“A Complete Denial of the Continuous”? Leibniz's 1

“A COMPLETE DENIAL OF THE CONTINUOUS”? LEIBNIZ'S LAW OF

CONTINUITY “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

1. Russell’s and Cantor’s charges of inconsistency

As has often been noted, Leibniz’s philosophy was a virtual hymn to the continuous. His axiom that nullam transitionem fieri per saltum, “no transition occurs by a leap,” was central to his critique of the Cartesian natural philosophy, as he made clear in his correspondence with the

Cartesian de Volder (G II168). And in the New Essays he writes:

Nothing takes place suddenly, and it is one of my great and best confirmed axioms that

nature never makes leaps. I call this the Law of Continuity… (1704/1981, 56)

In nature everything happens by degrees, and nothing by leaps, and this rule regarding

change is part of my law of continuity. But the beauty of nature, which insists upon

perceptions which are distinct from one another, requires the appearance of leaps and

musical cadences (so to speak) amongst phenomena … (1704/1981, 473; Russell 1900,

222)

For this, however, as in many other instances, Bertrand Russell takes Leibniz to task for want of consistency, charging that “In spite of the law of continuity, Leibniz’s philosophy may be described as a complete denial of the continuous” (Russell 1900, 111). Russell has a point. He documents Leibniz’s denial that anything actual is continuous, and also his oft-repeated doctrine that “the continuum is ideal because it has indeterminate parts, whereas in the actual everything is determinate” (111). He also gives the following passage from Leibniz’s letter to de Volder in the appendix of leading passages for this section:

Matter is not continuous but discrete, and actually infinitely divided, though no

assignable part of space is void of matter. But space, like time, is something not

substantial, but ideal, and consists in possibilities, or in an order of coexistents that is in

2 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

some way possible. And thus there are no divisions in it but such as are made by the

mind, and the part is posterior to the whole. In real things, on the other hand, unities are

prior to the multiplicity, and multiplicities exist only through unities. (The same holds of

changes, which are not really continuous.) (To De Volder, 11 Oct, 1704; G II 278-79;

Russell 1900, 245; Loemker 1976, 536)

So actuals—simple substances and their changes—are not really continuous. Nor are bodies or their changes, since as composites they are simply infinite aggregates or multiplicities of monads:

In actuals there is only discrete quantity, namely the multiplicity of monads or simple

substances, although this is in fact greater than any given for any aggregate that

is sensible or corresponds to phenomena. But continuous quantity is something ideal,

which pertains to possibles and to actuals insofar as they are possible. (To De Volder, Jan

19, 1706; G II 282; Russell 1900, 245; Loemker 1976, 539)

Yet this denial of continuity for substances and their changes seems to be incompatible with what Leibniz explicitly says elsewhere:

Not only is everything that acts an individual substance, but also every individual

substance acts continuously… (On Nature Itself, FW 215)

I also take it for granted that every created thing is subject to change, and therefore the

created monad as well; and indeed that such change is continuous in every one.

(, 1714, §10, FW 269)

3 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

But continuity is found in time, extension, qualities and motion, and in fact in all natural

changes, for these never take place by leaps (Initia Rerum Mathematicarum Metaphysica,

1715; GM VII 17-29)

In what follows I hope to show that the incompatibility between these various statements is only apparent. In so doing I shall try to shed some light on Leibniz’s philosophy of the continuous. I shall not be attending to subtle changes in Leibniz’s views in his later years; in any case, I do not believe that his basic position on these matters underwent any profound change after the early

1680s. This is especially so of Leibniz’s interpretation of as fictions, which I believe he had already formulated in 1676 as a consequence of his syncategorematic interpretation of the infinite. To be sure, Leibniz did not succeed in explaining this interpretation clearly to his contemporaries, but I do not read his later tactical hesitations as evidence for changes of heart on the status of infinitesimals.1 Thus I shall not be trying to trace changes in his description of his position, nor how they were occasioned by external factors. In defence of such an unhistorical approach, I plead economy; its justification must rest with whatever sense I am adjudged to have made of Leibniz’s apparently contradictory remarks by adopting it.

As starting point, I am going to examine a similar allegation of inconsistency concerning

Leibniz’s doctrine of the actual infinite. Indeed, one can see a strong parallel between Russell’s remark already quoted above, that “In spite of the law of continuity, Leibniz’s philosophy may

1 Thus I must reluctantly dissent from the analysis given by Douglas Jesseph in his (1998), who takes it as “clear that the most natural formulation of the Leibnizian makes it straightforwardly committed to the reality of infinitesimals”, and then sees Leibniz as developing the fictional treatment of infinitesimals “through the 1690s” in response to criticisms. Of course, I agree with Jesseph that in this period “Leibniz tries to settle on an interpretation of the calculus that can preserve the power of the new method while placing it on a satisfactory foundation”; I just do not see the fictional interpretation as new. I have similar disagreements with the way the Henk Bos frames the issue in his classic article (1974-74) on the Leibnizian calculus, as I shall explain below. Contra Bos (54), Leibniz did not introduce infinitesimals as “true quantities” that we was then obliged to treat as fictions; rather, he had already developed the syncategorematic interpretation, whose great advantage was that it did not commit him to actually existing infinitesimals, and yet provided an adequate foundation for the calculus.

4 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity be described as a complete denial of the continuous”, and a similar remark of Cantor’s about

Leibniz’s philosophy of the infinite:2

Even though I have ... quoted many places in Leibniz’s works where he comes out

against infinite , ... I am still, on the other hand, in the happy position of being

able to cite pronouncements by the same thinker in which, to some extent in contradiction

with himself, he expresses himself unequivocally for the actual infinite (as distinct from

the Absolute).3

Again, there is good textual evidence for the incompatibility, as Cantor documents. First there are the ecstatic pronouncements in favour, as in Leibniz’s letter to Foucher:

I am so much for the actual infinite that instead of admitting that nature abhors it, as is

commonly said, I hold that it affects it everywhere, the better to indicate the perfections

of its Author. Thus I believe that there is no part of matter which is not, I do not say

divisible, but actually divided; and that consequently the least particle ought to be

considered as a world full of an of different creatures (Leibniz 1693, G.I.416.)

