Matter and Two Concepts of Continuity in Leibniz

SAMUEL LEVEY

MATTER AND TWO CONCEPTS OF CONTINUITY INLEIBNIZ

(Received 20 November 1998)

Leibniz places the distinction between continuity and the structureof matter alongside his theory of substance and his philosophyof mathematics as one of the key elements required for findingan escape from the difficulties of the second great “labyrinth,”the labyrinth of the composition of the continuum. And nothinginforms his account of the whole fabric of nature so deeply as hisengagement with the second labyrinth. While in recent years ourunderstanding of this enshadowed end of his philosophy has beenimproving, still, it seems to me that our grasp of Leibniz’s thoughtabout continuity remains highly imperfect. And so too it seems tome that we remain very much in the dark about his account of thestructure of matter. For Leibniz articulates the second in terms ofits contrasts with the first: his analysis of matter is very much setout in counterpoint to his analysis of continuity. In this paper I shallinquire into Leibniz’s metaphysical account of reality by lookingmore closely at the concept of continuity and its influence on histheory of matter.Some fresh critical perspective is to be gained, I believe, by

examining Leibniz’s thought about matter and continuity in someof his earliest writings, in particular those of the early 1670s, aperiod in which his views about matter and continuity are still inturmoil and yet also a period in which those views essentially cometo rest. So it is an aim of my project here to uncover some of theearly development of Leibniz’s philosophy and to trace out some ofits connections to his later thought. About the philosophy itself, Iintend to show that in those early writings Leibniz’s thought is, in asubtle way, quite deeply divided between two different conceptionsof continuity: one based in the idea of potentiality, and the other a

Philosophical Studies 94: 81–118, 1999.© 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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sort of forerunner to the topological concept of connectedness. Eachof those two concepts of continuity leaves its mark on his emergingtheory of matter, and I think with both of them in mind we shall beable to achieve a much clearer view of his account of the structure ofmatter – and a clearer view also of some of the profound difficultiesfacing it. By no means do I intend my remark in this paper to offer afull reckoning of Leibniz’s theories of matter and continuity.1 But Ido hope to throw some new light on the subject, and to follow someleads that point in a rather new direction.

MATTER AND CONTINUITY IN 1669 (AND IN 1676 AND 1705)

1.1 Hylomorphism and Continuity: “Unbounded” Matter

As early as 1669 several of the main strands of Leibniz’s thoughtabout continuity, and about matter, are already visible in his writ-ings. At this point in time, Leibniz sponsors a “hylomorphic”analysis of material bodies as compounds of form and matter, ananalysis that hooks up directly with the contrast between the discreteand the continuous. In the background here is an early project toreconcile the Cartesian and Aristotelian paradigms concerning thenatural world; Leibniz aims to conjoin the two traditions by show-ing that “Aristotle’s theories of matter, form, and change can beexplained by magnitude, figure and motion” (A VI,2,434).In a 1669 letter to Jacob Thomasius, Leibniz writes that “primary

matter” – that is, matter taken abstractly as an undifferentiatedextensum – is a continuous quantity. Although it is extended inspace,

it is unbounded [interminatam] . . . ôr indefinite, for so long as it is continuous, itis not cut into parts and therefore boundaries [termini] are not actually assignedin it. (To Thomasius, 1669. A VI,2,435)2

In calling primary matter “unbounded”, Leibniz has especially inmind the idea that it lacks interior boundaries; that’s why the lackof boundaries is relevant to primary matter’s not being cut into parts.The properties that Leibniz ascribes to continuous primary matter

– namely, being “indefinite” and “unbounded” in the sense oflacking “actually assigned” interior boundaries – are in fact thekey elements in this earliest analysis of continuity. A continuous

MATTER AND TWO CONCEPTS OF CONTINUITY 83

quantity is one without any articulated inner structure; it lacks thedefiniteness or determinacy that would come along with an assign-ment of parts and boundaries. It is indifferently divisible into partsin any number of ways, but is not actually divided.

