
Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes
Among his achievements in all areas of learning, Leibniz’s contributions to the development of European mathematics stand out as especially influential. His idiosyncratic metaphysics may have won few adherents, but his place in the history of mathematics is sufficiently secure that historians of mathematics speak of the “Leibnizian school” of analysis and delineate a “Leibnizian tradition” in mathematics that extends well past the death of its founder. This great reputation rests almost entirely on Leibniz’s contributions to the calculus. Whether he is granted the status of inventor or coinventor, there is no question that Leibniz was instrumental in instituting a new method, and his contributions opened up a vast new field of mathematical research.
The foundations of this new method were a matter of some controversy, however. The Leibnizian calculus at least appears to violate traditional strictures against the use of infinitary concepts, and critics charged Leibniz with abandoning classical standards of rigor and lapsing into incoherence and error. In response, Leibniz argued that his methods were rigorous and that they did not suppose the reality of infinitesimal quantities. But the critics of the calculus were not the only ones who engaged Leibniz in a discussion of the nature of the infinitesimal. Even those sympathetic to the new method gave differing interpretations of the doctrine of infinitesimal differences, and Leibniz’s own doctrine evolved out of dissatisfaction with alternative foundations for the calculus, as well as a desire to satisfy the demands of his “traditionalist” critics.
The fundamental thesis in Leibniz’s mature account of the foundations of the calculus is that infinitesimals are wellgrounded fictions. Although it is not without its difficulties, the doctrine seems to solve or avoid problems inherent in two alternative treatments of the calculus, one (due [End Page 6] to Johann Bernoulli) that regards infinitesimals as real positive quantities and another (from John Wallis) that takes them as nothing, or “nonquanta.” At the same time the fictional infinitesimal allows a response to the traditionalists, whose criteria of rigor permit only finite magnitudes in mathematics. I do not wish to say that Leibniz’s theory arises out of a Hegelian synthesis of Wallis’s thesis and its Bernoullian antithesis. Nevertheless, the fictional treatment of the infinitesimal clearly appears designed 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 to settle on an interpretation of the calculus that can preserve the power of the new 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 outlines objections that Bernard Nieuwentijt and Michel Rolle raised to the calculus. The third considers Leibniz’s response to Nieuwentijt, and particularly the proposal that the calculus can be based on “incomparably small” magnitudes. Section four examines Leibniz’s correspondence with John Wallis and his rejection of Wallis’s claim that infintesimals lack quantity. The fifth section considers Leibniz’s correspondence with Bernoulli and its connection to the project of replying to Rolle. The final section then tries to make some sense of 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 the calculus has been told many times, both by Leibniz himself and by numerous commentators. ^{2} I do not propose to recount it in any detail, but it is important that we consider it at least in outline. Leibniz’s first mathematical investigations were apparently in the field of combinatorics; he reports in the essay Historia et origo calculus differentialis that he “took great delight in the properties and combinations of numbers” (GM V, p. 395), and these studies led him to publish his Dissertatio de Arte Combinatoria in 1666 (GM V, pp. 10–79). Leibniz became particularly interested in the [End Page 7] properties of numerical sequences and the sums or differences of the terms in such sequences. He noticed that the operations of addition and subtraction, when applied to the terms of a given sequence, will produce two new sequences, one of sums and another of differences. That is, starting with the sequence {a_{n}} = {a_{1},a_{2},a_{3},} we can generate a sequence {s_{n}}of sums and a sequence {d_{n}}of differences by making
Then, as Leibniz noted, there is an interesting reciprocity in the fact that the differences of the sum sequence and the sums of the difference sequence both yield the original sequence. ^{3}
This reciprocity is mirrored in another important concept at the core of the Leibnizian calculus, namely the idea that a curve is a polygon with an infinite number of infinitely small sides. Taking the curve ABC in figure 1 with Cartesian coordinate axes x and y, we can approximate the area beneath the curve by dividing the abscissa into a finite collection of equal subintervals {x_{0},x_{1},x_{2},x_{n}}. The sum of the rectangles x_{0}y_{1},x_{1}y_{2}, x_{2}y_{3},...x_{n1}y_{n} approximates the area, and this approximation can be systematically improved by taking ever larger collections of ever smaller [End Page 8] subintervals. Moreover, the tangent at any point on the curve can be approximated by taking differences between the y values associated with successive x coordinates of the abscissa. Thus, at point p the slope of the tangent is approximated by taking the difference y_{2}  y_{1}.
Leibniz invites us to consider the case (as in figure 2) where the finite approximations give way to exact results when the curve is treated as an infinitary polygon. ^{4} In the infinite case, the differences dx and dy become infinitesimal increments in 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, whose hypotenuse is an infinitesimal element of the curve. The area bounded by the curve and the axes can then be thought of as built up out of infinitely narrow parallelograms of the 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 more strikingly, the old reciprocity between sums and differences is maintained: the area under the curve is an infinite sum, differences between the terms of the sum are (the slope of) tangents, and the problem of quadrature is the inverse of the problem of drawing a tangent. The foundational role of these concepts is emphasized in the manuscript Elementa calculi novi pro differentiis et summis, tangentibus et quadraturis . . ., where Leibniz declares that the “foundation of the calculus” is the principle that “differences and sums are reciprocal to one another; that is the sum of differences of a sequence is a term of the sequence and the difference [End Page 9] of sums of a sequence is itself a term of the sequence. I express the former as the latter as “ (Leibniz 1855, p. 153).
The great power of Leibniz’s differential calculus is that it allows the problems of tangency and quadrature to be reduced to a relatively simple algorithmic procedure. Take as an example the curve with the analytic equation with variables x, y, and constants a, b, c, d. We then consider the “differential increment” of [1] obtained by replacing y and x with y + dy and x + dx respectively. The result is
Expanding the right side of equation [
Subtracting equation [
Dividing each side of [
But because dx is infinitely small in comparison with x, the terms containing it can be disregarded in the right side of equation [
The formula in [6] (known as the “derivative” of [1]) gives the slope of the tangent at any point on the curve in equation [
There is another important concept in the Leibnizian calculus that needs to be mentioned, namely that of higherorder differentials. In Leibniz’s presentation the differences dx and dy are themselves variable quantities, and they can be thought of as ranging over sequences of values of x and y that are infinitely close to one another. Depending upon the nature of the curve, the infinitesimal quantities dx and dy can stand in any number of different relations, and because these quantities are themselves variable, it makes sense to inquire into the rates at which they vary. The secondorder differences ddx and ddy appear as infinitesimal differences between values of the variables dx and dy, and similar considerations allow the construction of a sequence of differences of everhigher orders. Higherorder differences are employed to consider the behavior of curves that themselves are derived from the taking of a firstorder derivative, and this process can extend to a wide range of important cases. The means of introducing secondorder 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 firstorder difference as the firstorder difference stands to a finite quantity.
Even from the very cursory summary given here, it should be clear that the most natural formulation of the Leibnizian calculus makes it straightforwardly committed to the reality of infinitely small quantities. These are introduced to facilitate the study of continuous curves, and are treated like ordinary quantities when they are added, divided, or other algebraic operations are applied to them. Yet they can also be treated as “negligible” or “discardable” according to convenience, while the results obtained by their use are taken to be perfectly exact and not mere approximations. Although Leibniz’s first paper on the calculus is sufficiently vague to avoid any definite commitment to the reality of infinitesimals, his 1686 essay De geometria recondita speaks openly of “my differential calculus or analysis of indivisibles and infinities” and refers to the characteristic triangle “whose sides are indivisible (or, speaking more accurately, infinitely small), that is to say differential quantities” (GM V, pp. 230, 232). By 1694 Leibniz could offer “our new calculus of differences and sums, which involves the consideration of the infinite” as an example of reasoning which “extends beyond 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, it is worth considering one source of Leibniz’s ideas, namely Hobbes’s doctrine of conatus and its application to the problem of quadrature.
