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莱布尼兹论微积分的基础：关于无穷小量的真实性问题

1998年，美国学者发表研究论文，题为“Leibniz on theFoundations of the Calculus: The Question of the Reality of InfinitesimalMagnitudes”，史料充实，观点清晰，值得一读。

回到三百年前，查阅莱布尼兹未发表的手迹以及有关资料，理清思路，还原历史，说明了一个真实的故事：基于无穷小概念构建微积分是莱布尼兹对人类的一大贡献！

袁萌  11月27日

该文的全文如下

Among hisachievements in all areas of learning, Leibniz’s contributions to thedevelopment of European mathematics stand out as especially influential. Hisidiosyncratic metaphysics may have won few adherents, but his place in thehistory of mathematics is sufficiently secure that historians of mathematicsspeak of the “Leibnizian school” of analysis and delineate a “Leibniziantradition” in mathematics that extends well past the death of its founder. Thisgreat reputation rests almost entirely on Leibniz’s contributions to the calculus.Whether he is granted the status of inventor or co-inventor, there is noquestion that Leibniz was instrumental in instituting a new method, and hiscontributions opened up a vast new field of mathematical research.

The foundationsof this new method were a matter of some controversy, however. The Leibniziancalculus at least appears to violate traditional strictures against the use ofinfinitary concepts, and critics charged Leibniz with abandoning classicalstandards of rigor and lapsing into incoherence and error. In response, Leibnizargued that his methods were rigorous and that they did not suppose the realityof infinitesimal quantities. But the critics of the calculus were not the onlyones who engaged Leibniz in a discussion of the nature of the infinitesimal.Even those sympathetic to the new method gave differing interpretations of thedoctrine of infinitesimal differences, and Leibniz’s own doctrine evolved outof dissatisfaction with alternative foundations for the calculus, as well as adesire to satisfy the demands of his “traditionalist” critics.

The fundamentalthesis in Leibniz’s mature account of the foundations of the calculus is thatinfinitesimals are well-grounded fictions（注意这种说法！）. Although it isnot without its difficulties, the doctrine seems to solve or avoid problemsinherent in two alternative treatments of the calculus, one (due [End Page 6]to Johann Bernoulli) that regards infinitesimals as real positive quantitiesand another (from John Wallis) that takes them as nothing, or “non-quanta.” Atthe same time the fictional infinitesimal allows a response to thetraditionalists, whose criteria of rigor permit only finite magnitudes inmathematics. I do not wish to say that Leibniz’s theory arises out of aHegelian synthesis of Wallis’s thesis and its Bernoullian antithesis.Nevertheless, the fictional treatment of the infinitesimal clearly appearsdesigned in response to them and to the critics of the calculus. If I am right,we can see this doctrine take shape through the 1690s as Leibniz tries tosettle on an interpretation of the calculus that can preserve the power of thenew method while placing it upon a satisfactory foundation. 1

This essay is divided into six parts in roughly chronological order.The first is a brief overview of Leibniz’s formulation of the calculus,including its background in Hobbes’s doctrine of conatus. The second outlinesobjections that Bernard Nieuwentijt and Michel Rolle raised to the calculus.The third considers Leibniz’s response to Nieuwentijt, and particularly theproposal that the calculus can be based on “incomparably small” magnitudes.Section four examines Leibniz’s correspondence with John Wallis and hisrejection of Wallis’s claim that infintesimals lack quantity. The fifth sectionconsiders Leibniz’s correspondence with Bernoulli and its connection to theproject of replying to Rolle. The final section then tries to make some senseof the Leibnizian theory of the fictional infinitesimal.

1. Leibniz, Hobbes, and the Problem of Quadrature

The story of Leibniz’s invention (or, if you prefer, discovery) of thecalculus has been told many times, both by Leibniz himself and by numerouscommentators. 2I do not propose to recount it in any detail, but it is important that weconsider it at least in outline. Leibniz’s first mathematical investigationswere apparently in the field of combinatorics; he reports in the essay Historiaet origo calculus differentialis that he “took great delight in the propertiesand combinations of numbers” (GM V, p. 395), and these studies led him topublish his Dissertatio de Arte Combinatoria in 1666 (GM V, pp. 10–79). Leibnizbecame particularly interested in the [End Page 7] properties of numericalsequences and the sums or differences of the terms in such sequences. Henoticed that the operations of addition and subtraction, when applied to theterms of a given sequence, will produce two new sequences, one of sums andanother of differences. That is, starting with the sequence {an} = {a1,a2,a3,}we can generate a sequence {sn}of sums and a sequence {dn}of differences bymaking

Then, as Leibniz noted, there is an interesting reciprocity in the factthat the differences of the sum sequence and the sums of the differencesequence both yield the original sequence. 3

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Figure 1.

This reciprocity is mirrored in another important concept at the coreof the Leibnizian calculus, namely the idea that a curve is a polygon with aninfinite number of infinitely small sides. Taking the curve ABC in figure 1 with Cartesian coordinate axes x and y, we can approximatethe area beneath the curve by dividing the abscissa into a finite collection ofequal subintervals {x0,x1,x2,xn}. The sum of the rectangles x0y1,x1y2,x2y3,...xn-1yn approximates the area, and this approximation can besystematically improved by taking ever larger collections of ever smaller [EndPage 8] subintervals. Moreover, the tangent at any point on the curve can beapproximated by taking differences between the y values associated withsuccessive x co-ordinates of the abscissa. Thus, at point p the slope of thetangent is approximated by taking the difference y2 - y1.

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Figure 2.

Leibniz invites us to consider the case (as in figure 2) where the finite approximations giveway to exact results when the curve is treated as an infinitary polygon. 4In the infinite case, the differences dx and dy become infinitesimal incrementsin the abscissa and ordinate of the curve. At any given point on the curve,these increments form a “differential triangle” with sides dy and dx, whosehypotenuse is aninfinitesimal element of the curve. The area bounded by the curve and the axescan then be thought of as built up out of infinitely narrow parallelograms ofthe form ydx, so that infinite sums of such parallelograms give the area,while the ratio between dx and dy gives the slope of the tangent. Perhaps morestrikingly, the old reciprocity between sums and differences is maintained: thearea under the curve is an infinite sum, differences between the terms of thesum are (the slope of) tangents, and the problem of quadrature is the inverseof the problem of drawing a tangent. The foundational role of these concepts isemphasized in the manuscript Elementa calculi novi pro differentiis et summis,tangentibus et quadraturis . . ., where Leibniz declares that the “foundationof the calculus” is the principle that “differences and sums are reciprocal toone another; that is the sum of differences of a sequence is a term of the sequenceand the difference [End Page 9] of sums of a sequence is itself a term of thesequence. I express the former as the latter as “ (Leibniz 1855, p. 153).

The great power of Leibniz’s differential calculus is that it allowsthe problems of tangency and quadrature to be reduced to a relatively simplealgorithmic procedure. Take as an example the curve with the analytic equation1 with variables x, y, and constants a, b, c, d. We then consider the “differentialincrement” of [1] obtained by replacing y and x with y + dy and x + dxrespectively. The result is 2

Expanding the right side of equation [2] yields 3

Subtracting equation [1] from [3] gives the increment 4

Dividing each side of [4] through by dx results in an equationexpressing the ratio between dy and dx at any point on the curve, that is: 5

But because dx is infinitely small in comparison with x, the termscontaining it can be disregarded in the right side of equation [5], yielding 6

The formula in [6] (known as the “derivative” of [1]) gives the slopeof the tangent at any point on the curve in equation [1]. The algorithmiccharacter of this procedure is especially important, for it makes the calculusapplicable to a vast array of curves whose study had previously been undertakenin a piecemeal fashion, without an underlying unity of approach. In thisexample we have been concerned with a very simple third degree equation, butthe basic concepts can be extended to much more complex cases involvingfractional or irrational powers and exponents, as well as more difficult“transcendental” functions. 5 This is precisely the aspect of thecalculus that Leibniz trumpeted in 1684 with his first publication on thesubject, whose title promises a “new method for maxima and [End Page 10] minimaas well as tangents, which is not impeded by fractional or irrationalquantities; and a remarkable type of calculus for them.” 6

There is another important concept in the Leibnizian calculus thatneeds to be mentioned, namely that of higher-order differentials. In Leibniz’spresentation the differences dx and dy are themselves variable quantities, andthey can be thought of as ranging over sequences of values of x and y that areinfinitely close to one another. Depending upon the nature of the curve, theinfinitesimal quantities dx and dy can stand in any number of differentrelations, and because these quantities are themselves variable, it makes senseto inquire into the rates at which they vary. The second-order differences ddxand ddy appear as infinitesimal differences between values of the variables dxand dy, and similar considerations allow the construction of a sequence ofdifferences of ever-higher orders. Higher-order differences are employed toconsider the behavior of curves that themselves are derived from the taking ofa first-order derivative, and this process can extend to a wide range ofimportant cases. The means of introducing second-order differentials varies:sometimes they are introduced as products of differences of the first order,other times as magnitudes which stand in the same ratio to a first-orderdifference as the first-order difference stands to a finite quantity.