Then there are the denials that there can be such a thing as an infinite number, even juxtaposed with the embrace of the actual infinite, as in the New Essays:

It is perfectly correct to say that there is an infinity of things, i.e. that there are always

more of them than can be specified. But it is easy to demonstrate that there is no infinite

2 I should stress that Russell himself did not misinterpret Leibniz on the infinite. On the contrary, I believe he was the first to point out that there is no inconsistency between Leibniz’s upholding of the actual infinite and his denial of infinite number: “[the principle] that infinite aggregates have no number … is perhaps one of the best ways of escaping from the antinomy of infinite number” (Russell 1900, 117). He does, however, misconstrue Leibniz on infinite aggregates, and on his “deduction of monadism”: see (Arthur, 1989). 3 , “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (1883), in Cantor (1932, 179).

5 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

number, nor any infinite line or other infinite quantity, if these are taken to be genuine

wholes. (Leibniz 1704/1981, 157, Russell 1900, 244)

Cantor, of course, believed these views inconsistent. On the one hand he wished to call on

Leibniz as a historical precedent for upholding the actual infinite; on the other, Leibniz’s denial of infinite number seemed not only to contradict this, but to be belied by Cantor’s own theory of the transfinite. Others have agreed with him: Rescher says that Cantor’s theory of transfinite numbers, point- topology and measure theory “have shown that Leibniz’s method of attack was poor. Indeed, Galileo had already handled the problem more satisfactorily ...” (Rescher

1967, 111); Gregory Brown concurs (Brown 1998, 122-123; 2000, 23-24): Leibniz suffered a failure of nerve, and would have done better to have embraced infinite number and anticipated

Cantor in the process.

2. The syncategorematic infinite

Against this view, however, I have argued elsewhere (1999, 2001a, b, c) that there is no inconsistency in Leibniz’s account of the actual infinite. He holds that there is an of the parts into which matter can be divided, but that this infinite must be understood syncategorematically. On this view, any piece of matter is actually —not merely potentially— divided into further parts, but there is no totality or collection of all these parts. There are more parts than can be assigned any number finite N, even though there is no number that is greater than all N. This view is indebted to the distinction first formulated by Peter of Spain, and later elaborated by Jean Buridan, Gregory of Rimini and William of Ockham, who claimed that to assert that the continuum has infinitely many parts in a syncategorematic sense is to assert that

“there are not so many parts finite in number that there are not more” (partes non tot finitas numero quin plures, or non sunt tot quin sint plura). This is contrasted with the categorematic

6 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity sense of infinity, according to which to say that there are infinitely many parts is to say that there is a number of parts greater than any finite number, i.e. that there is an infinite number of parts.4

This distinction can be neatly expressed by means of predicate logic:

ÿ To assert an infinity of parts syncategorematically is to say that for any finite number x that

you choose to number the parts, there is a number of parts y greater than this: ("x)($y)Fx Æ

y > x, with Fx = x is finite, and x and y numbers.

ÿ But to assert their infinity categorematically would be to assert that there exists some one

number of parts y which is greater than any finite number x, i.e. that ($y)("x) Fx Æ y > x.

The first formula does not commit you to infinite number, since it merely asserts that there is a greater number y than the one you chose, and y may be finite. But the categorematic expression commits you to the existence of a number greater than all finite numbers; and this is denied by Ockham and Leibniz. Actually, whereas Ockham asserted that it is the nature of the continuum to be divided into infinitely many parts in this syncategorematic sense, Rimini, while recognizing the distinction, nevertheless denied that the continuum is so divided. Leibniz took

Ockham’s side, although there is a crucial : the “continuum” that is actually divided for Leibniz is more aptly termed a contiguum: “Matter is not continuous but discrete, and actually infinitely divided, though no assignable part of space is without matter.” It is matter, and only derivatively the continuous space in which it is imbedded, that is actually infinitely divided.

But in agreement with Ockham, Leibniz holds that there is no such number as infinity: an infinite number is not an entity, and nor is any infinite whole, just as he wrote in the passage from the

New Essays I quoted above.

4 For a brief but enlightening discussion of the categorematic/syncategorematic distinction, see A. W. Moore (1990), 51-52.

7 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

Admittedly, Cantorian prejudices are so strong that many today have trouble seeing how one could assert that there are infinitely many things without that entailing an infinite number of them. Why wouldn’t the of the parts Leibniz ascribes to his infinitely divided matter be ¿0, aleph null? Let me attempt to assuage such reluctance towards the syncategorematic conception by providing as a concrete example the case of the infinitude of the primes. This was first proved in Euclid’s Elements in one of the earliest and most beautiful examples of reasoning by reductio ad absurdum. As is well known, “Euclid” proceeds as follows.

He supposes that there is a greatest prime, i.e. that there exists some finite prime such that all primes are less than or equal to it: ($x)("y)(Fx & y £ x), where Fx = x is finite, and x and y range over prime numbers. He then constructs a number as follows: take the supposedly greatest prime, and multiply it successively by each prime smaller than it, then add 1 to the result. This is now a number which is greater than all the primes, but which is not divisible evenly by any of the primes; it is therefore itself a prime, and greater than the supposedly greatest prime, thus contradicting the starting supposition. Therefore (since this construction is completely general, and could be performed on any supposed greatest prime), it follows by reductio ad absurdum that the supposition was false, so that ~($x)("y)(Fx & y £ x). Turning the cranks of predicate logic, the latter expression is equivalent to ("x)($y)~(Fx & y £ x), and this in turn to

("x)($y)(Fx Æ y > x) —for every finite prime there is a greater prime.

But this last expression is the syncategorematic infinite, from which the categorematic infinite ($y)("x)(Fx Æ y > x) —there is a prime greater than every finite prime— does not follow. To think that by asserting the infinitude of the primes one thereby commits oneself to the categorematic infinite is to commit the Quantifier Shift Fallacy!

8 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

Of course, Cantor himself does not commit this fallacy. He would agree that there is no more a greatest prime than a greatest : w, the first infinite (transfinite ordinal) number, comes after all the natural numbers, and therefore after all the primes. Of course, he would still say that w and ¿0 (the first transfinite cardinal) are actually infinite, categorematic, numbers. My point, though, is that this is an extra assumption; it is not necessary to assume that there is an actually infinite number in order to assert an actual infinity of primes, if this infinity is understood syncategorematically.

Thus, I submit, there is no inconsistency in Leibniz’s account of the actual infinite. His account of the actually infinite division of matter is of a piece with his denial of

(categorematically) infinite number:

Created things are actually infinite. For any body whatever is actually divided into several

parts, since any body whatever is acted upon by other bodies. And any part whatever of a

body is a body by the very definition of body. So bodies are actually infinite, i.e. more

bodies can be found than there are unities in any given number. (A VI iv 1393; A235).

