# Leibniz on the indefinite as infinite.

I

Consider the natural numbers: 1, 2, 3,. ... Unreflectively, we typically take there to be infinitely many natural numbers. If pressed, we might offer something like the following reasoning. If there were only finitely many natural numbers, then there would be a largest natural number. If, however, we denote this supposed largest natural number by the symbol `n', then n+1 is also a natural number, contradicting the fact that n is assumed to be the largest natural number. Consequently there are not finitely many, but rather infinitely many, natural numbers.

It is not my intention to assess the status of this reasoning, which in any case is only meant to describe what I take to be (in the late twentieth century) some of our informal preconceptions about the nature of the finite and the infinite. Instead, I would like to set the picture suggested above against an argument that Leibniz uses to show that the number of finite (whole) numbers cannot be infinite. The text from which this argument is drawn, "On the Secrets of the Sublime, or on the Supreme Being," dates from early 1676, a period of tremendous intellectual upheaval in the life of Leibniz. Leibniz was living in Paris and had just invented the infinitesimal calculus.(1) He was engaged in an intensive reading of the Cartesians, especially Descartes and Malebranche, and was to meet Spinoza later that year on his way back to Hannover, where, apart from extended periods of travel, he would reside for the rest of his life. Here is the argument that Leibniz gives:

If the numbers are assumed to exceed each other continuously by one, the

number of such finite numbers cannot be infinite, for in that case the

number of numbers is equal to the greatest number, which is assumed to

be finite. It has to be replied that there is no greatest number. But

even if they were to increase in some way other than by ones, yet if

they always increase by finite differences, it is necessary that the

number of all numbers always has a finite ratio to the last number;(2)

further, the last number will always be greater than the number of all

numbers. From which it follows that the number of numbers is not

infinite; neither, therefore, is the number of units.(3)

A full analysis of this proof must wait for another time. The point I want to make here is that Leibniz distinguishes between there being no greatest number and the number of finite numbers being infinite. That is, Leibniz distinguishes between the indefinite progression of finite numbers, which he here takes to be the case, and the finite numbers being infinite in number, which he here takes to be impossible. This is made particularly clear when, directly following the passage cited above, Leibniz goes on to add to the conclusion he has just reached: "Therefore there is no infinite number, or, such a number is not possible."(4)

Leibniz discusses this distinction between the indefinitely progressing and the infinite explicitly earlier in the same writing, before reaching the conclusion that an infinite number is impossible. Here, focusing on the infinitely small, Leibniz remarks:

One must see if it can be proved that there exists something infinitely

small, but not indivisible. If this exists, wonderful consequences about

the infinite would follow. Namely: if one imagines creatures of another

world, which is infinitely small, we would be infinite in comparison with

them. ... From which it is evident that the infinite is--as indeed we

commonly suppose--something other than the unlimited.(5)

Leibniz goes on to remark, too, that since "the hypothesis of infinites and of infinitely small things is admirably consistent and is successful in geometry, this also increases the probability that they really exist."(6) As this last remark indicates, the metaphysical outpourings of 1676 should in large part be understood as motivated by the antecedent success of the newly invented infinitesimal calculus, which had washed over Leibniz's metaphysics with the force of a tidal wave.(7) However, as these two passages from "On the Secrets of the Sublime" attest, the conclusions to be drawn are far from clear. Leibniz has not yet developed a consistent metaphysical position; rather we see his views in flux. At least throughout the course of these reflections, however, Leibniz retains the belief that the indefinite is to be distinguished from any candidate for an infinite number.

The view that the indefinite is distinct from the infinite was not at all uncommon in the seventeenth century. In Galileo's Dialogues Concerning Two New Sciences, in response to Simplicio's traditional Aristotelian declaration that the quantified parts of the continuum are infinitely many potentially and finitely many actually, Galileo's representative Salviati responds:

To the question which asks whether the quantified parts in the continuum

are finite or infinitely many, I shall reply exactly the opposite of what

Simplicio replied; that is, [I shall say] "neither finite nor infinite."(8)

Salviati goes on to declare "that between the finite and the infinite there is a third, or middle term; it is that of answering to every designated number." On the basis of this term between the finite and the infinite, Salviati continues,

I concede to the distinguished philosophers that the continuum contains as

many quantified parts as you please; and I grant that it contains them

actually or potentially at the pleasure and to the satisfaction of those

gentlemen.(9)

Galileo's position thus insures the indefinite division of the continuum without enlisting the traditional Aristotelian distinction between the actual and the potential, and also without committing to the number of parts being either finite or infinite.(10)

Likewise we find Descartes adopting an idea of indefinite progression which is to be identified neither with the finite nor the infinite. In a letter of June 6, 1647, Descartes responds to Chanut concerning the extent of the world:

In the first place I recollect that the Cardinal of Cusa and many other

Doctors have supposed the world to be infinite without ever being censured

by the Church; on the contrary, to represent God's works as very great is

thought to be a way of doing him honour. And my opinion is not so difficult

to accept as theirs, because I do not say that the world is infinite, but

only that it is indefinite. There is quite a notable difference between

the two: for we cannot say that something is infinite without a reason to

prove this such as we can give only in the case of God; but we can say that

a thing is indefinite simply if we have no reason which proves that it has

bounds. ... Having then no argument to prove, and not even being able to

conceive, that the world has bounds, I call it indefinite. But I cannot deny

on that account that there may be some reasons which are known to God though

incomprehensible to me; that is why I do not say outright that it is

infinite.(11)

In contrast to Galileo's position, Descartes does seem to imply that there is a fact of the matter whether the world is finite or infinite, but given man's limited rational capacities the indefinite is reducible neither to the one nor the other.

As we will see, Leibniz's mature philosophical position will be, like Galileo's, that there are indefinitely many parts in the continuum. What distinguishes Leibniz's position from Galileo's, however, is that, modifying the position he took in "On the Secrets of the Sublime,"(12) Leibniz takes this to mean that there are indefinitely many whole numbers, or parts tn the continuum. That is, Leibniz takes the indefinite as infinite. My goal in this paper is to indicate some of the steps along the path Leibniz traversed in the exploration of this position, and some of the difficulties this position caused. I will begin, however, at the end, with an overview of the position Leibniz ultimately achieved.

