[OPE-L] Hegel, Marx, and the Differential Calculus

From: glevy@PRATT.EDU
Date: Sun Apr 09 2006 - 14:40:56 EDT

Paul B asked recently for literature on Marx and calculus.  Here's
a recent paper from Denmark on that subject.

Is there anybody on the list, with a greater knowledge of calculus
than I, who can comment on the following?

In struggling with differential calculus, was Marx struggling with how to
present his theory in a more dynamic form?

In solidarity, Jerry


An essay about


Raymond Swing, Copenhagen (Swing@mail.tele.dk)

This paper discusses aspects of H.–H. v. Borzeszkowski’s and R. Wahsner’s
paper Infinitesimalkalkül und neuzeitlicher Bewegungsbegriff oder Prozess
als Größe (Jahrbuch für Hegelforschung 2002) but first it presents its
author’s general view on the calculus and the limes transition thereby
underscoring the importance of comparability and steadiness of all
involved parameters and the real number system as well as of the
importance of  subjective moments like anticipation and decision for all
handling things and to define identity and continuity. On this basis the
author shows that one main difference between Hegel and Newton is their
different relation to time as independent variable. Thus Hegel attempts no
‘dynamisation’ of his social philosophy but rather recognises a ‘process
as magnitude’ as a specific systemic property of the young capitalist
society at his time. Marx in this respect in his Mathematical manuscripts
and Capital partly follows Hegel, even if he underscores the importance of
time as an essential factor of his value concept.

In a couple of books and essays Horst-Heino v. Borzeszkowski in cowork
with Renate Wahsner have discussed among other things G. W. F. Hegel’s
philosophical reflections upon physics and the differential calculus as
developed for functional analysis by Newton and Leibniz. Hegel in his
second edition of Wissenschaft der Logik. Teil I. Die objektive Logik.
Erster Band. Die Lehre vom Sein (1832) was especially concerned with this
issue and its supposed philosophical implications – and asserted
insufficiencies. These Hegelian reflections have been analysed by
Borzeszkowski and Wahsner in a recent paper Infinitesimalkalkül und
neuzeitlicher Bewegungsbegriff oder Prozess als Größe (Preprint no. 165
from the Max Planck Institute for the History of Sciences, 2001, also
published in Jahrbuch für Hegelforschung 2002). It is obvious that the
‘Modern Concept of Movement’ as well as the notion ‘Process as Magnitude’
mentioned in this title are central issue also to the present work. It
could therefore be of some interest to see how their common comments on
the Hegelian view relates to the ideas presented here and so to make more
clear also the relation between this one and other more general
philosophical problems and their history.

Resuming my own ideas about the philosophical content of the differential
calculus in the light of Hegel’s notion of movement as the ‘existing
contradiction’ (‘daseiender Widerspruch’) could perhaps most shortly be
done as follows.

Let us take our starting point in the two spatial locations of a moving
body x and x’, or x and x + Dx. The second location x’ must somehow relate
to the first one to express a realised movement or process, that is,
simply the (continuous) change from being at the location x to the other
location x’. So these two spatial determinations in the sense of
‘locations’, places, or more generally, ‘states’, are different but
nevertheless in some formal sense ‘equal’ (comparable). When x is
recognised to be the case we naturally anticipate x’ and we write the
above used anticipation notation <x|x’>. To be able to make such
anticipations is the general condition for all the following arguments. So
from the normal (‘scientific’) third-person stance to ‘confirm’ x is the
condition even for knowing the real (substantiated, persisting) object of
analysis as such, further of its being in (occupying) x and that
(‘actualised’) it later will be in the new location (‘state’) x’, thereby
just ‘having’ that changed location or ‘state’ as a variable property; or,
in another wording, ‘having’ the property of (spatially) being changing

Anticipation, in fact, is the real problem of the Hegelian ‘Existing
Contradiction’ contained in the ‘movement’ x ® x + Dx. Anticipation is
always being realised in ‘real (subjective) time’ of NOW and THEN.
However, such notions are neither expressible in mathematical nor physical
terms and therefore most often stay implicit. This temporal moment,
therefore, ‘time’ as such, had itself to be substantiated and
conceptualised, which for the first time was realised by Galileo by using
‘time’ as the independent variable in his equations. However, the real
‘independence’ here was – again implicitly – realised by the active
physicist himself deciding to make his experimental operations and
measurements at certain ‘timepoints’ or NOWs, say, exactly at the instants
of time t0 and t0 + Dt (t and t’). In this way – and only in this way –
the ‘movement’ x0 ® x0 + Dx could immediately be related to ‘time’, the
‘flow’ of which, so to say, itself makes the ‘movement’ t0 ® t0 + Dt.
These instants, then, were indicated by means of clocks readings, so that
the simultaneous movements of body and hands over the dial could be
immediately compared and numerically indicated by their coordinates and
angles, respectively; so the movement (process) and its ‘velocity’ could
in the registered time interval (in the average) be expressed by the
mathematical different quotient Dx/Dt.