Even from the very cursory summary given here, it should be clear thatthe most natural formulation of the Leibnizian calculus makes itstraightforwardly committed to the reality of infinitely small quantities.These are introduced to facilitate the study of continuous curves, and aretreated like ordinary quantities when they are added, divided, or otheralgebraic operations are applied to them. Yet they can also be treated as“negligible” or “discardable” according to convenience, while the resultsobtained by their use are taken to be perfectly exact and not mereapproximations. Although Leibniz’s first paper on the calculus is sufficientlyvague to avoid any definite commitment to the reality of infinitesimals, his1686 essay De geometria recondita speaks openly of “my differential calculus oranalysis of indivisibles and infinities” and refers to the characteristictriangle “whose sides are indivisible (or, speaking more accurately, infinitelysmall), that is to say differential quantities” (GM V, pp. 230, 232). By 1694Leibniz could offer “our new calculus of differences and sums, which involvesthe consideration of the infinite” as an example of reasoning which “extendsbeyond what the imagination can attain” (GM V, p. 307).

The rather obvious departure from classical standards of rigor implicit[End Page 11] in the calculus led to some considerable controversy. However,before moving to an account of the controversies surrounding the calculus, itis worth considering one source of Leibniz’s ideas, namely Hobbes’s doctrine ofconatus and its application to the problem of quadrature.

Hobbes first introduced the concept of conatus in his 1655 treatise DeCorpore—a work which he touted as the first part of the elements of philosophyand which contained his doctrines on the nature of body as well as histhoroughly materialistic philosophy of mathematics. As Hobbes defines it,conatus is essentially a point motion, or motion through an indefinitely smallspace: “conatus” he declares, “is motion through a space and a time less thanany given, that is, less than any determined whether by exposition or assignedby number, that is, through a point” (Hobbes [1839a–45] 1966a, I, p. 177).Hobbes employs his idiosyncratic conception of points here, in which a point isan extended body, but one sufficiently small that its magnitude is notconsidered in a demonstration. In explicating the definition of conatus hetherefore remarks that “it should be recalled that by a point is not understoodthat which has no quantity, or which can by no means be divided (for nothing ofthis sort is in the nature of things), but that whose quantity is notconsidered, that is, neither its quantity nor any of its parts are computed indemonstration, so that a point is not taken for indivisible, but for undivided.And as also an instant is to be taken as an undivided time, not an indivisibletime” (Hobbes [1839a–45] 1966a, I, pp. 177–78). The resultis that conatus is a kind of “tendency toward motion” or a striving to move ina particular direction.

This definition allows for a further concept of impetus, or theinstantaneous velocity of a moving point; the velocity of the point at aninstant can be understood as the ratio of the distance moved to the timeelapsed in a conatus. In Hobbes’s terms “impetus is this velocity [of a movingthing] but considered in any point of time in which the transit is made. And soimpetus is nothing other than the quantity or velocity of this conatus” (Hobbes [1839a–45] 1966a, I, p. 178).

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Figure 3.

The concepts of impetus and conatus can be applied to the case ofgeometric magnitudes as well as to moving bodies. Because Hobbes held thatgeometric magnitudes are generated by the motion of points, lines, or surfaces,he also held that it is possible to inquire into the velocities with whichthese magnitudes are generated, and this inquiry can be extended to the ratiosbetween magnitudes and their generating motions. 7 For example, we can think of a curve asbeing traced by the motion of a point, and at any given stage in the generationof the curve, the point will have a [End Page 12] (directed) instantaneousvelocity. This, in turn, can be regarded as the ratio between the indefinitelysmall distance covered in an indefinitely small time; this ratio will be afinite magnitude which can be expressed as the inclination of the tangent tothe curve at the point. Take the curve αβ as in figure 3. The conatus of its generating point atany instant will be the “point motion” with which an indefinitely small part ofthe curve is generated; the impetus at any stage in the curve’s production willbe expressed as the ratio of the distance covered to the time elapsed in theconatus. Represent the time by the x-axis and the distance moved by the y-axis.Then (assuming time to flow uniformly), the instantaneous impetus will be theratio between the instantaneous increment along the y-axis to the incrementalong the x-axis. We can represent these different increments somewhatfancifully by arrows in each direction of the impetus. The tangent to the curvecan then be analyzed as the diagonal of the parallelogram whose sides areproportional to the increments.

It is important to observe here that the tangent is constructed as afinite ratio between two quantities that, in themselves, are small enough to bedisregarded. That is to say, the ratio between two “inconsiderable” quantitiesmay itself be a considerable quantity. Hobbes emphasizes this feature of hissystem when he stresses that points may be larger or smaller than one another,although in themselves they are quantities too small to be considered in ageometric demonstration. Thus, in discussing the comparisons that may be madebetween one conatus and another, Hobbes declares: “as a point may be comparedwith a point, so a conatus can be compared with a conatus, and one may be foundto be greater or less than another. For [End Page 13] if the vertical points oftwo angles are compared to one another, they will be equal or unequal in theratio of the angles themselves to one another; or if a right line cuts manycircumferences of concentric circles, the points of intersection will beunequal in the same ratio which the perimeters have to one another” ([1839a–45] 1966a, I, p. 178).

Hobbes’s concepts of conatus and impetus can also be applied to thegeneral problem of quadrature by analyzing the area of a plane figure as theproduct of a moving line and time. Hobbes himself was eager to solve problemsof quadrature (most notably the quadrature of the circle), and it is here thathis concept of conatus is put most fully to work. Indeed, it is no exaggerationto say that the third part of De Corpore (which bears the title “On the Ratiosof Motions and Magnitudes”) is Hobbes’s attempt to furnish a general method forfinding quadratures. In the very simplest case, the whole impetus imparted to abody throughout a uniform motion is representable as a rectangle, one side ofwhich is the line representing the instantaneous impetus while the otherrepresents the time during which the body is moved. More complex cases can thenbe developed by considering nonuniform motions produced by variable impetus. Inchapters 16 and 17 of De Corpore Hobbes approached a variety of differentquadrature and tangency problems, and in so doing he presented a number ofimportant results that belong to the “prehistory” of the calculus. Of specialinterest in this context is Hobbes’s appropriation of important results fromBonaventura Cavalieri’s Exercationes Geometricae Sex, which he set forth inchapter 17 of De Corpore as an investigation into the area of curvilinearfigures. 8

It is well known that Leibniz was profoundly influenced by his readingof Hobbes, and he seems to have been particularly enamored of the Hobbesianconcept of conatus. In his famous 1670 letter to Hobbes, Leibniz declares theEnglish philosopher to be “wholly justified” in “the foundations [he has] laidconcerning the abstract principles of motion” (Leibniz to Hobbes, 22 July 1670;GP VII, p. 573 ). To the extent that the concept of conatus is the basis forHobbes’s analysis of motion, this endorsement suggests that Leibniz was readyto follow Hobbes in using the concept for the analysis of all phenomenaproduced by motion. Indeed, scholars today generally accept that “Leibniz’searly writings on natural philosophy are virtually steeped in De Corpore” (Bernstein 1980, p. 29). In particular, Leibniz’s readingof Hobbes appears to have been the source for much of [End Page 14] his(admittedly limited) mathematical knowledge before his stay in Paris in the1670s (Hoffman 1974, pp. 6–8).

The clearest evidence of Hobbes’s influence on Leibniz is in his essayTheoria motus abstracti, where Leibniz employs the concept of conatus toinvestigate the nature of motion and eventually arrives at the remarkableconclusion that every body is a momentary mind. 9 In a 1671 letter to Henry Oldenburg,Leibniz announced that his theory of abstract motion provides the basis for thesolution of any number of mathematical and philosophical puzzles. The theory,he claimed, “explains the hitherto unresolved difficulties of continuouscomposition, confirms the geometry of indivisibles and arithmetic ofinfinities; it shows that there is nothing in the realm of nature withoutparts; that the parts of any continuum are in fact infinite; that the theory ofangles is that of the quantities of unextended bodies; that motion is strongerthan motion, and conatus stronger than conatus—however, conatus is instantaneousmotion through a point, and so a point may be greater than a point” (Oldenburg 1965–77, vol. 8, p. 22).