1.2 Hylomorphism and Discreteness: Form, Bounded Matter andthe Contiguum

The early account of discreteness which is the counterpart to thatanalysis of continuity invokes the other side of Leibniz’s hylo-morphism: form is to fill in the ontological detail of “unbounded”matter and cut it into discrete bodies precisely by introducing actu-ally assigned boundaries and parts. Here the program of rewritingAristotle in Cartesian terms is particularly salient. “Form,” Leibnizsays, “is nothing but figure [figuram],” and “figure is the boundary ofa body” (A VI,2,435). Hence for primary matter to be compoundedwith form is nothing more than for boundaries to be assigned in it,marking off distinct bodies. And Leibniz holds that with the assign-ment of boundaries, the continuity of (primary) matter is broken,for

in order to have a variety of boundaries arising in matter a discontinuity of theparts is necessary. (A VI,2,435)

Thus secondary matter – matter taken concretely as invested withfigures – is not continuous but discrete; its parts are strictly discon-tinuous.It’s important to note that the discontinuity of the various parts of

matter that is introduced by form is not maintained by the presenceof intervening voids or loci vacui between neighboring parts. The1669 letter to Thomasius interestingly reveals a young Leibniz notyet the steady advocate he is soon to become of the “plenum hypo-thesis” (i.e. the claim that there are no absolutely empty spaces orvacua in nature but rather that everything is “full” or a plenum ofmatter), for in it he writes, “to me it seems that neither a vacuum ora plenum is necessary; the nature of things can be explained in eitherway” (A VI,2,434).3 Still, at this early date the plenum hypothesisalready looms large in his thought, and his account of the discon-tinuity of matter is precisely crafted, in the following way, to fit theplenum view of nature.4 Discontinuity in matter does not involve

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the existence of voids but arises instead due to the fact that at thelocus of a discontinuity between neighboring bodies, those bodiesare moving with different motions while still remaining in contact.“This happens,” he says,

when the parts stay together [immediata] but are moved in different directions.For example, two spheres, one included in the other, can be moved in differentdirections and yet remain contiguous, though they cease to be continuous. (1669.A VI,2,435–6)

Contiguity really is supposed to preserve the integrity of theplenum because immediately neighboring bodies will touch in thestrong sense that there is no empty space at all between them. AsLeibniz will announce explicitly when he sets down a definition ofcontiguity a few years later:

Contiguous things are those between which there is no distance. (“On theCohesiveness of Bodies.” 1672. A VI,3,94)

A last key point about continuity and discreteness also emerges inthe 1669 Thomasius letter. Leibniz holds that discrete ôr secondarymatter is in fact torn into separate parts each of which has its ownboundaries:

For by the very fact that the parts are discontinuous, each will have its ownseparate boundaries [terminos] (for Aristotle defines continuous things as hôn taeschata hen [i.e. those whose boundaries are one]). (1669. A VI,2,435).5

The thought here is presumably that discontinuous things, bycontrast with continuous ones, are those whose boundaries are two.On this account, then, pairs of boundaries always arise at the locusof a discontinuity between any two adjacent parts of (secondary)matter. That fact will come to be quite important later on in ouranalysis; but for the present I’ll just raise it to your attention.The metaphysical views so far recovered from this early writ-

ing stake out accounts of the structure of matter and the natureof continuity that set the two in sharp contrast. Continuity is akind of structural indeterminacy: a continuous quantity would besomething interminatum, “unbounded,” lacking an assignment ofinterior boundaries or parts. Matter, on the other hand, is something“bounded.” It’s a fully determinate, discrete quantity actually faultedthroughout with interior boundaries, and the material world is a


plenum of secondary matter that constitutes not a true continuum butrather a contiguum, so to speak, of discontinuous parts, all markedout by their various individual motions.6