II

In the brief chapter, "Of Infinity," in the New Essays on Human Understanding (1703-5), Theophilus, speaking for Leibniz, provides a survey of the Leibnizian infinite: "It is perfectly correct," Theophilus reports, "to say that there is an infinity of things, that is, that there are always more than one can specify."(13) Here Leibniz effects exactly that identification of the infinite and indefinite within the realm of the quantitative which Galileo explicitly denies. In Theophilus's next assertion we see what leaves Leibniz, unlike Galileo, in a position to identify the indefinite with the infinite: "But it is easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes."(14)

Accepting an infinity of things, but not infinite wholes, amounts, Leibniz goes on to say in the New Essays, to accepting a syncategorematic infinite but not a categorematic one.(15) It is infinite only in a negative sense, the original etymological sense of the word: it is unlimited, and consequently not finite. This indefinite quantitative infinite is to be contrasted with the "true infinite," which, "strictly speaking, is only in the absolute, which precedes all composition and is not formed by the addition of parts."(16) The syncategorematic infinity of things is not the true infinite, strictly speaking, and our attempts to speak of an infinite number, or an infinite line or other infinite quantity, taken as a whole, seem to result from a confusion of the two. In speaking of such infinite wholes, we would attempt incorrectly to attribute a true infinity to that which is not absolute. It is not absolute, hence not truly infinite, because it has parts, which are limitations. The absolute, however, is perfect in its lack of limitation. In a passage from Leibniz's correspondence with the Jesuit theologian and professor of mathematics Bartholomew des Bosses, in addition to distinguishing between syncategorematic and categorematic infinites (the latter again taken to be impossible), Leibniz adds that God as absolute is hypercategorematically infinite.(17) To take an infinite number as a whole is to attempt to treat a syncategorematic infinity in a way which is only appropriate to the hypercategorematically infinite.

It becomes clear, indeed, that to take the infinite plurality of things as a whole is, for Leibniz, to reverse the true order of precedence: the syncategorematic notion of infinity in fact derives from the hypercategorematic sense. This is made clear when Leibniz derives the thought of the syncategorematic infinite from the thought of likeness. Beginning with a straight line, then doubling it, "it is clear that the second line, being perfectly similar to the first, can be doubled in its turn to yield a third line which is also similar to the preceding ones. ..."(18) Because the similitude is perfect, that is, unlimited, it is impossible that the process can ever be hindered: "the same principle is always applicable. ..."(19) Leibniz goes on explicitly to see the thought of the infinite deriving from the thought of likeness, thereby sharing a common origin with universal necessary truths, which for Leibniz are themselves grounded in the thought of likeness. He then indicates that our error in considering an infinite quantity as a whole would be to take something absolute and attribute it to something limited: "The idea of the absolute, with reference to space, is just the idea of the immensity of God and thus [my emphasis] of other things. But it would be a mistake to try to suppose an absolute space which is an infinite whole made up of parts."(20) The implied criticism of Newton's description of space as the sensorium Dei here is unmistakable; in this context it is clear that Leibniz's opposition to the notion of space as absolute devolves from its incompatibility with Leibniz's most fundamental metaphysical commitment to God as absolute. "The idea of the absolute is internal to us, as is that of being: these absolutes are nothing but the attributes of God; and they may be said to be as much the source of ideas as God himself is the principle of beings."(21) The absolute attributes of God are that from which our ideas derive as God is that from which beings derive.

My concern in this essay is not, however, with Leibniz's notion of the absolute as infinite, but with the question of what room is left in Leibniz's metaphysics for the distinct idea of an indefinite, or syncategorematic, infinite.(22) In this regard we may begin by recognizing an issue which Leibniz has not addressed in Theophilus's report. If that which is absolute is understood to be that which is perfect, hence without the limitation of parts, it is clear that a whole made of parts cannot be absolute. This does not yet explain, however, why there cannot be an infinite whole made up of parts, unless we already have either an argument to show that an infinite whole must be absolute or else some direct proof that such an infinite whole made up of parts cannot exist. Even though Leibniz tells us, as mentioned above, that it is easy to demonstrate that there is no infinite number, and so forth, he does not do this here. Leibniz's commitment to a hypercategorematic infinite does not by itself explain the exclusion of a categorematic infinite. In order to understand what it means to be left with only an indefinite infinite in the realm of quantity, it would help to begin by considering Leibniz's proof of the impossibility of a categorematic infinite. Leibniz does not give us, however, such an impossibility proof in the New Essays; I would like to turn now to those contexts in which such a proof is supplied.

III

Leibniz originally arrived at the position that there is no largest number, or number of all numbers, in specific opposition to the position of Galileo, during an intensive reading of the Two New Sciences in 1672 or 1673. As we will see, this position is closely related to his declaration in the New Essays that there are no infinite wholes. In the dialogue of the first day of the Two New Sciences, Galileo recognizes that the infinite number of all (whole) numbers is equal in size to a proper subcollection of itself, namely the subcollection of all square numbers. Since the same holds for all other powers as well (cube numbers, and so forth), Galileo concludes that the infinite number must therefore be whatever number contains all its powers within itself. The only such number is one, or unity, from which indeed all other number is generated, and so Galileo takes the infinite number of all numbers to be one.(23)

Leibniz first responds in detail to Galileo's identification of infinity with the number one in the Accessio ad Arithmeticam Infinitorum of 1673.(24) By driving Galileo's point even further Leibniz notices a host of properties of the infinite whole which prevent it from being identified with unity. For example, Leibniz notices, the number of all numbers must also be equal in size to the number of all even numbers, or all multiples of three, and so forth. These properties are not shared by the number one, which, although it contains all its powers, does not contain all its multiples. The only number which satisfies all these requisites is not the number one, but, according to Leibniz, the number zero. Given Leibniz's understanding of the number zero, this means that the infinite is no number at all: "thus the infinite is impossible, not one, not the whole, but nothing. Thus the infinite number = 0."(25)

Leibniz interpreted his impossibility proof as a consequence of the universal axiom that the whole is greater than the part. He regarded this axiom as itself demonstrable in a syllogism taking as, its major a definition (of greater) and as its minor an identical proposition (a part is equal to a part of the whole); the syllogistic derivation of this axiom occurs already in 1671, prior to Leibniz's departure for Paris.(26) It is here, specifically, that Leibniz's disagreement with Galileo rests. For Galileo, "the attributes of greater, lesser, and equal do not suit infinities, of which it cannot be said that one is greater, or less than, or equal to, another."(27) Yet while Leibniz draws the conclusion that an infinite number of all numbers is absurd since its existence would contradict a universal axiom, Galileo concludes instead that such an axiom does not pertain universally.

