Karl Marx and Mathematics

Pradip Baksi, Pradip Baksi

• Éditeur: Routledge
• Année: 2020
• ISBN:
  9781000263480
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: Calculus

This collection of various texts on Karl Marx and Mathematics is the revised and extended second edition of the Special Supplement to Karl Marx, Mathematical Manuscripts (1994; Calcutta: Viswakos) titled Marx and Mathematics. The sources of the texts included in the three parts of this collection and, some biographical information about their respective authors have been indicated at the end of each text.

The emergence and development of the Ethnomathematics movement continue to change our understanding of the history of evolution of plural mathematics on planet earth since the Neolithic age. Rediscovery and study of some of the neglected source texts have further energized investigations on the subsequent history of mathematical cultures, including those on the histories of algebra and analysis in some of the ancient and medieval languages of Asia, like Sanskrit, Arabic and Malayalam. Consequently, it is now possible to indicate some of the larger gaps in the dominant understanding of history of mathematics not only in Marx's time, but also at the time of editing Marx's mathematical manuscripts in the twentieth century, and even today. Finally, the emergence and development of mathematical and statistical software packages are vigorously reshaping our ways of conceptualizing and doing mathematics towards an unknown future. It is time now for taking yet another look at all mathematical text from the past and that includes the mathematical manuscripts of Marx.

These texts have been divided into three parts. Part one contains some topical texts related to the history of emergence, development, editing, publication and reception of the mathematical manuscripts of Karl Marx. Part two contains a selection of five articles reflecting some of the investigations inspired by these manuscripts in Russia, India and France. Part three contains five articles on plural mathematics before and after Karl Marx (1818-1883). The texts in this collection are followed by two appendices containing two bibliographies: one on Hegel and mathematics and, the other on mathematics and semiotics.

Please note: This title is co-published with Aakar Books, Bew Delhi. Taylor & Francis does not sell or distribute the print edition in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan, Maldives or Bhutan).

Part One
History

On the History of Emergence, Development, Reception and Editing of the Mathematical Manuscripts of Marx

EXCERPTS FROM LETTERS

Marx’s mathematical investigations have been discussed in a number of letters of Marx and Engels. English translations of the relevant sections of some of the hitherto published relevant letters are being reproduced here.

Marx to Engles

In Manchester

[London,] 11 January [1858]

Dear Frederick,

* * *

In elaborating the PRINCIPLES of economics^1* I have been so damnably held up by errors in calculation that in DESPAIR I have applied myself to a rapid revision of algebra. I have never felt at home with arithmetic. But by making a detour via algebra, I shall quickly get back into the way of things.

* For the notes see pp. 13–14.

* * *

your
K.M.

[MECW, 40, 244]

Marx to Engels

In Manchester

London, 6 July 1863

Dear Engels,

* * *

… My spare time is now devoted to differential and integral calculus. Apropos, I have a superfluity of works on the subject and will send you one, should you wish to tackle it. I should consider it to be almost essential to your military studies. Moreover, it is a much easier branch of mathematics (so far as mere technicalities are concerned) than, say, the more advanced aspects of algebra. Save for a knowledge of the more ordinary kind of algebra and trigonometry, no preliminary study is required except a general familiarity with conic sections.

* * *

your
K.M.

[MECW, 41, 484]

Engels to Friedrich Albert Lange ²

In Duisburg

Manchester, 29 March 1865
7 Southgate

Dear Sir,

* * *

There is a remark about old Hegel which I cannot let pass without comment: you deny him any deeper knowledge of the mathematical sciences. Hegel knew so much mathematics that none of his disciples was capable of editing the numerous mathematical manuscripts that he left behind³. The only man who, to my knowledge, has enough understanding of mathematics and philosophy to be able to do so is Marx.

* * *

Yours very respectfully
Friedrich Engels

[MECW, 42, 138]

Marx to Engels

In London

[Manchester,] 31 May 1873
25 Dover Street

Dear Fred,

* * *

I have been telling Moore⁴ about a problem with which I have been racking my brains for some time now. However, he thinks it is insoluble, at least pro tempore, because of the many factors involved, factors which for the most part have yet to be discovered. The problem is this: you know about those graphs in which the movements of prices, discount rates, etc., etc., over the year etc., are shown in rising and falling zigzags. I have variously attempted to analyse crises by calculating these UPS AND DOWNS as irregular curves and I believed (and still believe it would be possible if the material were sufficiently studied) that I might be able to determine mathematically the principal laws governing crises⁵. As I said, Moore thinks it cannot be done at present and I have resolved to give it up FOR THE TIME BEING.

* * *

your
K.M.

[MECW, 44, 504]

Engels to Marx

In London

August 18, 1881

Dear Moor⁶,

… yesterday I found the courage at last to study your mathematical manuscripts⁷ even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing is as clear as daylight, so that we can wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put $\frac{dy}{dx}=\frac{0}{0}$, firmly and point blank, does not enter their skulls. And yet it is clear that $\frac{dy}{dx}$ can only be the pure expression of a completed process it the last trace of the quanta x and y has disappeared, leaving the expression of the preceding process of their change without any quantity.