However, to measure or mediate (model) such magnitudes in numerical it was
necessary, first, that the number system itself ‘resembled’ (was
isomorphic to) both spatial distances and time durations so that these
(the measured magnitudes) could be related to each other in the mentioned
proportion. But all this was in no way given. In the last instance this
was the problem of continuity. On the other hand, to go further to
indicate also velocity at a singe timepoint or at a single spatial
coordinatepoint it was necessary that the development of the number system
had advanced to include also irrational numbers (real number system)
making calculation of infinite number series possible, which was first
achieved in the course of the nineteenth century. This was needed for any
exact operation with infinitesimals and, consequently, to give a strict
mathematical definition of the limit. The recognition of an instantaneous
change of the (material or mathematical) function ¦ at the decided NOW
could then be defined by the limes transition of the differrence
quotientDx/Dt into the differential quotient dx/dt, Dx/Dt ® dx/dt.

These magnitudes x (and Dx or the infinitesimal dx) and t (and the
corresponding Dt or dt) had always to be expressed through concrete
numbers, i.e., by the unity of qualities and quantities; thus Dx/Dt and
dx/dt could never be viewed as ‘pure’ mathematical expressions but would
always be associated to some physical concepts, for instance as here to
the conceptualised relationship between distance and duration. However,
the fact in this connection is that calculating the differential quotient
we from the outset must have a theory or at least some elementary
knowledge about the ‘movement’ or change in question represented by the
function ¦; if not, we would not be able to anticipate anything at all. We
also had to suspect that this function or process ¦ could represent by
just that instantaneously measurable (respectively calculable) ‘quality’
at the ‘timepoint’ t0; missing such a knowledge, we would not even have
looked for it!

On this basis, eventually, we can write the equation in question referring
to all corresponding concepts ¦0, x0, and t0: ¦0 = dx/dt, ¦0 just being
the velocity in x0 at the time t0. As a consequence, not only the body’s
existence as such, also its location and an essential property (the
faculty of moving) has been defined as different aspects of its ‘state’.
It is this generalised ‘state’ that troubles Hegel. What Hegel and others
have characterised as the ‘existing contradiction’, this ‘contradiction’
between being in a state, for example, of a certain location and, at the
same time, being not in that ‘state’ but in a ‘state of change’, of
movement or processing, is at issue here. The second aspect of ‘being in
change’ could be viewed as a measurable process itself, which inspired
Hegel to talk of a ‘process as magnitude (‘Prozess als Größe’) based on
some ‘science of magnitude’. In this sense this process appears as a kind
of bodyinherent power (cf. the old notion of ‘Impetus’), which even Marx
in his Matehematical manuscripts referred to as an operative principle
with dx as its symbol (see below).

Now let us turn to the two essential problems to Hegel’s understanding of
the differential calculus discussed by Borzeszkowski & Wahsner. The first
one is that of the limes transition with its narrowing of Dx, Dy, Dt, etc.
to the corresponding differentials dx, dy, dt, etc. disappearing in the
limit and so at last, quantitatively, may be posited = 0. The first
condition for the numerical representation the magnitudes in question was
that of steadiness of the real number system had to be proven to give the
concept of limit its exact definition. This indeed problematic condition
for any numerical representation of physical processes, for instance a
spatial movement along a certain path, was to assure that all involved
developments in questions in themselves could be asserted ‘steady’
(isomorphic). This, however, is not at all a matter of course; ‘random
walk’, for example, is not differentiable. The calculus always depended on
the concept of steadiness valid as well to the number system, to lines in
space (or other trajectories of development) as also to the ‘time line’
(consequently been conceptualised as linear parameter).