In other contexts, Leibniz gives other arguments to show the absurdity of infinite wholes; we have seen one such argument at the beginning of this paper.(28) Yet the demonstration given above seems to continue to be the predominant argument behind Leibniz's assertion that an infinite whole such as the number of all numbers is impossible. One indication of this predominance is that it is this argument which Leibniz sketches in the context of one of his most intensive debates regarding the nature of the infinite in his correspondence with his mathematical protege, Johann Bernoulli.(29) This debate focuses on the status of infinitesimals, but the debate itself is delimited by the impossibility of a maximal or minimal quantity. In a letter from August/September 1698, Leibniz reports to Bernoulli his opposition to the opinions of de Volder and Gregory of St. Vincent that the axiom that the whole is greater than the part does not hold for the infinite.(30) Instead, Leibniz continues:

it appears to me that we must say either that the infinite is not truly

one whole, or else that if the infinite is a whole, and yet is not greater

than its part, then it is something absurd. Indeed I demonstrated many

years ago that the number of the multitude of all numbers implies a

contradiction if it is taken together whole. The same [holds] for a maximum

number and a minimum number, or fraction smaller than all others. This must

also be said about a fastest motion and all similar things.(31)

Leibniz goes on to confess that this does not rule out, however, the possibility of infinitesimals and infinitely large things, "since a maximum is different from the infinite and a minimum from the infinitely small."(32) He reports to Bernoulli that he cannot determine whether such infinite and infinitely small magnitudes are possible, and so he will "allow the matter to remain in the middle."(33) He does assert, however, that if these magnitudes can be demonstrated to be possible, then their existence will follow.(34)

Bernoulli believes that Leibniz is committed to the existence of infinitesimals because Leibniz asserts that the continuum is actually divided into infinitely many parts. He offers Leibniz the following argument in support of his contention:

Consider any determinate magnitude divided into parts according to this

geometric progression: 1/2+1/4+1/8+1/16+ and so forth. As long as the

number of terms is finite, I confess the singular terms will be finite; but

if all the terms actually exist there will surely be infinitesimals and all

of the following infinitely small magnitudes.(35)

Bernoulli goes on to point out that in bodies all such divisions are (according to Leibniz) actual, so that, Bernoulli argues, infinitesimals, that is, infinitely small magnitudes, would be necessary.

Like the position Leibniz took in 1676, Bernoulli's position is apparently that there are not an infinite number of finite terms in the series 1/2+1/4 +1/8+. ... He reaches this conclusion on the basis of an argument (from which the force of the "surely" in the passage quoted above derives) which runs as follows: suppose that a finite portion of matter is already actually divided into an infinite number of parts and yet none are forced to be infinite (by which Bernoulli presumably means "infinitely small;" this is at any rate how Leibniz understands him). Then the single parts are finite, and "if the single parts are finite, then all of them together make up an infinite magnitude, contrary to hypothesis."(36)

Despite Bernoulli being an accomplished practitioner of the infinitesimal calculus, it may perhaps be tempting to dismiss this as a quaint fallacy generated by an archaic understanding of the nature of the mathematical infinite. I believe this would be a mistake. Bernoulli's position was expressive of a pervasive concern in early modern discussions of the infinite; an almost identical proof occurs in Galileo's Two New Sciences.(37)

Leibniz responds to Bernoulli's example by admitting that if there were a finite term no smaller than every term in the infinite series then the sum would indeed be infinite,(38) but Leibniz does not grant Bernoulli's conclusion in general. Leibniz replies that, even if we take all the terms in the progression 1/2, 1/4, 1/8, ... actually to exist, "I do not hold anything to follow hence except that there are actually given finite assignable fractions of any given smallness."(39) Here again Leibniz's recourse is to a notion of the indefinite infinite not conflicting with the infinitude of parts actually being given.

Since Bernoulli goes on in a later letter (December 6, 1698) to say that between the finite and the infinite no third term is given,(40) it would seem that he is committed to saying that the number of finite terms in the series 1/2, 1/4, 1/8, ... is finite. In the same letter Bernoulli asserts that either all the terms in the series are not actually given, and then only finitely many are given and more could be given, or else all the terms are actually given, and there is an infinite number of them, hence infinitesimals. In the former case it is clear that the number of finite terms is finite; but in the latter case as well, it seems, the number of finite terms must be finite. For otherwise, according to Bernoulli, the series would then sum to an infinite magnitude, which is contrary to hypothesis.(41) What, then, in this latter case, could this number of finite terms be? Is it some determinate finite number, or is it some (indeterminate) finite number larger than any finite number we can supply?(42)

In the correspondence that follows, Bernoulli's opposition to Leibniz's concurrent understanding of infinity as indefinite and the infinitude of (physical) parts as actually given continues unabated. After reiterating his opposition to this combination of positions in his next letter and offering some objections more specifically directed to the physical (or as Leibniz would say, dynamical) end of Leibniz's position,(43) and receiving a reply in which Leibniz lays out his position systematically but with little supporting argument, Bernoulli replies to Leibniz that Leibniz's responses "are much too laconic, and are definitions rather than explications."(44) Leibniz's response to Bernoulli's charge in his subsequent reply is most telling: "But would that definitions might always be given, for they virtually contain the explanations."(45) So it is for Leibniz in general, and with regard to the infinite beginning already with the writings of 1672-3: the nature of the infinite is delimited by the proof of the impossibility of a largest magnitude, and this proof is but an application of the universal axiom that the whole is greater than the part.(46)

Bernoulli agrees with Leibniz that such a number of the multitude of all numbers is impossible, and yet he nonetheless believes that if Leibniz is committed to an actual division of magnitude into an infinity of parts then Leibniz must be committed to the existence of infinitesimals. Bernoulli makes one final, and his most direct, attempt to convince Leibniz that an actual infinity of terms implies the existence of infinitesimals:

If there are ten terms there certainly exists a tenth; if there are a

hundred terms there certainly exists a hundredth, if there are a

thousand terms there certainly exists at least a thousandth; thus if

there are an infinite number of terms there exists an infinitesimal.(47)

If we are allowed to reason by analogy from the finite to the infinite, the essential point of the argument must be granted. Of course, this analogical reasoning is precisely what Leibniz strictures in his reply. In response he points out that if such argumentation were allowed, we could as easily argue from the existence of a last term in a series of ten to the conclusion that "among all numbers there is a last, which is also the greatest of all numbers."(48) Yet from Bernoulli's perspective Leibniz's stricture against such analogical reasoning must appear arbitrary. If Leibniz admits an actual infinite division of matter, then he admits an infinite number, and why should this number be treated any differently than other numbers? Although Bernoulli does not make his objection so explicitly, I take this to be the spirit of his (final) response to Leibniz in his letter of February 11, 1699: "I am astonished that you refuse to admit an infinitely small magnitude when you are forced, however, to admit an infinite number, which indeed I recall you deny elsewhere."(49) At this point Bernoulli's position, if it is indeed anything more than an expression of frustration,(50) seems to be simply that Leibniz's commitment to an actual infinity of parts is in conflict with his impossibility proof.

For Bernoulli, there is no room to admit Leibniz's indefinite infinite, which, from Bernoulli's perspective, has the status of an impossible "third thing" between the finite and the infinite. On the other hand, since Leibniz rejects Galileo's conception of the infinite as a largest or absolute magnitude, nothing blocks him, as it does Galileo, from recognizing the indefinite as infinite; Leibniz thus avoids the need to posit an intermediate between the finite and the infinite.(51) Yet neither does the gap left by Leibniz's proof of the impossibility of an absolute magnitude in any positive way sanction his declaration that the indefinite is infinite.