1.3 Cohesion asMotus ConspiransAlthough I only mean here to sketch the barest outlines of Leibniz’svery early thought about matter and continuity, there is a detail Ihave been withholding and must now pause to introduce. It is thetopic of cohesion – the old issue about how particular pieces ofmatter manage to hold intact rather than dispersing or dissolvinginto the surrounding environment, and a well-known can of wormsthat I wish to open only as slightly as possible. But as we shall seeshortly, in Leibniz’s thinking the concept of cohesion is somewhatentangled with issues concerning continuity, and so I shall need tomake a few remarks about it.Probably the most important single fact about Leibniz’s early

views on the topic of cohesion is that they are shaped aroundthe intensely Cartesian thought that matter is unified into cohesivebodies by motion (cf. A VI,3,80, VE 2041). Adjacent parcels ofmatter form a cohesive whole in virtue of sharing a common motion– what Leibniz sometimes calls motus generalis (A VI,3,42) ormotus communis (VE 2047), or sometimes, most sweetly, motusconspirans (cf. VE 492, 495). But this common motion is consistentwith each parcel of matter inside the whole having a motion of itsown that divides and distinguishes it from the others. And likewisethere can be further differing motions within each parcel whichdistinguish its parts, and so on, as far as you please. The operativeideas in Descartes are visible in Principles of Philosophy, II, 25:

By a ‘body’ or a ‘part of matter’ I understand everything that is transferredtogether, even though this may consist of many parts which have different motionsrelative to one another. (Principles, II,25. AT VIII,53–4)

This idea about motion and matter – the “Cartesian motion thesis”that bodies are individuated by motion – crystallizes nicely inLeibniz’s account of cohesion as, for example, when a few yearsafter the Thomasius letter he writes:

Heterogeneous ôr agitating [turbans] matter is bound into one !body" by a motusgeneralis. (PQP, 1672. A VI,3,42)7,8

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Thus the binding of matter into cohesive bodies and the dividingof it into distinct parts emerge as two sides of the same coin:cohesion and division are the opposing faces of motion in thematerial plenum. One face is harmony, and the other, disorder.9Let me sum up a few conclusions about Leibniz’s views of matter

and continuity circa 1669. Continuity is taken to be a form ofindefiniteness and is characterized specifically as unboundedness– the lack of actually assigned interior termini or boundaries, andhence the lack of actually assigned parts. Matter, by contrast, isdiscrete, determinate, fully invested with shapes and interior bound-aries, and it is made so by motion. The differing motions withinthe plenum of matter assign a variety of boundaries in it, cutting itinto an infinite mosaic of discontinuous parts, each enclosed withinits own separate outer boundaries. Neighboring parts that share acommon motion form cohesive bodies, though such bodies are stillstrictly divided into smaller discontinuous parts by further internalmotions; and all the parts of matter in the universe are so packedtogether that the boundaries of contiguous parts are “indistant” fromone another, leaving no empty spaces anywhere.

1.4 Continuity as Potentiality and Leibniz’s Later Metaphysics,c. 1705

As I said before, being “indefinite” and lacking “actually assigned”parts and boundaries are the key elements in the 1669 analysis ofcontinuity, and being actually divided into parts in a definite wayis characteristic of matter. To fans of the later Leibniz those termsshould strike some familiar notes as well, for they are also keyelements in his most considered analyses of the structure of matterand the structure of continuity that he articulates in various contextsaround the turn of the century, notably in his “Note on Foucher’sObjection,” in his reply to Bayle’s Rorarius, in the letters to deVolder, and in the letters to Sophie.10I won’t give a full rehearsal of Leibniz’s account of continuity

and matter from the later writings here, but it will be useful to havea few of its details in mind. For I want to bring out a point abouthis view of continuity that is fairly clear in those later writings onthe continuum but which occurs rather more subtly in our piecefrom 1669. The concept at work in Leibniz’s account of continuity


is in fact an Aristotelian one much despised by Descartes and hisfollowers: it is the concept (or perhaps a concept) of potentiality. Ina passage from a 1702 text attacking the Cartesian physics, Leibnizdistinguishes the discrete from the continuous this way:

Every repetition (ôr multitude of similar things [eorundem]) is either discrete, asin numbered things where the parts of the aggregate are distinguished, or continu-ous, where the parts are indeterminate and can be obtained [possunt assumi] in aninfinity of ways [infinitis modis]. (G IV,394)

A continuous repetition or multitude has only “indeterminate parts,”and Leibniz’s gloss on that phrase is his usual one: a continuouswhole has no determinate assignment of parts actually made withinit but is rather merely indifferently assignable or divisible into partsin any of an infinity of different ways. In a very late writing thisview is still readily visible:

In the ideal or continuous the whole is prior to the parts, as the arithmetical unitis prior to the fractions that divide it, which can be assigned arbitrarily, the partsbeing only potential [le parties ne sont que potentielles]. (1714. G III,622)

The example in this passage of a continuous whole with indeter-minate parts is a favorite of Leibniz’s for clarifying his account,and a compelling one too, for the “arithmetical unit” can indeedbe understood as the sum of (and thus as “dividing into”) any ofan infinity of different series of fractions that can be extracted fromit. For instance, the “unit” or number 1 can be taken as the sum1/4 + 1/4 + 1/4 + 1/4, or with equal justice as the sum 2/3 + 2/9+ 2/27 + 2/81 + &c. ad infinitum, or perhaps instead as the sum1/2 + 1/4 + 1/8 + 1/16 + &c. ad infinitum. Obviously there is noend to such possible assignments of “parts” to the arithmetical unit,and Leibniz’s suggestion is that to identify any one of those assign-ments as the way the arithmetical unit falls into parts would bearbitrary, and conceptually misleading. For the unit is not investedwith any determinate substructure of actual parts at all, but rather –and in the last passage the reference to the concept of potentialityis unmistakable on the surface of his text – “the parts are onlypotential.”Such explicit reference to potentiality or potential parts is not

what one usually finds in his texts, however, and even in the laterwritings about continuity where there is far less concern to forge

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an alliance between Aristotle and Descartes, the concept of poten-tiality most often is not mentioned outright but rather only liesimplicit in the discussion, as Leibniz’s account of continuous quant-ity is instead delivered in terms of parts that are “indefinite” or“indeterminate” or “ideal” or “possible.”

[I]t is the same for the line, in which the whole is prior to the part because the partis only possible and ideal. (1695. G IV,491f.)

Indeed, a mathematical line is like the arithmetical unit: for both, the parts areonly possible and completely indefinite. (1704. G II,268)

If in Leibniz’s usual statements of his view the concept of potenti-ality is dressed in the language of indeterminacy, of “possible andindefinite parts,” the contrast between the structure of matter and thestructure of a continuous quantity nonetheless remains fully exposedand set out in terms of “actually assigned” parts and divisions inmatter:

But in real things, namely, in bodies, the parts are not indefinite (as they are inspace, a mental thing [re mentali]), but are actually assigned a certain way asnature has actually instituted divisions and subdivisions in accordance with thevarieties of motion. (1704. G II,268)

Thus one finds in these later writings that the main elements of hisview of matter and continuity that were first outlined in 1669 stillcommand the center of Leibniz’s account circa 1704–1705. Thatlater account trades on the idea that continuity is a form of indeter-minacy, a concept he repeatedly calls by name. He writes to Sophiein 1705 that in matter “nothing has remained indeterminate, whereasindeterminacy is the essence of continuity” (G VII,563). And theancient notion being unlimbered in that analysis of continuity, butmainly not being called upon by name, is the concept of potentiality.Looking back from this perspective at the very early hylomorphic

account of physical bodies, we can readily find marks of therole being played by the “indeterminist” or “potentialist” view ofcontinuity: for example, recall how Leibniz writes to Thomasius thatcontinuous matter is unbounded “ôr indefinite” [seu indefinitam](A VI,2,435). The terms primarily in play in the early discus-sions of continuity, however, are “part,” “motion” and “boundary”– ones, we might note, with impeccable Cartesian credentials. YetLeibniz still manages to capture a notion of potentiality in those


terms simply by denying that they apply to continuous quantity.To be continuous is thus precisely to lack an actual assignment ofparts and boundaries; and for a concrete quantity to be continuouswould be for it to lack a variety of differing internal motions, forit to be totally at rest through and through. Hence in this respecttoo the early writing are precursors of the later ones: when in1669 Leibniz describes continuous primary matter as “indefinite” or“unbounded,” underlying his description is the concept of poten-tiality with its metaphysical and distinctly Aristotelian pedigree.