It is perfectly correct to say that there is an infinity of things, i.e. that there are always more

of them than can be specified. 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. (New

Essays, 1704/1981, 157, Russell 1900, 244)

As I have argued elsewhere (Arthur 1998, 2001a), there are some very interesting metaphysical consequences that Leibniz draws from these views. It is because the multiplicity of parts in a given body is “a quantity which is in fact greater than any given number for any aggregate that is sensible or corresponds to phenomena” (G. II. 282; to De Volder) that bodies, taken as wholes,

9 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity are mere phenomena. The parts are actual, the whole, being infinite, is not. This is one half of

Leibniz’s formulaic solution to the continuum problem: “In real things, unities are prior to a multiplicity, and multiplicities exist only through unities”; as contrasted with ideal things, like space, time and other continua, where “the part is posterior to the whole.” (G II 279; to De

Volder)5

In the remainder of the paper, I want to show how the second half of that solution follows.

Briefly, in the continuous, which is an ideal whole, the parts that may be conceived in it are only fictitious. That is, corresponding to his syncategorematic interpretation of the infinite, Leibniz held to a syncategorematic interpretation of the infinitesimal, which I believe

Hidé Ishiguro was the first fully to recognize and explain (Ishiguro 1990, 79-100). Leibniz’s talk of infinitesimals, she explains, “is, as he says, syncategorematic and is actually about ‘quantities that one takes … as small as is necessary in order that the error should be smaller than the given error’”.6

But before detailing the syncategorematic interpretation, I think it will be useful to go over some of Leibniz’s ideas prior to his arriving at this mature view. For earlier in his career

Leibniz’s interpretation of infinitesimals was a categorematic one: he held that they were actual constituents of the continuum. Aside from its intrinsic interest, a brief digression on these earlier views should at least absolve Leibniz of the charge that he was in any way dogmatic in his attitude to the infinite and the continuous.

5 Cf. “In actuals, single terms are prior to aggregates, in ideals the whole is prior to the part.” (G II 379; to Des Bosses) 6 Ishiguro, 1990, 90; Leibniz, to Varignon, February 2nd, 1702, GM 4 92; Loemker, 1976, 543. For a careful summary of Leibniz’s mathematical techniques see also Bertoloni Meli (1993, 68).

10 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

3. Non-Archimedean infinitesimals

According to an analysis I have given elsewhere (“From Actuals To Fictions: Four Phases in

Leibniz’s Early Thought On Infinitesimals,” to appear in Studia Leibnitiana), Leibniz gives three more or less distinct accounts of infinitesimals as actual (i. e. non-Archimedean) entities in the continuum before arriving at his syncategorematic understanding.

The first is implicit in an early theory of motion. It is perhaps best described as a metaphysical or physical speculation, and does not constitute a sophisticated mathematical theory. On this theory, the motion of a body is characterized as its being created at an assignable point at each assignable instant of its motion, but as being at rest (and therefore, he claimed, non- existent) for unassignable intervals between the assignable instants. These unassignable intervals are “times smaller than any given”, in contradistinction to point-instants, and therefore correspond to his later definition of the infinitely small as a quantity smaller than any given. So an apparently continuous motion is analysed in terms of the occupation of differing points at differing instants, with infinitesimal intervals of rest or non-existence between them.

But Leibniz soon abandoned this account of the continuum. By 1670, this theory of motion as interrupted by infinitesimal rests had given way to a neo-Hobbesian account in which a true continuity of motion is established through its composition out of conatus or endeavours. This second theory is sketched in the Fundamenta Praedemonstrabilia of the Theoria Motus Abstracti

(hereafter TMA), a monograph sent by Leibniz to the Royal Society, the Academie Française and leading mathematicians in 1671, and in related unpublished documents of the same time.7

According to this account, the continuum is composed of indivisibles, defined as parts smaller

7 Pertinent selections from the TMA may be found in translation in Leibniz (2001), Appendix 1.

11 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity than any assignable, with no gaps or intervals of rest between them.8 Indivisibles have

“indistant” parts, but no extension, and stand in the ratio of 1 to ∞ with the continuum they compose.9 Having parts, they have magnitude, so that any two indivisibles of the same order and dimension have a finite ratio.

The main argument in the TMA for the existence of these unextended indivisibles is an inverted version of Zeno’s Dichotomy argument. The argument depends on assuming (with

Zeno) that a real motion must have an intelligible beginning, but also (against Zeno) that the phenomenon of motion is real. Leibniz then takes an interval ac, and divides it into ad and dc, with d the midpoint. The subinterval ad is also bisected, and so on to infinity. Assuming the beginning of the motion lies to the left, it cannot be ac since dc can be subtracted; nor in ad, since ed can be subtracted; and so on to infinity. It therefore cannot be any extended subinterval, but must be contained in the left side of each. “Therefore the beginning of a body, space, motion, or time, (namely, a point, and endeavour, or an instant) is either nothing, which is absurd, or is unextended, which as to be demonstrated.” (A339).

⋅ ⋅ ⋅ ⋅ ⋅

a f e d c

Leibniz then proceeds to outline a theory according to which each such “beginning” of a line is proportional to the endeavour of its generating motion. Thus if we take two points p and q that are the beginnings of two different lines described in time T by the unequal uniform motions

(whose speeds are) M and N, they will be proportional to the endeavours that are the beginnings

8 “Motion is continuous, i.e. not interrupted by any little intervals of rest.” (Fundamentum 7, TMA; A340). 9 A point, indivisible, or beginning is “that which has no extension, i.e. whose parts are indistant, whose magnitude is inconsiderable, unassignable, smaller than can be expressed by a ratio to another sensible magnitude unless the

12 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity of these motions, M/• and N/•, resp. Therefore even though they are infinitely small they will be in the ratio M:N, i.e. in the same ratio as their generating motions. An infinity of points of length MT/• will compose a line of length MT, just as an infinity of endeavours M/• will compose the motion M.

