Hegel on Calculus

History of Philosophy Quarterly Volume 34, Number 4, October 2017

Ralph M. Kaufmann and Christopher Yeomans

t is fair to say that Georg Wilhelm Friedrich Hegel’s philosophy of Imathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twenti- eth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel’s) the untimely appear- ance of naïveté. The common view was expressed by Bertrand Russell: The great [mathematicians] of the seventeenth and eighteenth cen- turies were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscuri- ties in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. . . .The resulting puzzles [of mathematics] were all cleared up during the nineteenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). Recently, however, interest in Hegel’s discussion of calculus has been awakened by an unlikely source: Gilles Deleuze. In particular, work by Simon Duffy and Henry Somers-Hall has demonstrated how close Deleuze and Hegel are in their treatment of the calculus as compared with most other philosophers of mathematics. We also believe that Hegel’s treatment of the calculus is worthy of serious examination, not least because the confusions he finds in early nineteenth-century analytical procedures are reproduced in virtually all of the current pedagogical presentations of calculus in the United States. Unlike Duffy and Somers-Hall, however, we believe that the value of Hegel’s discussion turns primarily not on its relation to the methodological foundations of metaphysics but on his attempts to forge a connection between analytical techniques and the theory of numbers generally.

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This connection does have a metaphysical payoff, but its beneficiary is mathematics rather than philosophy.

1. Objections to Hegel’s Interpretation of Calculus

According to Deleuze, Hegel makes two mistakes in his interpretation of the calculus. First, though he sees that there is a true infinite in the notion of the infinitesimal, he ruins this insight by forcing his interpretation into a dialectical logic of contradiction that works on principles at odds with those of the calculus (Deleuze 1995, 310n9).1 Second, Hegel thinks that mathematical procedures for working with differentials can never explain the relation between infinitesimals and reality; thus, he thinks that his own conceptual logic is required to show this relation. These two mistakes are related, of course, and, on Simon Duffy’s view, they are further connected with an historical fact. In Leibniz, integration is characterized as both (a) the inverse transformation of differentiation and (b) the summation of rectangles under the curve (which rectangles have infinitesimal sections of the curve as one side). Hegel follows his own contemporaries in viewing (a) as more promising and (b) as unten- able. But, subsequent to Hegel, Augustin-Louis Cauchy and Bernhard Riemann developed (b) with new mathematical techniques that are more powerful: sufficiently powerful, on Deleuze’s view, to explain the reality of vanishing infinitesimals without recourse to the Hegelian dialectical logic of contradiction (Duffy 2009, 566–73).2 To this same general way of viewing Hegel’s approach to calculus, Henry Somers-Hall adds the recognition that, in contrast to the infinitesimal interpretation of the calculus, Hegel takes Newton’s conception of fluxions and the ultimate ratio of evanescent quantities to provide the best available interpretation at his time. However, Newton’s presentation is afflicted with certain confusions, is thus insufficiently clear in its abstraction, and thus requires the dialectical logic of contradiction to demonstrate its import. Specifically, that logic tries to do so as a form of the unity of the finite and the infinite (Somers-Hall 2010, 561–62). But Hegel does not get beyond this opposition—he “only gets as far as the vanishing of the quantum [dx], and therefore leaves its status as vanished (from the realm of quanta at least) untouched” (Somers-Hall 2010, 567). For both Duffy and Somers-Hall, Deleuze then responds to this failure by building a metaphysics that depends on the distinctive and unique status of differentials. In response to these objections and that of Russell, we advance the following interpretation: neither contradiction nor anything specific to Hegelian dialectical method is playing any fundamental role here; instead, the crucial notion is quantity itself. Before coming to his ex- Hegel on Calculus 373 tended engagement with the calculus, Hegel had already advanced a conception of quantity very much in tune with later developments in the theory of real numbers. In particular, he advanced such a conception that embeds the notion of the limit of a series at the heart of quantitative determinacy.3 Hegel’s argument is more indirect than that of Deleuze: the calculus brings into greater relief problems that are inherent in any notion of quantitative determinacy, and so the same resources that must be brought to bear to make sense of the latter can be extended to make sense of the former.4 According to Hegel one must look through the infelicities of analytical techniques to conceptualize what is actually being done, and focusing on the figure of the infinitesimal is a barrier to doing so. Once one does pierce the veil of these infelicities, however, Hegel thinks he can show the way in which quantity necessarily realizes itself as materiality. This is how the metaphysical payoff is to mathematics itself, since Hegel’s interpretation seems to explain why mathematics has the powerful insight into the nature of reality that it does.

2. Hegel’s Interpretation A. Relation The core of Hegel’s analysis derives from the realization that derivatives are given by ratios (dy/dx) and, as such, they can be viewed as giving quality to the quantities that are the moments or relata of this relation (dy and dx). This depends on Hegel’s conception of quality as the simplest form of a relation, that is, as the necessity of contrast for any determinate character. Mathematically speaking, it is legitimate to take the limit of these ratios, but only if one uses the composite (that is, one can take the limit of dy/dx but not dy or dx). This binds the relata to the ratio as mere moments (that is, dependent or subsidiary elements), which by themselves thus lose their nature as quanta (since quanta are supposed to be independent of comparison or variation). In Hegel’s terminology, taking the limit (Grenze) of the ratio supersedes the qualitative character of the composite and transforms it back into a quantity: Quite generally: quantum is superseded quality; but quantum is infinite, it surpasses itself, is the negation of itself; this, its surpassing, is therefore in itself the negation of the negated quantity, the restoration of it; and what is posited is that the externality, which seemed to be a beyond, is determined as quantum’s own moment. (WL 21.235–36; italics in original)5 This whole process is taken to be definitive of any quantum or number as such, and Hegel’s description of the process is just the translation into his own jargon of an ordinary mathematical procedure central to calculus. 374 HISTORY OF PHILOSOPHY QUARTERLY

It is easy to read such jargon as invoking as a premise basic dialectical principles such as the purportedly inevitable negation of the negation, when, in fact, these are rather exemplified by the form of calculation that is being interpreted. The dialectical language provides a guiding thread rather than a driving force. Hegel’s point is simply that symbols like dx should only be considered in ratios, and, as such, they are relational in nature, which ascribes a qualitative character to them: Now the conception of limit does indeed imply the stated true category of the qualitatively determined relation of variable magnitudes, for the forms of it which occur, dx and dy, are supposed to be taken simply as only moments of , and itself ought to be regarded as one single indivisible symbol. (WL 21.265) The higher-order operations of taking the limit of the related terms is a way that the relation surpasses or transcends itself (über sich hinausgeht) and generates itself as quantity by dissolving and subsum- ing the independence of its constituents as specifiable quantities. The consideration of ratios naturally leads Hegel back to simple ratios of numbers as an illustration. The difference between the ratios of numbers and the ratios of infinitesimals is that hidden indx is a variable x. The variability of x is fundamental in the usual considerations, since, in order to take the limit, one lets x approach certain values, yet this variability is an ad hoc property. It follows from all of this that infinitesimals are not, by themselves, bona fide quantities.6 Indeed, it is very hard to say what a simple infinitesimal—usually symbolized by ε, i, or dx— is supposed to be.7 As a stand-alone object, they are suspect and certainly not numbers (as George Berkeley correctly pointed out). In InLeibniz Leibniz and andNewton Newton, there thereis a certain is a certainmysticism mysticism about this about infinitesimal this in - quantity,finitesimal which is why quantity, calculus whichwas at is first why just calculus a way to wascompute at first (thus just the aname), way to but not strictlycompute well- founded(thus, the even name), within but mathematics not strictly itself. well Leibniz’s founded version even ofwithin the Fundamentalmathematics Theorem itself. of Calculus Leibniz’s requires version the ofdifficult the Fundamental step of calculating Theorem the function of of a curveCalculus from itsrequires law of tangency,the difficult for example,step of calculating the curve x=ythe2 functionwith the lawof a ofcurve tangency x=y3/3:from its law of tangency, for example, the curve x=y2 with the law of 3 tangency x=y /3: ( + ) ( + ) = = 3 3 3 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑+ 3 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑+−3𝑑𝑑𝑑𝑑 ( 𝑑𝑑𝑑𝑑) +𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑( −) 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑3 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 2 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 3 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑( ) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑑𝑑 = + + = ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 2 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 What is suspect in the calculation is the last equality. To obtain the desired result one has to treat dy and hence dy2 as zero. On the other hand, if one does this in the first or second line, one obtains problematic expressions. The question that has to be dealt with is the following: what is the reason to keep only the first term y2 and drop the second y y two terms d and 2? Going back one equation this becomes the question, why ultimately one only𝑑𝑑𝑑𝑑 keeps𝑦𝑦𝑦𝑦 the term 3y2dy of the numerator and not the terms with 3 higher powers of dy. There is a different rendering of the computations separating dx and dy that goes as follows, if y=x3 + = ( + ) = + 3 + 3 + y=x3 y x2 3 3 2 2 3 Using that , one𝑑𝑑𝑑𝑑 arrives𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 at𝑑𝑑𝑑𝑑 d =3𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 dx, but𝑑𝑑𝑑𝑑 only𝑑𝑑𝑑𝑑 if 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑one drops𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 the terms𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 with higher powers of dx (3 + ) Rand keeps only the linear term, that is, the term linear in dx (3 ). This last2 step is3 the one that is under scrutiny, for if dx were a number, then dx2=0 would2 imply𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 that d𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑x itself is zero. Thus for Leibniz, this mysticism comes from the fact that𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 an infinitesimal squared is treated as zero, but the infinitesimal itself is not (Laubenbacher and Pengelley 2000, 135). And it is this feature that exposes early modern accounts to Berkeley’s criticism, namely that there is supposedly some quantity or quality that can be disregarded, and which suddenly appears or disappears (Berkeley, George 1999, 73). To legitimize this last step of the calculus different arguments had been presented. For instance, in Newton the arguments for limits involve the nebulous concept of fluxions, and Lagrange legitimizes the different treatment for the various terms in the sum by attributing physical meaning to each of them (such as speed, acceleration, etc.). Hegel decisively exposes the deficiency in both interpretations and hence the inadequacy of the arguments. What these approaches (and others like that of Carnot) have in common is that in order to treat these objects sensibly they are amalgamated with other properties to make their existence more credible. Hegel correctly points out that this amalgamation is an insufficient, confusing and ad hoc response. This is summarized nicely in Hegel’s criticism of Wolff’s presentation as