IV

As we have seen, in the correspondence with Bernoulli, there is a potentially competing candidate for the role played by the indefinite infinite in both the realms of the infinitely large and small: although he cannot demonstrate them Leibniz also cannot find a way to rule out the existence of infinitesimal (but not minimal) and infinitely large (but not maximal) magnitudes. If such were to exist, we may ask, what would the status of Leibniz's indefinite infinite be? Would it be supplanted by these true infinitesimal and infinite magnitudes?

These questions point in the direction of the final issue I would like to consider in this essay. By 1703-5, the period during which Leibniz wrote the New Essays, Leibniz has moved to the position that any infinite or infinitesimal quantities are impossible. In the New Essays he says, for example:

But it would be a mistake to try to suppose an absolute space which is an

infinite whole made up of parts. There is no such thing: it is a notion

which implies a contradiction; and these infinite wholes, and their

opposites, the infinitesimals, have no place except in geometrical

calculations, just like the use of imaginary roots in algebra.(52)

Here Leibniz takes the opposite of infinite wholes to be infinitesimals and declares that neither exist.(53) What motivates Leibniz to change his position in the period between 1699 and 1705 is, I believe, his continuing engagement in debates concerning the status of infinitesimals. Yet what most fundamentally underlay these changes is, I believe, Leibniz's deepening sense of his commitment to the indefinite as infinite. In order to point out some of these developments I would like to turn to one of the central documents regarding the status of infinitesimals which Leibniz composed during this intervening period.

In the "Letter to Varignon, with a Note on the `Justification of the Infinitesimal Calculus by That of Ordinary Algebra'" of 1702,(54) Leibniz's position is that we need not be committed to infinitesimals in any rigorous metaphysical sense. In this letter, Leibniz pleads 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 in contrast to our ordinary ones. ..."(55) Here Leibniz's focus is on the lack of a logical implication from the lack of a maximal or minimal finite whole to an infinitely large or small quantity--even as he fails here to rule such an implication out. It is precisely the unnecessariness of such an implication which permits the free conduct of mathematics without a prerequisite engagement in metaphysical disputes concerning the existence of the infinite. Not only do we not need to have a proof that infinite wholes are contradictory in order to make mathematical use of the infinitesimal calculus, equally we do not need to decide whether infinitely large and small quantities exist or not.(56) Here Leibniz is most interested in gaining support for the weaker position that such infinite wholes are unnecessary (as are infinitely large and small magnitudes) for the proper conduct of mathematics. On this weaker position, neither infinitely large nor infinitely small magnitudes are mathematically or metaphysically requisite for dealing with infinite pluralities.(57)

In the letter to Varignon and the accompanying note, Leibniz goes on to present in considerable detail the "mechanics" of working with the infinite without appealing to infinite magnitudes. Yet in doing so, he indicates just how weak a commitment to the infinite is required to work in the mathematical realm, even in the case of the infinitesimal calculus. What is perhaps most interesting in Leibniz's strategy here is that instead of relying on quantities as large or small as we wish, that is, quantities bearing respectively as small or large as we wish a (finite) ratio to some fixed quantity, Leibniz instead opts to consider quantities which are taken to have no finite relation: we take them, instead, to be "incomparably greater or smaller than ours."(58) This is effectively to say that the quantities involved are not in any finite relation to each other, but stops short of saying that they are in an infinite ratio to each other in any positive sense. Leibniz is working with quantities (although it is necessary to ask exactly what it means to speak of them as quantities) which are stipulated not to have relations of proportion with the quantities that we usually employ.(59) These quantities, then, are representatives of the fact that the failure of a finite ratio to obtain need not imply that an infinite ratio positively does obtain; logically they stand, so to speak, half way between the one and the other. Further, they are representatives of the fact that no such infinite ratio is needed to conduct the infinitesimal calculus: there are, or at least need be, no infinite or infinitesimal ratios in the infinitesimal calculus.

Leibniz goes on to show in the "Note" that there are always finite quantities which stand in the same proportions to each other as those incomparable quantities stand in relation among themselves, so that the finite quantities may, so to speak, serve as proxies for the incomparable ones. Thus Leibniz's recourse to these incomparable quantities is exclusively symbolic: they are mathematical symbols whose true interpretation can only be given in terms of their replacement,(60) in any given context, by other, relatively more freestanding symbols. The infinitesimal calculus, as advertised in the title to the note, is justified by "ordinary algebra." Beyond the use of certain auxiliary symbols, there appears to be no commitment to the infinite in the infinitesimal calculus. This makes the title of Leibniz's proposed, but never drafted, summa of the infinitesimal calculus, De Scientia Infinitii, ring with a certain irony.(61) The infinitesimal calculus is as much, or as little, about infinitesimals as the business of a variety shop is to sell varieties. Consequently Leibniz will later declare that infinitesimals and infinitely large magnitudes are "convenient fictions."(62) This position dovetails with Leibniz's general strategy of separating the operation of domains (mathematics, dynamics, and to a lesser extent logic) from their metaphysical foundations. What we see here is an instance of Leibniz's commitment to the infinite in the realm of mathematical and physical quantity in the weakest sufficient sense required, and it is in this light that Leibniz's commitment to the indefinite infinite should also be understood. This commitment is in fact so weak that we may wish to understand it as no commitment to the infinite at all; at any rate, this is at least how one of Leibniz's contemporaries understood Leibniz's position.(63)

To be sure, Leibniz is committed to infinite pluralities, such as the terms of an infinite series.(64) Such infinities are indefinite or, to fall back on the Scholastic terminology to which Leibniz has recourse, syncategorematic. So if we ask how many terms there are in an infinite series, the answer is not: an infinite number (if we take this either to mean a magnitude which is infinitely larger than a finite magnitude or a largest magnitude) but rather: more than any given finite magnitude. The force of this maneuver is to restrict the infinite, in the realm of the quantitative and consequently limited, to an adjectival role, and this will be precisely Leibniz's point in the chapter on infinity in the New Essays: "The idea of the absolute, with reference to space, is just the idea of the immensity of God and thus of other things."(65) The immensity of God can be referred to space; we thereby acquire the idea of space as infinite, in the sense of being unending. Infinite space is unlimited space; on the other hand, to elevate the unlimited directly from an adjectival to a substantival level would be to have an infinitely large magnitude, if indeed this were to be possible.