                      On the other hand, in real physical work the exact
limes transition Dx ® dx (quasi = 0) is not possible
at all. The practical problem is, of course, to
prepare the limes transitions of their proportion
Dy/Dx ® dy/dx, as Hegel wrote, by determining the
quantities in question with reasonable exactness.
Every measurement is in the last instance based on
identity between the value of the magnitude measured
and that of the magnitude indicated by the measuring
device (meter rule, balance, clock, etc.). But, how
to assess that identity? And even, how at all to
define identity as such as a theoretical concept?

                      Marx offered in the first chapter of his Capital an
interesting commodity or value form analysis (cf.
for example, chapter XI). Here a weaver meets the
tailor wanting to give some linen in reward for a
new coat (value form I). Most essentially, none of
them must feel cheated. So they bargain and come to
the result, 20 yards of linen in regard for the coat
is not too much (to the weaver) and not too little
(to the tailor). So the goods themselves are thereby
asserted ‘equivalent’ and the exchange can be
realised. In more elaborated terms we may say that
the ‘value’ of 20 yards than the ‘value’ of the coat
and, at the same time,  than that of the coat: LC &
LC. Under this – certainly subjective – condition
the two persons accept their commodities to be in
respect of their ‘value’ identical (this identity
definition is more thoroughly elaborated in the
relevant parts of the present work). On the abstract
market to accept this is the real condition for
breaking off further bargaining; after all both only
want to exchange their goods!

                      What does this mean to measurements in general?
Using our rules and other measureing devices we
really never will be able to assess exact equality,
not even a single value identity. We only assess
that under the given conditions of observation we
must be satisfied when the object is experienced to
be neither greater (higher, heavier, etc.) than the
notches of the rule or the hand of other devices
indicate, nor, at the same time, to be smaller, etc.
But that means that we have to make the (in itself
essential but not unproblematic, dialectical)
predicationas the result of the comparison process.
This ideally defines identity by posited exclusion
of uncertainties. Under modern measurement
conditions, indeed, such uncertainties can be made
small; but ‘identity’ as such will forever be an
abstract concept, an ideal construct, useful and
fertile for the mathematical sciences, but never
found in the real world.

                      So the problem of the disappearing infinitesimals in
the infinite limit will always be problematic to the
empirical sciences; and so it was, too, to the

Hegel reacts against the empirical uncertainty caused by the necessary
actions and valuations made by the scientist and he does not conceptualise
the necessary subjective moment of real participation inherent in the
experimental sciences (contrary to the theorising afterwards!) where
‘participation’ under the first-person aspect is exactly the dialectic
opposite to the concept of dominum (basic to all ‘alienation, ‘isolation’,
etc.) equally essential to society in general and to science in particular
treating all relevant issues under the third-person aspect. Both moments
are indispensable for determining functions of things; without forms of
‘objectifying’ on the basis of constant participation in the processes to
be studied (cf. Bohr and the ‘problems’ of modern quantum physics); not
even such (seemingly so simple) concepts like motion and change in general
can be conceptualised. Indeed, the ‘existing contradiction’ also implies
the paradox that all ‘objective’ observation is realised by self-conscious
scientists synchronically uniting both first- and third-person

The second problem raised by Hegel and mentioned by Borzeszkowski &
Wahsner is that of the missing concept of the independent variable. The
problem is here that the term dy/dx is taken as a simple proportion, not
as a term in an equation, that is, not explicitly naming a process being
at work. “He (Hegel) reads this expression not as ‘dy after dx’, that is,
not as the changing of the magnitude y with x, not as the change of the
dependent variable with regard to the independent variable of a structure
defined through the function ¦ in case” (p. 10). This means that in the
view of Hegel the changing magnitude of y is not seen under the condition
of a (possibly provoked) change of another variable x (measured under
certain conditions, for instance at a decided instant of time or a short
duration of observation). Thus Hegel had no longer to do only with ‘pure’
proportions but merely recognised the practical usefulness of the calculus
to the scientists. Therefore he also declares: “I eliminate here those
determinations which belong to the idea of motion and velocity … because
in them the thought does not appear in its proper abstraction but as a
concrete and mixed with nonessential forms. “ (Hegel, p. 255) And he
concludes that use of the differential calculus in relation to “the
elementary equations of motion” (Hegel, p. 294) as such is without any
real philosophical interest.