4 In Leibniz and Newton there is a certain mysticism about this infinitesimal quantity, which is why calculus was at first just a way to compute (thus the name), but not strictly well-founded even within mathematics itself. Leibniz’s version of the Fundamental Theorem of Calculus requires the difficult step of calculating the function of a curve from its law of tangency, for example, the curve x=y2 with the law of tangency x=y3/3:

( + ) ( + ) = = 3 3 3 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑+ 3 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑+−3𝑑𝑑𝑑𝑑 ( 𝑑𝑑𝑑𝑑) +𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑( −) 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑3 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 2 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 3 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑( ) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑑𝑑𝑑𝑑 H=egel + on Calculus+ = 375 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 3 2 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 2 What is suspect in the calculation𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ∙ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 is the last equality.𝑑𝑑𝑑𝑑 To obtain the Whatdesired is suspect result, in the one calculation has to treat is the dy last and equality. hence dyTo2 obtain as zero. the On desired the other result one hashand, to treat if onedy and does hence this dyin2 the as zero. first On or thesecond other line, hand, one if oneobtains does problematic this in the first or secondexpressions. line, one obtainsThe question problematic that expressions.has to be dealt The question with is thatthe hasfollowing: to be dealt with is thewhat following: is the reason what is to the keep reason only to the keep first only term they first2 and term drop y2 the and second drop the two second y y twoterms terms y ddy andand 2? Going Going back oneone equationequation, this this becomes becomes the the question, question why ultimatelyof why oneultimately only𝑑𝑑𝑑𝑑 keeps𝑦𝑦𝑦𝑦 one the only term keeps 3y2d they of termthe numerator 3y2dy of theand numeratornot the terms and with 3 highernot powersthe terms of d withy. There higher is a different powers renderingof dy. There of the is acomputations different rendering separating dx andof d ythe that computations goes as follows, separating if y=x3 dx and dy that goes as follows, if y=x3 y+ + dy == (x ++ dx)3) = =x3 + 3+x2dx3 + 3xdx+ 32 + dx3 + 3 2 3 3 2 2 3 y=x 3 y x 2 Using thatUsing that, one𝑑𝑑𝑑𝑑 y=x arrives𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑, one at 𝑑𝑑𝑑𝑑arrives d =3𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 dx, at butd𝑑𝑑𝑑𝑑y=3 onlyx dx,𝑑𝑑𝑑𝑑 if 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑one but drops only𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 theif one terms𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 drops with the higher powersterms of withdx (3 higher+ powers) Rand of keeps dx(3xdx only2 +the dx linear3) (and term, keeps that only is, thethe term linear linear in dx (term:3 that). This is, last the2 step term is3 linearthe one in that dx (is3 xunder2dx). This scrutiny, last forstep if disx thewere one a number, that then dx2is=0 under would2 scrutiny,imply𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 that for, d𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑x ifitself dx wereis zero. a number,Thus for Leibniz, then dx this2=0 mysticismwould imply comes that from the factd thatx𝑑𝑑𝑑𝑑 itself𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 an infinitesimalis zero. Thus, squared for Leibniz, is treated this as mysticismzero, but the comes infinitesimal from the itself fact is not (Laubenbacherthat an infinitesimal and Pengelley squared 2000, 135) is treated. And it isas this zero, feature but thethat infinitesimal exposes early modernitself accounts is not (Laubenbacher to Berkeley’s criticism, and Pengelley namely that2000, there 135). is supposedlyAnd this feature some quantity or qualityexposes that early can bemodern disregarded, accounts and towhich Berkeley’s suddenly criticism, appears or namely, disappears that (Berkeley, Georgethere 1999, is supposedly 73). some quantity or quality that can be disregarded and that suddenly appears or disappears (Berkeley 1999, 73). To legitimize this last step of the calculus different arguments had been presented. To legitimize For instance this ,last in Newto step ofn thethe calculus,arguments different for limits arguments involve the had nebulous been conceptpresented. of fluxions, For instance, and Lagrange in Newton, legitimizes the thearguments different for treatment limits involve for the thevarious termsnebulous in the sum concept by attributing of fluxions, physical and Joseph-Louismeaning to each Lagrange of them legitimizes (such as speed, the acceleration,different treatment etc.). Hegel for decisively the various exposes terms the in deficiency the sum by in attributingboth interpretations physi- and hencecal themeaning inadequacy to each of of the them arguments. (such as Whatspeed, these acceleration, approaches and (and so on). others Hegel like that of Carnot)decisively have in exposes common the is deficiencythat in order in to bothtreat interpretations these objects sensibly and, hence, they are the inadequacy of the arguments. What these approaches (and others like that amalgamated with other properties to make their existence more credible. Hegel of Lazare Carnot) have in common is that, in order to treat these objects correctly points out that this amalgamation is an insufficient, confusing and ad hoc sensibly, they are amalgamated with other properties to make their exis- response. This is summarized nicely in Hegel’s criticism of Wolff’s presentation as tence more credible. Hegel correctly points out that this amalgamation is an insufficient, confusing, and ad hoc response. This is summarized nicely in his criticism ofemploying Christian the Wolff’s latter’s presentation “customary as way employing of populari thezing latter’s things – in effect, by polluting the 4 “customary way conceptof popularizing and replacing things—in it with effect, false bysense polluting-representations” the concept (WL 21.256). and replacing it with false sense-representations” (WL 21.256). Subsequently Hegel offers his own argument for the selection of the linear term Subsequently, Hegel offers his own argument for the selection of the linear term by appealby appeal to the to naturethe nature of the of the relation relation . But . before But before going going into into the details of this 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 the details of thisargument argument, it will it be will convenient be convenient to recall to therecall modern the modern definition of a derivative and point 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 definition of a derivativeout the relevant and point features out forthe Hegel’srelevant philosophical features for analysis: Hegel’s philosophical analysis:

( ) ( ) ( ) = lim 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+𝜀𝜀𝜀𝜀 −𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0 0 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→0 𝜖𝜖𝜖𝜖 The right hand side contains two nested operations, first a ratio and then a limit. The left hand side is indeed an un-separated ratio; it is often written as , where one first sets = ( ). The realization that there is a second operation on the𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 previously formed 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ratio that needs to be considered is central to Hegel’s argument. 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 If we attend to the contrast between this modern way of proceeding and Newton’s, we can bring the distinction between Newton and Hegel into relief. Newton doesn’t yet have the concept of a limit (this comes from Cauchy), and so Newton’s version of the above formula is: = In this formula𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 the limit is already inherent in the quantities dx and dy, but only as 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 implicit. The problem is that in the calculation on the basis of this formula, higher order terms are generated that have to vanish to generate the desired result (as we saw above with respect to Leibniz). Without the explicit concept of a limit, Leibniz and Newton are denied the possibility of explaining why this vanishing is legitimate. Furthermore, they have to explain why, in the determination of the ratio of vanishing, higher-order terms are neglected. While Hegel embraces the fact that in Newton’s argument a certain quality of the differential takes shape as the last value of the ratio (the quacum evanescunt), Hegel rejects Newton’s reasoning for the negligibility of the higher order terms legitimizing the calculation. In fact, he ponders what “could have brought Newton to deceive himself about such a proof” (WL 21.261). Hegel thinks that had Newton only realized that the composite is a quality now – a ratio of the two quantities together that remains as the individual constituents vanish – then he would have had an adequate response to Berkeley and the other, more sympathetic critics who proposed alternative forms of analysis (WL 21.262-3). If the composite is essentially relational, then there is nothing strange in the idea that it might have a determinate character over and above whatever determinate character its relata have, and thus a determinate character that might remain or even first become clear in the relation between the changes between those relata undergo as they get smaller. In fact, the way Hegel explains this is the way we currently take the limit of the ratio as it goes to zero in contemporary mathematics:

5 employing the latter’s “customary way of popularizing things – in effect, by polluting the concept and replacing it with false sense-representations” (WL 21.256). Subsequently Hegel offers his own argument for the selection of the linear term by appeal to the nature of the relation . But before going into the details of this argument it will be convenient to recall𝑑𝑑𝑑𝑑 the𝑦𝑦𝑦𝑦 modern definition of a derivative and point 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 out the relevant features for Hegel’s philosophical analysis: employing( ) the( latter’s) “customary way of popularizing things – in effect, by polluting the ( ) = lim 376 HISTORY OF PHILOSOPHY QUARTERLY concept and replacing it with false sense-representations” (WL 21.256). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+𝜀𝜀𝜀𝜀 −𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0 0 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→0 Subsequently𝜖𝜖𝜖𝜖 Hegel offers The his right-hand own argument side for contains the selection two ofnested the linear operations: term first a ratio The right hand side contains two nested operations,and then first a ratiolimit. and The then left-hand a limit. Theside is indeed an unseparated ratio; it by appeal to the nature of the relation . But before going into the details of this left hand side is indeed an un-separated ratio; isit isoften often written written𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 as as , where oneone first first setsy = f(x) The realization that argument it will be convenient to recall the modern𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 definition of a derivative and point sets = ( ). The realization that there is a secondthere operationis a𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 second on operationthe previously on the formed previously formed ratio that needs out the relevant features for Hegel’s philosophical𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 analysis: ratio that needs to be considered is central to Hegelto be ’considereds argument. is central to Hegel’s argument. 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 If we attend to the contrast between this modern way of proceeding ( ) ( ) If we attend to the contrast( ) = betweenlim thisand modern Newton’s, way ofwe proceeding can bring andthe distinction between Newton and Hegel Newton’s, we can bring the𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 distinction between𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+𝜀𝜀𝜀𝜀 into−𝑑𝑑𝑑𝑑 Newton𝑑𝑑𝑑𝑑0 relief. and Newton Hegel doesinto relief. not yet Newton have the concept of a limit (this comes 0 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→0 𝜖𝜖𝜖𝜖 doesn’t yet have theThe concept right hand of a side limit contains (this comes fromtwo nestedfrom Cauchy), Cauchy), operations, so Newton’sand sofirst Newton’s a versionratio and of then the aabove limit. formulaThe is version of the above formula is: left hand side is indeed an un-separated ratio; it is often writtendy as= ,dx where one first = 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 sets = ( ). The realizationIn that this there formula, is a second the limit operation is already on the inherent previously in formedthe quantities dx and In this formula the limit is already inherent in the quantities dx and dy, but only as 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑ratio that needs to be considereddy, but is central only as to implicit. Hegel’s argument. The problem is that, in the calculation on the implicit. The problem is that in the calculation on the basis of this formula, higher order 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 basis of this formula, higher-order terms are generated that have to terms are generated that have to vanish to generate the desired result (as we saw If we attend to the contrastvanish between to generate this modernthe desired way ofresult proceeding (as we andsaw above with respect to above with respect to Leibniz). Without the explicit concept of a limit, Leibniz and Newton’s, we can bring the distinctionLeibniz). Withoutbetween Newtonthe explicit and Hegelconcept into of relief. a limit, Newton Leibniz and Newton Newton are denieddoesn’t the possibility yet have of the explaining conceptare ofwhy adenied limit this vanishing(this the cpossibilityomes is legitimate.from ofCauchy), explaining and sowhy Newton’s this vanishing is legitimate. Furthermore, theyversion have to of explain the above why, formula in the Furthermore,determination is: ofthey the have ratio to of explain vanishing, why, in the determination of the ratio higher-order terms are neglected. = of vanishing, higher-order terms are neglected. While Hegel embraces the fact that in Newton’s argument a certain quality of 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 the differential takesIn this shape formula as the the last limit value is already of theWhile ratio inherent Hegel(the quacumin embracesthe quantities evanescunt the dxfact and), that,Hegel dy, in but Newton’s only as argument, a certain 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 quality of the differential takes shape as the last value of the ratio rejects Newton’s reasoningimplicit. The for problem the negligibility is that in of the the calculation higher order on terms the basis legitimizing of this formula, the higher order (the quacum evanescunt), Hegel rejects Newton’s reasoning for the calculation. In fact,terms he ponde are generatedrs what “could that have have to brought vanish Newtonto generate to deceive the desired himself result (as we saw negligibility of the higher-order terms legitimizing the calculation. In about such a proof”above (WL with21.261). respect Hegel to thinksLeibniz). that Without had Newton the explicit only realizedconcept that of a thelimit, Leibniz and fact, he ponders what “could have brought Newton to deceive himself composite is a qualityNewton now are – a deniedratio of the the possibility two quantities of explaining together why that this remains vanishing as the is legitimate. Furthermore, they have to explainabout why, such in a the proof” determination (WL 21.261). of the Hegel ratio thinks of vanishing, that, had Newton only individual constituents vanish – then he wouldrealized have had that an adequate the composite response is a to quality now—a ratio of the two quanti- higher-order terms are neglected. Berkeley and the other, more sympathetic criticsties who together proposed that alternative remains asforms the ofindividual constituents vanish—then While Hegel embraces the fact that in Newton’s argument a certain quality of analysis (WL 21.262-3). If the composite is essentiallyhe would relational, have had then an there adequate is nothing response to Berkeley and the other, the differential takes shape as the last value of the ratio (the quacum evanescunt), Hegel strange in the idea that it might have a determinatemore charactersympathetic over criticsand above who whatever proposed alternative forms of analysis determinate characterrejects its Newton’s relata have, reasoning and thus for(WL a thedeterminate 21.262–63). negligibility character Ifof thethe compositehigher that might order is termsessentially legitimizing relational, the then there is remain or even firstcalculation. become clear In fact, in the he relation pondenothingrs between what “couldstrange the changeshave in the brought between idea Newtonthat those it might to deceive have himself a determinate character relata undergo as aboutthey get such smalle a proof”r. (WL 21.261).over Hegeland above thinks whateverthat had Newton determinate only realized character that the its relata have, and In fact, thecomposite way Hegel is explains a quality this now is the– thusa ratioway a we ofdeterminate thecurrently two quantities take character the limittogether thatof the that might remains remain as the or even first become ratio as it goes to individualzero in contemporary constituents mathematics: vanishclear – then in thehe would relation have between had an adequatethe changes response between to those relata undergo Berkeley and the other, moreas sympathetic they get smaller. critics who proposed alternative forms of analysis (WL 21.262-3). If the composite is essentially relational, then there is nothing In fact, the way Hegel explains this is the way we currently take the strange in the idea that it might have a determinate character over and above whatever 5 limit of the ratio as it goes to zero in contemporary mathematics: determinate character its relata have, and thus a determinate character that might remain or even first become clearSince in the at relation issue here between is a relationthe changes and betweennot a sum those, the differential is relata undergo as they get smallecompletelyr. given by the first term; and where there is the need of In fact, the way Hegel explainsfurther this terms, is the of way differentials we currently of highertake the orders, limit of their the determination ratio as it goes to zero in contemporarydoes not mathematics: involve the continuation of a series as sum, but the repetition rather of one and the same relation, the only one wanted and the one