The fact that indivisibles, although infinitely small and unextended, have non-zero magnitude is used by Leibniz to claim that, even though a circle may be regarded as a polygon with infinitely many indivisible sides, the circle and the polygon may differ in magnitude: “An arc smaller than any that can be given is still greater than its chord, although this is also smaller than can be expressed, i.e. consists in a point. But that being so, you will say, an infinitangular polygon will not be equal to a circle: I reply, it is not of an equal magnitude, even if it be of an equal extension: for the difference is smaller than can be expressed by any number.” (A342)

This theory has an interesting modern counterpart in John Conway’s theory of infinitesimals, defined in terms of his surreal numbers.10 Conway defines the infinitesimal e as an or kind of cut between two sets of numbers, one containing zero (the leftmost point of Leibniz’s Zenonian interval), the other containing Zeno’s dichotomy sequence written right to left, …1/8, 1/4, 1/2, 1:

e =def { 0 | …1/8, 1/4, 1/2, 1}

That is, e (also written 1/w) is a new number representing the “unassignable” gap between zero and the dichotomy sequence.

ratio is infinite.” (Fundamentum 5, TMA; 339-340). “Endeavour is to motion as a point is to space, i.e. as one to infinity, for it is the beginning and end of motion. (Fundamentum 10, TMA; A340). 10 I explore this and other parallels between Leibniz’s theories and modern theories of infinitesimals in an unpublished ms., “Leibniz’s Infinitesimals in Relation to Modern Theories”.

13 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

Thus beginning with an interval of magnitude M, the unassignable interval between 0 and

1 what is left when one has subtracted all the fractions of the original interval from the right ( /2M,

1 1 1 then /4M, then /8M, and so forth) —namely Leibniz’s “beginning”— will be /wM. Here, of course, ‘w’ is the corresponding to Cantor’s first transfinite ordinal, which is precisely the first (transfinite) number after the set of all natural numbers in order, equivalent to the number of terms in the sequence.

1 1 Now since w and /w are reciprocals in Conway’s theory, w ¥ /wM = M. This supports

Leibniz’s idea that the interval M is actually infinitely divided into an infinity of parts smaller than any assignable, each of whose magnitude is in the ratio 1:• to the magnitude M.

Despite this support from modern theory, however, there is a curious inconsistency at the heart of the TMA that commentators have not failed to point out. This is that if it is assumed that a line is composed of points—even ones, like Leibniz’s, that have magnitude but no extension—then they are susceptible to a well-known objection originating with Sextus

Empiricus: the “Diagonal Paradox”, that there will be as many in the diagonal of a square as in one of the sides. The reason this is curious is that in the TMA Leibniz presents this very objection as a refutation of Euclidean or partless points, one which he thinks his indivisibles can avoid.

By the Fall of 1673, however, Leibniz has come to see that this will apply equally to the indivisibles of his TMA, if they are assumed as fixed constituents of the continuum. But he thinks this objection can be avoided if the infinitely small elements of the continuum are instead defined only with respect to a generating motion. That is, only points generated by the same motion, the motion of the regula in Cavalieri’s scheme, may be compared. The “beginnings” or points composing a given line are generated by a given motion, as laid out in the TMA, but these

14 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity are infinitely small lines whose magnitude is proportional to the conatus of the generating motion. The saving of Cavalieri therefore requires the “points” to have an infinitely small extension, rather than to be unextended indivisibles. Each of these is equal to any other in the same line, but not necessarily to the points composing another line connecting them, which would be generated by a different motion.

Metaphysically, Leibniz interpreted this new foundation to entail that “there is no body without motion”, on the explicit grounds that “if there is some body distinct from motion, then indivisibles must be admitted. But this is absurd, and contrary to what has been demonstrated”

(A VI iii 99-100; A15). But if a body is understood as that which moves, then its beginning can be defined as an infinitely small line. He still maintained this view in the Spring of 1675, when he wrote to Malebranche: “it is necessary to maintain that the parts of the continuum exist only insofar as they are effectively determined by matter or motion” (Letter to Malebranche, March-

April 1675 (?): G I 322.

The final stage in the evolution of Leibniz’s thought is his abandonment (after much hesitation in the spring of 1676) of the idea that infinitesimals are actual parts of the continuum.

It is perhaps not a huge step from regarding each infinitesimal as relative to a particular generating motion to regarding it as not actually an element of the continuum. But as late as

February 1676, Leibniz could still write: “Since the hypothesis of infinites and the infinitely small (infinite parvorum) is splendidly consistent and successful in geometry, this also increases the likelihood that they really exist.” (A VI iii 475; A51). It is not easy to see precisely what forces his change of mind, but one candidate is a proof he devised on March 26th, 1676, to the effect that “the differential is not after all that which is infinitely small, but that which is nothing at all” (De infinite parvis, A VI iii 434; A65). Such a proof can be effected, Leibniz asserts, if a

15 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity polygon approximating a circle “can always be inflected to such an extent that, even with the difference assumed infinitely small, the error will be smaller. Granting this, it follows not only that the error is not infinitely small, but that it is nothing at all, since, of course, none can be assumed.” (ibid). Although one might wish for more perspicuous wording, Leibniz’s reasoning here seems very evocative of Newton’s in his justification of the method of first and last ratios:

Quantities, or ratios of quantities, which in a given time constantly tend to equality, and

before the end of that time approach so close to one another that their difference is less than

any given quantity, become ultimately equal. If you deny this, let their ultimate difference be

D. Then they cannot approach so close to equality that their difference is less than the given

difference D, contrary to hypothesis.11

In each case it is argued that no ultimate difference can be assumed, since a difference smaller than any given difference can always be supposed. This in turn depends on the so-called

“Archimedean Axiom” of Eudoxus: if a and b are any two segments, then a can always be added to itself so many times that na > b. This axiom guarantees that for any given quantity one can always find a smaller one. As we shall see below, Leibniz explicitly gives a foundation for both integration and differentiation based on this axiom. Thus there is a surprisingly strong agreement between Newton’s and Leibniz’s approaches, both of which are based on the

Archimedean Axiom. The difference is that from it Leibniz derives a license to use infinitesimals directly, even though they are a mere shorthand for the fact that a continuously variable quantity may always be chosen to have a value smaller than a given pre-assigned value. Leibniz gave a succinct formulation of this interpretation of infinitesimals in a letter to Des Bosses in 1706:

11 Newton (1999, 433); quoted in the version of the first edition of 1687. Ironically, when Leibniz saw Newton’s Principia for the first time he misconstrued this reductio, making the rather lame remark that “It can be doubted

16 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

Meanwhile I have shown that these expressions are of great use for the abbreviation of

thought, and thus for discovery, as they cannot lead to error. For it is sufficient to substitute

for the infinitely small a thing as small as one may wish, so that the error may be less than

any given amount; hence it follows that there can be no error. (G. II. 305; Ishiguro (1990,

85))

This is Leibniz’s syncategorematic interpretation of the infinitesimal, which I shall now discuss.