You need not fear that any mathematician has preceded you here. This kind of differentiation is indeed much simpler than all others, so that just now I applied it myself to derive a formula I had suddenly lost, confirming it afterwards in the usual way. The procedure must have made the greatest sensation, especially, as is clearly proved, since the usual method of neglecting dxdy etc. is positively false. And that is the special beauty of it; only if $\frac{dy}{dx}=\frac{0}{0}$, is the mathematical operation absolutely correct.

So old Hegel guessed quite correctly when he said that differentiation had for its basic condition, that the variables must be raised to different powers, and at least one of them to at least the second, or 1/2, power⁸. Now we also know why.

If we say that in y = f(x) the x and y are variables, then this claim has no further consequences, as long as we do not move on, and x and y are still pro tempore, in fact constants. Only when they really change, i.e., inside the function, do they indeed become variables, and only then can the relation still hidden in the original equation reveal itself—not the relation of the two magnitudes but of their variability. The first derivative $\frac{\Delta y}{\Delta x}$ shows this relation as it happens in the course of a real change, i.e., in each given change; the completed derivative $\frac{dy}{dx}$ shows it in its generality, pure, and hence we can come from $\frac{dy}{dx}$ toeach $\frac{\Delta y}{\Delta x}$, while the latter itself only covers the special case. However, to pass from the special case to the general relationship, the special case must be sublated (aufgehoben) as such.

Hence, after the function has passed through the process from x to x′ [in the notation now in use: to x₁—Ed.] with all its consequences, x′ can be allowed calmly to become x again; it is no longer the old x, which was variable in name only; it has passed through actual change, and the result of the change remains, even if we again sublate (aufheben) it.

At last we see clearly, what mathematicians have claimed for a long time, without being able to present rational grounds, that the (derivative or) differential quotient is the original, the differentials dx and dy are derived: the derivation of the formulae demands that both so-called irrational [here “irrational” means “non-rational”—Ed.] factors stand at the same time on one side of the equation, and only if you put the equation back into its first form $\frac{dy}{dx}=f^{\prime }(x)$, as you can, are you free of the irrationals [i.e. “non-rationals”—Ed.] and instead have their rational expression.

The thing has taken such a hold of me that it not only goes round my head all day, but last week in a dream I gave a chap my shirt-buttons to differentiate, and he ran off with them.

Yours
F.E.

[Mathematical Manuscripts of Karl Marx, London, 1983, XXVII-XXVIIL]

[For another translation of this letter, see: MECW, 46, 130–132]

Engels to Marx

In Ventnor

London, November 21, 1882

Dear Moor,

… enclosed a mathematical essay by [Samuel] Moore. The conclusion that “the algebraic method is only the differential method disguised” refers of course only to his own method of geometrical construction and is pretty correct there too. I have written to him that you place no value on the way the thing is represented in geometrical construction, the application to the equation of curves being quite enough. Further, the fundamental difference between your method and the old one is that you make x change to x′ [in the notation now in use: to x₁—Ed.], thus making them really vary, while the other way starts from x + h, which is always only the sum of two magnitudes, but never the variation of a magnitude. Your x therefore, even when it has passed through x′ and again became the first x, is still other than it was; while x remains fixed the whole time, if h is first added to it and then taken away again. However, every graphical representation of the variation is necessarily the representation of the completed process, of the result, hence of a quantity which became constant, the line x; its supplement is represented as x + h, two pieces of a line. From this it already follows that a graphical representation of how x′ again becomes x, is impossible …

Your
F.E.

[Mathematical Manuscripts of Karl Marx, London, 1983, p. XXIX]

[For another translation of this letter, see: MECW, 46, 378–379]

Marx to Engels

In London

November 22, 1882
1, St. Boniface Gardens,
Ventnor.

Dear Fred,

… Sam, as you saw immediately, criticises the analytical method applied by me by just pushing it aside, and instead busies himself with the geometrical application, about which I said not one word. In the same way, I could get rid of the development of the proper so-called differential method—beginning with the mystical method of Newton and Leibniz, and then going on to the rationalistic method of d’Alembert and Euler, and finishing with the strictly algebraic method of Lagrange (Which, however, always begins from the same original basic outlook [as that of] Newton-Leibniz)—I could get rid of this whole historical development of analysis by saying that practically nothing essential has changed in the geometrical application of the differential calculus, i.e., in the geometrical representation.

The Sun is now shining, so the moment for going for a walk has come, so no more pro nunc of mathematics, but I’ll come back later to the differential methods occasionally in detail …

Yours
K.M.

[Mathematical Manuscripts of Karl Marx, London, 1983, p. XXX]

[For another translation of this letter, see: MECW, 46, 380]

EXCERPTS FROM REMINISCENCES

Marx’s friends and comrades have mentioned his mathematical studies in their reminiscences or elsewhere. English translations of the relevant portions of some such writings are being published here.

Engels’ Speech At Marx’s Funeral (Excerpts)

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… in ev...