5 Hegel on Calculus 377

which is, therefore, already completely determined in the first term (WL 21.264; italics in original). What is important is not the series qua sequence or quantitative sum, but rather the basic form of the relation. That basic form is revealed by the operation of taking the limit of the ratio itself as the quantitative values of both numerator and denominator to zero. More concretely, Hegel argues that the linear term (3x2dx) constitutes the relation between dx and dy, while the higher terms represent merely repeated applications and introduce no new quality or relation. Hegel’s solution holds that a quality is given by the ratio but then denies that this in any independent way survives when the quantities vanish by computing modulo higher-order terms (for example, as a Newtonian fluxion). Instead, the quality given by a ratio (Verhältnis) becomes the sought-after quantity when taking the limit. A surviving independent quality would be something with which the quantity was amalgamated (whether that is conceived as fluxion or something else). The key is that, for Hegel, the surviving quality is internal to the quantity generated by the limit-taking process rather than something that survives the destruction of the quantity in the infinitesimal. Thus (as Somers-Hall notes), Hegel does not follow Newton’s argument in full, only that to which it points (the ratio with which they vanish); Newton just does not know what that means (Somers-Hall 2010, 562). Hegel explicitly says that he cannot endorse the fluxions orquacum evanescent—these are still the “something else” with which the differential is amalgamated to make it accessible to intuition. Thus, two features distinguish Hegel’s view from Newton’s: first, this qualitative nature is something intrinsic to the mathematical phenomenon (WL 21.310–22), not something with which it is amalgamated in an ad hoc manner. Second, it is something essentially conceptual or logical, rather than intuitive or physical. But note that the qualitative nature is a conceptual or logical feature required to do justice to the mathematical phenomenon (that is, it is what is rendered as a limit in contemporary mathematics), not for idiosyncratically Hegelian methodological reasons (such as the purported inevitability of dialectical contradiction). There is, of course, a sense in which this conceptual or logical feature is added to the notion of quantity by means of a dialectical process. That process, however, works not by forcing the mathematical phenomenon into a preconceived pattern but rather by noticing that a widely recognized confusion about the phenomenon can be traced back to the tension between qualitative and quantitative aspects of that phenomenon. This is the sense in which contradiction functions here for Hegel as at most a heuristic thread to follow rather than a methodological premise, contra Deleuze. 378 HISTORY OF PHILOSOPHY QUARTERLY

We would do well to pause here and regard Hegel’s analysis in the light of modern mathematical practice. Hegel realizes that nesting operations are necessary and that the ratio has to be understood as qualitative, and it is certainly true that we only get the derivative after taking the limit of the ratio as a whole. Before such an operation, we find only a quotient, and there is no way to take the limit just as a limit of the numerator divided by a limit of the denominator. This would yield the nonsensical and derided as Hegel and all critics have pointed out. This Gordian knot is cut by modern mathematics almost tautologically by defining the limit of a Cauchy sequence to be the class of the sequence itself, just like a decimal expansion defines a real number. In this connection, we should read Hegel as offering a philosophical concept (Begriff) to capture the nature of this class and an answer to the question of its justification. This answer is to insist on the qualitative nature of that composite, or its essential character of being a ratio. It then follows that, to take the limit, there must be brackets around the ratio, treating it as a whole. And, indeed, it is the case that the sequence of ratios is subjected to another ratio like equivalence.8 Furthermore, the fact that a derivative is a becoming or nascent quantity is also reflected in the modern treatment, as the fact that a derivative can only be computed locally. One needs a neighborhood of a point to be able to define the derivative at that point; this neighborhood can be arbitrarily small, but it is necessary. Technically, this is called a germ, which suggests its nascent character.9 Again, there are important philosophical and mathematical issues involved in conceptualizing the practice of calculus, but Hegel does manage to show that it can be done by extending the same logical resources already required to understand number as such—here, resources relating to ratios.

B. Variability Resources related to ratios can be extended to eliminate the ad hoc character of the disregarding of all but the linear term, but the ad hoc nature of the variability at the heart of calculus remains. Philosophi- cally, we would like to know why we have the two terms dy and dx in the ratio in the first place rather than determinate quantities. ButHegel realizes that quanta are themselves inherently variable, and, if it is in the basic nature of all quantitative determinacy to be variable, then the use of such variability in the methods of differential calculus is not an arbitrary and ungrounded novelty in mathematical practice but rather an extension of other mathematical uses of numbers that brings into relief one of their essential properties. Thus, Hegel gives a philosophical meaning to Newton’s fluxions by reinterpreting it: on this interpretation, the quacum evanescent is not a vanishing but a creation; it is a becoming (Werden) that is inherent in quanta. But if that is true, then the logical Hegel on Calculus 379

form expressed so confusingly by infinitesimals should be amenable to treatment by the techniques that Hegel had already developed in his discussions of number. Here, unfortunately, we can no longer avoid one of the classic tropes of Hegelian metaphysics, namely, the distinction between true and false infinites; but we can also illuminate it from the modern mathematical perspective. Hegel investigates the role of the infinite in both ratios and derivatives, frequently making use of his distinction between false and true infinities. The main examples for the former are series, whereas true infinities have to include a certain aspect of transcendence or change from mere quantity to quality and vice versa. The series do not have this property; they are merely truncated quantities. Here one would naturally be in- clined to criticize Hegel for not realizing that the real numbers are about convergent sequences (to be precise, Cauchy sequences). However, one would miss an important point: the real numbers are not sequences, but convergent sequences (toclasses be precise: of sequences, Cauchy whichsequences). is a quality However, of the one sequences would miss in an Hegel’s terms. important point, namelyAnd that so the it isrea quitel numbers right areof Hegel not sequences, to point out,but classesfor instance, of that a decimal sequences, which is a qualityexpansion, of the of, sequences for example, in Hegel’s one-third terms. or Andone-seventh so it is quite is not right a true infinity, of Hegel to point out forbut instance that actually that a decimal the ratio expansion, is much of closer,, for example, since it 1/3 contains or 1/7, a quality. In is not a true infinity, butmodern that actually mathematical the ratio isterminology, much closer, this since is encoded it contains by thea quality. fact that rational 10 In modern mathematicalnumbers terminology are classes. this is encoded As Hegel by theexplains fact that it, quotients rational numbers as rational numbers are classes.10 As Hegel areexplains constituted it, quotients of three as rational parts (the numbers two relata are constituted and the value of of their relation, which Hegel calls the exponent); this gives them a qualitative aspect. three parts (the two relata and the value of their relation, which Hegel calls the Their value, the class, is then again a quantity, but, for this, one needs an exponent); this gives them a qualitative aspect. Their value, the class, is then again a infinite process of taking classes of ratios. Hegel then simply applies this quantity, but for this one needs an infinite process of taking classes of ratios). Hegel result from his number theory to infinitesimals by observing that onemust then simply applies thisadd result the from variability his number of the theory constituent to infinitesimals (here, the by variability observing of x in dx). The that one must add the variabilityvariability of is the thus constituent conceptually (here, necessary the variability rather of xthan in dx ad). Thehoc. This insis- variability is thus conceptuallyemployingtence theon necessary variability latter’s “customaryrather resolves than ad waythe hoc ofnature. populariThis insistenceof quantityzing things on of –the in effect,constituents by polluting of the variability resolvesconcept the naturethe andcompound of replacing quantity object it of with the given falseconstituents by sense the- relation.representations” of the compound The constituents (WL object 21.256). are in these given by the relation. Theconsiderations constituents arejust in moments these considerations (WL 21.325). just Only moments after the (WL infinite process Subsequently Hegel offers his own argument for the selection of the linear term 21.325). Only after the infiniteof taking process classes of oftaking ratios classes of do ofwe ratios find of a quantity. do we find a by appeal to the nature of the relation . But before𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 going into the details of this quantity. In modern terms, it is true that, before taking the limit of the ratio argument it will be convenient to recall𝑑𝑑𝑑𝑑 the𝑦𝑦𝑦𝑦 modern𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 definition of a derivative and point , the ratio is that of functions 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑and, in that sense, they are not finite In modern terms,out thequanta it relevant is true or that realfeatures before numbers; for taking Hegel’s they the philosophical become limit of thesuch ratio analysis: only ,after the ratio evaluation. is After that of functions and in that sense they are not finite quanta or real numbers;𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 they this intermediate step of evaluation, one again𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 gets a function. Unac- become such only aftercountably, evaluation. Hegel After this (tries intermediate) to( bypass) thisstep point.of evaluation, This is one a shame again because it ( ) = lim gets a function. Unaccountably, Hegel tries to bypass this point. This is a shame, would𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 have helped𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0 +him𝜀𝜀𝜀𝜀 −𝑑𝑑𝑑𝑑 bring𝑑𝑑𝑑𝑑0 out some consequences of the necessity because it would have helpedof that him variability.0 to bring outSpecifically, some consequences it is actually of the not necessity true that ofdx contains a 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→0 𝜖𝜖𝜖𝜖 that variability. Specifically,The rightfree ithandvariable is actually side xcontains innot any true twoway; that nested otherwise, dx contains operations, if a yfree is firstviewed variable a ratio as x insomeand any then other a limit. vari -The way; otherwise if yleft is viewed handable, side asthere some is indeed would other an be variable un no-separated relationship there would ratio; of itbe dxis no often to relationship dy written. Indeed as of , iswhere actually one first dx to dy. Indeed sets is actually an= archaic( an) .archaic The albeit realization albeit widely widely that used thereused way wayis ofa secondofexpressing expressing operation g . Theon the 𝑑𝑑𝑑𝑑intermediate𝑦𝑦𝑦𝑦 previously formed 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 intermediate step ratioleft out that by ne Hegedsel to and be other considered users ofis centralthe notation to Hegel is’s argument.= ( ), 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 which expresses y as a dependent variable. This only makes𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 the depth of the relational If we attend to the contrast between𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 this modern𝑑𝑑𝑑𝑑 way𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 of proceeding and tie between dy and dx (and thus their qualitative nature) clearer. Thus the variability of Newton’s, we can bring the distinction between Newton and Hegel into relief. Newton quantity is deeply tied to its essentially relational nature. doesn’t yet have the concept of a limit (this comes from Cauchy), and so Newton’s C. Powers version of the above formula is: So far, we have seen the= application of the resources Hegel developed to 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 comprehend ratiosIn and this variability formula the to thelimit case is already of the ratioinherent and in the its varyingquantities numerator dx and dy , but only as 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 and denominator inimplicit. particular. The problemIn another is thatextension, in the calculationHegel𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 takes on up the his basisunderstanding of this formula, higher order 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 powers (WL21.197terms-203) andare generatedapplies them that to have calculus. to vanish In fact, to generateHegel claims the desiredthat result (as we saw derivatives can be aboveexhaustively with respect analyzed to Leibniz).into powers Without and thus the explicittreated concept algebraicially. of a limit,11 Leibniz and There are two pointsNewton of contact are denied between the derivatives possibility ofand explaining powers; onewhy isthis mathematical, vanishing is legitimate. and the other is moreFurthermore, philosophical. they have to explain why, in the determination of the ratio of vanishing, higher-order terms are neglected. The mathematical Whilepoint isHegel that embracesdifferentiation the fact picks that out in the Newton’s information argument about a certain quality of the exponent. Thisthe is obviousdifferential in the takes equation shape thatas the Hegel last usesvalue for of histhe demonstration: ratio (the quacum evanescunt), Hegel = (WLrejects 21.273). Newton’s As Hegel reasoning correctly for pointsthe negligibility out, this isof not the as higher simple order as it terms legitimizing the 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛−1 calculation. In fact, he ponders what “could have brought Newton to deceive himself 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑛𝑛𝑛𝑛𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 about such a proof” (WL 21.261). Hegel thinks that had Newton only realized that the composite is a quality now8 – a ratio of the two quantities together that remains as the individual constituents vanish – then he would have had an adequate response to Berkeley and the other, more sympathetic critics who proposed alternative forms of analysis (WL 21.262-3). If the composite is essentially relational, then there is nothing strange in the idea that it might have a determinate character over and above whatever determinate character its relata have, and thus a determinate character that might remain or even first become clear in the relation between the changes between those relata undergo as they get smaller. In fact, the way Hegel explains this is the way we currently take the limit of the ratio as it goes to zero in contemporary mathematics:

employing( ) the( )latter’s “customary way of popularizing things – in effect, by polluting the ( ) = lim 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 concept𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+ 𝜀𝜀𝜀𝜀and−𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑replacing0 it with false sense-representations” (WL 21.256). 0 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→3800 𝜖𝜖𝜖𝜖 HISTORY OF PHILOSOPHY QUARTERLY The right hand side containsSubsequently two nested Hegeloperations, offers first his owna ratio argument and then for a thelimit. selection The of the linear term left hand side is indeedbystep appeal an left un outto-separated the by natureHegel ratio; ofand the it other is relation often users written .of But the as before , notation where going one into is firsty the= f ( detailsx), which of this 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 argument it will be convenient to recall the modern𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 definition of a derivative and point sets = ( ). The realizationexpresses thaty as there a dependent is a second variable. operation𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 This on onlythe previously makes the formed depth of the 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ratio that needs to outberelational considered the relevant tie is between featurescentral to fordy Hegel Hegel’sand’s argument.dx philosophical (and thus theiranalysis: qualitative nature) 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 clearer. Thus, the variability of quantity is deeply tied to its essentially relational nature. ( ) ( ) If we attend to the contrast( ) = betweenlim this modern way of proceeding and Newton’s, we can bring the𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 distinction between𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+𝜀𝜀𝜀𝜀 − Newton𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0 and Hegel into relief. Newton C. Powers 0 doesn’t yet have the concept𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 of𝑑𝑑𝑑𝑑 a limit𝜀𝜀𝜀𝜀→ (this0 comes𝜖𝜖𝜖𝜖 from Cauchy), and so Newton’s version of the aboveTheSo formula rightfar, we hand is: have side seen contains the applicationtwo nested operations, of the resources first a ratio Hegel and developed then a limit. The = leftto comprehendhand side is indeed ratios an and un -variabilityseparated ratio; to the it iscase often of writtenthe ratio as , andwhere its one first 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 setsvarying = numerator( ). The realization and denominator that there inis aparticular. second operation In another on the extension, previously formed In this formula the limit is already inherent in the quantities dx and dy, but only as 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Hegel takes up his understanding powers (WL21.197–203) and applies implicit. The problemratio is thatthat inne theeds calculationto be considered on the isbasis central of this to Hegelformula,’s argument. higher order them𝑑𝑑𝑑𝑑 to𝑓𝑓𝑓𝑓 calculus.𝑑𝑑𝑑𝑑 In fact, he claims that derivatives can be exhaustively terms are generatedanalyzed that have intoto vanish powers to generate and thus the treated desired algebraically. result (as we saw11 There are two above with respect to Leibniz).If we Without attend to the the explicit contrast concept between of a this limit, modern Leibniz way and of proceeding and Newton’s,points of wecontact can bring between the distinction derivatives between and powers; Newton one and is Hegel mathematical, into relief. Newton Newton are denied theand possibility the other of is explaining more philosophical. why this vanishing is legitimate. Furthermore, they doesn’thave to explainyet have why, the conceptin the determination of a limit (this of c omesthe ratio from of Cauchy),vanishing, and so Newton’s higher-order termsversion are Theneglected. of mathematical the above formula point is: is that differentiation picks out the information about= the exponent. This is obvious in the equation that Hegel uses While Hegel embraces the fact that in Newton’sn n argument–1 a certain quality of for his demonstration:𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 dx = nx dx (WL 21.273). As Hegel correctly the differential takesIn shapethis formula as the thelast limit value is of already the ratio inherent (the quacum in the quantities evanescunt dx), and Hegel dy , but only as points𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 out, this𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 is not as simple as it sounds, because it pertains to the rejects Newton’s reasoningimplicit. Thefor the problem negligibility is that ofin thethe highercalculation order on terms the basis legitimizing of this formula,the higher order relation of the variable x of the function and the power n that defines the calculation. In fact, he ponders what “could have brought Newton to deceive himself termsfunction. are generated Hegel reduces that have this tospecial vanish relation to generate to binomials the desired and result then (as gives we saw about such a proof”above a(WL clever 21.261). with argument respect Hegel to thinksassimilating Leibniz). that Without had it Newton to the his explicit previousonly realized concept discussion that of athe limit, of powers. Leibniz and composite is a qualityNewtonA nowpayoff –are a for ratio denied Hegel of thethe is twothat,possibility quantities in this of explainingline together of reasoning, whythat thisremains hevanishing can as constructthe is legitimate. his individual constituentsFurthermore,own vanis interpretationh – then they he have would of to the explain have fundamental had why, an in adequate the step determination ofresponse picking to ofout the the ratio linear of vanishing, Berkeley and the other,higherterm more -and,order sympathetic moreover, terms are neglected.criticsits coefficient. who proposed The alternativeargument formsproceeds of in several analysis (WL 21.262-steps.3). If theWhile composite Hegel embraces is essentially the fact relational, that in Newton’sthen there argument is nothing a certain quality of strange in the idea thethat differential it might have takes a determinate shape as the character last value over of the and ratio above (the whatever quacum evanescunt), Hegel (1) He first observes that, in a variable quantity, it is inherently al- determinate characterrejects its relataNewton’s have, reasoning and thus for a determinate the negligibility character of the thathigher might order terms legitimizing the ways possible to consider adding to them and, hence, one inherently is remain or even first become clear in the relation between the changes between those calculation.led to reflect In fact, upon he the ponde functionrs what of “could a sum have as in brought x( + a) nNewton. This expression to deceive himself relata undergo as theyaboutcan get be such smalle expanded a proof”r. by(WL considering 21.261). Hegel the thinks binomial that expansionhad Newton ( xonly + a realized)n = xn + that the In fact, the waycompositenxn Hegel–1a + … explainsis. Thisa quality is this crucially now is the – away ratioinherent we of currently the in two quantity quantitiestake the itself limit together as of quantities the that remains can as the ratio as it goes to zeroindividualbe intaken contemporary constituents to their nth mathematics: vanis power,h – asthen argued he would previously. have had an adequate response to Berkeley and the other, more sympathetic critics who proposed alternative forms of (2) The linear term nxn–1a can now be taken to be a relation between analysis (WL 21.262-3). If the composite is essentially relational, then there is nothing a and x. Hegel now asks: Why should we consider this linear term to be strange in the idea that it5 mig ht have a determinate character over and above whatever fundamental? The first answer is that this is the fundamental relation determinate character its relata have, and thus a determinate character that might between powers and the leading and following terms are all iterates of remain or even first become clear in the relation between the changes between those this fundamental relation (this step is the more developed version of relatathe argument undergo as from they ratios get smalle we consideredr. in § IIA). Indeed, as he points out, formallyIn fact, the this way follows Hegel from explains the thisproduct is the rule. way we currently take the limit of the ratio as it goes to zero in contemporary mathematics: (3) Hegel’s second answer is that it is not the position of the terms in the expansion that is fundamental, but rather the relation that each 5 Hegel on Calculus 381 term and its successor stand. This relationship is fully contained in the power being considered: the power n for the first two terms, the power n-1 for the second and third term, and so on. Here, there is a role reversal in which each term is essential for its successor. (4) In calculus, usually a is replaced by an infinitesimal dx and expanded only to the first order, but Hegel posits that one may as well set a equal to 1, or the unit. This is, in a sense, more natural, since now one is augmenting by units (Einheiten). This circumvents the introduction of infinitesimals and directly yields the expressionx n + nxn–1a + …, which only involves different powers of x and thus eliminates the need for an infinitesimal. (5) The coefficients are now the relations between the different powers of x and in particular the coefficientn of the linear term, which gives the relation between the nth power and the n-1st power. This is why we are justified in picking out the linear term and discarding the rest: which relation one picks depends on the power one is considering, but, for the function at hand, that power is n, which, therefore, constitutes the primate. The first successor is then fundamental and the further successors higher-derived quantities. This is the meaning of his identification of “the reduction of the magnitude to the next lowest power [die Herabsetzung der Größe auf die nächst niedrigere Potenz]” (WL 21.282.12–13) as the main driver of the analytical technique. It would be good at this point to connect this positive argument with Hegel’s earlier criticism of Newton and others for their amalgamation of sensible or otherwise nonmathematical qualities with the derivative that remains as the ratio goes to zero. The argument Hegel provides replaces this nonmathematical quality with a mathematical one: namely, the relation between a power and the next-lowest power. The argument from ratios tells you why the series qua series is not that important (that is, it tells you why what is important is the relation at which the series points), and the argument from powers tells you that that relation is to be found in the linear term (that is, in the relation between the nth and n-1th powers). So one can consider Hegel to have an overarching a two-step argument. The first step tells you the point of the mathematical formalism, and the second tells you the part of the formalism to isolate to determinate its content. This connects with the earlier point that the progression of the ratio to 0/0 is the mathematical form of the divergence of measures (for example, in phase transitions), which is itself the way that Hegel introduces a nonmathematical concept, namely, essence (Wesen). What Hegel might have seen given his own analysis of powers is that the taking of the limit is itself a way to generate the concrete conception 382 HISTORY OF PHILOSOPHY QUARTERLY of the relation between dramatically different scales that is built into the notion of an essence. This would be a different route to a far more Deleuzian metaphysics, in which strictly mathematical tools do more work than the more traditionally metaphysical concepts Hegel em- ploys in the Doctrine of Essence. But, instead of using the divergence of measures, he turns to a different mathematical resource developed earlier in his number theory: the relation between unit and amount. This leads us to the philosophical point of contact between derivatives and powers, which concerns the coincidence of the two conceptual aspects of number (Einheit and Anzahl, or unit and amount) (WL 21.275). This is given by noticing that, in a multiplication one factor is the unit) and the other is the amount. When these two coin- cide, then their difference in quality is resolved, and this transforms multiplication to power and to a quantity. Thus, in computing squares, there is a similar transcendence as in ratios. In this treatment, power (namely, squares) and, with it, higher powers now become an inherent property of quanta (by their variability and their relation to units), namely, that they can be units, thus giving rise to powers, and that they can be augmented by units, thus giving rise to the binomial expansion and the arguments above. Here the mechanism that produces a quantity from the ratio becomes the underlying principle for the definition of powers. Thus, calculus becomes all about powers, which are inherent in the variability of quanta, which was inherent in quanta to start with. Nothing in the way of fundamental principles has to be added that is specific to the calculus itself. The power analysis thus works together with the ratio analysis to show that calculus can be handled by extension of the same basic techniques required for defining real numbers. But, in the same way that no additional mathematical principles have to be added, it is important to Hegel that neither do any geo- metrical or physical ideas have to be added. This was also important to Lagrange, as one can see from the subtitle of his “Théorie des Fonctions Analytiques: Principes du Calcul différentielle, dégagés de toute consi- dération d’infiniment petits, d’évanouissantes, de limite et de fuxions, réduits à l’analyse algébrique des quantités finies” (Theory of Analytic Functions: The Principles of Differential Calculus, Set apart from All Considerations of Infinitesimals, Effervescents, Limits and Fluxtions and Reduced to an Algebraic Analysis of Finite Algebraic Quantities). Hegel also rids calculus of these superfluous terms. Furthermore, he establishes algebra’s primacy over geometry, which is also the approach of modern mathematics.12 As mentioned before, the algebraic approach both Hegel and Lagrange pursue is only valid for analytic functions. Hegel on Calculus 383