Hobbes first introduced the concept of conatus in his 1655 treatise De Corpore—a work which he touted as the first part of the elements of philosophy and which contained his doctrines on the nature of body as well as his thoroughly materialistic philosophy of mathematics. As Hobbes defines it, conatus is essentially a point motion, or motion through an indefinitely small space: “conatus” he declares, “is motion through a space and a time less than any given, that is, less than any determined whether by exposition or assigned by 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 is an extended body, but one sufficiently small that its magnitude is not considered in a demonstration. In explicating the definition of conatus he therefore remarks that “it should be recalled that by a point is not understood that which has no quantity, or which can by no means be divided (for nothing of this sort is in the nature of things), but that whose quantity is not considered, that is, neither its quantity nor any of its parts are computed in demonstration, 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 indivisible time” (Hobbes [1839a–45] 1966a, I, pp. 177–78). The result is that conatus is a kind of “tendency toward motion” or a striving to move in a particular direction.
This definition allows for a further concept of impetus, or the instantaneous velocity of a moving point; the velocity of the point at an instant can be understood as the ratio of the distance moved to the time elapsed in a conatus. In Hobbes’s terms “impetus is this velocity [of a moving thing] but considered in any point of time in which the transit is made. And so impetus is nothing other than the quantity or velocity of this conatus” (Hobbes [1839a–45] 1966a, I, p. 178).
The concepts of impetus and conatus can be applied to the case of geometric magnitudes as well as to moving bodies. Because Hobbes held that geometric magnitudes are generated by the motion of points, lines, or surfaces, he also held that it is possible to inquire into the velocities with which these magnitudes are generated, and this inquiry can be extended to the ratios between magnitudes and their generating motions. ^{7} For example, we can think of a curve as being traced by the motion of a point, and at any given stage in the generation of the curve, the point will have a [End Page 12] (directed) instantaneous velocity. This, in turn, can be regarded as the ratio between the indefinitely small distance covered in an indefinitely small time; this ratio will be a finite magnitude which can be expressed as the inclination of the tangent to the curve at the point. Take the curve αβ as in figure 3. The conatus of its generating point at any instant will be the “point motion” with which an indefinitely small part of the curve is generated; the impetus at any stage in the curve’s production will be expressed as the ratio of the distance covered to the time elapsed in the conatus. Represent the time by the xaxis and the distance moved by the yaxis. Then (assuming time to flow uniformly), the instantaneous impetus will be the ratio between the instantaneous increment along the yaxis to the increment along the xaxis. We can represent these different increments somewhat fancifully by arrows in each direction of the impetus. The tangent to the curve can then be analyzed as the diagonal of the parallelogram whose sides are proportional to the increments.
It is important to observe here that the tangent is constructed as a finite ratio between two quantities that, in themselves, are small enough to be disregarded. That is to say, the ratio between two “inconsiderable” quantities may itself be a considerable quantity. Hobbes emphasizes this feature of his system 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 a geometric demonstration. Thus, in discussing the comparisons that may be made between one conatus and another, Hobbes declares: “as a point may be compared with a point, so a conatus can be compared with a conatus, and one may be found to be greater or less than another. For [End Page 13] if the vertical points of two angles are compared to one another, they will be equal or unequal in the ratio of the angles themselves to one another; or if a right line cuts many circumferences of concentric circles, the points of intersection will be unequal 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 the general problem of quadrature by analyzing the area of a plane figure as the product of a moving line and time. Hobbes himself was eager to solve problems of quadrature (most notably the quadrature of the circle), and it is here that his concept of conatus is put most fully to work. Indeed, it is no exaggeration to say that the third part of De Corpore (which bears the title “On the Ratios of Motions and Magnitudes”) is Hobbes’s attempt to furnish a general method for finding quadratures. In the very simplest case, the whole impetus imparted to a body throughout a uniform motion is representable as a rectangle, one side of which is the line representing the instantaneous impetus while the other represents the time during which the body is moved. More complex cases can then be developed by considering nonuniform motions produced by variable impetus. In chapters 16 and 17 of De Corpore Hobbes approached a variety of different quadrature and tangency problems, and in so doing he presented a number of important results that belong to the “prehistory” of the calculus. Of special interest in this context is Hobbes’s appropriation of important results from Bonaventura Cavalieri’s Exercationes Geometricae Sex, which he set forth in chapter 17 of De Corpore as an investigation into the area of curvilinear figures. ^{8}
It is well known that Leibniz was profoundly influenced by his reading of Hobbes, and he seems to have been particularly enamored of the Hobbesian concept of conatus. In his famous 1670 letter to Hobbes, Leibniz declares the English philosopher to be “wholly justified” in “the foundations [he has] laid concerning 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 for Hobbes’s analysis of motion, this endorsement suggests that Leibniz was ready to follow Hobbes in using the concept for the analysis of all phenomena produced by motion. Indeed, scholars today generally accept that “Leibniz’s early writings on natural philosophy are virtually steeped in De Corpore” (Bernstein 1980, p. 29). In particular, Leibniz’s reading of Hobbes appears to have been the source for much of [End Page 14] his (admittedly limited) mathematical knowledge before his stay in Paris in the 1670s (Hoffman 1974, pp. 6–8).
The clearest evidence of Hobbes’s influence on Leibniz is in his essay Theoria motus abstracti, where Leibniz employs the concept of conatus to investigate the nature of motion and eventually arrives at the remarkable conclusion 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 the solution of any number of mathematical and philosophical puzzles. The theory, he claimed, “explains the hitherto unresolved difficulties of continuous composition, confirms the geometry of indivisibles and arithmetic of infinities; it shows that there is nothing in the realm of nature without parts; that the parts of any continuum are in fact infinite; that the theory of angles is that of the quantities of unextended bodies; that motion is stronger than motion, and conatus stronger than conatus—however, conatus is instantaneous motion 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” to which Leibniz refers are, I take it, the works of Cavalieri and Wallis. Cavalieri’s method of indivisibles is mentioned explicitly in section six of the Theoria motus abstracti, as a theory whose “truth is obviously demonstrated so that we must think of certain rudiments, so to speak, or beginnings of lines and figures, as smaller than any given magnitude whatever” (GP IV, p. 228). Wallis’s 1655 treatise Arithmetica Infinitorum, although not mentioned explicitly in the text, is evidently referred to in the letter to Oldenburg when Leibniz refers to the “arithmetic of infinities.” In light of this, it is no great interpretive leap to see Leibniz connecting the doctrine of conatus with the classic problem of quadrature, just as Hobbes had done, and thus to find part of the origin of the calculus in Leibniz’s close reading of De Corpore.