Seen in the perspective of Hegel was at issue not so much the breaking off
the infinite number series refusing exactness (absolute identification) of
the measurements but rather proposed these members of the number series
not “be regarded as parts of a sum, but rather as qualitative moments of a
whole determined by the concept.” (Hegel, p. 264) According to him,
therefore, the notion of limit was developed on the basis of a mere
qualitative relation, dx and dy themselves viewed only as the moments of
this so that the composed term dy/dx could be read as a single sign naming
a thereby specified quality, a certain property of a thing or phenomenon
as such. On this basis, of course, the quality at issue could be made
subject to a ‘science of magnitudes’ to be measured under the given

                      The essential point of ‘dynamising’ the world
picture through the new temporal concepts of
movement, change, function, process, etc. can as
mentioned above only be understood on the historical
basis of the most essential material conditions for
the common life in that era always having vital (but
different) meaning to the societies in question.
These concepts, therefore, are (often implicitly) to
be recognised as reflections on the social work, for
instance in the manufactures, later on in the
factories. In this perspective, in fact, capitalist
factories just realise a certain new ‘quality’
represented by the economic proportion between the
magnitudes of money investments (capital) and its
outcome (surplus value, eventually as profit).
However, more essential than this difference in
their proportion, so that the proportion dy/dx can
be read as the productivity measured at some certain
point of time. Indeed, capitalists are primarily not
interested in real (material and ideal) processes
causing this productivity, only in this single
proportion as such. In this sense Hegel, more or
less implicitly, understood what subliminally was
developing in the years about 1830 and which became
rather obvious shortly after his own lifetime, just
in the time of Marx. So Hegel could not yet see the
need for going behind this mere proportion-thinking
of his ‘science of magnitude’ just being a ‘science
of value’ (the specific capitalist science of
money). In this view his term dy/dx not at all aimed
at any dynamism but merely stated a certain
proportionality of economical values as a specific
quality or property of these values as such.

              In this sense Hegel expresses an essential difference
between the ideas of philosophers and physicists. This
difference was caused by the simple fact that the physicists
had an other job to do than the philosophers and economists;
they were just the persons preparing this capitalist
development creating the necessary ‘scientific’ technology.
So they had to conceptualise the material processes
underlying these economical proportions that to them were
without special interest. Marx too analysed the developing
social system but explicitly reflected the real
(quasi-organismic) functioning of capital as an economic
whole and so also had to reflect the real work (labour) to
be done by the workers. So also he in his Mathematical
Manuscripts is concerned with dx (in this connection, in
fact, dy) that he (atemporally) characterises as an
‘operational symbol’ referring to a ‘process which must be
carried out...’ (cf. Marx 1983, p. 21). Not even Marx is yet
able to operate with the temporal differential quotient
dx/dt even if he in his value theory explicitly includes
time as the essential factor of real labour.

Conceptualising of real organismic wholes was eventually made explicit by
Robert Rosen by his relational analyses proposing a minimal organismic
structure. On this analysis the  more elaborated concepts of anticipation,
circularity, complex time, etc. could be grounded, thereby explicitly
conceptualising the very notion of ‘subjectivity’ (as the dialectic
opposite to ‘objectivity’).

Indeed, Marx came rather near – nearer than Hegel – to transgress the
ideological limits of the ‘exact sciences’ to explicate the true character
of real ‘becoming’ – including just notions like productivity, creativity,
even of life itself. Exactly such notions are more extensively to be
analysed in the time to come and will then surely cause of new
mathematical, natural scientific, and philosophical problems to emerge
calling for new arguments in a presumably much wider field of inquiry than
the classical problems of ‘dynamisation’ of the old world picture could


von Borzeszkowski, H.-H. and Wahsner, R. (2002): Infinitesimalkalkül und
neuzeitlicher Bewegungsbegriff oder Prozess als Größe. Jahrbuch für
Hegelforschung. (Also as Preprint no. 165 from the Max Planck Institute
for the History of Sciences, 2001.)

Hegel’s Science of Logic (1969). Translated by A.V. Miller, foreword by
prof. J. N. Findlay. Humanities Press International, INC., Atlantic
Highlands, NJ.

Marx, Karl (1983): Mathematical Manuscripts of Karl Marx. New Park
Publications Ltd.

¾ (1990): Capital, Vol. I , transl. by Ben Fowkes, Penguin Books (Penguin

                      Rosen, Robert (1991): Life Itself. A Comprehensive
Inquiry Into the Nature, Origin, and Fabrication of
Life (Columbia University Press, New York)

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