The “geometry of indivisibles” and the “arithmetic of infinities” towhich Leibniz refers are, I take it, the works of Cavalieri and Wallis.Cavalieri’s method of indivisibles is mentioned explicitly in section six ofthe Theoria motus abstracti, as a theory whose “truth is obviously demonstratedso that we must think of certain rudiments, so to speak, or beginnings of linesand figures, as smaller than any given magnitude whatever” (GP IV, p. 228).Wallis’s 1655 treatise Arithmetica Infinitorum, although not mentionedexplicitly in the text, is evidently referred to in the letter to Oldenburgwhen Leibniz refers to the “arithmetic of infinities.” In light of this, it isno great interpretive leap to see Leibniz connecting the doctrine of conatuswith the classic problem of quadrature, just as Hobbes had done, and thus tofind part of the origin of the calculus in Leibniz’s close reading of DeCorpore.

It would doubtless be going too far to claim that the whole ofLeibniz’s calculus is simply the application of Hobbes’s ideas. It is wellknown that Leibniz’s mathematical thought was also strongly influenced byGalileo’s approach to the geometry of indivisibles, for example, and theinfluence of [End Page 15] Huygens cannot be overlooked. 10 Nevertheless, we can agree that Hobbeswas one among many whose writings stimulated the development of the Leibnizianapproach to the calculus. 11 However, there is one importantdifference between the Leibnizian and Hobbesian conceptions of conatus that issignificant for my present purposes: Leibniz’s language (at least in theTheoria motus abstracti) requires that conatus be a literally infinitesimalquantity, while Hobbes regards it as having finite magnitude, but one so smallas to be disregarded. It was the introduction of infinitesimal magnitudes intothe foundations of the calculus that involved Leibniz in importantphilosophical disputes to which we can now turn.

2. Nieuwentijt, Rolle, and the Case against the Calculus

Whatever the full story of its origins, the differential calculus didnot make its way in the world without a struggle. For some ten years after thepublication of Leibniz’s first paper on the calculus in 1684, there was littlecriticism of the method, although there was also little comprehension of it. 12The circle around Leibniz—including the brothers Johannn and Jacob Bernoulli,Pierre Varignon, and the Marquis de l’Hôpital—extended and developed the methodduring this period, but by the mid-1690s they were faced with the challenge ofjustifying the foundations of the new calculus. Two critics stand out asespecially significant: Bernard Nieuwentijt and Michel Rolle. Both criticizedthe calculus as ill-founded and unrigorous, although their arguments haveimportant differences. These criticisms are important enough in Leibniz’saccount of the foundations of the calculus that we must make a brief overviewof them.

Nieuwentijt was a Dutch mathematician of some note who attacked theLeibnizian calculus in his 1694Considerationes circa analyseos ad quantitatesinfinite parvas applicatae principia et calculi differentialis usum inresolvendis [End Page 16] problematibus geometricis. 13 He argued that in rejecting infinitesimalmagnitudes the practitioners of the calculus effectively treated them as zeroor nothing. He further argued that, even if infinitesimals are admitted, therecan be no basis for accepting the higher-order infinitesimal. In essence,Nieuwentijt tried to develop a rival version of the Leibnizian calculus thatwould not discard infinitesimal quantities and would be confined to theconsideration of infinitesimals of the first order.

In support of the thesis that first order infinitesimals are zeromagnitudes, Nieuwentijt takes an example from the Lectiones Geometricae ofIsaac Barrow, where an infinitesimal quantity is rejected during thedetermination of a tangent. 14 Nieuwentijt observes that “[t]hecelebrated author sets forth this thesis in his reasoning: If a determinatequantity has a ratio greater than any assignable to any other quantity, thelatter will be equal to zero” (Nieuwentijt1694, p. 6). Following essentially the classical conception ofrigor, Nieuwentijt further insists that only those magnitudes are equal whosedifference is zero; as he puts it: “I declare that this proposition isindubitable and carries with it most evidently the certain signs of truth: Onlythose quantities are equal whose difference is zero, or is equal to nothing” (Nieuwentijt 1694, p. 10).

Although he was convinced that infinitesimal differences were treatedas zero-magnitudes in the calculus, Nieuwentijt was prepared to acceptinfinitesimals as long as they were treated as positive quantities and notdiscarded. Thus, where Leibniz and his followers freely dropped terms dx or dyfrom equations, Nieuwentijt would require that they be retained. Moreimportant, he insisted that the higher-order differentials must be banned fromthe calculus altogether, since he saw no prospect of developing a coherenttheory of positive quantities less than an infinitesimal. He ultimately heldthat the infinite divisibility of geometric magnitudes guarantees thelegitimacy of first-order infinitesimals, but regarded any extension of thedoctrine to higher orders of infinity as unwarranted.

The second attack on the calculus began in the Paris Académie Royaledes Sciences in July of 1700, when Michel Rolle voiced opposition to the use ofinfinitesimal magnitudes. 15 Rolle was not alone in this project, forhe allied himself with several mathematical conservatives, including the AbbéJean Gallois and the Abbé Thomas Gouye, both of whom venerated the Greekstandards of rigor and had significant reservations about the use of [End Page17] infinitesimal methods. Rolle’s criticisms were later published in thememoir Du nouveau systême de l’infini, which he opened by declaring that

We have always regarded geometry as an exact science, and also as thesource of the exactness which is spread throughout all the other parts ofmathematics. We see among its principles only true axioms: all the theorems andall the problems proposed here are either solidly demonstrated or capable of asolid demonstration. And if it should happen that any false or less certainprinciples slip in, they should be at once banished from this science.

But it seems that this character of exactitude no longer reigns ingeometry, ever since we became entangled in the new system of the infinitelysmall. For myself, I do not see that it has produced any new truth, and itseems to me that it often leads to error.

(Rolle 1703, p. 312)

The “new system of the infinitely small” mentioned here is thedifferential calculus, particularly as formulated in L’Hôpital’s 1696 Analyse des infiniment petits. This treatise wassomething of an official statement of the methods and philosophical foundationsof the calculus that, in the words of L’Hôpital “penetrates into infinityitself” by comparing the ratios of infinitesimal quantities (L’Hôpital 1696, p. iii).According to L’Hôpital “this analysis extends beyond the infinite: for it doesnot rest with infinitely small differences, but discovers the relations anddifferences of such differences, and again of third differences, fourth, and soon without finding an end. So that it embraces not only the infinite but theinfinite of the infinite, or an infinity of infinities” (1696, p. iv). InL’Hôpital’s presentation, infinitesimal magnitudes are introduced in the formof an axiomatic system, complete with definitions, postulates, and theorems.The first definition, for example, declares that “[v]ariable quantities arethose which increase or diminish continually; and constant quantities are thosewhich remain the same while others change” (L’Hôpital 1696, p. 1). Thesecond stipulates that “[t]he infinitely small portion by which a variablequantity continually increases or diminishes is called its difference” (L’Hôpital 1696, p. 2). Themost important postulate in this system is the first, which declares that “onecan take indifferently for one another two quantities which differ from oneanother by an infinitely small quantity; or (which is the same thing) that aquantity which is augmented or diminished by another infinitely less than it,can be considered as if it remained the same” ( L’Hôpital1696, pp. 2–3). Itshould be evident that, whatever reservations others may have had about thereality and intelligibility of infinitesimals, L’Hôpital was not one to balk atthem. It should also be [End Page 18] evident that, taking his first postulateat face value, L’Hôpital seems committed to the thesis that x + dx = x, whichimplies that dx = x - x = 0.

Rolle’s charges against this system are serious indeed. He not onlyalleges that its foundations are themselves incoherent, but he also claims thatit leads to falsehood, and adds that in any case it is incapable of discoveringor proving new truths. The defense of the calculus within the Académie wasundertaken by Varignon, who sought to provide proofs of the reality ofinfinitesimals while also responding to technical criticisms, in which Rolleclaimed to show that the use of infinitesimals led to false results. JohannBernoulli, L’Hôpital, and others assisted in addressing Rolle’s challenges, andthe next six years saw an extended debate over the metaphysics of the calculuswhich ended with the triumph of the new methods.

Both of these criticisms of the calculus are significant for thedevelopment of Leibniz’s theory of the fictional infinitesimal. He resistsNieuwentijt’s conclusion that the infinitesimal is effectively treated asnothing, while at the same time defending the practice of discardinginfinitesimal quantities from equations. Similarly, he cannot accept Rolle’scharge that the calculus is unrigorous, and he must try to show that the onlyreal quantities required in his method are finite and positive. In examiningLeibniz’s replies to his critics, we can see the evolution of his doctrines,and we find him (as ever) trying to reconcile conflicting viewpoints in acoherent synthesis.