It would doubtless be going too far to claim that the whole of Leibniz’s calculus is simply the application of Hobbes’s ideas. It is well known that Leibniz’s mathematical thought was also strongly influenced by Galileo’s approach to the geometry of indivisibles, for example, and the influence of [End Page 15] Huygens cannot be overlooked. ^{10} Nevertheless, we can agree that Hobbes was one among many whose writings stimulated the development of the Leibnizian approach to the calculus. ^{11} However, there is one important difference between the Leibnizian and Hobbesian conceptions of conatus that is significant for my present purposes: Leibniz’s language (at least in the Theoria motus abstracti) requires that conatus be a literally infinitesimal quantity, while Hobbes regards it as having finite magnitude, but one so small as to be disregarded. It was the introduction of infinitesimal magnitudes into the foundations of the calculus that involved Leibniz in important philosophical 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 did not make its way in the world without a struggle. For some ten years after the publication of Leibniz’s first paper on the calculus in 1684, there was little criticism of the method, although there was also little comprehension of it. ^{12} The circle around Leibniz—including the brothers Johannn and Jacob Bernoulli, Pierre Varignon, and the Marquis de l’Hôpital—extended and developed the method during this period, but by the mid1690s they were faced with the challenge of justifying the foundations of the new calculus. Two critics stand out as especially significant: Bernard Nieuwentijt and Michel Rolle. Both criticized the calculus as illfounded and unrigorous, although their arguments have important differences. These criticisms are important enough in Leibniz’s account of the foundations of the calculus that we must make a brief overview of them.
Nieuwentijt was a Dutch mathematician of some note who attacked the Leibnizian calculus in his 1694Considerationes circa analyseos ad quantitates infinite parvas applicatae principia et calculi differentialis usum in resolvendis [End Page 16] problematibus geometricis. ^{13} He argued that in rejecting infinitesimal magnitudes the practitioners of the calculus effectively treated them as zero or nothing. He further argued that, even if infinitesimals are admitted, there can be no basis for accepting the higherorder infinitesimal. In essence, Nieuwentijt tried to develop a rival version of the Leibnizian calculus that would not discard infinitesimal quantities and would be confined to the consideration of infinitesimals of the first order.
In support of the thesis that first order infinitesimals are zero magnitudes, Nieuwentijt takes an example from the Lectiones Geometricae of Isaac Barrow, where an infinitesimal quantity is rejected during the determination of a tangent. ^{14} Nieuwentijt observes that “[t]he celebrated author sets forth this thesis in his reasoning: If a determinate quantity has a ratio greater than any assignable to any other quantity, the latter will be equal to zero” (Nieuwentijt 1694, p. 6). Following essentially the classical conception of rigor, Nieuwentijt further insists that only those magnitudes are equal whose difference is zero; as he puts it: “I declare that this proposition is indubitable and carries with it most evidently the certain signs of truth: Only those quantities are equal whose difference is zero, or is equal to nothing” (Nieuwentijt 1694, p. 10).
Although he was convinced that infinitesimal differences were treated as zeromagnitudes in the calculus, Nieuwentijt was prepared to accept infinitesimals as long as they were treated as positive quantities and not discarded. Thus, where Leibniz and his followers freely dropped terms dx or dy from equations, Nieuwentijt would require that they be retained. More important, he insisted that the higherorder differentials must be banned from the calculus altogether, since he saw no prospect of developing a coherent theory of positive quantities less than an infinitesimal. He ultimately held that the infinite divisibility of geometric magnitudes guarantees the legitimacy of firstorder infinitesimals, but regarded any extension of the doctrine to higher orders of infinity as unwarranted.
The second attack on the calculus began in the Paris Académie Royale des Sciences in July of 1700, when Michel Rolle voiced opposition to the use of infinitesimal magnitudes. ^{15} Rolle was not alone in this project, for he allied himself with several mathematical conservatives, including the Abbé Jean Gallois and the Abbé Thomas Gouye, both of whom venerated the Greek standards of rigor and had significant reservations about the use of [End Page 17] infinitesimal methods. Rolle’s criticisms were later published in the memoir Du nouveau systême de l’infini, which he opened by declaring that
We have always regarded geometry as an exact science, and also as the source of the exactness which is spread throughout all the other parts of mathematics. We see among its principles only true axioms: all the theorems and all the problems proposed here are either solidly demonstrated or capable of a solid demonstration. And if it should happen that any false or less certain principles slip in, they should be at once banished from this science.
But it seems that this character of exactitude no longer reigns in geometry, ever since we became entangled in the new system of the infinitely small. For myself, I do not see that it has produced any new truth, and it seems to me that it often leads to error.
(Rolle 1703, p. 312)
The “new system of the infinitely small” mentioned here is the differential calculus, particularly as formulated in L’Hôpital’s 1696 Analyse des infiniment petits. This treatise was something of an official statement of the methods and philosophical foundations of the calculus that, in the words of L’Hôpital “penetrates into infinity itself” by comparing the ratios of infinitesimal quantities (L’Hôpital 1696, p. iii). According to L’Hôpital “this analysis extends beyond the infinite: for it does not rest with infinitely small differences, but discovers the relations and differences of such differences, and again of third differences, fourth, and so on without finding an end. So that it embraces not only the infinite but the infinite of the infinite, or an infinity of infinities” (1696, p. iv). In L’Hôpital’s presentation, infinitesimal magnitudes are introduced in the form of an axiomatic system, complete with definitions, postulates, and theorems. The first definition, for example, declares that “[v]ariable quantities are those which increase or diminish continually; and constant quantities are those which remain the same while others change” (L’Hôpital 1696, p. 1). The second stipulates that “[t]he infinitely small portion by which a variable quantity continually increases or diminishes is called its difference” (L’Hôpital 1696, p. 2). The most important postulate in this system is the first, which declares that “one can take indifferently for one another two quantities which differ from one another by an infinitely small quantity; or (which is the same thing) that a quantity which is augmented or diminished by another infinitely less than it, can be considered as if it remained the same” ( L’Hôpital 1696, pp. 2–3). It should be evident that, whatever reservations others may have had about the reality and intelligibility of infinitesimals, L’Hôpital was not one to balk at them. It should also be [End Page 18] evident that, taking his first postulate at face value, L’Hôpital seems committed to the thesis that x + dx = x, which implies that dx = x  x = 0.
Rolle’s charges against this system are serious indeed. He not only alleges that its foundations are themselves incoherent, but he also claims that it leads to falsehood, and adds that in any case it is incapable of discovering or proving new truths. The defense of the calculus within the Académie was undertaken by Varignon, who sought to provide proofs of the reality of infinitesimals while also responding to technical criticisms, in which Rolle claimed to show that the use of infinitesimals led to false results. Johann Bernoulli, L’Hôpital, and others assisted in addressing Rolle’s challenges, and the next six years saw an extended debate over the metaphysics of the calculus which ended with the triumph of the new methods.
Both of these criticisms of the calculus are significant for the development of Leibniz’s theory of the fictional infinitesimal. He resists Nieuwentijt’s conclusion that the infinitesimal is effectively treated as nothing, while at the same time defending the practice of discarding infinitesimal quantities from equations. Similarly, he cannot accept Rolle’s charge that the calculus is unrigorous, and he must try to show that the only real quantities required in his method are finite and positive. In examining Leibniz’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 a coherent synthesis.