Not only the passages from the New Essays cited previously, but the tenor of the entire discussion there indicates that for Leibniz it is in fact not possible. In the letter to Varignon Leibniz is not yet so definitive; nonetheless, there are two regards in which the discussion in the letter to Varignon presages that given in the New Essays. First, there is a revealing parenthetical remark which Leibniz makes in his draft of the letter, but which was removed before the letter was sent. In the letter as sent Leibniz remarks 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 in contrast to our ordinary lines,

or as a result that there are lines infinitely greater than our ordinary

ones.(66)

So ends the sentence in the letter as sent and as published. In Leibniz's copy, however, there is added parenthetically:

(yet with ends; this is important inasmuch as it has seemed to me that

the infinite, taken in a rigorous sense, must have its source in the

unterminated; otherwise I see no way of finding an adequate around for

distinguishing it from the finite).(67)

On the basis of this parenthetical declaration it becomes much easier to see why Leibniz was already pessimistic about the existence of infinitesimals in the correspondence with Bernoulli,(68) and why he would shortly rule them impossible. It is because the only source we can locate for the infinite in the realm of quantity is the unlimited, which in the quantitative realm is precisely the indefinite. Yet clearly, once the infinite and the indefinite are determined to coincide, it is a short step (if any) to the conclusion that infinitesimals are impossible. To be sure, Leibniz does not offer any proof here that the only source of the quantitative infinite is in the unterminated, but the passage makes it clear that he is strongly inclined in that direction.

There is, I believe, a second indication in the letter to Varignon that Leibniz is moving in the direction of denying infinitesimals outright as well. Toward the end of the letter Leibniz describes the relation between the operation of the finite and the infinite in a way which bears an interesting resemblance to Leibniz's many descriptions of the preestablished harmony:

Yet one can say in general that though 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 the

abstract and that the rules of the finite are found to succeed in the

infinite--as if there were atoms, that is, elements of an assignable size

in nature, although there are none because matter is actually divisible

without limit. And conversely the rules of the infinite apply to the

finite, 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.(69)

There is much to be said about this passage, but I will limit myself to one remark concerning the imminent denial of infinitesimals. Leibniz describes here two "counterfactuals": we have rules which allow us to treat matter as if it were extended yet indivisible and rules which allow us to treat finite extension as if it were composed of infinitely small extended metaphysical beings. In fact the only matter there is is divisibly extended and the only metaphysical beings there are are indivisibly unextended. This picture effectively leaves no ground for a third thing which could mediate metaphysically between these two: like Descartes, Leibniz takes the distinction between the extended and the unextended to be categorical and complete. We are left with nothing more than the phraseology of "as if," which is our way of understanding the preestablished harmony of the ideal and the real. My point here is not so much logical as it is "architectonic": existing infinitesimals, whether ideal or real, fit nowhere. This is the ultimate consequence of Leibniz's understanding that in the realm of quantity there are no infinite or infinitesimal wholes: the quantitative infinite is identified with the indefinite or unlimited. The consequence is that the proof which was first taken to prove the impossibility of maximal or minimal units may now be taken more broadly to show the impossibility of all infinite or infinitesimal quantitative wholes. In a letter to Varignon shortly following the one just discussed, Leibniz in fact reports that he believes he can prove "that there are not, nor could there be, any infinitely small things."(70)

As Leibniz declared in a letter to the Princess Electress Sophie in 1696, "My fundamental meditations circle around two things, namely unity and infinity."(71) Leibniz ultimately accepted that in the realm of quantity infinity could in no way be construed as a unified whole. As we have seen, the cleft which this impossibility established between these two most fundamental of Leibnizian concerns left at least one of them with a particularly delicate, if not indeed problematic, status.(72)

Correspondence to: Department of Philosophy, University of Georgia, 107 Peabody Hall, Athens, GA 30602-1627.

(1) The term "infinitesimal calculus" is that most commonly used by Leibniz to refer to his invention, but it is unfortunately misleading, since the fundamental object of the Leibnizian calculus is not the infinitesimal but rather the differential. See the excellent presentation in H. J. M. Bos, "Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus," Archive for History of Exact Sciences 14 (1974): 1-90, especially 16.

(2) Here Leibniz later added a note: "Rather (N.B.) one proves by this only that a series is endless."

(3) G. W. Leibniz, De Summa Rerum: Metaphysical Papers, 1675-1676, trans. G. H. R. Parkinson (New Haven: Yale University Press, 1992), 31-3.

(4) Leibniz, De Summa Rerum, 33. The end of this reading ("possible") is conjectural, but in any case the point is clear from the first clause of the sentence.

(5) Leibniz, De Summa Rerum, 31.

(6) Ibid.

(7) Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments ..."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another mayor motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premieres Animadversions sur les `Principes' de Descartes," reprinted in his Etudes leibniziennes (Paris: Editions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.

(8) Galileo Galilei, Dialogue Concerning Two New Sciences, trans. Stillman Drake (Madison: University of Wisconsin Press, 1974), 43. I make use of this translation despite the greater availability and felicity of the translation of Henry Crew and Alfonso de Salvio (New York: Dover, 1954) due to the greater sensitivity of the former translation to nuances of Galileo's linguistic usage. For ease of reference, however, I will refer to the Galileo Opera numbers which are given in both translations; for example, the above reference will now be given as Galileo, Opera, 81.

(9) Ibid.

(10) In his article, "Galileo's Theory of Indivisibles: Revolution or Compromise," Journal of the History of Ideas 37 (1976), 571-88, A. Mark Smith asserts that Galileo's position regarding the continuum represents a modified Aristotelianism. In a critical, but unfortunately obscure, passage Smith seems to interpret Galileo's declaration that the number of parts in the continuum is indefinite as indicating a potentially indefinite division of the continuum, concluding that "an indefinite `intermediate term' (every assigned number), neither finite nor infinite, must serve to describe the Aristotelian continuum of spatial and temporal processes" (577). Here Smith omits, however, Salviati's explicit declaration that the parts may be taken to be either actual or potential; so far as I can see this passage greatly vitiates his interpretation.

(11) Rene Descartes, Correspondence, in The Philosophical Writings of Descartes, 3 vole., trans. John Cottingham, Robert Stoothoff, Dugald Murdoch, and Anthony Kenny (Cambridge: Cambridge University Press, 1991), 3:319-20. The original letter may be found in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, new presentation in 12 vols. (Paris: J. Vrin, 1964-76), 5:51-2. This passage is briefly discussed with reference to the doctrine of Nicholas of Cusa in Alexandre Koyre, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins University Press, 1957), 6. Descartes's conception of the indefinite is discussed at length in the fifth chapter of Koyre's book, "Indefinite Extension or Infinite Space," 110-24.

