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 > > > <http://home20.inet.tele.dk/swing/Hegel_Marx_and_the_Differential_Calculus.htm> > ------------------------------------------------------ > > An essay about > > > > HEGEL, MARX > AND THE DIFFERENTIAL CALCULUS > > > > > 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 > (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) ___________________________________________________________ To help you stay safe and secure online, we've developed the all new Yahoo! Security Centre. http://uk.security.yahoo.com
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