4. Syncategorematic infinitesimals

Leibniz’s attempts to provide a justification for his infinitesimals have often been taken to be essentially diplomatic. What I am arguing here (following Ishiguro’s lead) is that there is a distinct sense to Leibniz’s characterization of infinitesimals as fictions; it is not a counsel of despair, a stratagem invented to save his embarrassment at the fact that the calculus of infinitesimals had worked so well despite a supposed lack of foundation. Even Henk Bos

(1974/75, 54-56) takes Leibniz to have provided two different approaches to interpreting infinitesimals. One is finitist and Archimedean, in which differentials are interpreted as finite differences that may be taken so small as to lead to an error less than any assignable. The second is based on the Law of Continuity: it accepts infinitely small quantities as “true quantities of their own sort”, but insists on interpreting them as fictions. But as Ishiguro has argued, these approaches are in fact two sides of the same coin. To say that dx is a fiction is not to say that there exist “fixed entities with non-Archimedean magnitudes, the introduction of which shortens proofs” (Ishiguro 1990, 83). “The word infinitesimal does not designate a special kind of magnitude. It does not designate at all.” (83) This is the point of calling the interpretation

whether there is an ultimate difference” (Bertoloni Meli 1993, 226), when Newton was in fact proving that there cannot be one.

17 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity syncategorematic: terms involving infinitesimals are “ostensibly designating expressions which follow certain sui generis rules” (83) and whose introduction shortens proofs; but they do not in fact designate real entities. The syncategorematic interpretation explains how it is possible to treat infinitesimals as if they are infinitely small actuals under certain well-defined conditions.

As Ishiguro puts it,

When we make reference to infinitesimals in a proposition, we are not designating a fixed

magnitude incomparably smaller than our ordinary magnitudes. Leibniz is saying that

whatever small magnitude an opponent may present, one can assert the existence of a

smaller magnitude. (87)

We can achieve some clarification, I believe, by showing how the syncategorematic interpretation of infinitesimals is connected with Leibniz’s philosophy of the actual but syncategorematic infinite. This connection is explained by Leibniz in his correspondence with

Johann Bernoulli, where he writes to Bernoulli in February 1699:

[S]ince an infinite plurality does not constitute a number or a single whole, it follows that

even given an infinite series, there need not be an infinitesimal term. The reason is that we

can conceive an infinite series consisting merely of finite terms ordered in a decreasing

geometric progression. (GM III 575; Ishiguro (1990, 85))

Leibniz’s meaning can be illustrated by reference to the dichotomy series discussed above:

1 1 1 1 1 n Sn = /2 + /4 + /8 + /16 + … + ( /2)

This series has a sum that can easily be calculated. For since

1 1 1 1 1 n + 1 /2Sn = /4 + /8 + /16 + … + ( /2) , on subtracting we have

18 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

1 1 1 n + 1 Sn – /2Sn = /2 – ( /2)

1 n so that Sn = 1 – ( /2) . Clearly, the greater the number of terms one takes, the closer the sum of the resultant series is to 1. Now if, when one has subtracted all the halves from a magnitude M,

1 ∞ there still remains an unassignable or actually infinitely small magnitude ( /2) —as Leibniz had previously claimed in the TMA— this corresponds to interpreting • as a number. But Leibniz’s thinking on the infinite had developed fast, and by the winter of 1672-73 he had already objected to the idea that an infinite number of entities could be treated as one whole, since taking • as a number would lead to a contradiction.12 On the other hand, though, the laws that Leibniz had discovered relating finite series of sums and differences worked equally well for the continuous case if one were allowed to treat an infinite decreasing as if it had last or infinitieth terms, and to treat dx as a non-zero, infinitieth part of the continuum “incomparably smaller than” any finite part. Indeed, in his later accounts, Leibniz made it clear that it was an extension of this kind of reasoning about infinite series that had led him to the differential calculus, by an extrapolation from the case of discrete numerical differences to continuous

(variable) geometric ones:

1 1 1 1 1 For example, /3 + /8 + /15 + /24 + /35 etc. or Údx/(xx – 1), with x equal to 2, 3, 4, etc., is a

series which, taken wholly to infinity, can be summed, and dx is here 1. For in the case of

numbers the differences are assignable. But if x and y are not discrete terms but continuous

ones, i.e. not numbers that differ by an assignable interval but the abscissas of a straight line,

12 In his notes on Galileo’s Two New Sciences made in Fall 1672 Leibniz wrote: “It follows either that in the infinite the whole is not greater than the part, which is the opinion of Galileo and Gregory of St. Vincent, and which I cannot accept; or that infinity itself is nothing, i.e. that it is not one and not a whole…” (A iii no.11, 168); and in Non datur minimum… : “The infinite number of all unities is not one whole, but is comparable to nothing. For if the infinite number of all unities, or what is the same thing, the infinite number of all numbers, is a whole, it will follow that one of its parts is equal to it; which is absurd.”(A iii no.5, 98)

19 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

increasing continuously or by elements, that is, by unassignable intervals, so that the series

of terms constitutes the figure; … (Leibniz 1702, GM V 356-357; Bos 1974/75, 14)

Thus in early 1676 Leibniz’s problem with respect to sums of converging infinite series was how to define them if there really is no last or infinitieth term. In Numeri infiniti (first week of April,

1676) he offers this surprisingly modern solution:

Whenever it is said that a certain infinite series of numbers has a sum, I am of the opinion

that all that is being said is that any finite series with the same rule has a sum, and that the

error always diminishes as the series increases, so that it becomes as small as we would like.

For numbers do not in themselves go absolutely to infinity, since then there would be a

greatest number. But they do go to infinity when applied to a certain space or unbounded

line divided into parts. (A VI 3, 503; Leibniz 2001, 98-99)

This is a very close anticipation of the modern definition of the sum of a converging infinite series as the limit of its partial sums Sn as n Æ •. It is also in keeping with the syncategorematic infinity of the series’ terms: to say that there are infinitely many is to say that, no matter how great a number of terms one takes, there are yet more. But one cannot strictly take all the terms, so that likewise there is no sum of an infinite series, strictly speaking. One can, however, specify a number that is closer to the sum than any arbitrary pre-assigned error: i.e. one can find an n

1 n sufficiently great that ( /2) < e, where e is the pre-assigned error.