(12) There is, however, a subtle, but critical, point to be made here. In "On the Secrets of the Sublime," Leibniz declares that the number of finite numbers cannot be infinite; what he does not consider is the possibility of nonetheless admitting that there is an (indefinite) infinity of finite numbers. That is, although Leibniz recognizes the finite whole numbers as progressing indefinitely he does not consider understanding this indefinite progression as infinite. From the perspective of the Leibniz of the late 1690's maintaining that there is an infinity of finite numbers is weaker than maintaining that there is an infinite number; as we will see, while in the late 1690's Leibniz will accept that there is an infinity of finite numbers he will remain agnostic on whether there are infinite numbers. These issues will be considered in more detail below in the context of Leibniz's correspondence with Bernoulli.

(13) G. W. Leibniz, New Essays on Human Understanding, trans. and ed. Peter Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1981), 157.

(14) Ibid.

(15) The distinction is standard in the medieval scholastic tradition. Ockham, for example, says: "Categorematic terms have a definite and fixed signification, as for instance the word `man' (since it signifies all men) and the word `animal' (since it signifies all animals), and the word `whiteness' (since it signifies all occurrences of whiteness). Syncategorematic terms, on the other hand, as `every', `none', `some', `whole', `besides', `only', `in so far as', and the like, do not have a fixed and definite meaning, nor do they signify things distinct from the things signified by categorematic terms. Rather, just as, in the system of numbers, zero standing alone does not signify anything, but when added to another number gives it a new signification; so likewise a syncategorematic term does not signify anything, properly speaking, but when added to another term, it makes it signify something or makes it stand for some thing or things in a definite manner, or has some other function with regard to a categorematic term"; William of Ockham, Philosophical Writings: A Selection, trans. Philotheus Boehner, O.F.M., rev. Stephen Brown (Indianapolis: Hackett, 1990), 51. Thus Leibniz's indefinite infinite is syncategorematic in the sense that the term "infinite" only signifies when applied to finite numbers, that is, "more than any given finite number." Leibniz also aligns the distinction between syncategorematic and categorematic with the distinction between distributive and collective uses of the term "infinite": Leibniz's position is that there is no collective quantitative infinite. See, for example, G. W. Leibniz, Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, (Berlin, 1875-90; reprint, Hildesheim: Olms, 1961) 2:314. The attempt to use the distinction between the syncategorematic and the categorematic to resolve sophismata pertaining to the infinite is traditional as well. An excellent discussion of one such account, that of Albert of Saxony, is given in Joel Biard's article, "Albert de Saxe et les sophismes de l'infini," in Sophisms in Medieval Logic and Grammar, ed. Stephen Read (Dordrecht: Kluwer, 1993), 288-303 (hereafter cited as "Albert de Saxe"). Biard focuses on the sophisma, "Infinite sunt finite" ("Infinites are finite"), treated by Albert of Saxony but going back at least to the De solutionibus sophismatum, circa 1200 (see Biard, "Albert de Saxe," 288). Henri de Gand proves this sophism as follows: "Infinite sunt finite. Probatio: duo sunt finite, tria sunt finiea, et sic in infinitum; ergo infinite sunt finite." ("Infinites are finites. Proof: two are finite, three are finite, and thus to infinity; therefore infinites are finite"): for references, see Biard, "Albert de Saxe," 291 n. 18. A close analogue of this proof recurs in Leibniz's correspondence with Bernoulli, and will be discussed below.

(16) Leibniz, New Essays, 157.

(17) Translated in G. W. Leibniz, Philosophical Papers and Letters, ed. and trans. Leroy Loemker (Boston: Reidel, 1969), 31. The original passage may be found in Leibniz, Die Philosophische Schriften, 2:314. See also the discussion in Loemker's introduction to Philosophical Papers, 31, and also 514 n. 2 and 541 n. 21.

(18) Leibniz, New Essays, 158.

(19) Ibid.

(20) Ibid.

(21) Ibid.

(22) For a fuller discussion of Leibniz's chapter on infinity in the New Essays, and especially on Leibniz's opposition to Locke's account of the infinite, see Antonio Lamarra, "Leibniz on Locke on infinity," in L'Infinito in Leibniz: Problemi e Terminologia, ed. Antonio Lamarra (Rome: Edizioni dell'Ateneo, 1990), 173-91. For another discussion of Leibniz on the indefinite, see Hans Poser, "Die Idee des Unendlichen und die Dinge. Infinitum und immensum bei Leibniz," in L'Infinito in Leibniz, 225-33. Poser's excellent discussion is delimited, however, by his exclusion of the debates regarding infinitesimals from his purview. (23) Galileo, Opera, 78-85. It is important to note that Galileo has Salviati repeatedly deliver a series of disclaimers along with his remarks about the infinite. Salviati, for example, says he is "going to produce a fantastic idea of mine which, if it concludes nothing necessarily, will at least by its novelty occasion some wonder"; Galileo, Opera, 73. In another passage Salviati speaks of "marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the natures of these have no necessary relation between them"; Galileo, Opera, 83. See also passages at Galileo, Opera, 96 and 105.

(24) G. W. Leibniz, Samtliche Schriften und Briefe, 2d ser. (Berlin: Akademie Verlag, ongoing), 1:226.

(25) Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.

(26) Leibniz, Samtliche Schriften, 6th ser., 2:482-3. For a discussion of this proof in the Demonstratio propositionum primarum see Ezequiel de Olaso, "Scepticism and the infinite," in L'Infinito in Leibniz, 95-118, especially 107 and the following pages. See also the footnote below regarding an analogous presentation of this argument in the correspondence with Bernoulli.

(27) Galileo, Opera, 78.

(28) In those contexts in which Leibniz wishes to suppress the more subtle, and technical, issues surrounding the consideration of the infinite, he often substitutes for the proof that there is no greatest number a proof that a fastest motion is absurd. For example, in the Meditations on Knowledge, Truth, and Ideas, Leibniz proves the absurdity of a fastest motion as follows: "For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail's motion is not the fastest, contrary to the hypothesis"; G. W. Leibniz, Philosophical Essays, trans. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), 25. The original text may be found in Leibniz, Die Philosophische Schriften, 4:424. This proof is closely related to debates about what would happen were space to be limited. For a discussion in a variety of early modern contexts, see Koyre, The Infinite Universe.

(29) For an alternative account of Leibniz's debate with Bernoulli, centered on the question of the existence of infinitesimals, see George MacDonald Ross, "Are There Real Infinitesimals in Leibniz's Metaphysics?" in L'Infinito in Leibniz, 125-41.

(30) Leibniz presents Bernoulli with a proof of this axiom in a part of the correspondence earlier than that under consideration here. See Leibniz's letter of 23 August 1696 in Leibniz, Mathematische Schriften, 3:321-2. This presentation largely follows the argument given in 1671 cited above. For a logical analysis of the argument as presented in this letter to Bernoulli, see H. G. Knapp, "Some Logical Remarks on a Proof by Leibniz," Ratio 12 (1970): 125-37.

