Marx and mathematics-1:
Marx and the calculus

C. K. Raju

Marx’s relation to mathematics has always been a troubling question. However, when looking into these issues in a “historical context”¹ one needs a history based on solid evidence, else one can draw all the wrong conclusions. This involves the history of mathematics as much as it does the history of Marx’s engagement with mathematics. As a teacher of mathematics and its history and philosophy, perhaps I can shed some light on the issue, hopefully without generating too much heat.

Much has been written on Marx and mathematics, particularly on Marx and the calculus, which will be my sole concern here. Some key sources are the articles by Struik,² Kennedy,³ and the book on the mathematical manuscripts of Karl Marx⁴ by Dr Baksi whose scholarship is meticulous though I must confess I have yet to read its 2^nd edition.

My aim in these articles will not be to provide a rehash of the much that has already been written, but to add a significant new point which has been missed by ALL Western scholars till date. Basically, the questions which will concern me are (1) Why did Marx have so much difficulty in understanding the calculus? (And why did he refer to the calculus of Newton and Leibniz as “the mystical calculus”?) (2) What effect, if any, does the new understanding articulated here have on Marxism?

Struik’s thesis

Before beginning on my thesis, however, let me summarise the purva paksa of Struik’s thesis. As Struik (a noted Marxist historian of math) pointed out (in the pre-McCarthy era):

"Marx studied the calculus from textbooks which were all written under the direct influence of the great mathematicians of the late seventeenth and the eighteenth centuries, notably of Newton, Leibniz, Euler, D'Alembert and Lagrange. He was not so much interested in the technique of differentiation and integration as in the basic principles on which the calculus is built, that is, in the way the notions of derivative and differential are introduced. He soon found out that a considerable difference of opinion existed among the leading authors concerning these basic principles, a difference of opinion often accompanied by confusion....Different answers were given on such questions as whether the derivative is based on the differential or vice versa, whether the differential is small and constant, small and tending to zero, or absolutely zero, etc. Marx felt the challenge offered by a problem which had attracted some of the keenest minds of the past and which dealt with the very heart of the dialectical process, namely the nature of change.”

This descriptive part of Struik’s thesis is quite acceptable. Struik further goes on to correctly point out that

"Marx’s...feeling of dissatisfaction with the way the calculus was introduced was shared by some of the leading younger professional mathematicians of his day. In the same year (1858) in which Marx resumed his study of mathematics, Richard Dedekind at Zürich felt similar dissatisfaction, in his case while teaching the calculus. Writing in 1872, he first stated that in his lessons he had recourse to geometrical evidence to explain the notion of limit; then he went on: “But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a...perfectly rigorous foundation for the principles of infinitesimal calculus.” This led Dedekind [1878] to a new axiomatic concept of the continuum [formal real numbers]....”

Struik’s conclusion is that “This work [including the work of Cauchy, Weierstrass etc.] appeared too late to influence Marx and Engels...The result is that Marx' reflections on the foundations of the calculus must be appreciated as a criticism of eighteenth century methods.”

I completely agree with the first part of Struik’s thesis that Marx had difficulty in understanding the calculus, not due to any personal shortcoming, but because European mathematicians collectively did not properly understand the calculus in Marx’s time. The point of disagreement is the second part of Struik’s thesis whether such a proper understanding emerged at the end of the 19^th c. with Dedekind, Cauchy etc.

Now it is true that Dedekind cuts (as they are called, or formal real numbers) are today taught as the correct basis of the calculus.⁵ (Formal real numbers can also be constructed as equivalence classes of Cauchy sequences.⁶) However, Struik wrongly supposes, as per the stock Western myth, that Dedekind cuts (or Cauchy sequences) had resolved the problems of the foundations of calculus at the end of the 19^th c.

Thus, for example, “Cauchy” sequences alone won’t do: the “Cauchy” sequence of successive approximations to square root of 2, 1.4, 1.41, 1.414..., will not converge in the system of rational numbers. It will also NOT converge to a unique limit in a (non-Archimedean) system of numbers larger than the “real” numbers. So, to ensure the convergence of all Cauchy sequences, the exact system of “real” numbers (neither smaller nor larger) is needed. And the fact is that set theory (a metaphysics of infinity) is essential to construct formal real numbers as Dedekind cuts (or as equivalence classes of Cauchy sequences).

The only set theory available in the 19^th c. was Cantor’s set theory which was full of all sorts of paradoxes. That is, using Cantorian set theory one could easily derive a contradiction (as in Russell’s paradox), hence draw any nonsense conclusion whatsoever. Therefore, Cantorian set theory was abandoned. That is, in the 19^th c., in prospect (rather than in Struik’s retrospective view), Dedekind cuts (or Cauchy sequences) did NOT result in a rigorous solution of the problem of calculus. That had to await (at least) the coming of axiomatic set theory in the 20^th c. The best that could be said of Dedekind cuts (or equivalence classes of Cauchy sequences) , then, was that they pushed the doubts and confusion about calculus into the doubts and confusion about set theory. This made many Western mathematicians socially more comfortable. But that degree of social comfort is no indicator of truth.

Thus, in fact, the term Dedekind “cut” arises in relation to the first proposition of “Euclid’s” Elements. The proposition is to construct an equilateral triangle on a given line segment. The proof involves the construction of two arcs of equal radius, and with centers at either end of the line segment. (This is called the fish-figure in Indian tradition, used to decide cardinal directions, and the fish symbol was sacred to Christians in the 4^th c., not solely because it embodies a cross.⁷) Anyway, the two arcs are seen to intersect. But this may be an illusion: on a pixelated computer screen, the two arcs though seen to intersect may intersect or may not intersect in a common point. There is no axiom in the book “Euclid’s” Elements from which the intersection of the two arcs can be deduced. But ALL Western mathematicians were comfortable with the proof of the proposition in the book as a model of “deductive” proof for some seven centuries, declaring it to be both axiomatic and infallible! Dedekind woke up the Western world, and it was only after that that Russell declared the proofs in the book “Euclid’s” Elements to be a bundle of errors.⁸

Dedekind cuts supposedly provided the correction to this “error”, to make the proof of the first proposition axiomatic, but Dedekind’s work was incomplete without the axiomatic set theory which came about only in the 20^th c.

It would, however, be only fair to point out that there are still doubts even about axiomatic set theory, since its consistency has NOT been established, and it still results in paradoxes such as the Banach-Tarski paradox that a ball of gold can be subdivided into a finite number of pieces which can be reassembled into two balls of gold identical to the first! (That is, if this applied to the real world, one could get infinitely rich just using set theory! This is the same axiomatic set theory needed for “real” numbers.)

Anyway, setting aside doubts about axiomatic set theory etc., the definite fact is that contrary to what Struik (and many others) asserted, the calculus was NOT properly understood in the West even by the end of the 19^th c., or the beginning of the 20^th.

Thus, to understand the source of Marx’s difficulties with the calculus, the key question we need to ask is one which no Westerner has asked so far: how exactly could Newton and Leibniz have “discovered” the calculus without understanding? For, there was certainly no clear (European) understanding of calculus (by all accounts) even two centuries after them in the 19^th c.

(To be continued)

Notes