But this in turn is equivalent to saying that the unassignable or infinitely small “beginning”

1 1 1 1 that is the difference between 1 and /2 + /4 + /8 + /16 + … is not an actually infinitely small

1 n quantity —as Leibniz had claimed in the TMA— but a finite quantity ( /2) so small that, by taking n sufficiently large, it can be made smaller than any pre-assigned error e, so that the

20 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

1 1 1 1 13 difference between 1 and /2 + /4 + /8 + /16 + … is “incomparably small” —i.e. negligible within any pre-assigned margin of error— compared with any finite quantity. This is the syncategorematic interpretation of infinitesimals.

The exact point at which Leibniz’s thinking on infinitesimals crystalizes into his syncategorematic interpretation is hard to determine, but the latter appears to be firmly in place in April of 1676. In his notes on Spinoza’s Letter on the Infinite, probably written that month,

Leibniz explicitly adopts the syncategorematic definition of infinite number as “a multiplicity so great that is exceeds any number, that is, any number assignable by us.” (A VI 3, 280; Leibniz

2001, 111) He uses this notion to counter Spinoza’s claim that certain things, such as the parts in the actually infinite division of matter, cannot be equated with any number “not … because of the multiplicity of their parts, but because the nature of the thing cannot admit a number without manifest contradiction.” (111) “Why shouldn’t they [exceed every number because of the multiplicity of their parts],” Leibniz objects, “if indeed it is obvious that they are more numerous than can bear an assignable number?” (111) Granting Spinoza the Cartesian thesis that “in every case of a solid moving in a perfect liquid plenum” matter is indefinitely but actually divided by the motions in it, he urges that what should be concluded from this is that matter is “in fact so divided into all the parts into which it can be divided” (113). He then adds: “Indeed, there emerge difficulties whose resolution occasions certain splendid theorems, and if Descartes had happened to discover them, he would have corrected certain of his opinions”. This is an

13 Ishiguro objects to Leibniz’s use of the expressions ‘incomparably small’ or ‘incomparably smaller’ on the grounds that “smaller is already a notion involving comparison.” (87) But in the following passage quoted by Bos (1974/75, 14) Leibniz makes his meaning clear: “I agree with Euclid (Book V, Def. 5) that only those homogeneous quantities are comparable of which the one can become larger than the other if multiplied by a number, that is, a finite number.” —i.e. if the Archimedean Axiom holds. Thus if an “incomparably smaller line” dx is added to a finite line x, the two are equal: x = x + dx. “This is precisely what is meant by saying that the difference is smaller than any given quantity” (Leibniz 1695a, GM V 322). Thus there is strictly no ratio between an infinitely small difference or differential and the quantity whose differential it is: they are incomparable.

21 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity enigmatic remark. What are the theorems referred to here, and what is their relation to infinite division and the syncategorematic infinite? We can throw some light on it by noting that almost the same words appear in a passage from Leibniz’s masterwork on the differential calculus, the

De quadratura arithmetica, finished by the summer of 1676 (but not published until 1993):14

Nor does it matter whether there are such quantities [as and infinitesimals] in the

nature of things, for it suffices that they be introduced by a fiction, since they allow

abbreviations of speech and thought in discovery as well as in demonstration. Nor is it

necessary always to use inscribed or circumscribed figures, and to infer by reductio ad

absurdum, and to show that the error is smaller than any assignable; although what we have

said in Props. 6, 7 & 8 establishes that it can easily be done by those means. Moreover, if

indeed it is possible to produce direct demonstrations of these things, I do not hesitate to

assert that they cannot be given except by admitting these fictitious quantities, infinitely

small or infinitely large (see above, Scholium to Prop 7.) … [M]y readers ... will sense how

much the has been opened up when they correctly perceive this one thing, that every

curvilinear figure is nothing but a polygon with an infinite number of sides, of an infinitely

small magnitude. And if Cavalieri or even Descartes himself had considered this

sufficiently, they would have produced or anticipated more. (Scholium to Prop. 23, Leibniz

1993, 69)

Thus the “splendid theorems” Descartes might himself have discovered relate to the decomposition of figures into infinitely many elementary figures, and of curves into infinite- sided polygons. This brings us to a discussion of Leibniz’s Proposition 6 of the De quadratura

14 Leibniz 1993. Its editor Eberhard Knobloch states that this manuscript “was already begun in 1675, but certainly completed before Fall 1676.” (p11). It was likely nearly finished by February, since Leibniz refers to it in “Numeri progressionis harmonicae”, N.54 of A VII 3, dated 8th February 1676.

22 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity arithmetica, referred to in the above passage. For, as Eberhard Knobloch (2002, 61-67) has shown in detail, in this theorem Leibniz gave a rigorous justification for what is now known as

Riemannian Integration. The demonstration proceeds as follows.

1B 1D 1N

2B 1E 1P 2D

x 3B 1L 2E 2P 3D

y Leibniz shows that if an interval containing the area under a curve is partitioned into a number of unequal subintervals, the difference between the sum of elementary rectangles approximating the area under the curve (the Riemannian sum) and the area itself is less than a quantity 1L3D ¥ hm, where 1L3D is the difference between the greatest and smallest ordinate, and

15 hm is the greatest height of any of the elementary rectangles. But “this greatest height (an abscissa) can be chosen smaller than any given quantity, because the curve is continuous” (65). It therefore follows that the difference between the Riemannian sum and the area under the curve is smaller than any assignable, and therefore null. As Leibniz himself rightly remarks about this

15 Here I am following Knobloch’s exposition. In the original, Leibniz denotes the greatest ordinate 4D and the greatest height of any of the elementary rectangles y4D, so that the sum of the elementary rectangles £ 1L4D ¥ y4D (Leibniz 1993, 29-32).

23 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity theorem, “it serves to lay the foundations of the whole method of indivisibles in the soundest way possible” (1993, 24); it is, as Knobloch says (2002, 72), a “model of mathematical rigour”.

Similar considerations apply to the conception of curves as infinitangular polygons.

Previously, in the TMA, the infinitely small straight sides were interpreted as indivisible points.

But if they are instead interpreted as arbitrarily small finite sides then, with respect to some property such as , by taking sufficiently many of sufficiently small size, the difference of the sum of their lengths from the arc length of a section of a continuous curve can be made less than any pre-assigned error, and one has the condition for treating a curve by means of the differential calculus. In a way, this is the same thing he had said in the TMA: the curve differs in magnitude from the infinite-sided polygon, but this difference is smaller than any that can be assigned. The difference is that in the TMA the unassignable differences are regarded as real, non-Archimedean elements of the continuum; whereas, on the syncategorematic view, such differences may be treated as if they are infinitely small actuals; they are actually finite and obey the Archimedean axiom, but can be made so small as to make no difference within an arbitrary preset margin of error. Thus, by the argument we reviewed above from De infinite parvis, the difference in magnitude between the curve and the infinite-sided polygon is in fact null.