1.5 Matter and Continuity in the 1676 Metaphysics

To this point I have been tracing out connections betweenLeibniz’s earliest thoughts about matter and continuity and his mostconsidered later views about them (c. 1705), trying to show that thesame basic elements are in play in his thought at both ends of hisphilosophical career. It is going to be important to note as well thatthose elements are also the key players in his most considered earlyviews about matter and continuity – not exactly the views of 1669(that are to be challenged severely) but rather those of 1676, whichfind their best and most fully articulated expression in the dialoguePacidius Philalethi . A proper study of those 1676 views will haveto await another occasion, but a look at a few telling passages shouldbe enough to raise the salience those core elements of Leibniz’sviews on matter and continuity that we have observed already inthe earlier and later writings.For example, unmistakably occurring in the 1676 texts is the

trademark thought about the internal structure of matter and its linkto motion:

Matter is a discrete being [ens discretum], not a continuous one; it is onlycontiguous, and is united by motion. (“On the Secrets of the Sublime,” 1676.A VI,3,474).

Evident also is the potentialist view of continuous quantity, whichsurfaces in the following passage, as Leibniz – in the voice of thecharacter Charinus – sums up a long discussion of whether thecontinuum could be composed of points:

We have concluded that the continuum can neither be dissolved into points norcomposed of them, and that there is no fixed and determinate [certum ac deter-

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minatum] number (either finite or infinite) of points assignable in it. (1676. AVI,3,555)

The claim here that points are not determinately assigned in acontinuum fills out a view about the parts of a continuum andthe “extrema” or boundaries of those parts (i.e. points, lines andphrases) that is detailed a few pages earlier in the dialogue. AgainCharinus is speaking:

[T]here are no points before they are designated. If a sphere touches a plane, thelocus of contact is a point; if a body is intersected by another body, or a surfaceby another surface, then the locus of intersection is a surface or a line, respec-tively. But there are no points, lines or surfaces anywhere else, and in general [inuniversum] there are no extremes except those that are made by a dividing [fiuntdividendo]: nor are there any parts in the continuum before they are produced bya division [divisione]. But not all the divisions that can be made are ever made.(A VI,3,552–3)

Leibniz’s view in this passage is what we might by now expect. Acontinuum is divisible but is not in its own right actually divided intoparts and boundaries; rather, it is only by divisions being made intothe continuum that points or other “extrema” come to be assignedin it. As he will often note, points, lines and planes are not parts ofa continuum at all but are mere “modes” of it: they are the vertices,edges and surfaces of extensa (cf. A VI,3,555, VE 2040, C 523, AVI,6,152, G II,520). But the view Leibniz offers here is not simplyabout what particular ontological status points, lines and planes areto have, whether that of part of mode. It is about the conditionsunder which points, lines or planes exist at all in a continuum: theyexist not prior to but only dependently upon an actual division ofthe continuum into parts. And with that view of the structure of thecontinuum firmly in hand, later on in the dialogue Leibniz is able torespond to an objection that presupposes points to exist even in anundivided continuum with the following riposte:

When the matter is adequately weighed, it seems to me that – as I also said withyour approval on another occasion above – these points of yours do not pre-existbefore an actual division, but are brought about by division. (1676. A VI,3,562)

The 1676 metaphysics of matter and continuity evidently engagesmuch the same conceptual machinery that we find at work both inthe early writings of 1669 and in the later ones of 1705 as well. As


we shall soon see, however, the transition from the earliest accountto his account of 1676 is interrupted by fairly dramatic changesof mind. But before turning to consider what happens in Leibniz’sthought during that turbulent interval between 1669 and 1676, I wantto look one last time to the 1705 account of matter and continuityand draw out a few more of its relations to those earlier accounts.