3. Leibniz’s Rebuttal to Nieuwentijt

Shortly after the publication of Nieuwentijt’s Considerationes, Leibnizanswered with an essay Responsio ad nonnullas difficultates a Dn. BernardoNiewentiit circa methodum differentialem seu infinitesimalem motas (GM V, pp.321–26) that appeared in the Leipzig Acta Eruditorum in July of 1695 and wasintended to disarm Nieuwentijt’s objections. This reply seems to have been puttogether hastily, since Leibniz felt the need to print a short appendix to itin the next month’s issue of the Acta (GM V, pp. 327–28). Although he was notdirectly involved in responding to Rolle, Leibniz kept up a correspondence withBernoulli, Varignon, and other principals in the controversy in the Académie.The effect of Leibniz’s pronouncements in these matters was not, as one mightexpect, to vindicate the use of infinitesimals, but instead to muddy the watersso that it is difficult to discern exactly what Leibniz held about the statusof the infinitesimal. However, I think that some sense can be made of Leibniz’sapparently contradictory pronouncements on the nature of infinitesimals andtheir status in his mathematics. In particular, I think that the doctrine ofthe fictionality of the infinitesimal develops out of Leibniz’s reaction tocriticisms [End Page 19] of the calculus and his correspondence with Wallis andBernoulli. We can see the beginnings of this process in the reply toNieuwentijt.

Leibniz opens his response to Nieuwentijt with what looks like adefense of the reality of infinitesimals, although he chooses to speak ofquantities “incomparably small” where one might expect reference to theinfinitely small. To Nieuwentijt’s requirement that only those quantities areequal whose difference is zero, Leibniz appears to ally himself with L’Hôpitalby insisting that equal quantities can still differ from one another. He firstadmits his admiration for those who desire to see all things demonstrated fromundeniable first principles, but cautions that an excess of scruple may impedethe art of discovery and deny us its fruits. Regarding the question of whetherequal quantities can differ from one another, Leibniz declares:

I think that those things are equal not only whose difference isabsolutely nothing, but also whose difference is incomparably small; andalthough this difference need not be called absolutely nothing, neither is it aquantity comparable with those whose difference it is. Just as when you add apoint of one line to another line or a line to a surface you do not increasethe magnitude, it is the same thing if you add to a line a certain line, butone incomparably smaller. Nor can any increase be shown by any suchconstruction.

(GM V, p. 322)

We may note, in passing, that the reference here to “incomparablysmall” elements of lines or surfaces has a strongly Hobbesian ring to it, forit is exactly the hallmark of Hobbes’s points that—though finite—they are toosmall to be considered in any demonstration. Leibniz’s preference here for thelanguage of the incomparable rather than the infinitesimal raises the questionof whether such incomparable magnitudes are to be thought of as literallyinfinitesimal or whether they should be treated as finite but negligiblequantities in the manner of Hobbes’s points.

At first sight, it seems natural to take the unassignable orincomparably small as just the infinitesimal in a different guise, perhapsseeing the term “incomparably small” as a kind of euphemism for“infinitesimal.” But Leibniz balks at such an identification. 16 Instead, he indicates that it is enoughto show that incomparably small quantities can be justly neglected in acalculation, and he refers to certain “lemmas communicated by me in [End Page20] February of 1689” for the full justification of this procedure (GM V, p.322).

These lemmas of 1689 are contained in Leibniz’s Tentamen de motuumcoelestium causis (GM VI, pp. 144–60). But when we turn to them forenlightenment it is evident that they were intended explicitly to avoidreferences to infinitesimal quantities and instead to replace infinitesimalmagnitudes with finite differences sufficiently small to be ignored inpractice. The paragraph expounding these lemmas opens with the declaration that

I have assumed in the demonstrations incomparably small quantities, forexample the difference between two common quantities which is incomparable withthe quantities themselves. Such matters as these, if I am not mistaken, can beset forth most lucidly in what follows. And then if someone does not want toemploy infinitely small quantities, he can take them to be as small as hejudges sufficient to be incomparable, so that they produce an error of noimportance and even smaller than any given [error]. Just as the Earth is takenfor a point, or the diameter of the Earth for a line infinitely small withrespect to the heavens, so it can be demonstrated that if the sides of an anglehave a base incomparably less than them, the comprehended angle will beincomparably less than a rectilinear angle, and the difference between thesides will be incomparable with the sides themselves; also, the differencebetween the whole sine, the sine of the complement, and the secant will beincomparable to these differences.

(GM VI, pp. 150–51)

The use intended for such incomparably small magnitudes is to avoiddisputes about the nature or existence of infinitesimal quantities, and Leibnizholds that “it is possible to use ordinary [communia] triangles similar to theunassignable ones, which have a great use in finding tangents, maxima, minima,and for investigating the curvature of lines” (GM VI, p. 150). In other words,the lemmas on incomparable magnitudes are to serve as a foundation for thecalculus which permits the talk of infinitesimals to be reinterpreted in termsof incomparable (but apparently finite) differences. These lemmas loom large inLeibniz’s writings on the foundations of the calculus, since he frequentlyrefers back to them in later discussions on the nature of the infinitesimal. Itis also significant that the incomparably small satisfies Hobbes’s definitionof a geometric point—it is a quantity sufficiently small that its magnitudecannot be regarded in a demonstration.

In the appendix to his reply to Nieuwentijt, Leibniz returns to thetheme that infinitesimals can be avoided by using finite lines that stand in[End Page 21] the same ratio as the differential increments. The idea here isthat talk of infinitesimal increments of ordinate and abscissa can bereinterpreted in terms of finite ratios between finite lines that express theratio of the ordinate to the abscissa at any given point. “In order to removeall disputes about the reality of differences of any order,” Leibniz writes,“they can always be expressed in proportional finite right lines [rectisordinariis proportionalibus]” (GM V, p. 327).

The project of replacing ratios of infinitesimals with ratios of finitequantities should, according to Leibniz, satisfy the demands of the rigorists:if they do not care for infinitesimals whose ratios are investigated by thecalculus, they can retain the ratios and replace their (infinitesimal) termswith finite quantities. The weakness of this approach, of course, is that theratios were originally acquired by manipulating infinitesimal quantities. Ithardly satisfies the standards of geometrical rigor to work with ratios thatcan only be obtained by the introduction of infinitesimals and then to pretendthat these ratios are legitimate because they can be expressed in terms offinite quantities.

In the end, Leibniz’s reply to Nieuwentijt falls well short of aspirited defense of the infinite in mathematics, nor is it a particularlycompelling or satisfying attempt to reinterpret the infinitesimal out of thecalculus. Leibniz neither affirms nor denies the real existence of theinfinitely small in this exchange, and he goes out of his way to insist thatinfinitary considerations can be avoided by making use of finite (butnegligible) quantities. These facts suggest that Leibniz was not athoroughgoing realist about infinitesimal magnitudes, and at the very least hisviews on the nature of the infinitely small were not fixed and settled in themid-1690s.

4. Leibniz and Wallis on the Infinitely Small

In 1695, the year of his published reply to Nieuwentijt, Leibniz beganto correspond with Wallis. His principal motives for undertaking thiscorrespondence were to keep abreast of mathematical developments in England andto promote his own calculus differentialis, which he saw as a fundamentalextension of Wallis’s methods. The priority dispute with Newton eventually ledLeibniz to end his contact with Wallis and the English mathematical world, butbefore the correspondence ended in 1700 the two exchanged twenty letters thatcovered a wide variety of topics. Key among these were the nature of the infinitelysmall and the difference between their respective approaches to the mathematicsof the infinite. 17 [End Page 22]

For his part, Wallis was concerned with maintaining a claim for theoriginality and extensiveness of his methods. This led him to assert thatLeibniz’s calculus was really little more than a notational variant of his ownarithmetic of infinities. Wallis was also intent upon defending the rigor ofhis approach, and in the course of this defense he compared his work to theclassical methods. Inevitably, this project led Wallis to clarify his ownconception of the infinitesimal and to give his own account of the developmentof seventeenth century mathematics.

Responding to Leibniz’s request for a further account of his methods,their background, and their foundations, Wallis explained that hisinvestigations had their origin in the problem of the angle of contact betweena circle and its tangent. This problem had been the source of much controversyin the preceding century, most notably between Jacques Peletier and ChristopherClavius. 18As Wallis tells the story, a proper understanding of the angle of contact leadsimmediately to a method of rectifying curvilinear arcs and finding the area ofcurved surfaces. He explains:

I had long since claimed that the angle of contact to a circle is of nomagnitude; nor was I the first to do this, but vindicated the opinion ofPeletier, which had been opposed by the authority of Clavius. By the samereasoning we may conclude that the angle of contact to any curve is of nomagnitude . . . It follows at once that any point of any curve has thedirection, obliquity, [or] inclination . . . of the right line tangent to thesame curve. Thus the point can be considered as an infinitesimal part of thisright line. The whole doctrine of rectifying [curves] takes its origin fromthis . . . while the same can be extended to the quadrature of curved surfaces.