The same conception of curves as infinitangular polygons is found in Numeri infiniti of early April 1676, where Leibniz makes explicit the character of the curve as the of polygons (—actually, two sequences, corresponding to the inscribed and circumscribed polygons by which Archimedes approximated the circumference of the circle16):

16 This is conceivably connected with Leibniz’s belief that Archimedes himself used arguments involving infinitesimals to determine the value of p.

24 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

The circle —as a polygon greater than any assignable, as if that were possible— is a fictive

entity, and so are other things of that kind. So when something is said about the circle we

understand it to be true of any polygon such that there is some polygon in which the error is

less than any assigned amount a, and another polygon in which the error is less than any

other definite assigned amount b. However, there will not be a polygon in which this error is

less than all assignable amounts a and b at the same time, even if it can be said that

polygons somehow approach such an entity in order. And so if certain polygons are able to

increase according to some law, and something is true of them the more they increase, our

mind imagines some ultimate polygon; and whatever it sees becoming more and more so in

each single polygon, it declares to be perfectly so in this ultimate one. And even though it

does not exist in the nature of things, an expression for it can still be produced, for the sake

of giving propositions in abbreviated form. (A VI 3, 498; Leibniz 2001, 88-89)

5. Infinitesimals, limits and the Law of Continuity

Crucial in all such reasoning, of course, is the existence of a limit, such as, in the above case, the circle or “ultimate polygon”. Ishiguro (1990, 90-91) has raised the question whether Leibniz’s reasoning in such cases is lacking in rigour, in that he does not establish the existence of such limits. But as she responds, it is not to be expected that Leibniz should have given existence proofs of the kind Weierstraß and Dedekind were later to provide, since he regarded such limits as fictitious last terms of the transitions considered. Still, he was well aware of the need to justify this kind of transition to a limit. Indeed, as Ishiguro observes, this was the point of his Law of

Continuity:

25 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

If any continuous transition is proposed that finishes in a certain limiting case (terminus),

then it is possible to formulate a general reasoning which includes that final limiting case.17

In the wording of its first publication in 1688, “A Certain General Principle, Useful not only in

Mathematics but in Physics”,

When the difference between two instances in a given series, or in whatever is presupposed,

can be diminished until it becomes smaller than any given quantity whatever, the

corresponding difference in what is sought, or what results, must of necessity also be

diminished or become less than any given quantity whatever. (A VI 4, 371, 2032)18

As Bos observes, Leibniz explains this principle more clearly through his examples. A reasoning about ellipses, for instance, could be extended to parabolas by introducing a second focus infinitely distant from the first:

… [For example,] we know that a given ellipse approaches a parabola as closely as desired,

so that when the second focus of the ellipse is removed far enough away from the first

focus, the difference between the ellipse and the parabola becomes less than any given

difference, since then the radii from that distant focus differ from parallel lines by an

amount as small as desired. (2032)

Thus provided one knows there is a number (albeit irrational) to which (in the case of the circles above) the inscribed and circumscribed polygons approach arbitrarily closely, or that there is a figure to which the ellipse approaches arbitrarily closely as one of its foci is removed indefinitely far away, the limit —which lies as a fictional entity outside the transition itself— can be

17 “Proposito quocunque transitu in aliquem terminum desinente, liceat ratiocinationem communem instituere, qua ultimus terminus comprehendatur.” (1701, 40).

26 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity fictitiously included as the terminus of the transition. Such extensions, as Bos explains, “involve the use of terms or symbols which become meaningless in the limiting case, while the argument they describe remains applicable, and in such cases the terms and symbols can be kept as

‘fictions’. According to Leibniz, the use of infinitesimals belongs to this kind of argument.” (Bos

1974-75, 57).

In his manuscript of c.1701, Cum prodiisset…(first published in Gerhardt’s edition 1846),

Leibniz presents an explicit foundation for his infinitesimals on the basis of this Law of

Continuity, as Bos has explained with admirable lucidity. It proceeds as follows. dx and dy are finite, arbitrarily small, and variable: they are neither fixed quantities, nor infinitely small ones.

In the characteristic triangle,

ds dy

dx ds can be treated as the side of an inscribed finite polygon, which can vary in length to become arbitrarily small. Now let (d)x be a fixed finite line segment. For all finite dx and dy, (d)y may now be defined by the proportion

(d)y:(d)x = dy:dx (1)

18 Leibniz adds: “Or, to put it more commonly, when the cases (or given quantities) continually approach one another, so that one finally passes over into the other, the consequences or events (or what is sought) must do so

27 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

dy d(x)

y

x dx

s

But the same (d)y can also be given an interpretation in the limit when the variable dx = 0, namely through the proportion

(d)y:(d)x = y:s (2) where s is the subtangent to the curve.

Now, for all finite dx ≠ 0, (d)y:(d)x can be substituted for dy:dx in any formula. But since the resulting formula is still interpretable even in the case where dx = 0, the Law of Continuity asserts that this limiting case may also be included in the general reasoning: dy:dx can be substituted for (d)y:(d)x in the resulting formulas even for the case where dx = 0, with dy and dx in this case interpreted as fictions. If a third variable v is involved, which varies with x, (d)v:(d)x can be defined in an entirely analogous way.

