Ralph Dumain rdumain at igc.apc.org
Fri Apr 7 00:36:38 MDT 1995

Long ago and far away in cyberspace I uploaded a short
bibliography relating to Marx's mathematical manuscripts,
particularly his philosophy of the calculus.  If anyone is
interested, I will try to find this information and upload it
here.  In the meantime, here is an abstract I just wrote of an old
article on the subject I recently came across.

Struik, Dirk J.  "Marx and mathematics", in A CENTENARY OF
MARXISM, ed. by Samuel Bernstein et al; New York: Science &
Society, 1948; p. 181-196.

Marx produced about 900 pages of manuscript material on
mathematics.  He began to develop his own views after 1870 and
pursued this interest to the time of his death.  Marx shared some
of his researches with Engels and Sam Moore.  Marx wrote on
algebra and analytical geometry, but Struik describes his work on
the calculus.

Marx learned the calculus from numerous textbooks inspired by the
various approaches of the 17th- and 18th century mathematicians.
Marx was unsatisfied with the presentation of the subject matter
in the available textbooks.  (He was not alone in this; leading
mathematicians agreed on this point.)  Marx apparently was not
acquainted with the work of contemporaneous developments in
mathematics (esp. of Cauchy and Dedekind) which began to
straighten out the difficulties in the foundations of calculus,
and in any event this work had not yet been incorporated into
textbooks.  Hence Marx should be seen as a critic not of his
contemporaries but of 18th century mathematics.

Marx classified the various approaches to differentiation into
three basic categories: the mystical (Newton, Leibniz), the
rational (D'Alembert), and the algebraic (Lagrange).  Struik goes
into these approaches to differentiation and Marx's criticisms of
them in great detail, which I will not reproduce here.  Marx was
not satisfied that the process of differentiation was adequately
clarified by various mathematicians.  He favored the algebraic
approach of Lagrange.

Let me just quote a couple of the more readable passages.

"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 .... Marx felt the challenge offered
by a problem .... which dealt with the very heart of the
dialectical process, namely the nature of change.  Not finding any
satisfying answer in the books, he tried to reach an answer for
himself in his own typical way ...." [p. 185]

"This last remark of Marx shows affinity with that of Dedekind,
who also endeavored to build up the calculus independent of the
geometrical representation of the derivative.  We can consider
this as one of the characteristics of Marx' [sic] analysis, in
which it agreed with our modern approach.  Another important
feature was his insistence on the operational character of the
differential and on his search for the exact moment where the
calculus springs from the underlying algebra as a new doctrine.
"infinitesimals" do not appear in Marx' work at all.  In his
insistence on the origin of the derivative in a real change of the
variable he takes a decisive step in overcoming the ancient
paradox of Zeno -- by stressing the task of the scientist in not
denying the contradictions in the real world but to establish the
best mode in which they can exist side by side.  Here his position
is directly opposite to that taken by Du Bois Reymond, who thought
that the increments dx, dy have to be taken as being at rest,
invariable, or of the modern mathematician Tarski, who denies the
existence of variable quantities altogether.  Marx' position in
this respect will be appreciated by most mathematicians." [p.

[Struik, a noted Marxist scholar and historian of mathematics, is
still alive, by the way, and he is now 100 years old.  His latest
article on mathematics appears in the current MONTHLY REVIEW.]

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