Marx and mathematics1:
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”
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,
Kennedy,
and the book on the mathematical manuscripts of Karl Marx
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 preMcCarthy 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.
(Formal real numbers can also be constructed as equivalence
classes of Cauchy sequences.)
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
(nonArchimedean) 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 fishfigure
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.)
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.
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
BanachTarski 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