1.6 From 1669 to 1705: New Elements in Leibniz’s Later Views,and the Priority Thesis

Leibniz’s views develop over the fifty years of his philosophicalcareer, and I should not want to imply that his final thoughts onthese topics simply emerge fully formed in their initial appearancein his writings. On several important points the 1705 metaphysicsof matter is not already mapped out in the 1669 writings, and mostnotable among the developments, of course, is the famous doctrineof the monads: the thesis that the only true substances are indi-visible, immaterial mind-like units whose properties are variousperceptual states and states of “force.” But even quite apart fromthe theory of substance, Leibniz’s 1705 views on motion, matterand the continuum have evolved significantly from the ancestors ofthem that we find in 1669. Just for example, three elements of the1705 account that set it apart from the views formulated in 1669are: (1) the modalized spin on continuity, according to which thecontinuity of continuous magnitudes is due to the fact that theyencode or “express” possibilities (cf. G II,268–9, 276, 278–9, 282,IV,568, VII,562–3); (2) the denial of the Cartesian motion thesis thatmotion is (or even could be) what individuates bodies in a plenum,which Leibniz famously and powerfully registers in Article 13 ofDe Ipsa Natura (1698); and (3) what I have elsewhere called the“mereological priority thesis” and shall say more about in a momentbelow.11 None of these three facets of Leibniz’s later thought ispresent in the 1669 account; and the first two are ideas that reallycome into being only starting around 1680 or well later. But the lastof the three that I mentioned is making itself apparent already by1676, and few remarks about it are in order.It is an important strand of Leibniz’s later thought about con-

tinuity that continuous wholes are “prior” to their parts, by which hehas in mind ontological rather than, say, temporal priority; and this

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strand was entirely visible to us once already in a passage from 1704quoted earlier, where he writes, “In the ideal or continuous the wholeis prior to the parts” (G III,622). This thesis about continuouswholesand their parts forms one half of a staple feature of his account ofthe distinction between the discrete and the continuous, namely, themereological priority thesis that in a continuous quantity the wholeis prior to the parts, whereas in a discrete one, such as matter, theparts are prior to the whole.

In matter and in actual realities the whole is a result of the parts; but in ideas or inpossibles . . . the indeterminate whole is prior to the divisions. (1705. G VII,562)

The contrast between the continuous and the discrete laid out inthe priority thesis ties neatly together with Leibniz’s distinctionbetween indeterminate quantities, whose parts are only potential,and determinate ones, whose parts are actually assigned.

The continuum, that is, involves indeterminate parts, whereas in actuals thereis nothing indefinite – indeed, in them any division that can be made, is made.Actuals are composed as a number is composed out of unities [e.g., 6 = 1 + 1 +1 + 1 + 1 + 1], ideals as a number out of fractions: the parts are actual in the realwhole, not in the ideal whole. (1706. G II,282)

Indeed as he tells us at G IV,492 – also quoted above – the continu-ous whole is prior to its parts “because the part is only possible andideal.”The priority thesis becomes something of a slogan in the later

writings for Leibniz’s complex analysis of the source in our reason-ing of the paradoxes concerning the composition of the continuum,and one might be tempted to consider it the distinctive stamp of hislater views about matter and continuity. It is therefore worth keepingin mind that the priority thesis has in fact made its way into hiswritings already by 1676, where he tells us straight out in one placethat “in the continuum, the whole is prior to its parts” (A VI,3,502),and, in another, that in “discrete things” – his example is “number,”presumably conceived as an aggregate composed out of unities –“the whole is not prior to the parts, but the converse” (A VI,3,520).At this point our running chronicle of Leibniz’s views about

matter and continuity as they appear in three distinct periods of histhought should be complete enough to show that a striking unity ofmind presides over his account in each of those three periods, as he


hammers out successively refined versions of a single theory whosebasic contents were first advanced in 1669. Given that unity of mind,one might naturally come to suspect that there is also a constancy tohis opinions on these topics as well, one that holds the same theoryintact throughout the course of four decades; but in fact Leibniz willdefy expectations of that sort, as we are about to see. All I mean todo here is to put the separate writings of 1669, 1676, and 1705 allon a line and say that they express essentially the same core view: asingle deep current of thought about matter and continuity and thedistinction between them.And now the plot thickens.