(Wallis to Leibniz, 30 July 1697; GM IV, p. 30)

The key point here is that the angle of contact is of no magnitude,although at the point of contact there is an inclination or tendency todirection which exists abstracted from all magnitude. 19 Wallis extends this doctrine to the caseof tangency, and in doing so he makes explicit what was merely implicit in hisdiscussion of the angle of contact: namely that [End Page 23] the infinitesimalis really not a magnitude at all. In computing tangents, Wallis had introduceda minute increment designated a, which is diminished in the course of thecomputation and ultimately discarded. The magnitude a might appear to be aninfinitesimal, but Wallis declares that it is really nothing. This leads him todraw a distinction between the Leibnizian calculus of the infinitely small andhis own arithmetic of infinities. As he describes the situation:

You see that my methods for tangents were summarily set out in thePhilosophical Transactions for the month of March 1672, and again inProposition 95 of my Algebra, which I had earlier applied throughout myTreatise of Conic Sections of 1655, and these methods plainly rest on the sameprinciples as your differential calculus, but in a different form of notation.For my quantity a is the same as your dx, except that my a is nothing and yourdx infinitely small. Then when those things are neglected which I hold shouldbe neglected in order to abbreviate the calculation, that which remains is yourminute triangle, which according to you is infinitely small, but according tome is nothing or evanescent.

(Wallis to Leibniz, 30 July 1697; GM IV, p. 37)

This rather startling declaration should not be taken to mean thatWallis everywhere regards the magnitude a as nothing. It is first introducedinto a calculation as a finite positive increment, but then “infinitelydiminished” to become nothing, and therefore dropped out of a calculation,although results obtained under the hypothesis that a is a positive incrementare retained, living on after the demise of the increment itself.

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Figure 4.

Wallis returned to this theme nearly a year later. Again discussing theproblem of finding the tangent to a curve, he insisted that the “foundation ofthe whole procedure” is to move from an approximation to the tangent to anexact value by letting a secant to the curve become a tangent. Consider, forexample, the problem of finding the tangent to the curve αβ at the point F, asin figure 4. Wallis’s procedure begins by taking a secant that cuts thecurve at the points F and G; he then brings them into coincidence by rotatingthe secant about F until the points coincide and the difference between them(marked by the letter a) vanishes. Terms containing a are then cancelled fromthe equation representing the tangent. Although the increment vanishes intonothingness, it leaves something behind in the form of a triangle abstractedfrom all magnitude. As Wallis explains:

When the simplification of the calculation I teach is applied, thatwhich remains is in fact your differential calculus (for it is not so [End Page24] much a new thing as a new way of speaking, although perhaps you were notaware of it). My term a is everywhere the same as your segment of the abscissax or y, with this one difference: namely that your quantity dx is infinitelysmall, mine is simply nothing. Then when it is deleted or (to shorten thecalculation) all those terms are dismissed which should be deleted, that whichremains is your minute deferential triangle, formed between two adjacent ordinates.. . . But in your presentation it is infinitely small, in mine it is clearlynothing. Of course the species of the triangle is retained, but abstracted frommagnitude. That is to say, the form of a triangle remains, but of nodeterminate magnitude.

(Wallis to Leibniz, 22 July 1698; GM IV, p. 50)

On this scheme, the product of two or more infinitesimals will also berejected, on the grounds that nothing multiplied by nothing always remainsnothing. Wallis argues that it is precisely on this point that his methods arepreferable to Leibniz’s, because “I have no need of any of your postulatesabout the infinitely small multiplied by itself . . . (which must be appliedwith some caution), since it is self-evident . . . that nothing multiplied anynumber of times is still nothing” (Wallis to Leibniz, 22 July 1698; GM IV, p.50).

This doctrine did not find favor with Leibniz, and it is not difficultto see why. Taking the infinitely small as nothing does have the advantage ofjustifying the rejection of terms containing infinitesimal factors, since suchterms are by definition equal to zero. But, by the same token, it would seem torequire that when quantities are divided by infinitesimals, or when ratiosbetween infinitesimal increments are compared, the result is a division by zeroor the comparison of ratios of nothings. Moreover, the infinitesimal itself isoften treated as a quantity with its own infinitesimal [End Page 25] parts—theinfinitesimal of the second order—and it is difficult to see how the apparatusof higher-order infinitesimals can be justified if the infinitesimal of thefirst degree is itself simply nothing. Finally, the idea that the “differentialtriangle” is the persistence of the form of a triangle without magnitude seemsat least as metaphysically problematic as the infinitesimal itself.

Leibniz was quick to point out these problems, and in his response toWallis he laid out the case for denying that infinitesimal magnitudes arenothing:

I think it is better if elements or instantaneous differentials areconsidered as quantities according to my fashion, rather than their being takenfor nothing. For they in their turn have their own differences, and these caneven be represented by proportional assignable lines. I do not know whether itis intelligible to take this inassignable triangle as nothing . . ., in whichthere is nevertheless retained the species of a triangle abstracted frommagnitude, so that it is the species of a given figure but of no magnitude.This certainly seems to introduce an unnecessary obscurity. Who acknowledges afigure without magnitude? Nor do I see how magnitude can be removed, when tosuch a given triangle another can be understood similar to it but much smaller.

(Leibniz to Wallis, 29 December 1698; GM IV, p. 54)

Leibniz further points out that many applications of the calculusrequire the use of second-order infinitesimals. These must be infinitely lessthat infinitesimals of the first order, and thus require that first-orderinfinitesimals not be taken for nothing. He writes:

Although you say that you have no need of the infinitely smallmultiplied into itself, see whether it does not to a certain extent arise againout of oblivion: are not the elements of a curve to be represented by assuming thatthe right line dx is the element of the abscissa x, and the right line dy isthe element of the ordinate y? Thus I observe in these things a kind of new lawof homogeneity for the infinitesimal calculus: for the square of a differentialor element of the first degree is homogeneous to a rectangle made from a finiteright line multiplied by a difference of the second degree, or dxdx ishomogeneous to ddx. Since this is the case, then the element of the firstdegree is a mean proportional between a finite right line and a difference ofdifferences, so far is this from being taken for nothing.

(Leibniz to Wallis, 29 December 1698; GM IV, p. 55) [End Page 26]

Wallis tried to defend himself against these charges of incoherentmetaphysics and inadequate mathematics; he argued that by the abstraction ofthe form of a triangle from its magnitude he did not mean “a triangle which hasno magnitude, but that the species or form of a triangle can be consideredabstracted from magnitude” (Wallis to Leibniz, 16 January 1699; GM IV, p. 58).Later he adds, “If this does not please you, then where the species of thetriangle is mentioned, you can say the degree of inclination or declination ofthe curve at the point of contact, or the angle the curve makes with theordinate it touches, for indeed this is what is sought” (Wallis to Leibniz, 16January 1699; GM IV, p. 58). In a somewhat conciliatory tone, Wallis adds thathis doctrine can be made consistent with Leibniz’s theory by treating thecoincidence of two points as a degenerate case where there is an infinitelysmall distance between the two points.

Leibniz concluded this exchange on the nature of the infinitesimal witha similarly conciliatory tone, although he was adamant that the infinitesimalnot be regarded as nothing. He writes:

Of course, the form of the characteristic triangle can be rightlyexplained by the degree of declination, but for the calculus it is useful toimagine [fingere] quantities infinitely small, or as Nicholas Mercator calledthem, infinitesimal: and such things cannot be taken for nothing when theassignable ratio among them is sought. On the other hand they are rejectedwhenever they are added [adjiciuntur] to quantities incomparably greater,according to lemmas on incomparable quantities I once proposed in the ActaEruditorum of Leipzig, which foundation the Marquis de L’Hôpital also uses. . .. It is simpler, I admit, as you say that nothing multiplied by anything isstill nothing, but this does not have the use of the system we have proposed.

(Leibniz to Wallis, 30 March 1699; GM IV, p. 63)

In these exchanges with Wallis, Leibniz appears to play the role of thedefender of the reality of the infinitesimal. He consistently opposes Wallis’sequation of the infinitesimal with nothing, and is at great pains to point outthat the calculus depends for its coherence upon the assumption thatinfinitesimal (or incomparable) magnitudes are positive quantities. He evenclaims that his principles are the same as those of L’Hôpital. Nevertheless, itis clear that by 1699 he had come to have doubts about the reality ofinfinitesimal quantities, although he may not have intended to make them clearto Wallis. In a passage from the letter of 30 March 1699, Leibniz added asentence that hints at his theory of the fictional [End Page 27] infinitesimal,but the sentence was apparently not included in the letter as sent. It reads:

In the end, I do not dispute whether these inassignable quantities aretrue or fictive; it suffices that they serve for the abbreviation of thought,and they always bring with them a demonstration in a different style; and so Iobserved that if someone substitutes the incomparably small or that which issufficiently small for the infinitely small, I would not oppose it.