3. Leibniz’s Rebuttal to Nieuwentijt
Shortly after the publication of Nieuwentijt’s Considerationes, Leibniz answered with an essay Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas (GM V, pp. 321–26) that appeared in the Leipzig Acta Eruditorum in July of 1695 and was intended to disarm Nieuwentijt’s objections. This reply seems to have been put together hastily, since Leibniz felt the need to print a short appendix to it in the next month’s issue of the Acta (GM V, pp. 327–28). Although he was not directly involved in responding to Rolle, Leibniz kept up a correspondence with Bernoulli, Varignon, and other principals in the controversy in the Académie. The effect of Leibniz’s pronouncements in these matters was not, as one might expect, to vindicate the use of infinitesimals, but instead to muddy the waters so that it is difficult to discern exactly what Leibniz held about the status of the infinitesimal. However, I think that some sense can be made of Leibniz’s apparently contradictory pronouncements on the nature of infinitesimals and their status in his mathematics. In particular, I think that the doctrine of the fictionality of the infinitesimal develops out of Leibniz’s reaction to criticisms [End Page 19] of the calculus and his correspondence with Wallis and Bernoulli. We can see the beginnings of this process in the reply to Nieuwentijt.
Leibniz opens his response to Nieuwentijt with what looks like a defense of the reality of infinitesimals, although he chooses to speak of quantities “incomparably small” where one might expect reference to the infinitely small. To Nieuwentijt’s requirement that only those quantities are equal whose difference is zero, Leibniz appears to ally himself with L’Hôpital by insisting that equal quantities can still differ from one another. He first admits his admiration for those who desire to see all things demonstrated from undeniable first principles, but cautions that an excess of scruple may impede the art of discovery and deny us its fruits. Regarding the question of whether equal quantities can differ from one another, Leibniz declares:
I think that those things are equal not only whose difference is absolutely nothing, but also whose difference is incomparably small; and although this difference need not be called absolutely nothing, neither is it a quantity comparable with those whose difference it is. Just as when you add a point of one line to another line or a line to a surface you do not increase the magnitude, it is the same thing if you add to a line a certain line, but one incomparably smaller. Nor can any increase be shown by any such construction.
(GM V, p. 322)
We may note, in passing, that the reference here to “incomparably small” elements of lines or surfaces has a strongly Hobbesian ring to it, for it is exactly the hallmark of Hobbes’s points that—though finite—they are too small to be considered in any demonstration. Leibniz’s preference here for the language of the incomparable rather than the infinitesimal raises the question of whether such incomparable magnitudes are to be thought of as literally infinitesimal or whether they should be treated as finite but negligible quantities in the manner of Hobbes’s points.
At first sight, it seems natural to take the unassignable or incomparably small as just the infinitesimal in a different guise, perhaps seeing 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 enough to show that incomparably small quantities can be justly neglected in a calculation, and he refers to certain “lemmas communicated by me in [End Page 20] 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 motuum coelestium causis (GM VI, pp. 144–60). But when we turn to them for enlightenment it is evident that they were intended explicitly to avoid references to infinitesimal quantities and instead to replace infinitesimal magnitudes with finite differences sufficiently small to be ignored in practice. The paragraph expounding these lemmas opens with the declaration that
I have assumed in the demonstrations incomparably small quantities, for example the difference between two common quantities which is incomparable with the quantities themselves. Such matters as these, if I am not mistaken, can be set forth most lucidly in what follows. And then if someone does not want to employ infinitely small quantities, he can take them to be as small as he judges sufficient to be incomparable, so that they produce an error of no importance and even smaller than any given [error]. Just as the Earth is taken for a point, or the diameter of the Earth for a line infinitely small with respect to the heavens, so it can be demonstrated that if the sides of an angle have a base incomparably less than them, the comprehended angle will be incomparably less than a rectilinear angle, and the difference between the sides will be incomparable with the sides themselves; also, the difference between the whole sine, the sine of the complement, and the secant will be incomparable to these differences.
(GM VI, pp. 150–51)
The use intended for such incomparably small magnitudes is to avoid disputes about the nature or existence of infinitesimal quantities, and Leibniz holds that “it is possible to use ordinary [communia] triangles similar to the unassignable 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 the calculus which permits the talk of infinitesimals to be reinterpreted in terms of incomparable (but apparently finite) differences. These lemmas loom large in Leibniz’s writings on the foundations of the calculus, since he frequently refers back to them in later discussions on the nature of the infinitesimal. It is also significant that the incomparably small satisfies Hobbes’s definition of a geometric point—it is a quantity sufficiently small that its magnitude cannot be regarded in a demonstration.
In the appendix to his reply to Nieuwentijt, Leibniz returns to the theme 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 is that talk of infinitesimal increments of ordinate and abscissa can be reinterpreted in terms of finite ratios between finite lines that express the ratio of the ordinate to the abscissa at any given point. “In order to remove all disputes about the reality of differences of any order,” Leibniz writes, “they can always be expressed in proportional finite right lines [rectis ordinariis proportionalibus]” (GM V, p. 327).
The project of replacing ratios of infinitesimals with ratios of finite quantities should, according to Leibniz, satisfy the demands of the rigorists: if they do not care for infinitesimals whose ratios are investigated by the calculus, they can retain the ratios and replace their (infinitesimal) terms with finite quantities. The weakness of this approach, of course, is that the ratios were originally acquired by manipulating infinitesimal quantities. It hardly satisfies the standards of geometrical rigor to work with ratios that can only be obtained by the introduction of infinitesimals and then to pretend that these ratios are legitimate because they can be expressed in terms of finite quantities.
In the end, Leibniz’s reply to Nieuwentijt falls well short of a spirited defense of the infinite in mathematics, nor is it a particularly compelling or satisfying attempt to reinterpret the infinitesimal out of the calculus. Leibniz neither affirms nor denies the real existence of the infinitely small in this exchange, and he goes out of his way to insist that infinitary considerations can be avoided by making use of finite (but negligible) quantities. These facts suggest that Leibniz was not a thoroughgoing realist about infinitesimal magnitudes, and at the very least his views on the nature of the infinitely small were not fixed and settled in the mid1690s.
4. Leibniz and Wallis on the Infinitely Small
In 1695, the year of his published reply to Nieuwentijt, Leibniz began to correspond with Wallis. His principal motives for undertaking this correspondence were to keep abreast of mathematical developments in England and to promote his own calculus differentialis, which he saw as a fundamental extension of Wallis’s methods. The priority dispute with Newton eventually led Leibniz to end his contact with Wallis and the English mathematical world, but before the correspondence ended in 1700 the two exchanged twenty letters that covered a wide variety of topics. Key among these were the nature of the infinitely small and the difference between their respective approaches to the mathematics of the infinite. ^{17} [End Page 22]
For his part, Wallis was concerned with maintaining a claim for the originality and extensiveness of his methods. This led him to assert that Leibniz’s calculus was really little more than a notational variant of his own arithmetic of infinities. Wallis was also intent upon defending the rigor of his approach, and in the course of this defense he compared his work to the classical methods. Inevitably, this project led Wallis to clarify his own conception of the infinitesimal and to give his own account of the development of seventeenth century mathematics.
