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

From: Philip Dunn (hyl0morph@YAHOO.CO.UK)
Date: Sun Apr 09 2006 - 14:50:43 EDT

Here is a whole book. The Mathematical Manuscripts of Karl Marx. New
Park 1983. ISBN 0 86151 028 3. As I recall Marx questioning of whether
0/0 made is mathematically valid. It does not always make sense.
Derivatives do not always exist.

On Sun, 2006-04-09 at 14:40 -0400, 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
> <>
>  ------------------------------------------------------
> 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)

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