(GM IV, p. 63)

Leibniz’s reservations about the reality of infinitesimals can be seenmore readily in his correspondence with Johann Bernoulli during the same timeas his correspondence with Wallis. To complete the picture, we must thereforemake a brief excursion into the Leibniz-Bernoulli correspondence.

5. The Leibniz-Bernoulli Correspondence and the Reply to Rolle

Johann Bernoulli was one of the first converts to the Leibniziancalculus, and it is largely due to his efforts and those of his brother Jacobthat a vast array of new results were discovered in the 1690s. The calculus asunderstood by Bernoulli, L’Hôpital, and other continental mathematicians wasnothing less than a true science of the infinite, and Bernoulli was quiteuntroubled by its seemingly paradoxical nature. Traditional strictures againstthe infinite held that there can be no clear conception of a quantity greaterthan nothing yet infinitely small, and even less of an ordered structure ofever more infinitesimal magnitudes, each infinitely less than its predecessor.Bernoulli, however, was prepared to accept the whole apparatus ofinfinitesimals as both fully real and completely intelligible.

By the mid-1690s Leibniz was unhappy with a metaphysics of theinfinitesimal which accords it such genuine reality. As a result of theseworries he explicitly raised the suggestion that infinitesimal magnitudes aremere fictions whose use is justified by their utility in developing thecalculus. In June of 1698 he wrote to Bernoulli:

I recognize . . . that you have written some profound and ingeniousthings concerning various infinite bodies [de corporibus varie infinitis]. Ithink that I understand your meaning, and I have often thought about thesethings, but have not yet dared to pronounce upon them. For perhaps theinfinite, such as we conceive it, and the infinitely small, are imaginary, andyet apt for determining real things, just as imaginary roots are customarilysupposed to be. These things are among the ideal reasons by which, as it were,things are ruled, although they are not in the parts of matter. For if [EndPage 28] we admit real lines infinitely small, it follows also that lines areto be admitted which are terminated at either end, but which nevertheless areto our ordinary lines, as an infinite to a finite. Which things being posited,it follows that there is a point in space which can not be reached in anassignable time by uniform motion. And it will similarly be required toconceive a time terminated on both sides, which nevertheless is infinite, andeven that there can be given a certain kind of eternity (as I may expressmyself) which is terminated. Or further that something can live so as not todie in any assignable number of years, and nevertheless die at some time. Allwhich things I dare not admit, unless I am compelled by indubitabledemonstrations.

(GM III, pp. 499–500)

This passage is significant because it evinces not only Leibniz’sreservations about the coherence of the infinitesimal, but actually indicatesthe line of argument he takes to show the problems inherent in the concept. Totake the infinitely small as real we must think of something that is bothlimited and unlimited: a determinate, bounded space that is smaller than anyfinite space; and similarly, we must acknowledge the reality of the infinitelylarge, which is a limited and yet unlimited quantity. Although Leibniz doesadmit that there might be “indubitable demonstrations” that compel his assentto the reality of the infinitesimals, he is clearly not prepared to accept themas real entities without an argument.

The theme of the fictionality of infinitesimal magnitudes recurs in theexchanges with Bernoulli through the summer of 1698, with Leibniz insistingthat it is “useful in the calculus to assume” that there are lines which standto ordinary lines in the ratio of finite to infinite (Leibniz to Bernoulli, 22July 1698; GM III, p. 516) and again arguing that “it suffices for the calculusthat they are represented in thought [finguntur], just as imaginary roots inalgebra” (Leibniz to Bernoulli, 29 July 1698; GM III, p. 524). This last remarkalso appears along with Leibniz’s earlier claim that the new calculus canultimately be founded on the basis of Archimedean exhaustion proofs, since“whatever is concluded by means of these infinite or infinitely smallquantities can always be shown through reductio ad absurdum, by my method ofincomparables (the lemmas of which I once published in the Acta)” (Leibniz toBernoulli, 29 July 1698; GM III, p. 524).

His reaction to Rolle and the dispute in the French Academy seems tocomplete Leibniz’s retreat from a commitment to the reality of infinitesimalmagnitudes to an explicit fictionalism about the infinite. In a famous letterto M. Pinson, parts of which were published in the Journal des Sçavans in 1701,Leibniz responded to an anonymous criticism of the [End Page 29] infinitesimalwhich Abbé Gouye had published in the Journal. Leibniz argued that

there is no need to take the infinite here rigorously, but only as whenwe say in optics that the rays of the sun come from a point infinitely distant,and thus are regarded as parallel. And when there are more degrees of infinity,or infinitely small, it is as the sphere of the earth is regarded as a point inrespect to the distance of the sphere of the fixed stars, and a ball which wehold in the hand is also a point in comparison with the semidiameter of thesphere of the earth. And then the distance to the fixed stars is infinitelyinfinite or an infinity of infinities in relation to the diameter of the ball.For in place of the infinite or the infinitely small we can take quantities asgreat or as small as is necessary in order that the error will be less than anygiven error. In this way we only differ from the style of Archimedes in theexpressions, which are more direct in our method and better adapted to the artof discovery.

(GM IV, pp. 95–96)

These remarks are of a piece with Leibniz’s earlier claims about theeliminability of infinitesimal magnitudes: he denies that the calculus reallyneeds to rely upon considerations of the infinite and again insists that it canbe based on a procedure of taking finite but “negligible” errors that can bemade as small as desired; moreover, the fact that such errors can be madearbitrarily small sets the stage for the reductio proofs characteristic ofclassical exhaustion methods, since if one supposes that the error has a fixedmagnitude m, the error can be made less than m. His comments are, however, moredefinite than his earlier remarks, for he no longer claims that theinfinitesimal “might” be imaginary.

The more ardent partisans of the infinitesimal (notably Bernoulli,Varignon, and L’Hôpital) were deeply concerned by Leibniz’s apparent concessionto the critics of the calculus, and Varignon wrote to Leibniz in November of1701 requesting a clarification of Leibniz’s views on the reality ofinfinitesimals. He remarked that the publication of the letter to M. Pinson haddone harm to the cause, and that some had taken him to mean that the calculuswas inexact and capable only of providing approximations. He thereforerequested “that you send us as soon as possible a clear and precise declarationof your thoughts on this matter” (Varignon to Leibniz, 28 November 1701; GM IV,p. 90).

In his reply to Varignon, Leibniz issued a summary statement of hisviews on the infinite and its role in the calculus. This statement bringstogether themes we have already seen: the fictional nature of infinitesimals,the possibility of basing the calculus upon a science of incomparably small[End Page 30] (but still finite) differences, and the equivalence of the newmethods and the Archimedean techniques of exhaustion. After assuring Varignonthat his intention was “to point out that it is unnecessary to make mathematicalanalysis depend on metaphysical controversies or to make sure that there arelines in nature which are infinitely small in a rigorous sense” (Leibniz toVarignon, 2 February 1702; GM IV, p. 91), Leibniz once again suggests thatincomparably small magnitudes be taken in place of the genuine infinite. Theseincomparables would provide “as many degrees of incomparability as one maywish;” and although they are really finite quantities they can still beneglected, in accordance with the notorious “lemmas on incomparables” from theLeipzig Acta (Leibniz to Varignon, 2 February 1702; GM IV, pp. 91–92).

For our purposes, the most important part of the reply to Varignon isLeibniz’s frank admission that infinitesimal magnitudes are fictions, althoughfictions sufficiently well-grounded that “everything in geometry and even innature takes place as if they were perfect realities” (Leibniz to Varignon, 2February 1702; GM IV, p. 93). The treatment of infinitesimal magnitudes aswell-grounded fictions recalls Leibniz’s remark to Bernoulli that, although theinfinite may be imaginary, infinitesimal magnitudes are among the “idealreasons by which, as it were, things are ruled, although they are not in theparts of matter” (GM III, p. 499). But in elaborating his doctrine in theletter to Varignon, Leibniz goes somewhat further in justifying his doctrinesby making reference to his metaphysical principle of continuity. He insiststhat both infinitesimal magnitudes and imaginary roots have a foundation in thenature of things, and cites as evidence

not only our geometrical analysis of transcendental curves, but also mylaw of continuity, in virtue of which it is permitted to consider rest as aninfinitely small motion (that is, as equivalent to a species of itscontradictory), and coincidence as infinitely small distance, equality as thelast inequality, etc . . . Yet one can still say in general that althoughcontinuity is something ideal and there is never anything in nature withperfectly uniform parts, the real, in turn, never ceases to be governedperfectly by the ideal and abstract. The rules of the finite are found tosucceed in the infinite—as if there were atoms (that is to say, assignableelements in nature), although there are none because matter is actuallysubdivided without end, and conversely the rules of the finite succeed in theinfinite, as if there were infinitely small metaphysical beings, although wehave no need of them, and the division of matter never does proceed toinfinitely small particles. This is because everything is governed by reason;otherwise there could be no science, nor rule, and this [End Page 31] would notconform at all with the nature of the sovereign principle.