Responding to Leibniz’s request for a further account of his methods, their background, and their foundations, Wallis explained that his investigations had their origin in the problem of the angle of contact between a circle and its tangent. This problem had been the source of much controversy in the preceding century, most notably between Jacques Peletier and Christopher Clavius. ^{18} As Wallis tells the story, a proper understanding of the angle of contact leads immediately to a method of rectifying curvilinear arcs and finding the area of curved surfaces. He explains:
I had long since claimed that the angle of contact to a circle is of no magnitude; nor was I the first to do this, but vindicated the opinion of Peletier, which had been opposed by the authority of Clavius. By the same reasoning we may conclude that the angle of contact to any curve is of no magnitude . . . It follows at once that any point of any curve has the direction, obliquity, [or] inclination . . . of the right line tangent to the same curve. Thus the point can be considered as an infinitesimal part of this right line. The whole doctrine of rectifying [curves] takes its origin from this . . . 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 to direction which exists abstracted from all magnitude. ^{19} Wallis extends this doctrine to the case of tangency, and in doing so he makes explicit what was merely implicit in his discussion of the angle of contact: namely that [End Page 23] the infinitesimal is really not a magnitude at all. In computing tangents, Wallis had introduced a minute increment designated a, which is diminished in the course of the computation and ultimately discarded. The magnitude a might appear to be an infinitesimal, but Wallis declares that it is really nothing. This leads him to draw a distinction between the Leibnizian calculus of the infinitely small and his own arithmetic of infinities. As he describes the situation:
You see that my methods for tangents were summarily set out in the Philosophical Transactions for the month of March 1672, and again in Proposition 95 of my Algebra, which I had earlier applied throughout my Treatise of Conic Sections of 1655, and these methods plainly rest on the same principles 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 your dx infinitely small. Then when those things are neglected which I hold should be neglected in order to abbreviate the calculation, that which remains is your minute triangle, which according to you is infinitely small, but according to me is nothing or evanescent.
(Wallis to Leibniz, 30 July 1697; GM IV, p. 37)
This rather startling declaration should not be taken to mean that Wallis everywhere regards the magnitude a as nothing. It is first introduced into a calculation as a finite positive increment, but then “infinitely diminished” to become nothing, and therefore dropped out of a calculation, although results obtained under the hypothesis that a is a positive increment are retained, living on after the demise of the increment itself.
Wallis returned to this theme nearly a year later. Again discussing the problem of finding the tangent to a curve, he insisted that the “foundation of the whole procedure” is to move from an approximation to the tangent to an exact value by letting a secant to the curve become a tangent. Consider, for example, the problem of finding the tangent to the curve αβ at the point F, as in figure 4. Wallis’s procedure begins by taking a secant that cuts the curve at the points F and G; he then brings them into coincidence by rotating the secant about F until the points coincide and the difference between them (marked by the letter a) vanishes. Terms containing a are then cancelled from the equation representing the tangent. Although the increment vanishes into nothingness, it leaves something behind in the form of a triangle abstracted from all magnitude. As Wallis explains:
When the simplification of the calculation I teach is applied, that which remains is in fact your differential calculus (for it is not so [End Page 24] much a new thing as a new way of speaking, although perhaps you were not aware of it). My term a is everywhere the same as your segment of the abscissa x or y, with this one difference: namely that your quantity dx is infinitely small, mine is simply nothing. Then when it is deleted or (to shorten the calculation) all those terms are dismissed which should be deleted, that which remains is your minute deferential triangle, formed between two adjacent ordinates. . . . But in your presentation it is infinitely small, in mine it is clearly nothing. Of course the species of the triangle is retained, but abstracted from magnitude. That is to say, the form of a triangle remains, but of no determinate magnitude.
(Wallis to Leibniz, 22 July 1698; GM IV, p. 50)
On this scheme, the product of two or more infinitesimals will also be rejected, on the grounds that nothing multiplied by nothing always remains nothing. Wallis argues that it is precisely on this point that his methods are preferable to Leibniz’s, because “I have no need of any of your postulates about the infinitely small multiplied by itself . . . (which must be applied with some caution), since it is selfevident . . . that nothing multiplied any number 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 difficult to see why. Taking the infinitely small as nothing does have the advantage of justifying the rejection of terms containing infinitesimal factors, since such terms are by definition equal to zero. But, by the same token, it would seem to require that when quantities are divided by infinitesimals, or when ratios between infinitesimal increments are compared, the result is a division by zero or the comparison of ratios of nothings. Moreover, the infinitesimal itself is often treated as a quantity with its own infinitesimal [End Page 25] parts—the infinitesimal of the second order—and it is difficult to see how the apparatus of higherorder infinitesimals can be justified if the infinitesimal of the first degree is itself simply nothing. Finally, the idea that the “differential triangle” is the persistence of the form of a triangle without magnitude seems at least as metaphysically problematic as the infinitesimal itself.
Leibniz was quick to point out these problems, and in his response to Wallis he laid out the case for denying that infinitesimal magnitudes are nothing:
I think it is better if elements or instantaneous differentials are considered as quantities according to my fashion, rather than their being taken for nothing. For they in their turn have their own differences, and these can even be represented by proportional assignable lines. I do not know whether it is intelligible to take this inassignable triangle as nothing . . ., in which there is nevertheless retained the species of a triangle abstracted from magnitude, so that it is the species of a given figure but of no magnitude. This certainly seems to introduce an unnecessary obscurity. Who acknowledges a figure without magnitude? Nor do I see how magnitude can be removed, when to such 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 calculus require the use of secondorder infinitesimals. These must be infinitely less that infinitesimals of the first order, and thus require that firstorder infinitesimals not be taken for nothing. He writes:
Although you say that you have no need of the infinitely small multiplied into itself, see whether it does not to a certain extent arise again out of oblivion: are not the elements of a curve to be represented by assuming that the right line dx is the element of the abscissa x, and the right line dy is the element of the ordinate y? Thus I observe in these things a kind of new law of homogeneity for the infinitesimal calculus: for the square of a differential or element of the first degree is homogeneous to a rectangle made from a finite right line multiplied by a difference of the second degree, or dxdx is homogeneous to ddx. Since this is the case, then the element of the first degree is a mean proportional between a finite right line and a difference of differences, 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 incoherent metaphysics and inadequate mathematics; he argued that by the abstraction of the form of a triangle from its magnitude he did not mean “a triangle which has no magnitude, but that the species or form of a triangle can be considered abstracted 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 the triangle is mentioned, you can say the degree of inclination or declination of the curve at the point of contact, or the angle the curve makes with the ordinate it touches, for indeed this is what is sought” (Wallis to Leibniz, 16 January 1699; GM IV, p. 58). In a somewhat conciliatory tone, Wallis adds that his doctrine can be made consistent with Leibniz’s theory by treating the coincidence of two points as a degenerate case where there is an infinitely small distance between the two points.
Leibniz concluded this exchange on the nature of the infinitesimal with a similarly conciliatory tone, although he was adamant that the infinitesimal not be regarded as nothing. He writes:
Of course, the form of the characteristic triangle can be rightly explained by the degree of declination, but for the calculus it is useful to imagine [fingere] quantities infinitely small, or as Nicholas Mercator called them, infinitesimal: and such things cannot be taken for nothing when the assignable ratio among them is sought. On the other hand they are rejected whenever they are added [adjiciuntur] to quantities incomparably greater, according to lemmas on incomparable quantities I once proposed in the Acta Eruditorum of Leipzig, which foundation the Marquis de L’Hôpital also uses. . . . It is simpler, I admit, as you say that nothing multiplied by anything is still 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 the defender of the reality of the infinitesimal. He consistently opposes Wallis’s equation of the infinitesimal with nothing, and is at great pains to point out that the calculus depends for its coherence upon the assumption that infinitesimal (or incomparable) magnitudes are positive quantities. He even claims that his principles are the same as those of L’Hôpital. Nevertheless, it is clear that by 1699 he had come to have doubts about the reality of infinitesimal quantities, although he may not have intended to make them clear to Wallis. In a passage from the letter of 30 March 1699, Leibniz added a sentence 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 are true 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 I observed that if someone substitutes the incomparably small or that which is sufficiently small for the infinitely small, I would not oppose it.
