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

From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Wed Apr 12 2006 - 11:28:23 EDT

glevy@PRATT.EDU wrote:

> 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
> <>
The problem of the continuum and the physical reality or unreality
thereof has some indirect political economy relevance via the link of the
hypothetical hypercomputation.
Witold Marciszewski argues that the idea of hypercomputation
undermines Lange's argument against Hayek.
for arguments against see:
Paulo Cotogno in
or Cockshott and Michaelson in

One issue at stake is whether or not Aristotle's
distinction between potential and actual infinites is justified.
I would argue that it is, and the there are no physically
realisable actual infinities. The proposed hyper-computational
or hyper-rational methods mostly depend upon the possible
existence of completed inifities.

The link to the calculus comes through some work by
Pour El on whether there exist systems of differential
equations whose solutions are uncomputatble. The connection
is then made between such systems and possible physical systems.
It is questionable however whether the formalism of
the calculus can be applied to reality at all scales, at a sufficiently
small scale of subdivision it can be expected to break down
as a picture of reality, as argued for instance by Smolin.

Such possibilities of a breakdown of the calculus at small
scales were as far as I know not being considered in the
19th century.

> ------------------------------------------------------
> An essay about
> Raymond Swing, Copenhagen (
> 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
> (moving).
> 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
> philosophers.
> 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
> perspectives.
> 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
> conditions.
> 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
> evoke.
> Literature:
> 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
> Classics).
> Rosen, Robert (1991): Life Itself. A Comprehensive
> Inquiry Into the Nature, Origin, and Fabrication of
> Life (Columbia University Press, New York)

Paul Cockshott
Dept Computing Science
University of Glasgow

0141 330 3125

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