Marx's Mathematical Manuscripts and Labour Theory of Value

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Manuscripts and Labour Theory of Value Andrea Ricci

1 University of Urbino From the SelectedWorks of Andrea Ricci September 12, 2018 Marx's Mathematical Manuscripts and Labour Theory of Value Andrea Ricci Available at:

POLITICAL ECONOMY September 12, 2018, Juraj Dobrila University of Pula,

2 Marx s Mathematical Manuscripts and Labour Theory of Value. Andrea Ricci (University of Urbino, Italy) 9TH ANNUAL CONFERENCE IN POLITICAL ECONOMY September 12, 2018, Juraj Dobrila University of Pula, Croatia FET Juraj Dobrila University of Pula Faculty of Economics and Tourism Dr. MijoMirkovic

3 Some literature on Marx and Maths. D. Struik, Marx and Mathematics, Science and Society, 12(1), pp , Winter L. Smolinski, Karl Marx and Mathematical Economics, Journal of Political Economy, Vol. 81, No. 5 (Sep. - Oct., 1973), pp A. Guerraggio e F. Vidoni, Nel laboratorio di Marx: scienze naturali e matematica, Milano, Franco Angeli editore, P. Gerdes, Marx demystifies calculus, Studies in Marxism, Vol. 16, Minneapolis (USA), MEP Publications, University of Minnesota, 1985; P. Baksi (ed.), Special Supplement, in K. Marx, Mathematical Manuscripts, Calcutta (India), Viswakos Parisad, 1994; P.H. Matthews, The Dialectics of Differentiation, Middlebury Vermont, Middlebury College Economics Discussion Paper n , 2002; H. Kennedy, Negation of the Negation: Karl Marx and Differential Calculus, Concord California, Peremptory Publications, A. Alcouffe, Marx, Hegel et le Calcul, in K. Marx, Les Manuscrits Mathematiques, Paris, Economica, Carchedi, Dialectics and Temporality in Marx's Mathematical Manuscripts, Science and Society, vol. 72(4), October 2008, pp A. Alcouffe e J. Wells, Marx, maths and MEGA2, MPRA Paper n , April A. Ricci, La matematica di Marx, Lettera Matematica, forthcoming, 2018.

4 Edition of Marx s Mathematical Manuscripts (MMM) 1883: In every single field, wherever Marx has conducted investigations, even in mathematics, he has obtained independent results (Engels funeral speech for Karl Marx). 1885: Engels preface to the second edition of the Anti-Dühring 1920s: Gumbel s project of MMMs publication 1931: The edition of the manuscripts was charged to a team of Soviet mathematicians, coordinated by Yanovskaya 1968:Yanovskaya s edition of MMM. 1994: English translation of MMM s Yanovskaya s edition. Now: We are waiting for MMM s publication within the MEGA 2 project.

5 Marx s study of mathematics 1835, Marx s graduation at the Lyceum Gymnasium in Trier: good knowledge of mathematics In London, during the 1850s, Marx felt the need to resume and deepen the young mathematical knowledge. Marx s new interest in mathematics coincided with the rediscovery of Hegelian dialectics. In the last years of his life, mathematics assumed an increasing role in Marx's theoretical work. The vast majority of mathematical manuscripts were written in 1870s and 1880s. He had a special kind of inclination to mathematics. Algebra even provided him with mental solace, be used to study this branch of mathematics during the most painful moments of his eventful life. He could not concentrate upon his usual scientific activities during the last days of his wife's illness, in those days he could forget the pain that seared his mind owing to her illness, only by studying mathematics (Lafargue, Memories,1890)

6 The contents of MMM The mathematical manuscripts vary from simple notes and extracts, to essays ready for publication. They are primarily dedicated to differential calculus, for a total of over a thousand handwritten pages. The texts used by Marx, in the study of differential calculus, were those used at the University of Cambridge. Marx was never aware of the new developments in the field of differential analysis in continental Europe, on the basis of the concept of limit of Cauchy. Marx's method is historical-genetic, identical to that used in the critique of political economy. It starts from the critical analysis of the theoretical development of the differential calculus, to enucleate the internal logical contradictions.

7 The history of differential calculus The problem of Marx is the dissatisfaction with the logical and conceptual foundation of differential calculus. Marx identifies three different methods in the theory of differential calculus. The "mystic" method of Newton and Leibniz: differentials (dy and dx) as infinitely small quantities. Astonishing results were obtained with an incorrect mathematical procedure (Marx, MMM). The "rational" method of D'Alembert and Eulero: finite increments (Δy and Δx) and the hell of (0/0). The "purely algebraic" method of Lagrange: a technique of calculation.