(GM IV, p. 63)
Leibniz’s reservations about the reality of infinitesimals can be seen more readily in his correspondence with Johann Bernoulli during the same time as his correspondence with Wallis. To complete the picture, we must therefore make a brief excursion into the LeibnizBernoulli correspondence.
5. The LeibnizBernoulli Correspondence and the Reply to Rolle
Johann Bernoulli was one of the first converts to the Leibnizian calculus, and it is largely due to his efforts and those of his brother Jacob that a vast array of new results were discovered in the 1690s. The calculus as understood by Bernoulli, L’Hôpital, and other continental mathematicians was nothing less than a true science of the infinite, and Bernoulli was quite untroubled by its seemingly paradoxical nature. Traditional strictures against the infinite held that there can be no clear conception of a quantity greater than nothing yet infinitely small, and even less of an ordered structure of ever more infinitesimal magnitudes, each infinitely less than its predecessor. Bernoulli, however, was prepared to accept the whole apparatus of infinitesimals as both fully real and completely intelligible.
By the mid1690s Leibniz was unhappy with a metaphysics of the infinitesimal which accords it such genuine reality. As a result of these worries he explicitly raised the suggestion that infinitesimal magnitudes are mere fictions whose use is justified by their utility in developing the calculus. In June of 1698 he wrote to Bernoulli:
I recognize . . . that you have written some profound and ingenious things concerning various infinite bodies [de corporibus varie infinitis]. I think that I understand your meaning, and I have often thought about these things, but have not yet dared to pronounce upon them. For perhaps the infinite, such as we conceive it, and the infinitely small, are imaginary, and yet apt for determining real things, just as imaginary roots are customarily supposed 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 [End Page 28] we admit real lines infinitely small, it follows also that lines are to be admitted which are terminated at either end, but which nevertheless are to 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 an assignable time by uniform motion. And it will similarly be required to conceive a time terminated on both sides, which nevertheless is infinite, and even that there can be given a certain kind of eternity (as I may express myself) which is terminated. Or further that something can live so as not to die in any assignable number of years, and nevertheless die at some time. All which things I dare not admit, unless I am compelled by indubitable demonstrations.
(GM III, pp. 499–500)
This passage is significant because it evinces not only Leibniz’s reservations about the coherence of the infinitesimal, but actually indicates the line of argument he takes to show the problems inherent in the concept. To take the infinitely small as real we must think of something that is both limited and unlimited: a determinate, bounded space that is smaller than any finite space; and similarly, we must acknowledge the reality of the infinitely large, which is a limited and yet unlimited quantity. Although Leibniz does admit that there might be “indubitable demonstrations” that compel his assent to the reality of the infinitesimals, he is clearly not prepared to accept them as real entities without an argument.
The theme of the fictionality of infinitesimal magnitudes recurs in the exchanges with Bernoulli through the summer of 1698, with Leibniz insisting that it is “useful in the calculus to assume” that there are lines which stand to ordinary lines in the ratio of finite to infinite (Leibniz to Bernoulli, 22 July 1698; GM III, p. 516) and again arguing that “it suffices for the calculus that they are represented in thought [finguntur], just as imaginary roots in algebra” (Leibniz to Bernoulli, 29 July 1698; GM III, p. 524). This last remark also appears along with Leibniz’s earlier claim that the new calculus can ultimately be founded on the basis of Archimedean exhaustion proofs, since “whatever is concluded by means of these infinite or infinitely small quantities can always be shown through reductio ad absurdum, by my method of incomparables (the lemmas of which I once published in the Acta)” (Leibniz to Bernoulli, 29 July 1698; GM III, p. 524).
His reaction to Rolle and the dispute in the French Academy seems to complete Leibniz’s retreat from a commitment to the reality of infinitesimal magnitudes to an explicit fictionalism about the infinite. In a famous letter to 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] infinitesimal which Abbé Gouye had published in the Journal. Leibniz argued that
there is no need to take the infinite here rigorously, but only as when we 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 in respect to the distance of the sphere of the fixed stars, and a ball which we hold in the hand is also a point in comparison with the semidiameter of the sphere of the earth. And then the distance to the fixed stars is infinitely infinite 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 as great or as small as is necessary in order that the error will be less than any given error. In this way we only differ from the style of Archimedes in the expressions, which are more direct in our method and better adapted to the art of discovery.
(GM IV, pp. 95–96)
These remarks are of a piece with Leibniz’s earlier claims about the eliminability of infinitesimal magnitudes: he denies that the calculus really needs to rely upon considerations of the infinite and again insists that it can be based on a procedure of taking finite but “negligible” errors that can be made as small as desired; moreover, the fact that such errors can be made arbitrarily small sets the stage for the reductio proofs characteristic of classical exhaustion methods, since if one supposes that the error has a fixed magnitude m, the error can be made less than m. His comments are, however, more definite than his earlier remarks, for he no longer claims that the infinitesimal “might” be imaginary.
The more ardent partisans of the infinitesimal (notably Bernoulli, Varignon, and L’Hôpital) were deeply concerned by Leibniz’s apparent concession to the critics of the calculus, and Varignon wrote to Leibniz in November of 1701 requesting a clarification of Leibniz’s views on the reality of infinitesimals. He remarked that the publication of the letter to M. Pinson had done harm to the cause, and that some had taken him to mean that the calculus was inexact and capable only of providing approximations. He therefore requested “that you send us as soon as possible a clear and precise declaration of 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 his views on the infinite and its role in the calculus. This statement brings together 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 new methods and the Archimedean techniques of exhaustion. After assuring Varignon that his intention was “to point out that it is unnecessary to make mathematical analysis depend on metaphysical controversies or to make sure that there are lines in nature which are infinitely small in a rigorous sense” (Leibniz to Varignon, 2 February 1702; GM IV, p. 91), Leibniz once again suggests that incomparably small magnitudes be taken in place of the genuine infinite. These incomparables would provide “as many degrees of incomparability as one may wish;” and although they are really finite quantities they can still be neglected, in accordance with the notorious “lemmas on incomparables” from the Leipzig Acta (Leibniz to Varignon, 2 February 1702; GM IV, pp. 91–92).