(Leibniz to Varignon, 2 February 1702; GM IV, pp. 93–94)

The full scope of this “fictionalist” reading of the infinite was notmade widely known, largely because Leibniz and his associates had reason tofear that any public retreat from a full commitment to the reality of theinfinitesimal would complicate the already difficult battle for the acceptanceof the calculus. 20 Despite his relative silence on thematter I take it as established that, at least in his mature thought, Leibnizdid not believe in the reality of infinitesimal magnitudes. 21 The language of infinitesimal differencesis not, however, unacceptable in the Leibnizian scheme of things: it has manyuses in mathematical matters, enabling proofs to be shortened and fostering theart of discovery. Indeed, the calculus cannot even be stated without referenceto fictional entities like dx and dy, but the principles of true (that is,Leibnizian) metaphysics show that the indulgence in such fictions does notdetract from the truth of the results.

6. The Place of Mathematics in the Leibnizian Scheme of the Sciences

In concluding, I would like to offer a few remarks on the place ofLeibniz’s calculus in his more general conception of philosophy and thesciences. It should be clear from the foregoing that Leibniz more or lessexplicitly denies the real existence of infinitesimal magnitudes, andnevertheless allows himself to employ the concept of infinity quite freely indeveloping the calculus. This might well be thought to pose a problem: how,after all, can Leibniz speak of a curve as a polygon with an infinite number ofinfinitesimal sides, if he does not really believe in the infinitesimal? Moreto the point, how can an apparently real curve be literally composed of [EndPage 32] fictional parts? One reaction to this apparent difficulty is to takeLeibniz as holding a “secret doctrine” of the reality of the infinitesimal,even if he occasionally professes not to accept the infinite. Another possibleway around the problem would be to see Leibniz as a mendacious propagandist ofthe new mathematics who was happy to tell any story that might promote thehigher goal of advancing mathematical learning, even though he never took anyof his supposed justifications of the calculus seriously. Both of thesestrategies are unappealing, and I take the hypothesis of disingenuity to be aninterpretive principle of absolute last resort.

Taking Leibniz’s pronouncements on the infinitesimal seriously does,however, require that we give at least some content to the concept of afictional entity and have some kind of account of how such fictions can be partof a properly developed mathematical theory. 22 Leibniz is less informative on this pointthan one might wish, but it is possible to construct a fairly complete pictureof his doctrines. Infinitary magnitudes are not, after all, the only things heregards as fictions, and it is worth asking how the infinitesimal compares withother fictional entities.

One particularly important kind of fiction is the ens per aggregatum or“being through aggregation” which arises when individuals are grouped togetherto form a non-substantial unit. A flock of sheep, to take Leibniz’s stockexample, is not a real thing in its own right: it is an assemblage of realsubstances, but there is no substantial flock over and above the individualsheep. Nevertheless, it is convenient to regard the flock as a unitary thingwithout making specific reference to the individuals out of which it iscomposed. In a letter to Bartholomeus Des Bosses in 1706, Leibniz considersaggregates in the context of the Scholastic dictum ens et unum convertitur, or“being and one are convertible.” He agrees that “[B]eing and one areconvertible, but when there is a being by aggregation, so also is there unity[by aggregation], and this being and unity are semi-mental” (Leibniz to DesBosses, 11 March 1706; GP II, p. 304). As Leibniz explains, the semi-mentalnature of such things derives from the fact that their unity (and hence theirbeing) is imposed upon them by the mind. They are literally fictions, i.e.,things made up by the mind, but not answering to anything in the real world,since a catalog of the world’s contents would not include a flock over andabove the individual sheep.

In the same letter Leibniz argues that infinite totalities arise froman analogous kind of mental imposition. In reality there can be no infinitequantities, for quantity must be essentially limited and the infinite is bydefinition unlimited. Nevertheless: [End Page 33]

It is thus for the sake of convenience of speech when we say that thereis one where there are more than can be comprehended in an assignable whole,and we bring forth [efferimus] something like a magnitude which neverthelessdoes not have its properties. Just as it cannot be said of an infinite numberwhether it is even or odd, neither can it be said of an infinite line whetheror not it is commensurable to a given line. And so all of these expressionstaking the infinite as a magnitude are improper; and although they are foundedin a certain analogy, they still cannot be maintained if you examine the mattermore carefully.

(Leibniz to Des Bosses, 11 March 1706; GP II, p. 305)

This doctrine is extended to the case of infinitely small magnitudes:

Philosophically speaking, I no more admit magnitudes infinitely smallthan infinitely great. . . . I take both for mental fictions, as moreconvenient ways of speaking, and adapted to calculation, just like imaginaryroots are in algebra. I once demonstrated that these expressions have a greatuse both in abbreviating thought and aiding discovery, and that they cannotlead to error, since in place of the infinitely small one may substitute [aquantity] as small as one wishes, and since any error will always be less thanthis, it follows that no error can be given. But the Reverend Father Gouyé, whoobjected, seems not to have understood me adequately.

(Leibniz to Des Bosses, 11 March 1706; GP II, p. 305)

The fictionality of the infinitesimal does not therefore make it lessadmissible into mathematical calculations. Indeed, in strict metaphysicaltruth, all mathematical objects are in some measure unreal, as Leibniz openlydeclares in a letter to Burchard De Volder: “from the fact that a mathematicalbody cannot be resolved into first constituents we can at least infer that itis not real, but rather something mental, indicating only the possibility ofparts, but nothing actual” (Leibniz to De Volder, 30 June 1704; GP II, p. 268).23

Furthermore, Leibniz held (at least in his later metaphysics) that many[End Page 34] of the concepts employed in the study of nature do not match upwith anything at the ultimate level of the metaphysically real. 24Bodies, for example, are ultimately phenomenal, as are space and time. Even theapparent causal interactions of bodies is merely phenomenal: the principle ofpre-established harmony dictates that the course of events in the world unfoldswithout such causal interactions, although everything is arranged to appear asif bodies actually had causal powers. Physics, however, need not concern itselfwith the ultimately real monadic constituents of the universe. It is enough forthe physicist to describe the workings of the phenomenal world with laws ofmotion grounded in the principles of true metaphysics.

One feature common to all Leibnizian fictions is that they are “wellfounded” at the level of real substances. A fiction is well-founded when itreliably enables us to investigate the properties of real things, so thatindulgence in the fiction cannot lead us into error. Leibniz’s standard exampleof such a fiction is the “imaginary” root . Roots ofnegative numbers are not themselves possible; nevertheless, they can be invokedto generalize algebraic laws, and the results obtained by their use arecompletely reliable. The stress on the well-foundedness of such fictionsinvites us to contrast them will ill-founded fictions. An ill-founded fictionin the case of physics would be something both unintelligible and ill-suited tothe task of mechanistic physics: the void, action at a distance, or otherNewtonian monstrosities are presumably such. Similarly, an ill-foundedmathematical fiction would be a concept at once incoherent and useless: theround square, for example, is a purely fictional entity whose definition includesa contradiction and which cannot be used to elucidate anything interesting inmathematics.

Infinitesimal magnitudes, although they may ultimately be impossible oreven unimaginable, are nevertheless well-founded fictions precisely because therealm of mathematical objects is structured just as it would be if [End Page35] the infinitesimal existed. 25 The calculus delivers true and importantresults, and these illuminating results are just what one would expect if therereally were infinitely small quantities. As it happens, there are none, but akind of mathematical “pre-established harmony” guarantees that in using them wewill never be led astray.

Leibniz’s famous “law of continuity” is another example of a usefulfiction, one with applications in both mathematical and physical reasoning.Strictly speaking, the law is false because it assimilates contradictories:rest is motion, but infinitely slow motion; parallel lines are inclined toward oneanother, but the inclination is infinitely small; and equal quantities areunequal, but the inequality is infinitely little. Despite its apparentabsurdity, the law is acceptable because it can be used to establish truthsabout real motions, angles, or other quantities. It is therefore grounded inthe nature of such real things and its use is both to facilitate discovery ofnew truths and to abbreviate otherwise laborious reasoning.

We can thus see Leibniz operating with a conception of metaphysics thatgrants a fair degree of autonomy to the individual sciences. Physics should usethe principles most convenient for the physicist’s purposes, even if the truemetaphysical account requires that they be granted merely fictional status.Mathematics is also free from the burden of satisfying all the constraints ofmetaphysical rigor, since “it is not necessary to make mathematical analysisdepend on metaphysical controversies” (Leibniz to Varignon, 2 February 1702; GMIV, p. 91). This does not imply that metaphysics has no relevance to physics ormathematics. Leibniz plainly thought that serious metaphysical errors (such asDescartes’s mistaken notion that the essence of body is extension) can lead toscientific errors. Nevertheless, his philosophy holds that the truths ofmetaphysics guarantee the rationality of the world, and it is this rationalitywhich—perhaps paradoxically—makes it possible to disregard strict metaphysicaltruth for the sake of an interesting mathematical story.