(31) Leibniz, Mathematische Schriften, 3:535. Unless otherwise noted, translations from this source are my own.

(32) Leibniz, Mathematische Schriften, 3:536.

(33) Ibid.

(34) The situation regarding the existence of infinitesimals bears considerable structural analogy to the situation concerning the ontological proof of the existence of God. Leibniz criticized the Cartesian ontological argument on the grounds that it merely proves that if God's existence is possible, then it is actual, but it does not give a proof of the possibility of God's existence. Leibniz was much preoccupied with such a proof of the possibility of God's existence during 1676, and returns to it later in writings of 1678, the mid-1680's, and 1714. The issues involved in the provision of such a proof are extremely intricate and consequently well beyond the bounds of this paper. The reader is referred to the detailed discussion of these issues in Robert Merrihew Adams's Leibniz: Determinist, Theist, Idealist (Oxford: Oxford University Press, 1994). See especially 141 and the following pages.

(35) Leibniz, Mathematische Schriften, 3:529.

(36) Leibniz, Mathematische Schriften, 3:529.

(37) This argument is in fact precisely the one that Galileo uses to motivate his position that the number of parts in the continuum is neither finite nor infinite: "... the quantified parts in the continuum, whether potentially or actually there, do not make its quantity greater or less. But it is clear that quantified parts actually contained in their whole, if they are infinitely many, make it of infinite magnitude; whence infinitely many quantified parts cannot be contained even potentially except in an infinite magnitude. Thus in the finite, infinitely many quantified parts cannot be contained either actually or potentially"; Galileo, Opera, 80-1. Galileo commits to the existence of infinite magnitudes in a way that Leibniz will not, but he cannot see any way to account for the number of parts in the continuum in terms of such an infinite magnitude. Consequently he assigns to them a magnitude intermediate between the finite and the infinite, analogous to what Leibniz refers to as the indefinite, but which I suggest might more appropriately be referred to in Galileo's case as the parafinite. Leibniz, on the one hand, cannot accept the Galilean infinite, since it fails the axiom of identity, but on the other hand understands the indefinite as infinite. Why though, we may ask, would Galileo believe that infinitely many quantified parts cannot be contained (either potentially or actually) except in an infinite magnitude? Consider, for example, the case of a line segment one unit long divided into successive parts, disjoint except for their endpoints, of successive lengths 1/2, 1/4, 1/8,. ... Is this not a perfectly good example of a finite magnitude containing an infinite number of quantified parts? The answer, presumably, is "no," because the number of parts given is, in fact, neither finite nor infinite, but indefinite.

(38) This is of course a sufficient, but not a necessary condition, as Leibniz was well aware. Already in 1673 Leibniz took the series 1/1+1/2 +1/3+1/4+ ... to diverge. For a discussion see Hofmann, Leibniz in Paris, 21.

(39) Leibniz, Mathematische Schriften, 3:536.

(40) Leibniz, Mathematische Schriften, 3:555.

(41) The other choice is to assume that Bernoulli changes his position in this particular regard. However, Bernoulli does not give any explicit indication that he concedes Leibniz's point that an infinitude of finite parts need not necessarily sum to an infinite magnitude. As I point out in a later note, there are other regards in which Bernoulli's position does seem to shift--specifically, regarding the probability that infinitesimals exist. But it would do him little good to shift in this regard, for if he were to admit that an infinite number of finite magnitudes may sum to a finite magnitude, then there would be little reason to insist that an infinity of terms being given requires that infinitesimals exist. This, of course, is precisely what Leibniz is trying to get Bernoulli to agree to. Since he does not do it, this should serve as indirect confirmation that Bernoulli would take the number of finite terms to be finite. One might also cite Bernoulli's assertion in his letter of 1698 that "if there are no infinitesimals in nature, then certainly the number of terms will be finitely many [tantum finitus] ..."; Bernoulli, in Leibniz, Mathematische Schriften, 3:555. Although this could be taken as supporting evidence, it does not tell us directly what the situation would be with the finite terms were there indeed to exist infinitesimals in nature. As such, it is not thoroughly conclusive. A more serious objection, I believe, is that it may not have occurred to Bernoulli to ask how many finite terms there are: the emphasis, after all, is on the infinitesimals filling out the actual infinity of terms. I believe this latter assertion is true; yet if Bernoulli did not consciously consider the question concerning the number of finite terms, it is nonetheless quite close to the surface of the issues being debated. I am not suggesting that it makes no difference whether Bernoulli recognized this question explicitly--in fact, I think it makes a great deal of difference, and it is probably quite significant that he does not broach this issue explicitly. But on the other hand, posing the question explicitly may help us to focus more clearly on Bernoulli's conception of the infinite, and at worst we are filling out Bernoulli's position in a way that would not have occurred to him.

(42) This latter would be close in certain regards to the Galilean notion of the indefinite, which in a note above I have suggested could appropriately be characterized as parafinite. Yet as I point out below, Bernoulli is unwilling to admit any tertium quid between the finite and the infinite. On this basis I find it unlikely that Bernoulli would be willing to countenance such an indeterminate finite number.

(43) The article by Ross, "Are there real infinitesimals?" considers these dynamical issues in detail.

(44) Bernoulli, in Leibniz, Mathematische Schriften, 3:545.

(45) Leibniz, Mathematische Schriften, 3:551.

(46) As Hofmann points out, Leibniz's correspondents failed to see the effectiveness of this axiom, and "not one of the correspondents to whom Leibniz sent his demonstration [of this axiom] approved of it; in particular Johann Bernoulli made it clear in his reply of September 22, 1696 that he considered the conclusion to be circular" (Leibniz in Paris, 14 n. 13). Since it depends on the use of this axiom, it is small wonder, then, that Leibniz's assertion that a largest magnitude is impossible is so infrequently accompanied by an explicit proof.

(47) Leibniz, Mathematische Schriften, 3:563.

(48) Leibniz, Mathematische Schriften, 3:566.

(49) Leibniz, Mathematische Schriften, 3:571.

(50) The shortness of Bernoulli's response is indeed a likely indication of frustration. In his previous letter, Leibniz defers his response regarding the argument of Bernoulli cited in the text above until a postscript, in which Leibniz begins by declaring that he "almost forgot" to respond to Bernoulli! After Bernoulli's brief response (the passage just cited in the text), Leibniz again responds to Bernoulli, but Bernoulli does not continue the discussion. Leibniz attempts a second time to rejuvenate the discussion, but without any success.

(51) In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resume of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Samtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Samtliche Schriften, 6th ser., 2:264. I discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz' Einfuhrung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.

(52) Leibniz, New Essays, 158.

