheets of paper in no sort of order). In the vast majority of cases it is known from which source Marx drew his excerpts, or which he summarised. By comparison with the original works all of Marx’s own comments have been identified in the summaries; all of Marx’s independent work and notes have been translated into Russian.
The task of separating the personal opinions of Marx from his summaries and excerpts involved a series of difficulties. Marx wrote his summaries for his own benefit, in order to have at hand the material he needed. As always, he made use of a large collection of the most varied sources, but if he did not consider the account worth special attention, if it was, for example, a contemporary textbook compiled and widely distributed in England, then Marx very frequently did not accompany his excerpts with an indication of from where they were drawn. The task is complicated still further by the fact that the majority of the books which Marx utilised are now bibliographical rarities. In the final analysis all this work could only be completed at first hand in England, where, in order to resolve this problem, were studied and investigated in detail the stocks of the extant literature in these libraries: the British Museum, London and Cambridge universities, University College London, Trinity and St. James’s Colleges in Cambridge, the Royal Society in London, and finally the private libraries of the eminent 19th century Englishmen de Morgan and Graves. Inquiries were made in other libraries as well, such as that of St. Catherine’s College. For those manuscripts which by nature were prepared from German sources, the German historian of mathematics Wussing, at the request of the Institute, investigated the bibliographical resources of the German Democratic Republic.
Photocopies of several missing pages of the manuscripts were kindly provided by the Institute of Social History in Amsterdam, where the originals of the mathematical manuscripts of K. Marx are preserved.
Since the manuscripts are of the nature of rough drafts, one encounters omissions and even errors in the copied excerpts. The corresponding insertions or corrections are enclosed in square brackets. As a result the square brackets of Marx himself are indicated with double square brackets. Words which Marx abbreviated are written out in full, but the text Is basically unchanged. In places obsolete orthography is even preserved.
The primary language of the manuscripts is German. If a reference in the manuscripts is in French or English, Marx sometimes writes his comments in French or English. In such cases Marx’s text turns out to be so mixed that it becomes hard to say in what particular language the manuscript is written.
The dating of the manuscripts also entailed great difficulties. A detailed description of these difficulties is presented in the catalogue of manuscripts. This last lists the archival number of the manuscript, its assigned title, and the characteristics of either its sources or its content. Where the title or subtitle is Marx’ own it is written in quotation marks in the original language and in Russian translation. In the first part of the book the titles not originating with Marx are marked with an asterisk.
The inventory of the manuscripts is given in the sequence of the arrangement of the archival sheets. Marx’s own enumeration, by number of letters, is given in the inventory together with the indication of the archival sheets. An indication of the archival sheets on which they are found accompanies the published texts. All the manuscripts stem from fond 1, . opuscule 1.
The language of Marx’s mathematical manuscripts in many cases departs from our usual contemporary language, and in order to understand his thought it is necessary to refer to the sources he used, to make clear the meaning of his terms. In order not to interrupt Marx’s text, we place such explanations in the notes at the end of the book. Then, where more detailed information about the subject-matter of the sources consulted by Marx is found necessary, it is given in the Appendix. All such notes and references are of a purely informational character.
In Marx’s text are a great number of underlinings, by means of which he emphasised the points of particular importance to him. All these underlinings are indicated by means of italics.
The book was prepared by S.A. Yanovskaya, professor of the M.V. Lomonosov Moscow Government University, to whom also are due the Preface, the Inventory of mathematical manuscripts (compiled with the assistance of A.Z. Rybkin), the Appendices and the Notes. Professor K.A. Rybnikov took part in the editing of the book, performing among other tasks the greater part of the work of researching the sources used by K. Marx in this work on the ‘Mathematical Manuscripts’. In the preparation of the present edition the comments and advice of Academicians A.N. Kolmogorov and I.G. Petrovskii were carefully considered.
A.Z. Rybkin, chief editor for the physical-mathematical section of Nauka Press, and O.K. Senekina, of the Institute for Marxism-Leninism of the Central Committee of the Communist Party of the Soviet Union, directed all the work of editing the book, preparing it for publication and proof-reading it. The book includes an index of references quoted and consulted, as well as an index of names. References in Marx’s text are denoted in the indices by means of italics.
This volume contains a translation of the first part only.
*2 By ‘finite increment’ the literature which Marx consulted understood a non-zero finite increment - S.A. Yanovskaya
*3 MacLaurin’s Theorem can be regarded - as it was by Mar (pp.111,112) - as a special case of Taylor’s Theorem. - Ed.