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 co-inventor, 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 well-grounded
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 "non-quanta." 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_{n-1}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 co-ordinates
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

y = ax^{3 }+ bx^{2} + cx + d
[1]

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

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

dy/dx = 3ax^{2}
+ 2bx + c
[6]

The formula in [6] (known as the "derivative" of [1]) gives
the slope of the tangent at any point on the curve in equation [1].
The algorithmic character of this procedure is especially important, for
it makes the calculus applicable to a vast array of curves whose study had
previously been undertaken in a piecemeal fashion, without an underlying
unity of approach. In this example we have been concerned with a very
simple third degree equation, but the basic concepts can be extended to
much more complex cases involving fractional or irrational powers and
exponents, as well as more difficult "transcendental" functions.
^{5}
This is precisely the aspect of the calculus that Leibniz trumpeted in
1684 with his first publication on the subject, whose title promises a
"new method for maxima and
[End Page 10]
minima as well as tangents, which is not
impeded by fractional or irrational quantities; and a remarkable type
of calculus for them."
^{6}

There is another important concept in the Leibnizian calculus that
needs to be mentioned, namely that of higher-order 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 second-order
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
ever-higher orders. Higher-order differences are employed to consider
the behavior of curves that themselves are derived from the taking of
a first-order derivative, and this process can extend to a wide range
of important 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-order difference as the first-order 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 [1839-45] 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 [1839-45] 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 [1839-45] 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 _{} 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 x-axis and the distance moved by the
y-axis. Then (assuming time to flow uniformly), the instantaneous impetus
will be the ratio between the instantaneous increment along the y-axis
to the increment along the x-axis. 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"
( [1839-45] 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 mid-1690s
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
ill-founded 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 1694 Considerationes 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 higher-order 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 zero-magnitudes 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 higher-order
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 first-order
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 mid-1690s.

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 _{} 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 self-evident
. . . 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 higher-order
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 second-order infinitesimals. These must be infinitely less
that infinitesimals of the first order, and thus require that first-order
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 Leibniz-Bernoulli
correspondence.

5. The Leibniz-Bernoulli 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 mid-1690s 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 well-grounded 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 well-grounded 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 non-substantial 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 semi-mental" (Leibniz
to Des Bosses, 11 March 1706; GP II, p. 304). As Leibniz explains,
the semi-mental 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 pre-established 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 well-founded
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 well-foundedness of such fictions invites us to contrast them will
ill-founded fictions. An ill-founded fiction in the case of physics would
be something both unintelligible and ill-suited to the task of mechanistic
physics: the void, action at a distance, or other Newtonian monstrosities
are presumably such. Similarly, an ill-founded 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 well-founded 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 "pre-established 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.

North Carolina State University

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.

Notes

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 self-serving 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 Leibniz-Wallis 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
"well-founded 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 mind-like 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
sub-intervals 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.

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