For our purposes, the most important part of the reply to Varignon is Leibniz’s frank admission that infinitesimal magnitudes are fictions, although fictions sufficiently wellgrounded that “everything in geometry and even in nature takes place as if they were perfect realities” (Leibniz to Varignon, 2 February 1702; GM IV, p. 93). The treatment of infinitesimal magnitudes as wellgrounded fictions recalls Leibniz’s remark to Bernoulli that, although the infinite may be imaginary, infinitesimal magnitudes are among the “ideal reasons by which, as it were, things are ruled, although they are not in the parts of matter” (GM III, p. 499). But in elaborating his doctrine in the letter to Varignon, Leibniz goes somewhat further in justifying his doctrines by making reference to his metaphysical principle of continuity. He insists that both infinitesimal magnitudes and imaginary roots have a foundation in the nature of things, and cites as evidence
not only our geometrical analysis of transcendental curves, but also my law of continuity, in virtue of which it is permitted to consider rest as an infinitely small motion (that is, as equivalent to a species of its contradictory), and coincidence as infinitely small distance, equality as the last inequality, etc . . . Yet one can still say in general that although continuity is something ideal and there is never anything in nature with perfectly uniform parts, the real, in turn, never ceases to be governed perfectly by the ideal and abstract. The rules of the finite are found to succeed in the infinite—as if there were atoms (that is to say, assignable elements in nature), although there are none because matter is actually subdivided without end, and conversely the rules of the finite succeed in the infinite, as if there were infinitely small metaphysical beings, although we have no need of them, and the division of matter never does proceed to infinitely small particles. This is because everything is governed by reason; otherwise there could be no science, nor rule, and this [End Page 31] would not conform 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 not made widely known, largely because Leibniz and his associates had reason to fear that any public retreat from a full commitment to the reality of the infinitesimal would complicate the already difficult battle for the acceptance of the calculus. ^{20} Despite his relative silence on the matter I take it as established that, at least in his mature thought, Leibniz did not believe in the reality of infinitesimal magnitudes. ^{21} The language of infinitesimal differences is not, however, unacceptable in the Leibnizian scheme of things: it has many uses in mathematical matters, enabling proofs to be shortened and fostering the art of discovery. Indeed, the calculus cannot even be stated without reference to fictional entities like dx and dy, but the principles of true (that is, Leibnizian) metaphysics show that the indulgence in such fictions does not detract 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 of Leibniz’s calculus in his more general conception of philosophy and the sciences. It should be clear from the foregoing that Leibniz more or less explicitly denies the real existence of infinitesimal magnitudes, and nevertheless allows himself to employ the concept of infinity quite freely in developing 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 of infinitesimal sides, if he does not really believe in the infinitesimal? More to the point, how can an apparently real curve be literally composed of [End Page 32] fictional parts? One reaction to this apparent difficulty is to take Leibniz as holding a “secret doctrine” of the reality of the infinitesimal, even if he occasionally professes not to accept the infinite. Another possible way around the problem would be to see Leibniz as a mendacious propagandist of the new mathematics who was happy to tell any story that might promote the higher goal of advancing mathematical learning, even though he never took any of his supposed justifications of the calculus seriously. Both of these strategies are unappealing, and I take the hypothesis of disingenuity to be an interpretive 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 a fictional entity and have some kind of account of how such fictions can be part of a properly developed mathematical theory. ^{22} Leibniz is less informative on this point than one might wish, but it is possible to construct a fairly complete picture of his doctrines. Infinitary magnitudes are not, after all, the only things he regards as fictions, and it is worth asking how the infinitesimal compares with other fictional entities.
One particularly important kind of fiction is the ens per aggregatum or “being through aggregation” which arises when individuals are grouped together to form a nonsubstantial unit. A flock of sheep, to take Leibniz’s stock example, is not a real thing in its own right: it is an assemblage of real substances, but there is no substantial flock over and above the individual sheep. Nevertheless, it is convenient to regard the flock as a unitary thing without making specific reference to the individuals out of which it is composed. In a letter to Bartholomeus Des Bosses in 1706, Leibniz considers aggregates in the context of the Scholastic dictum ens et unum convertitur, or “being and one are convertible.” He agrees that “[B]eing and one are convertible, but when there is a being by aggregation, so also is there unity [by aggregation], and this being and unity are semimental” (Leibniz to Des Bosses, 11 March 1706; GP II, p. 304). As Leibniz explains, the semimental nature of such things derives from the fact that their unity (and hence their being) 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 and above the individual sheep.
In the same letter Leibniz argues that infinite totalities arise from an analogous kind of mental imposition. In reality there can be no infinite quantities, for quantity must be essentially limited and the infinite is by definition unlimited. Nevertheless: [End Page 33]
It is thus for the sake of convenience of speech when we say that there is one where there are more than can be comprehended in an assignable whole, and we bring forth [efferimus] something like a magnitude which nevertheless does not have its properties. Just as it cannot be said of an infinite number whether it is even or odd, neither can it be said of an infinite line whether or not it is commensurable to a given line. And so all of these expressions taking the infinite as a magnitude are improper; and although they are founded in a certain analogy, they still cannot be maintained if you examine the matter more 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 small than infinitely great. . . . I take both for mental fictions, as more convenient ways of speaking, and adapted to calculation, just like imaginary roots are in algebra. I once demonstrated that these expressions have a great use both in abbreviating thought and aiding discovery, and that they cannot lead to error, since in place of the infinitely small one may substitute [a quantity] as small as one wishes, and since any error will always be less than this, it follows that no error can be given. But the Reverend Father Gouyé, who objected, 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 less admissible into mathematical calculations. Indeed, in strict metaphysical truth, all mathematical objects are in some measure unreal, as Leibniz openly declares in a letter to Burchard De Volder: “from the fact that a mathematical body cannot be resolved into first constituents we can at least infer that it is not real, but rather something mental, indicating only the possibility of parts, 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 up with anything at the ultimate level of the metaphysically real. ^{24} Bodies, for example, are ultimately phenomenal, as are space and time. Even the apparent causal interactions of bodies is merely phenomenal: the principle of preestablished harmony dictates that the course of events in the world unfolds without such causal interactions, although everything is arranged to appear as if bodies actually had causal powers. Physics, however, need not concern itself with the ultimately real monadic constituents of the universe. It is enough for the physicist to describe the workings of the phenomenal world with laws of motion grounded in the principles of true metaphysics.
One feature common to all Leibnizian fictions is that they are “well founded” at the level of real substances. A fiction is wellfounded when it reliably enables us to investigate the properties of real things, so that indulgence in the fiction cannot lead us into error. Leibniz’s standard example of such a fiction is the “imaginary” root . Roots of negative numbers are not themselves possible; nevertheless, they can be invoked to generalize algebraic laws, and the results obtained by their use are completely reliable. The stress on the wellfoundedness of such fictions invites us to contrast them will illfounded fictions. An illfounded fiction in the case of physics would be something both unintelligible and illsuited to the task of mechanistic physics: the void, action at a distance, or other Newtonian monstrosities are presumably such. Similarly, an illfounded mathematical fiction would be a concept at once incoherent and useless: the round square, for example, is a purely fictional entity whose definition includes a contradiction and which cannot be used to elucidate anything interesting in mathematics.
Infinitesimal magnitudes, although they may ultimately be impossible or even unimaginable, are nevertheless wellfounded fictions precisely because the realm of mathematical objects is structured just as it would be if [End Page 35] the infinitesimal existed. ^{25} The calculus delivers true and important results, and these illuminating results are just what one would expect if there really were infinitely small quantities. As it happens, there are none, but a kind of mathematical “preestablished harmony” guarantees that in using them we will never be led astray.
Leibniz’s famous “law of continuity” is another example of a useful fiction, 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 one another, but the inclination is infinitely small; and equal quantities are unequal, but the inequality is infinitely little. Despite its apparent absurdity, the law is acceptable because it can be used to establish truths about real motions, angles, or other quantities. It is therefore grounded in the nature of such real things and its use is both to facilitate discovery of new truths and to abbreviate otherwise laborious reasoning.