Hegel, however, differentiates himself from Lagrange by claiming that the introduction of power series is unnecessary and misleading; only individual powers, and not power series, are required. Hegel is right on this point, which deserves some untangling. The power series are only necessary to make taking a derivative an algebraic operation, not to define derivatives. They are key to extend differential to the wider class of analytic functions, which contains exponentials sine and cosine for instance, in contrast to polynomials. In this setting, one ad- ditionally needs theorems to say that one can interchange summation and differentiation, effectively separating the two concepts. These are developments in analysis, as even the modern definition of a function, which were not quite worked out to a modern level or rigor yet at Hegel and Lagrange’s time. Correctly, these theorems deal with two types of infinities that Hegel bemoans as being confounded. The modern mathematical analysis actually does not confound them, but rather realizes in a precise way that they are two different limits whose interchangeability, as order of operations, is not always guaranteed. Thus, Hegel is right that the series are not the reason for the negligibility of higher orders. Their interplay is, however, tantamount to establishing the algebraic nature of differentiation for a suitable class of functions. Furthermore, Lagrange does make the mistake that Hegel could have foreseen of defining the functions as power series instead of using classes of these. The fact that different power series can give different functions and not all functions are represented by power series was the reason the Cauchy wrote his textbook and refuted Lagrange’s view.

D. Integration Hegel completes his interpretation of calculus with a treatment of integration. This serves a two-fold purpose of demonstrating the applicability of his reasoning that the “reduction” of powers is the fundamental concept (which is inverse to the coincidence of amount and unit in the forming of powers) while also showing the primacy of algebraic thought over geometry. This inversion of the reduction and the coincidence of unit and amount ties into the property of the integral being an antiderivative. Using the method of Bonaventura Cavalieri,13 he discusses integration as a sum of infinitesimals,14 and he observes that, in these manipulations, the dx in ∫fdx naturally has different flavors depending on whether one is using them to compute area, length, or volume. In Cavalieri’s principle, it is not so much that one is multiplying lines by lines to get an area, but that one is again reflecting upon the ratio of two areas (usually to say that they are equal). This ties back into the application of Hegel’s primate of the next leading power, which is most apparent in the calculation of area as the integral over the function giving the boundary. Here one imagines 384 HISTORY OF PHILOSOPHY QUARTERLY

