Thus, as a contribution to the recent postings on Marx's Mathematical
Manuscripts, I'd like to add the following initial impressions on this
subject and on the monograph of Paulus Gerdes. But by way of introduction,
I'll share part of a personal correspondence with Prof. Struik. His
response dated August 24th read, in part,
"Those who are interested in the foundations of the
calculus will find interest in the Marxian manuscripts of
mathematics. After all, it has remained a field in which
even today there is discussion, as the non-standard
analysis of Robinson shows."
He continued with a suggestion to read the monograph of Paulus Gerdes
as well as two articles by Hubert Kennedy in Historia Mathematica,
material cited earlier by participants of this list.
That Karl Marx wrote anything on mathematics, let alone on the foundations
of differential calculus, may come as a surprise to many. In numerous
correspondences with Engels, he cites his growing interest with technology,
the sciences and mathematics, each of which he subjected to his unique
form of critique. One possibly overlooked aspect of his work may be his
appreciation of the invention of analytical geometry and the function as
part of the transformation which mathematics underwent from a science of
constant quantities to one of varying quantities; certainly an explication
of his interpretation of dialectics. From an historical perspective, he
argued that this process received foundation in, and was in turn, a source
for further development of the productive forces; whether they were material
as in the case of the production of capital, or abstract as in scientific
theory and philosophy. Ultimately, this was all part of a larger
interconnected process; each aspect with its own particular limits and
contradictions. That Marx sought to develop a quantitative treatment of
the economic laws of capitalism should not surprise anyone with a
familiarity of his critique of political economy, certainly as organized
and documented in his volumes of Capital.
The Gerdes booklet, 'Marx Demystifies Calculus', MEP Publications,
Minneapolis, 1985, is a short history which outlines many of the issues
addressed in the Manuscripts. Most of this study summarizes Marx's
characterization of three interpretations of the differential calculus,
i.e., the so-called mystical interpretation one (of Newton and Leibniz),
the rational one (of d'Alembert and Euler), and the algebraic one (of
Lagrange). Suffice it to say, many of the concerns identified and
discussed by Marx had either been the subject of extensive polemics or
remained undecided [1].
What he brought to the discussion was his critical training as a
philosopher and the desire to capture the process of motion through a
logic of concepts, part of a methodology which began with his critique
of Hegel [2]. Certainly, the logic of any process producing expressions
such as "dy/dx = 0/0" raised much concern, if not from the philosophical
community, of whom Berkeley was probably their most logical and persuasive
voice, then from the scientific community, though the latter ultimately
sought validation through scientific and analytical practice [3]. In a
footnote to a selection from Berkeley's Analyst, David Smith writes,
"Berkeley's explains that the calculus of Leibniz leads
from false principles to correct results by a "Compensation
of errors." The same explanation was advanced again later
by Maclaurin, Lagrange, and independently by L. Carnot in
his 'Reflexions sur la metaphysique du calcul infintesimal,
1797'." (A Sourcebook in Mathematics, NY: McGraw-Hill, 1929).
As a student of the French revolutionary movements and admirer of the
early Republic's system of its 'Grand' Ecoles, Marx was probably familiar
with Carnot's critique. Unfortunately, as he was not a mathematician,
it's unlikely that he was aware with anything beyond the standard calculus
texts of Lacroix, Bourcharlat, Hind, etc. Struik reports that though the
Ecole Polytechnique published Cauchy's ideas in 1823, it's unlikely Marx
was aware of them.
A reading of Paulus Gerdes' monograph is recommended as an introduction
to Marx's Mathematical Manuscripts. Though abbreviated in content and
overly polemical at points, it appears to have been the definitive summary
among non-Soviet Western scholarship at least during the 1980's. In
addition, the author offers numerous references from the former Soviet
Union and DDR that many may be unfamiliar with (this writer included).
One note of interest concerns the translations of these manuscripts. The
1983 English version of Aronson and Meo was a translation from the Russian
edition edited by the Soviet scholar S.Yanovskaya. Gerdes used the
original German work. There is also an Italian version which was the
subject of a review by Hubert Kennedy in Historia Mathematica (3), 1976,
pp 491-494 and in 'Karl Marx and the Foundations of Differential Calculus',
Historia Mathematica (4), 1977, pp 303-318. It was translated from the
German original as well.
By way of conclusion the following observation from the late Soviet
mathematician Andrei Kolomogorov is offered for thought:
"In an especially detailed manner, Marx analyzed the
question of the concept of the differential. He proposed
the concept of the differential as an 'operational symbol',
anticipating an idea that came forward again only in the
20th century".
So while Marx may not have invented any new mathematics, per se, his
contributions to the discussion of the foundations of the differential
calculus warrant a place in the history of the subject's philosophy. In
this regard, his mathematical commentaries deserve nothing short of a
place alongside those of Hegel or Kant.
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Notes:
[1] xxxxxxxxxxx
[2] though a 'traditional' philosophical concern dating to the
paradoxes of Zeno of Elea. Hegel earlier addressed some
of the same philosophical concerns of the calculus. See,
for example, the section, "Quantum" in his Science of
Logic, Miller translation, London, 1976. Surprisingly,
there is considerable discussion of the contributions of
Lagrange as well.
[3] Rolle's greatest concern was that the calculus was
flawed mathematically, i.e., that it would produce
erroneous results. Despite his polemics, Berkeley never
held this position.
More recently, Abraham Robinson observed, "after
awhile the glaring contradictions of the theory led to the
realization that alternative foundations were required."
(Nonstandard Analysis, Princeton, 1996).
Al Barron
Metuchen, New Jersey