8 Marx s foundation of differential calculus Marx wants to find a method for calculating the derivative, directly from the process of variation of the function, in such a manner that its algebraic, real origin is met. A new starting point. Previous methods: x 1 = x + Δx the increment is added as a separate quantity, and has the nature of a constant. Marx s method: x 1 - x = Δx the increment is the product of the change, and represents a variable quantity. Two consequences: a) the immediate application of the binomial theorem is no longer possible; b) the definition of the increment, in terms of negation, sets in motion a dialectical development. In the final equation, the increments, in the form of the incremental ratio (Δy / Δx), appear in the left-hand side of the equation, while the right-hand side contains only expressions in x 1 and x.

9 Marx s foundation of differential calculus Qualitative difference between the two sides of the equation: the left member has symbolic nature, while the right member has algebraic nature. The first represents the symbolic expression of the real process of change, that takes place entirely in the right part of the equivalence. The zeroing of the increments has effect only on the left-hand side of the equation, which is reduced to the (0/0) expression, leaving the right-hand member unchanged. The (0/0) ratio is a purely symbolic operator, which can be replaced by the differential ratio (dy / dx), without any logical contradiction. The differential ratio is a unitary operational symbol, indicating an ordered set of logical and algebraic operations, necessary to calculate the derivative of a function. The differential as an operational symbol presents strong analogies with the modern concept of algorithm, and this makes Marx a precursor of modern computational mathematics.

10 MMM and Labour Theory of Value The general task of differential calculus is how to find the real equivalent for the symbolic differential coefficient which has become an independent point of departure, as shadow without the body (Marx, MMM). In the same way, the task of labour theory of value is to find the real equivalent for the exchange value, which appears as an independent starting point in the surface of capital circulation. Conceptual analogies: o function and value; o independent variable x and socially necessary labour or abstract labour; o dependent variable y and exchange value; o functional change and value creation; o functional increment of x and living labour; o functional increment of y and net product; o derivative and magnitude of value; o derivative in one point and real measure of value.

11 MMM and Labour Theory of Value The function (value) is a relation between independent variable x (necessary labour) on the right side, and dependent variable y (exchange value) on the left side. Qualitative difference: left member (exchange value) is the the symbolic form of manifestation of the real algebraic right member (necessary labour). The change in x (necessary labour) causes the change in y (exchange value) and not vice versa. Since y is a dependent variable, it can by no means accomplish any independent movement. (Marx, MMM) The (exchange) value of a commodity varies directly as the quantity of the labour incorporated in it. (Marx, Capital, Volume 1, First chapter)

12 MMM and Labour Theory of Value In labour theory of value, under the assumption of wages paid ex-post, the increment of y corresponds to the new exchange value created in a period, or net product. It derives from the difference between original and increased quantities of x, corresponding to living labour: Δx = x 1 x Δy = f (x 1 x) Surplus came from a subtraction, and not from an addiction. The derivative (magnitude of value) quantitatively expresses the relationship of dependency, by the differential ratio (net product / living labour). It represents the power of x (necessary labour) to change (create) y (exchange value): (dy/dx) = f (x)

13 MMM and Labour Theory of Value The derivative (magnitude of value) is a symbolic operator representing quantitatively the real process of change (value production). The differentials dx and dy (net product and living labour) are indissolubly interlinked as numerator and denominator in the derivative. The particular, numerical value of the derivative in one point corresponds to the real measure of value in a given period, represented by the Monetary Expression of Labour Time in a given period: MELT = (monetary value added / living labour time) Numerator and denominator (money and living labour) cannot be studied in isolation, outside of their symbiotic relationship.

14 Conclusions Logical foundation of MELT. My opinion is that in the last years of his life Marx was looking for expressing the logical foundations of the labour theory of value in more formal terms. Hence, his interest on differential calculus. In higher mathematics Marx saw the most consistent and at once the most simple expression of dialectical movements. He was of the opinion, that so long as a science does not get used to the use of mathematics, it can not be called a truly mature science. (Paul Lafargue, son-in-law of Karl Marx)

Critical Analysis of the Mathematical Formalism of Theoretical Physics. I. Foundations of Differential and Integral Calculus

Critical Analysis of the Mathematical Formalism of Theoretical Physics. I. Foundations of Differential and Integral Calculus Temur Z. Kalanov Home of Physical Problems, Pisatelskaya 6a, 100200 Tashkent,