vertical lines making up the area; these are then “summed” together. The integral then is the transition of a sum of lengths to an area, which is exactly going up one power from length to area=(length)2. leadingleading power, power, which which is is most most apparent apparent in in the the calculation calculation of of area area as as the the integral integral over over the the From a modern perspective, there are two correct results here. When functionfunction giving giving the the boundary. boundary. Here Here one one im imaginesagines vertical vertical lines lines making making up up the the area, area, these these defining integrals, one is actually defining areas or lengths, again a areare then then “summed” “summed” together. together. The The integral integral then then is is the the transition transition of of a asum sum of of lengths lengths to to point missed in most explanations and textbooks. The area of a square 2 2 or volume of a higher dimensional version is definedanan byarea, area, an whichintegral, which is is exactly exactlyand going going up up one one power power from from length length to to area=(length) area=(length). . Hegel holds this identification to be a real insight: “TheFrom trueFrom a amodernmerit modern of perspective, perspective, there there are are two two correct correct results results here. here. When When defining defining mathematical acumen is that, from results alreadyintegrals,integrals, known oneelsewhere, one is is actually actually defining defining areas areas or or length, length, again again a apoint point missed missed in in most most it has found that certain sides of a mathematical objectexplanationsexplanations stand andto and each textbooks. textbooks. The The area area of of a asquare square or or volume volume of of a ahigher higher dimensional dimensional other in the relationship of original and derived function,versionversion is is definedand defined it hasby by an an integral, integral, and and Hegel Hegel holds holds this this identification identification to to be be a areal real insight: insight: found which these sides are” (WL 21.295). The way “Theto“The obtain true true meritthe merit results of of mathematical mathematical acumen acumen is is that, that, from from results results already already known known elsewhere, elsewhere, “known” from geometry is that one fixes the normalizationitit has has found found that that certain certainthe sides sides of of a amathematical mathematical object object stand stand to to each each other other in in the the length of the interval from zero to one is indeed one.relationship Therelationship other integralsof of original original and and derived derived function, function, a ndand it it has has found found which which these these sides sides are” are” then can be viewed as products and transformations(WL of(WL this 21.295). 21.295). integral The The and,way way to to obtain obtain the the results results “known” “known” from from geometry geometry is is that that one one fixes fixes the the hence, as ratios. Again, the power analysis, thus, worksnormalizationnormalization together that that with the the length length of of the the interval interval from from 0 0to to 1 1is is indeed indeed 1. 1. The The other other integrals integrals the ratio analysis to show that calculus can be handled by extension of thenthen can can be be viewed viewed as as products products and and tra transformationsnsformations of of this this integral integral and and hence hence as as ratios. ratios. the same basic techniques required for defining real numbers. Again,Again, the the power power analysis analysis thus thus works works together together with with the the ratio ratio analysis analysis to to show show that that The second correct result of Hegel’s construal ofcalculus calculusCavalieri’s can can be methodbe handled handled by by extension extension of of the the same same basic basic techniques techniques required required for for defining defining is one that Hegel himself misses (but would havereal beenreal numbers. numbers. accessible for him at this point): the resolution of Cavalieri’s paradox by means of the chain rule. The paradox pertains to the area of the two subtrianglesTheThe second second of correct acorrect r esultresult of of Hegel’s Hegel’s construal construal of of Cavalieri’s Cavalieri’s method method is is one one that that triangle created by an altitude of its largest angle. HegelTheHegel lines himself himself parallel misses misses to (but (but would would have have been been accessible accessible for for him him at at this this point): point): the the the height are in 1–1 correspondence, and, under thisresolutionresolution correspondence, of of Cavalieri’s Cavalieri’s paradox paradox by by means means of of the the chain chain rule. rule. The The paradox paradox pertains pertains to to the the line segments of equal height are paired. So, one couldareaarea argue, of of the the twothe two areassubtrian subtrian glesgles of of a atriangle triangle created created by by an an altitude altitude of of its its largest largest angle. angle. The The of the two subtriangles should be the same.15 But linesthislines isparallel parallel only theto to the thecase height height are are in in 1 -11- 1correspondence correspondence and and under under this this correspondence correspondence for isosceles triangles, and not triangles in which thelineline base segments segments is otherwise of of equal equal height height are are paired. paired. So, So, one one could could argue, argue, that that the the areas areas of of the the two two cut into two unequal parts, ab and bc. This is illustratedsubtrianglessubtriangles in Figure should should 1. be be the the same. same.1515 But But this this is is only only the the case case for for isosceles isosceles triangles, triangles, and and notnot triangles triangles in in which which the the base base is is otherwise otherwise cut cut into into two two unequal unequal parts, parts, and and . .This This isis illustrated illustrated in in the the following following figure: figure: �𝑎𝑎𝑎𝑎�𝑎𝑎𝑎𝑎�𝑏𝑏𝑏𝑏��𝑏𝑏𝑏𝑏� �𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏�𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏��

Figure 1 This can be resolved using the chain rule. If one ThissegmentThis can can be abbe resolved resolvedis taken using using the the chain chain rule. rule. If If one one segment segment is is taken taken as as the the unit, unit, the the as the unit, the other bc is some multiple of it, say otherkother, and hence is is some some if dxmultiple multiple is of of it, it, say say k ,k and, and hence hence if if dx dx is is the the infinitesimal infinitesimal (or (or measure) measure) on on �� the infinitesimal (or measure) on the first anddy isthe thethe first infinitesimalfirst and and dy dy is is the the on infinitesimal infinitesimal on on the the other other then then the the chain𝑎𝑎𝑎𝑎 chain𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 rule rule says says that that dx=kdy dx=kdy and and 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏��𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏�� the other, then the chain rule says that dx=kdy andhence hencehence dx === kdy is is the the correct correct identity. identity. is the correct identity. 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 ∫This𝑎𝑎𝑎𝑎∫This𝑎𝑎𝑎𝑎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 resolution resolution∫𝑏𝑏𝑏𝑏∫𝑏𝑏𝑏𝑏𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 makes makes use use of of several several of of the the features features that that are are important important to to Hegel. Hegel. This resolution makes use of several of the featuresFirst,First, that k karerepresents represents important the the difference difference in in normalization normalization and and hence hence a aratio ratio of of areas. areas. S econdSecond, in, in to Hegel. First, k represents the difference in normalizationthethe chain chain and, rule rule hence, it it becomes becomes a apparent apparent that that dx dx and and dy dy a reare actually actually bound bound together together by by their their ratioratio and and it it is is this this ratio, ratio, which which is is k ,k that, that is is meant meant by by . Third. Third, they, they should should not not exist exist 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 alone,alone, but but are are bound bound by by the the integral integral sign, sign, which which lends lends𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦𝑑𝑑𝑑𝑑 𝑦𝑦𝑦𝑦them them sense. sense. Finally Finally, as, as in in the the case case ofof powers, powers, one one can can see see that that the the “ infinite“infinite sum” sum” represented represented by by the the integral integral can can have have

1212 employing the latter’s “customary way of popularizing things – in effect, by polluting the concept and replacing it with false sense-representations” (WL 21.256). Subsequently Hegel offers his own argument for the selection of the linear term by appeal to the nature of the relation . But before going into the details of this argument it will be convenient to recall𝑑𝑑𝑑𝑑 the𝑦𝑦𝑦𝑦 modern definition of a derivative and point 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 out the relevant features for Hegel’s philosophical analysis:

( ) ( ) Hegel on Calculus 385 ( ) = lim 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0+𝜀𝜀𝜀𝜀 −𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑0 0 ratio of areas. Second, in the chain rule, it becomes apparent that dx and 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀→0 𝜖𝜖𝜖𝜖 The right hand side contains two nested operations,dy are first actually a ratio and bound then together a limit. The by their ratio; it is this ratio, which is k, left hand side is indeed an un-separated ratio; it thatis often is meant written by as ., Third,where onethey first should not exist alone but are bound by sets = ( ). The realization that there is a secondthe integraloperation sign, on thewhich𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 previously lends them formed sense. Finally, as in the case of powers, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ratio that needs to be considered is central to Hegelone’ scan argument. see that the “infinite sum” represented by the integral can have 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 three interpretations: (1) the sum giving the unit and the lines giving the If we attend to the contrast between thisdifferent modern wayamounts, of proceeding as a product and (as such this is an area, albeit normalized Newton’s, we can bring the distinction between differentlyNewton and in Hegel the two into triangles); relief. Newton or (2) regarding this as a sum, we would doesn’t yet have the concept of a limit (this comesobtain from a Cauchy), line composed and so Newton’s of all the pieces, which is not meant, neither in version of the above formula is: algebra nor in geometry; but (3), finally, the integral sign gives the magnitude of this line, which is a number. Again, there is a normalization (in = Hegel’s terminology, this is the degree [Grad]). In this formula𝑑𝑑𝑑𝑑𝑦𝑦𝑦𝑦 the limit is already inherent in the quantities dx and dy, but only as 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 implicit. The problem is that in the calculation on the basis of this formula, higher3. C orderonc lusion terms are generated that have to vanish to generate the desired result (as we saw above with respect to Leibniz). Without the explicitOne concept of the ofgreat a limit, difficulties Leibniz and in interpreting Hegel’s understanding of calculus and of mathematics generally is that he wrote right on the Newton are denied the possibility of explaining why this vanishing is legitimate. cusp of some of the great technical and conceptual breakthroughs in Furthermore, they have to explain why, in the determination of the ratio of vanishing, mathematics in the mid-nineteenth century, breakthroughs that Rus- higher-order terms are neglected. sell rightly credits with having resolved difficulties that had lingered While Hegel embraces the fact that in Newton’sthroughout argument the history a certain of quality mathematics of and in sharpened form through the differential takes shape as the last value of thethe ratio early (the modern quacum period. evanescunt Chief), Hegel among these fundamental problems rejects Newton’s reasoning for the negligibility ofwas the thehigher problem order termsof understanding legitimizing the mathematical continuity. Although calculation. In fact, he ponders what “could havewe brought have manyNewton intuitive to deceive presentations himself of continuity (for example, in about such a proof” (WL 21.261). Hegel thinks thatgeometrical had Newton figures), only realized it hadthat thealways seemed difficult to understand composite is a quality now – a ratio of the two quantitiessuch continuity together inthat terms remains of theas the discrete numbers that seem, equally individual constituents vanish – then he would haveintuitively, had an adequate to be paradigmatic response to (for example, the natural numbers). For Berkeley and the other, more sympathetic criticsexample, who proposed the attempt alternative to formsbuild ofup a continuous line out of extension- analysis (WL 21.262-3). If the composite is essentiallyless points relational, seems then incoherent. there is nothing Hegel thought that, on this point and on strange in the idea that it might have a determinatethe characterattempts over to demonstrate and above whatever the reality of infinitesimals by means of determinate character its relata have, and thus athe determinate amalgamated character properties that might such as fluxions, mathematics foundered remain or even first become clear in the relation onbetween the contradictions the changes between of its ownthose practice and it fell to philosophy to relata undergo as they get smaller. explain the objective reality of mathematical concepts (WL 21.237, In fact, the way Hegel explains this is the 252,way we256–57, currently and take 272). the As limit strong of the as Hegel’s analysis is, nonetheless it ratio as it goes to zero in contemporary mathematics:is at this point perhaps at its weakest, since he denies mathematics even the basic capability to treat these topics adequately. Of course, as Russell rightly points out, this has been proven wrong by subsequent developments. Modern mathematics has solved the problems 5 of continuity by using limits and an appropriate definition of the real numbers, yet Hegel’s position on calculus still provides constructive insights into the subject that remain relevant. This hinges on the fact that the modern mathematical treatment is at heart very technical and in a sense tautological—a fact that is glossed over in the usual education where recourse is taken to older forms of justifications of 386 HISTORY OF PHILOSOPHY QUARTERLY calculus—and such a technical and tautological account is in no position to explain the relation of calculus to reality. The fact that the solutions developed to the problem of the continuum are tautological and thus not intrinsically informative is glossed over in the standard way most of us learned calculus, at least in English-speaking countries. To exemplify this, consider from a Hegelian perspective what is now used as the standard high school and early college definition of a derivative of a function as the slope of the tangent line to graph. This tangent is considered as a limit of secant lines. This is a plausible geometric construction, but how is this legitimized? The key question is what this limit is. In modern mathematical terms, one chooses a local parameter- ization, that is, a representation in terms of functions and then takes the derivative of these functions. The derivative itself is a limit. If this limit exists, then it gives the direction of the tangent line. The exact mathematical background is actually not that important for the discussion at hand; we only need that the calculation uses a nongeometric representation and a limit. At this point, one should recall that historically there was a struggle to define the real numbers, and one of the two definitions of real numbers that come to be used is Cauchy sequences modulo null sequences. The limit is then a real number by virtue of representing such a sequence, which is the tautological part of the definition mentioned above. Bypass- ing these modern achievements, we still are used to handling the real number line as a basically pre-existing geometric object. The derivation used today for derivative in calculus instruction either uses spontaneous velocity (already debunked by Zeno) or the characteristic triangle of which Hegel rightfully says does not hold up to rigorous scrutiny. These, then, also enter into both common and intellectual debate. More than that: in the guise of infinitesimal variations, as in Jean Le Rond d’Alembert’s principle, they are alive and well in physics. One of the struggles of mathematical physics has been to make sense of these variations and integrals in infinite dimensional spaces as needed for a rigorous definition of path integrals. These older forms of justification are susceptible to valid criticisms, which have been voiced from Zeno to Berkeley and in a very detailed way by Hegel. But Hegel does more than criticize: he draws attention to the key issue, namely, how infinitesimals as limits are employed, and offers a fundamental philosophical treatment that has a metaphysical payoff for mathematics itself. Though Hegel is clearly opposed to the amalgamation of conceptual structures of mathematics with physical or dynamic notions intended to make those structures more intuitive, this very opposition has a materi- alist motivation. So long as such physical or dynamic notions are taken as given and then merely combined with the conceptual structures, Hegel on Calculus 387

Hegel thinks that the problem of showing the reality of such structures is insoluble; the question has simply been begged. In contrast, Hegel at several points provides an argument that what we mean by physical or material objects has largely to do with the integrity and continuity that such conceptual structures describe. This is clearest in his discussion of measure, where he argues that to be a physical object just is for something to define its own scale and metric (WL 21.345–46). But Hegel pursues a similar argument in his discussions of calculus (WL 21.255 and 279), which date from the same late revision to the Science of Logic. There is no space here to pursue the issue, but it is clear that Hegel means to provide his own argument for the objectivity of mathematics that circumvents the Kantian reference via intuition to space and time. Space and time provide the privileged contexts in which mathematical relations can be observed and tested with greatest clarity, but the objectivity of those relations is secured more directly, by the way that they constitute the very core of our conception of a material object as such.

Purdue University

Keywords: Hegel, calculus, Newton, mathematics, Deleuze

NOTES

1. Though more sympathetic, similar interpretations of Hegel’s view as driven by extraquantitative methodological concerns are found in Lacroix (2000) and Moretto (2013). 2. Or, as Henry Somers-Hall puts it, which are sufficient to “finesse the paradoxes resulting from [the infinitesimal interpretation of the calculus]” (2010, 560). 3. See Kaufmann and Yeomans (2017). Deleuze takes the view that “the limit must be conceived not as the limit of a function but as a genuine cut” ( 1995, 172), and Hegel appears to agree (see WL 21.265). 4. Compare Somers-Hall: “For [Hegel], the difficulty of differentials ap- pearing in the resultant formulae is resolved . . .through recognizing that the status of the nascent ratio differs from that of normal numbers” (2010, 570–71). 5. Citations to Hegel’s Science of Logic (WL) are to Hegel 1968, vol. 21; English translations are modified from Hegel (2010). 6. Of course, there is nonstandard analysis, which gives a mathematical home to infinitesimals, but only in a very construed or particular way. There is also an algebraic way to counter the problem of the square of an object being zero, with the object itself not being zero, in the ring of so-called dual numbers. 388 HISTORY OF PHILOSOPHY QUARTERLY

The latter is again a local technique that does not undermine the general point that dx is not a number. 7. As dx, there is another mathematical interpretation, namely, that of a one form as used in integrals. Hegel treats this interpretation in the third remark added to the discussion of the infinite (WL 21.299–309). 8. The equivalence in mathematics goes through the definition of function, which is given by evaluation. The differentiation is defined pointwise. Here Hegel did not foreshadow this development but discards it (WL 21.259). 9. This is the extra structure of a topology on the real numbers that is needed to capture continuity. See Kaufmann and Yeomans (2017). 10. Hegel’s argument for this as a conceptual truth is one of the high points of his philosophy of mathematics. See Kaufmann and Yeomans (2017) 11. This is not quite true, though it arguably applies to so-called analytic functions. 12. Its independence and yet relevance for physics is an entirely different matter, but the current view within mathematics is one that fits well with Hegel’s general analytical framework. 13. Cavalieri’s principle in two dimensions states that two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. An example is two parallelograms with the same height and base.

Figure 2 14. The integral sign invented by Leibniz as an elongated S stands, indeed, for summa. 15. This conclusion is prohibited by Cavalieri’s principle, since he assumes that the line segments have to be cut by the same line.

REFERENCES

Berkeley, George. 1999. “The Analyst.” In The Works of George Berkeley, edited by A. A. Luce and T. E. Jessop, 1:65–103. Past Masters. Charlottesville, VA: InteLex Corporation. Deleuze, Gilles. 1995. Difference and Repetition. Translated by Paul Patton. New York: Columbia University Press. Hegel on Calculus 389

Duffy, Simon S. 2009. “The Role of Mathematics in Deleuze’s Critical Engage- ment with Hegel.” International Journal of Philosophical Studies : IJPS. 17, no. 4: 563–82. Hegel, Georg Wilhelm Friedrich. 1968. Gesammelte Werke. Hamburg: Meiner. ———. 2010. The Science of Logic. Translated and edited by George Di Giovanni. Cambridge: Cambridge University Press. Cited to as “WL” (Wissenschaft der Logik) in text. Kaufmann, Ralph M., and Christopher Yeomans. 2017. “Math by Pure Thinking: R First and the Divergence of Measures in Hegel’s Philosophy of Mathemat- ics.” European Journal of Philosophy: http://dx.doi.org/10.1111/ejop.12258. Accessed November 15, 2017. Lacroix, Alain A. 2000. “The Mathematical Infinite in Hegel.”The Philosophical Forum 31, nos. 3 and 4: 298–327. Laubenbacher, Reinhard, and David Pengelley. 2000. Mathematical Expeditions: Chronicles by the Explorers. Cor. ed. New York: Springer. Moretto, Antonio. 2013. “Hegel on Greek Mathematics and the Modern Cal- culus.” In Hegel and Newtonianism, edited by Michael John Petry, 149–65. Dordrecht: Kluwer. Russell, Bertrand. 1956. “Logical Positivism.” In Logic and Knowledge: Essays, 1901–1950, 365–82. London: Allen & Unwin. Somers-Hall, Henry. 2010. “Hegel and Deleuze on the Metaphysical Intepreta- tion of the Calculus.” Continental Philosophy Review 42, no. 4: 555–72. x