That this foundation suffices for first-order differentials and the rules of the calculus is best shown by an example. In Cum prodiisset… Leibniz offers the following proof of the rule for the

too.” (A VI 4, 371, 2032)

28 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity differentiation of a product d(xv) = xdv + vdx. He lets ay = xv (here the purpose of the constant a is to conserve the homogeneity of the equation), and then allows x, y, and v all to increase infinitesimally. Following Bos, I quote his demonstration:

Proof: ay + dy = (x + dx)(v + dv)

= xv + xdv + vdx + dxdv,

and, subtracting from each side the equals ay and xv, this gives

ady = xdv + vdx + dxdv

or ady/dx = xdv/dx + v + dv

and transposing the case as far as possible to lines that never vanish, this gives

a(d)y/(d)x = x(d)v/(d)x + v + dv

so that the only term remaining which can vanish is dv, and in the case of vanishing

differences, since dv = 0, this gives

a(d)y = x(d)v + v(d)x

as was asserted. … Whence also, because (d)y:(d)x always = dy:dx one may assume this in

the case of vanishing dy, dx and put

ady = xdv + vdx.19

As Bos argues, this approach to securing the foundations of the calculus leads very naturally to the concept of the , and even to the introduction (later in the history of the calculus) of the concept of . Indeed, Leibniz’s definition in (1) of (d)y as (dy/dx) ¥ (d)x , coupled with the stipulation in (2) that when dx = 0 the secant becomes the , implies that (d)y is

29 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity identical to the differential of Cauchy, defined as f¢(x) (d)x. For Cauchy defines f¢(x) as the limit of {f(x + Dx ) – f(x)}/ Dx as Dx Æ 0. The approach leads (through Euler and others) to the idea of differential quotients, but crucially depends on the recognition that one has to choose one of the variables as independent. Even though Leibniz’s difficulties and mistakes in the dispute about second order differentials with Nieuwentijt stems from his confusion on this, as evidenced also in his own attempt to extend the above type of reasoning to justify second order differentials, Bos shows that a correct justification for second order differentials can be given along these lines. It is just that they will depend on a certain choice of the progression of the variables, i.e. on a choice of which differential is taken to be constant. In essence, Leibniz’s justification of infinitesimals is (in its own terms) a valid one.

I now wish to propose that this justification of infinitesimals using the syncategorematic interpretation can be extended to continuous change in general.

6. General Application

It seems very evident that Leibniz intended his Law of Continuity to have universal application.

As he wrote in the New Essays: “In nature everything happens by degrees, and nothing by leaps, and this rule regarding change is part of my law of continuity.” (1704/1981, 552) Here he seems to have in mind cases such as a body instantaneously receiving a motion contrary to its preceding one, or our being woken by a sudden noise. In each case, processes that seemed to be discontinuous will, on closer inspection, be found after all to be continuous. At first sight, though, these explanations seem to pull in the wrong direction: apparently discontinuous things are really continuous, whereas the cases we need to explain seem to be rather the opposite: we

19 Leibniz, “Cum prodiisset…”, pp. 46-47; quoted from Bos (1974-75, 58). The ellipsis omits an obvious error in calculation not important for the argument.

30 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity want to explain how matter, although apparently continuous, is in fact discrete, and how “in actuals there is only discrete quantity”.

The key to resolving this, I suggest, is to see that what are really continuous for Leibniz are in all cases processes or transitions, and that the justification for the continuity of these processes is given in terms similar to his justification of infinitesimals. In support of this reading,

Leibniz explicitly models the transitions among states of his monads on infinite series. As he writes to De Volder, “The succeeding substance will be considered the same as the preceding substance as long as the same law of the series or of simple continuous transition persists, which makes us believe in the same subject of change, or monad.” (Jan 21, 1704; G. II. 264)

Granted, on a modern philosophy of mathematics, there would be a difficulty here. For if, in addition to the appetition taking it from one state to the next, the monad consists in an infinite series of states (we would say ‘sequence’) then the states can indeed be said to be discrete, just like the terms of an infinite series. But if that is so, the changes from one state to the next will alike be discrete, and the transition could not be said to be continuous.

Given Leibniz’s understanding of the mathematics of continuity, however, things are otherwise. As we saw above, the passage from discrete series to continuous ones, from sums to integrals, is obtained by replacing finite differences of the terms in a series by infinitesimals: he

1 1 1 1 1 equates the series /3 + /8 + /15 + /24 + /35 etc. with Údx/(xx – 1), where x is equal to 2, 3, 4, etc. and dx equal to 1, and then proceeds to the case where “x and y are not discrete terms but continuous ones, i.e. not numbers that differ by an assignable interval but the abscissas of a straight line, increasing continuously or by elements, that is, by unassignable intervals”. (Leibniz

1702, GM. V. 356-357). On the syncategorematic interpretation, though, such an “unassignable

31 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity interval” stands for the fact that an interval may be chosen so small that the resultant error is smaller than any that is pre-assigned. Analogously, then, this analysis can be applied to any continuous transition: all changes are in actual fact discrete, but there are always some smaller than others, and it is this is that allows us to define a fictionally smallest change, which we can treat as if unassignable. That is, even though the terms or states are discrete, so that their differences are too, there are always differences so small as to be smaller than any given.

Consequently the process is continuous. In a way, this is the same as assimilating the second method of justifying infinitesimals that Bos sees in Leibniz to the first: in reality there are only finite differences. This is not a different conception to the one that says that infinitesimals are fictions whose use can be justified. They are two sides of the same coin.

Thus the same syncategorematic interpretation that justifies treating a curve that is really an infinitangular polygon as a curve composed (fictionally) of infinitesimal line segments can be applied to any infinitesimal changes, that is, arbitrarily small changes taking place in arbitrarily small times, making all changes, as Leibniz says, continuous. This, I suggest, is Leibniz’s meaning in the passage from the letter to De Volder quoted earlier:

in actuals there is only discrete quantity, namely the multiplicity of monads or simple

substances, although this is in fact greater than any given number in any aggregate that is

sensible or corresponds to phenomena. But a continuous quantity is something deal

which pertains to possibles and to actuals only insofar as they are possible… Meanwhile

the knowledge of the continuous, that is, of possibilities, contains eternal truths which are

never violated by actual phenomena, since the difference is always less than any given

assignable difference. (G. II. 282-283)

32 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity

7. Conclusion

I have argued that Russell’s criticisms of Leibniz’s account of continuity can be met.

Despite their initial plausibility, in the end they no more do justice to Leibniz’s subtle thought than do Cantor’s allegations of inconsistency in Leibniz’s account of infinity. Just as an actual infinity of terms can be understood syncategorematically as more terms than can be assigned a number, without there being any infinite numbers, so too the infinitely small can be given a syncategorematic interpretation by means of the Law of Continuity, without there existing any actual infinitesimals.

I have argued further that this syncategorematic interpretation is also applicable to series of changes, and thus exonerates Leibniz from Russell’s criticism: on this interpretation, all naturally occurring transitions are continuous in that the difference between neighbouring states is smaller than any assignable. This means not that there exists a least difference, but that for any assignable , there exists a smaller one. Thus there is a true continuous transition, even though the states themselves and all assignable differences between them are actually discrete.

Acknowledgements

[to be added in the proofs].

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