(53) It might be objected that what Leibniz refers to here is not infinitely small magnitudes, but minimal magnitudes in opposition to maximal wholes. This interpretation is, I believe, extremely implausible, however, because Leibniz goes on to speak of just these infinite wholes and their infinitesimal counterparts as what do have a place in geometrical calculations sub specie imaginationis. I believe the conclusion is inescapable that Leibniz is here declaring the impossibility of the existence of infinitesimals.

(54) Leibniz, Philosophical Papers, 542-6; original text in Leibniz, Mathematische Schriften, 4:91-5 and 4:104-6.

(55) Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. (56) This strategy of arguing for a relative independence of a given area from metaphysical intrusion while maintaining nonetheless the fundamental need for a proper metaphysical foundation is a pervasive one in Leibniz's philosophy, with the emphasis on independence generally increasing during his later years. On this issue, see my 1995 University of Chicago dissertation, "Labyrinthus de Compositione Continue: The Origins of Leibniz' Solution to the Continuum Problem," (Ph.D diss., University of Chicago, 1995), 149-51

(57) Even in pieces postdating the New Essays, Leibniz often emphasizes this weaker position; but even in such contexts he demonstrates more definite commitments than he does in the passages considered above from the correspondence with Bernoulli in the late 1690's. Here I consider only one such instance. In the "Conversation of Philarete and Ariste" (1712), Philarete, speaking for Leibniz, says: "It may be said that we can conceive, for example, that every straight line can be lengthened, or that there is always a straight line greater than any given one; but however, we do not have any idea of an infinite straight line, or of one greater than all other lines that can be given"; Leibniz, Philosophical Essays, 267; original text in Leibniz, Die Philosophische Schriften, 6:592. Here, to begin with, the force of the passage depends on whether `an infinite straight line' and `one greater than all other lines' are to be equated. However, on either reading it is important to note that what Leibniz says here is simply that we do not have any such idea; this is important because the dialogue constitutes a response to the Cartesian Malebranche, and the Cartesian conception of the indefinite depends critically on the limitations of our ability to conceive when the infinite is at issue. In any case, the weakest way we can construe what Leibniz says here is that we cannot conceive of such lines. This would mean that their possibility remains unavailable to us, and so we can not conclude their existence. This is stronger than simply leaving the matter "in the middle," as Leibniz did in the correspondence with Bernoulli, because here Leibniz is committing to their inconceivability. If, however, the two phrases are to be identified, then Leibniz is effectively saying that the only infinite straight line which is conceivable is one greater than all other lines, and this would effectively conform with his position in the New Essays, where he (also) refuses to make any distinction. It is important to note that in general in the context of responding to Cartesians, Leibniz is less likely to underline the outright denial of the existence of infinitesimals for the reasons indicated above: to do so requires us to identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols `dx' and `dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch "L' infinie dans les mathematiques de Leibniz," in L'Infinito in Leibniz, 33-51.

(58) Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.

(59) On the basis of such passages John Earman proposes that Leibniz is in fact speaking of two different sorts of infinitesimals, and that his denial of one sort is in fact a "cover" for his commitment to the other. See his "Infinities, Infinitesimals, and Indivisibles: The Leibnizian Labyrinth," in Studia Leibnitiana 7 (1975): 236-51. Although I disagree with many of the details of Earman's analysis, and although I ultimately reject the distinction he attempts to make, it nonetheless seems to me that the point of his distinction is closely related to many of the issues which I am attempting to discuss in this paper.

(60) Failing, that is, a proof that infinite magnitudes exist.

(61) Pierre Costabel has tried to explain why this project was never drafted in his "De Scientia Infiniti," in Leibniz 1646-1716, aspects de l'homme et de l'oeuvre (Paris: Aubier-Montagne, 1966), 105-17. Although I would tend to agree with Costabel that the greatest barriers to this project were of a practical (and perhaps, as Costabel also suggests, psychological) nature, I would suggest that there may also have been metaphysical reasons why this project was never pursued past the preliminary stages, much less completed.

(62) For example, in a letter from 1716 to Samuel Masson: "The infinitesimal calculus is useful with respect to the application of mathematics to physics; however, that is not how I claim to account for the nature of things. For I consider infinitesimal quantities to be useful fictions"; Leibniz, Philosophical Essays, 230; original text in Leibniz, Die Philosophische Schriften, 6:629.

(63) In his Eloge de M. Leibnitz (1716, edited 1718), Fontenelle writes: "He [Leibniz] understood this infinity of orders of the infinitely small always infinitely smaller the one than the other, and that in geometrical rigor, and the greatest geometers have adopted this idea in all its rigor. It seems however that he then scared himself, and that he believed that these different orders of the infinitely small were only incomparable magnitudes due to their extreme inequality, as are a grain of sand and the globe of the earth, the earth and the sphere including the planets, and so forth. But this would only be a great inequality, but not infinite, such as one establishes in this system ..." (translation mine). I have used the citation of this passage given in Michel Blay's article, "Du fondement du calcul differential au fondement de la science du mouvement dans les <<Elemens de la geometric de l'infini>> de Fontenelle," in Studia Leibnitiana Sonderheft 17 (1989): 99-122. The quotation from Fontenelle translated above is given at 100 n. 6.

(64) In the letter to Varignon, Leibniz uses this to defend his calculus as a calculus of the infinite: "Yet we must not imagine that this explanation debases the science of the infinite and reduces it to fictions, for there always remains a `syncategorematic' infinite, as the Scholastics say"; Leibniz, Philosophical Papers, 542; original in Leibniz, Mathematische Schriften, 4:93.

(65) Leibniz, New Essays, 158.

(66) Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.

(67) Ibid.

(68) See, for example, Leibniz, Philosophical Papers, 511; original text in Leibniz, Mathematische Schriften, 3:551. Bernoulli, on the other hand, is inclined to think that infinitesimals do exist, although he agrees with Leibniz that they have not been demonstrated to exist. See his letter of August/September 1698 in Leibniz, Mathematische Schriften, 3:539. Bernoulli's declaration in this letter that it is more probable that infinitesimals exist seems to mark a shift from the position he took in his previous letter of 16/26 August 1698. There he reports astonishment at the fact that he takes Leibniz to suggest that "it [is] possible for such as act among us to be infinite and infinitely small ..."; Bernoulli, in Leibniz, Mathematische Schriften, 3:529. Here too in this earlier letter Bernoulli grants neither a proof for or against such infinite and infinitely small things, but here his attitude toward them seems, on the whole, skeptical.

(69) Leibniz, Philosophical Papers, 544; original in Leibniz, Mathematische Schriften, 4:93-4.

(70) Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, however: my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systemiques et Idealite Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, Andre Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.

(71) Leibniz, Die Philosophische Schriften, 6:542.

(72) An earlier version of this paper was presented at the Department of Philosophy, Boston University, February 14, 1997. I would like to thank the members of the audience for their many helpful questions and comments.
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