Douglas M. Jesseph

North Carolina State University

Douglas M. Jesseph

Douglas Jesseph is associate professor of philosophy at North CarolinaState University. His current research interests are in the history andphilosophy of mathematics. Among his publications are Berkeley’s Philosophy ofMathematics (1993) and Squaring the Circle: The Mathematical War between Hobbesand Wallis (forthcoming). He is also currently editing Hobbes’s mathematicalworks for the Clarendon Edition of the Works of Thomas Hobbes.

Footnotes

1. There are a number of other authors who haveinvestigated these issues, in particular Breger (1990a, 1990b, 1992), Horvath (1982, 1986), Knobloch (1990), and Wurtz (1989).

2. Leibniz’s fullest statement of the history is inhis essay Historia et origo calculus differentialis (GM V, pp. 392–410). Thispiece is not without its difficulties, as it was written during the height ofthe priority dispute with Newton and is a self-serving account designed toestablish Leibniz’s claim to being the true inventor of the calculus. The bestoverview of Leibniz’s calculus and its background is Bos (1974), which can be supplemented by Parmentier (1989), Hoffman (1974) and the papers in Heinekamp(1986). See Hall (1980) for an account of the dispute between Newton and Leibniz.

3. Strictly speaking, the sums of the differencesequence differ from the original by the term a1, but this minor complicationcan be overlooked.

4. There are obvious difficulties in attempting torepresent infinitely small quantities in a diagram. You should think of thematerial enclosed by the dotted lines as having been magnified by an“infinitary microscope” to reveal the relationships among the variousinfinitesimal quantities which make up the curve.

5. For the background to Leibniz’s concept of thetranscendental in mathematics and its introduction into the calculus see Breger (1986).

6. The paper, Nova Methodis pro Maximis et Minimis,itemque Tangentibus, quae nec fractas nec irrationales quantitates moratur, etsingulare pro illis calculi genus, appeared in the Leipzig Acta Eruditorum in1684. It is reprinted in GM V, pp. 220–26.

7. For a fuller account of Hobbes’s philosophy ofmathematics and its relationship to the seventeenth century mathematicalcontext, see Jesseph (1993a, 1993b).

8. These results and their roots in Cavalieri’sExercitationes are discussed in Jesseph (1993a). For Cavalieri’s work, see Cavalieri (1647), which can be supplemented by Andersen (1985), De Gandt(1991), Giusti (1980), and Mancosu (1996).

9. Leibniz asserts that “No conatus without motionlasts longer than a moment except in minds. This opens the door to the truedistinction between body and mind, which no one has explained heretofore. Forevery body is a momentary mind, or one lacking recollection, because it doesnot retain its own conatus and the other contrary one for longer than a moment.For two things are necessary for sensing pleasure or pain—action and reaction,opposition and then harmony . . . Hence body lacks memory; it lacks theperception of its own actions and passions; it lacks thought” (GP IV, p. 230).

10. See Hoffman(1974) for atreatment of Leibniz’s early mathematical development. See also Knobloch’seditorial comments in Leibniz (1993) for more on the backgroundto the development of the calculus.

11. One might ask why, if he owes such a debt toHobbes, Leibniz never mentions him explicitly in his accounts of the origin ofthe calculus. Two plausible reasons would mitigate against such anacknowledgement: first, Hobbes’s mathematical reputation had been completelyobliterated in the course of his mathematical controversy with Wallis; second,Hobbes’s bitter public controversies with Wallis and Boyle had made himanathema to the British scientific establishment by 1670. Although he likelyfound much in Hobbes that could be called influential or even inspiring,Leibniz was certainly shrewd enough to forego any public acknowledgement of anintellectual debt to a figure as unpopular as Hobbes.

12. There was one early (and inept) criticism of thecalculus by the German Detleff Clüver, but it is of no interest to the presentinvestigation. For a study of this critique see Mancosu andVailati (1990).

13. For more complete accounts of Nieuwentijt’s attackon the calculus, see Mancosu (1996, Chapter 6), Petry (1986), Vermij (1989) and Vermeulen(1986).

14. For an account of Barrow’s mathematics and its rolein the development of the calculus see Feingold(1993) and Mahoney (1990).

15. See Blay (1986), Mancosu(1989), andMancosu (1996,Chapter 6) for more extended studies of this controversy.

16. See Knobloch (1990) for an account of Leibniz’sconception of the infinite which pays particular attention to his distinctionbetween the infinitely small and the indefinitely small.

17. See Hoffman(1973) for anotherstudy of the Leibniz-Wallis correspondence, concentrating more on the prioritydispute between Newton and Leibniz.

18. See Maierù (1984) and (1990) for a more detailed account ofthese controversies.

19. Wallis gave a fuller account of this part of hisdoctrine in the Defense of the Treatise of the Angle of Contact (1684). In particular,the sixth chapter of the Defense (entitled “Inceptives of Magnitude”) declaredthat “There are some things, which tho, as to some kind of Magnitude, they arenothing; yet are in the next possibility of being somewhat. They are not it,but tantum non; they are the next possibility to it; and the Beginning of it:Tho’ not as primum quod sit, (as the Schools speak) yet as ultimum quod non.And may very well be called Inchoatives or Inceptives, of that somewhat towhich they are in such possibility” (Wallis 1684, p. 96).

20. This is brought out nicely in a very late letterfrom Leibniz to M. Dangicourt, where Leibniz remarks that “[w]hen our friendswere disputing in France with the Abbé Gallois, father Gouye and others, I toldthem that I did not believe at all that there were actually infinite or actuallyinfinitesimal quantities; the latter, like the imaginary roots of algebra ( ) were onlyfictions, which however could be used for the sake of brevity or in order tospeak universally . . . But as the Marquis de l’Hôpital thought that by this Ishould betray the cause, they asked me to say nothing about it, except what Ialready had said in the Leipzig Acta” (Leibniz to Dangicourt, 11 September1716; Leibniz 1768, III, pp. 500–501).

21. This should not be taken to mean that Leibniz’sreservations about the infinite only appear in the 1690s. His early essay Dequadratura arithmetica circuli ellipseos et hyperbolae . . ., for example, treadscautiously over this terrain and uses the technique of unlimited approximationsto deliver the central result (Leibniz 1993, pp. 28–33), even while noting that therigorous form of the proof makes it seem excessively long and difficult.Moreover, even as late as 1702, Leibniz still seems to have been unsure of justexactly what to make of the infinitesimal. Pasini (1988) has drawn attention to manuscripts thatshow Leibniz to have engaged in a “controversy with himself” over the status ofthe infinitesimal in 1702.

22. See Costabel(1988) for a briefdiscussion of the concept of a “well-founded fiction” in Leibniz’s philosophy.

23. Ross (1990, p.133) makes thispoint in a manner that fits well with my account of these issues when heobserves that “[t]here is in fact an ambiguity in the notion of reality asLeibniz applies it to mathematical concepts. In one sense, even straightforwardgeometrical concepts, such as the concept of a perfect circle, are “unreal”,since there are no realities exactly corresponding to them. They are entiarationis, or “mental entities”, or “incomplete things”. In another sense, alllogically coherent mathematical concepts are “real”, as contrasted with“imaginary” ones, which contain a contradiction, and therefore cannot properlybe concepts at all. Of these last, some are useless, like the notion of thehighest number; whereas others, such as the notion of the square root of minusone, or of the limit of an infinite series, or of an infinitesimal quantity,are at least useful at the level of symbolic manipulation.”

24. This is not the place for a detailed discussion ofthe complexities of Leibniz’s physics and its relationship to the metaphysicsof the late period. The metaphysical picture of the Monadology, where onlymind-like simple substances are ultimately real, is difficult to fit togetherwith Leibniz’s pronouncements on the nature of force and motion. As Garbernotes, “it is not clear exactly how the world of the dynamics, primitive andderivative, active and passive forces is supposed to fit into Leibniz’s largermetaphysical picture. But then, what uncertainty there is derives fromLeibniz’s own uncertainties about the details of that metaphysics, as itevolved from the 1680s to the end of his life” (Garber 1995, p. 298). The significant point for mypurposes is that Leibnizian metaphysics dictates that not everythingpresupposed in the physicist’s account of nature is ultimately real.

25. As Herbert Breger has argued, the reliability ofthe infinitesimal is connected with Leibniz’s conception of the continuum. Inparticular, it is because the continuum is not composed of points that we canonly remove sub-intervals from any continuous magnitude, and this view ofcontinuity makes the infinitesimal appear as a quite natural fiction. SeeBreger (1990a) for a more detailed account of these matters.

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