We can thus see Leibniz operating with a conception of metaphysics that grants a fair degree of autonomy to the individual sciences. Physics should use the principles most convenient for the physicist’s purposes, even if the true metaphysical account requires that they be granted merely fictional status. Mathematics is also free from the burden of satisfying all the constraints of metaphysical rigor, since “it is not necessary to make mathematical analysis depend on metaphysical controversies” (Leibniz to Varignon, 2 February 1702; GM IV, p. 91). This does not imply that metaphysics has no relevance to physics or mathematics. Leibniz plainly thought that serious metaphysical errors (such as Descartes’s mistaken notion that the essence of body is extension) can lead to scientific errors. Nevertheless, his philosophy holds that the truths of metaphysics guarantee the rationality of the world, and it is this rationality which—perhaps paradoxically—makes it possible to disregard strict metaphysical truth for the sake of an interesting mathematical story.
Douglas Jesseph is associate professor of philosophy at North Carolina State University. His current research interests are in the history and philosophy of mathematics. Among his publications are Berkeley’s Philosophy of Mathematics (1993) and Squaring the Circle: The Mathematical War between Hobbes and Wallis (forthcoming). He is also currently editing Hobbes’s mathematical works for the Clarendon Edition of the Works of Thomas Hobbes.
Footnotes
1. There are a number of other authors who have investigated 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 in his essay Historia et origo calculus differentialis (GM V, pp. 392–410). This piece is not without its difficulties, as it was written during the height of the priority dispute with Newton and is a selfserving account designed to establish Leibniz’s claim to being the true inventor of the calculus. The best overview 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 difference sequence differ from the original by the term a_{1}, but this minor complication can be overlooked.
4. There are obvious difficulties in attempting to represent infinitely small quantities in a diagram. You should think of the material enclosed by the dotted lines as having been magnified by an “infinitary microscope” to reveal the relationships among the various infinitesimal quantities which make up the curve.
5. For the background to Leibniz’s concept of the transcendental 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, et singulare pro illis calculi genus, appeared in the Leipzig Acta Eruditorum in 1684. It is reprinted in GM V, pp. 220–26.
7. For a fuller account of Hobbes’s philosophy of mathematics and its relationship to the seventeenth century mathematical context, see Jesseph (1993a, 1993b).
8. These results and their roots in Cavalieri’s Exercitationes 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 motion lasts longer than a moment except in minds. This opens the door to the true distinction between body and mind, which no one has explained heretofore. For every body is a momentary mind, or one lacking recollection, because it does not 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 the perception of its own actions and passions; it lacks thought” (GP IV, p. 230).
10. See Hoffman (1974) for a treatment of Leibniz’s early mathematical development. See also Knobloch’s editorial comments in Leibniz (1993) for more on the background to the development of the calculus.
11. One might ask why, if he owes such a debt to Hobbes, Leibniz never mentions him explicitly in his accounts of the origin of the calculus. Two plausible reasons would mitigate against such an acknowledgement: first, Hobbes’s mathematical reputation had been completely obliterated in the course of his mathematical controversy with Wallis; second, Hobbes’s bitter public controversies with Wallis and Boyle had made him anathema to the British scientific establishment by 1670. Although he likely found much in Hobbes that could be called influential or even inspiring, Leibniz was certainly shrewd enough to forego any public acknowledgement of an intellectual debt to a figure as unpopular as Hobbes.
12. There was one early (and inept) criticism of the calculus by the German Detleff Clüver, but it is of no interest to the present investigation. For a study of this critique see Mancosu and Vailati (1990).
13. For more complete accounts of Nieuwentijt’s attack on the calculus, see Mancosu (1996, Chapter 6), Petry (1986), Vermij (1989) and Vermeulen (1986).
14. For an account of Barrow’s mathematics and its role in the development of the calculus see Feingold (1993) and Mahoney (1990).
15. See Blay (1986), Mancosu (1989), and Mancosu (1996, Chapter 6) for more extended studies of this controversy.
16. See Knobloch (1990) for an account of Leibniz’s conception of the infinite which pays particular attention to his distinction between the infinitely small and the indefinitely small.
17. See Hoffman (1973) for another study of the LeibnizWallis correspondence, concentrating more on the priority dispute between Newton and Leibniz.
18. See Maierù (1984) and (1990) for a more detailed account of these controversies.
19. Wallis gave a fuller account of this part of his doctrine in the Defense of the Treatise of the Angle of Contact (1684). In particular, the sixth chapter of the Defense (entitled “Inceptives of Magnitude”) declared that “There are some things, which tho, as to some kind of Magnitude, they are nothing; 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 to which they are in such possibility” (Wallis 1684, p. 96).
20. This is brought out nicely in a very late letter from Leibniz to M. Dangicourt, where Leibniz remarks that “[w]hen our friends were disputing in France with the Abbé Gallois, father Gouye and others, I told them that I did not believe at all that there were actually infinite or actually infinitesimal quantities; the latter, like the imaginary roots of algebra ( ) were only fictions, which however could be used for the sake of brevity or in order to speak universally... But as the Marquis de l’Hôpital thought that by this I should betray the cause, they asked me to say nothing about it, except what I already had said in the Leipzig Acta” (Leibniz to Dangicourt, 11 September 1716; Leibniz 1768, III, pp. 500–501).
21. This should not be taken to mean that Leibniz’s reservations about the infinite only appear in the 1690s. His early essay De quadratura arithmetica circuli ellipseos et hyperbolae..., for example, treads cautiously over this terrain and uses the technique of unlimited approximations to deliver the central result (Leibniz 1993, pp. 28–33), even while noting that the rigorous 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 just exactly what to make of the infinitesimal. Pasini (1988) has drawn attention to manuscripts that show Leibniz to have engaged in a “controversy with himself” over the status of the infinitesimal in 1702.
22. See Costabel (1988) for a brief discussion of the concept of a “wellfounded fiction” in Leibniz’s philosophy.
23. Ross (1990, p. 133) makes this point in a manner that fits well with my account of these issues when he observes that “[t]here is in fact an ambiguity in the notion of reality as Leibniz applies it to mathematical concepts. In one sense, even straightforward geometrical concepts, such as the concept of a perfect circle, are “unreal”, since there are no realities exactly corresponding to them. They are entia rationis, or “mental entities”, or “incomplete things”. In another sense, all logically coherent mathematical concepts are “real”, as contrasted with “imaginary” ones, which contain a contradiction, and therefore cannot properly be concepts at all. Of these last, some are useless, like the notion of the highest number; whereas others, such as the notion of the square root of minus one, 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 of the complexities of Leibniz’s physics and its relationship to the metaphysics of the late period. The metaphysical picture of the Monadology, where only mindlike simple substances are ultimately real, is difficult to fit together with Leibniz’s pronouncements on the nature of force and motion. As Garber notes, “it is not clear exactly how the world of the dynamics, primitive and derivative, active and passive forces is supposed to fit into Leibniz’s larger metaphysical picture. But then, what uncertainty there is derives from Leibniz’s own uncertainties about the details of that metaphysics, as it evolved from the 1680s to the end of his life” (Garber 1995, p. 298). The significant point for my purposes is that Leibnizian metaphysics dictates that not everything presupposed in the physicist’s account of nature is ultimately real.
25. As Herbert Breger has argued, the reliability of the infinitesimal is connected with Leibniz’s conception of the continuum. In particular, it is because the continuum is not composed of points that we can only remove subintervals from any continuous magnitude, and this view of continuity makes the infinitesimal appear as a quite natural fiction. See Breger (1990a) for a more detailed account of these matters.