2. COHESION, TOPOLOGY AND TWO CONCEPTS OFCONTINUITY

2.1 What Leibniz Adds to Descartes’ Theory of Cohesion, c. 1671

While the history of Leibniz’s analysis of matter and continuitybegins and ends with essentially the same account in ascendance,a defining piece of action occurs near the outset of those early yearsthat involves a rather different set of ideas and puts Leibniz’s wholeview of the structure of matter in a different light. At the center ofthe action is the topic of cohesion.Let’s consider another passage from Propositione Quaedem

Physicae (1672), where the Cartesian ideas about motion andcohesion are particularly apparent, but in which hints of somethingelse in Leibniz’s account of cohesion begin to surface:

It is manifest that some body is constituted as definite, one, particular, distinctfrom others, by some motion of its own, or a particular endeavor [conatus], andif it is lacking this it will not be some separate body, but that by whose motionalone one continuous body coherent with it is moved. And this is what I havesaid elsewhere, that cohesion comes from endeavor or motion, that those thingswhich move with one motion be understood to cohere with one another. (1672. AVI,3,28)

Besides the obvious links to the Cartesian views about how varyingmotions divide matter into separate parts while common motionsbind it into cohesive bodies, two features of this passage are espe-cially noteworthy. First, observe how Leibniz allows his claims

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about cohesion to flow over into claims about continuity. Taken atface value, the suggestion here seems to be that parcels of mattermoving with one motion will be not only cohesive but also continu-ous; and while that claim seems to stand in outright contradictionto his 1669 account of matter as a strictly discrete quantity, it’s nofluke. Leibniz really is claiming that cohesive matter is continu-ous, and we shall find him saying so time and again in the yearsbetween 1669 and 1676. Secondly, note the role being played hereby the term conatus or “endeavor.” While in this sort of contextthat term could be intended to convey nothing more than the ideaof an individual motion, we shall see below that in fact it signals thepresence of something extra in Leibniz’s account, something thatis not merely a feature of the Cartesian view.12 The bottom line isthat on Leibniz’s c. 1671 view of cohesion, it’s more than justmotusconspirans, it’s a common motion plus conatus ôr endeavor. Andthat difference turns out to have metaphysical implications.The passage at A VI,3,28 just quoted above comes directly from

Leibniz’s early physics of motion, and mixed into that physics is anontology of conati or infinitesimal elements of motion – infinitelysmall “strivings” scattered throughout the material plenum that aresupposed to be the very beginnings of true motions (cf. A VI,3,79–80, 95–96). This early conatus physics is somewhat fleeting; itdoesn’t survive past the end of March, 1676 – if indeed, it lastseven that long. Without going into the details, once Leibniz comesto reject the possibility of an infinitesimal quantity, which he doessometime in March of 1676, the early conatus physics simply fallsthrough.13Still, the early conatus physics proves very illuminating, for it

carries a point of special importance concerning Leibniz’s under-standing of the topic of continuity. During a brief period whilethe conatus physics is in force, Leibniz seems totally to reversehis view of the relationship between the concept of continuityand the idea of an actual assignment of parts. Sometime in 1670,and for a few years after that, he comes to hold that continuousthings can have actually assigned parts; and then just as abruptly hereverts back to his original view that an actual assignment of partsrequires discontinuity. In that small episode I think it’s possible todiscern that Leibniz runs together two different concepts under the


single heading of ‘continuity’. Some stage-setting will be neededin order to bring these two concepts and my point about them intosufficiently high relief.

2.2 Continuity and Topology

Early on especially, but fairly well across his writings, questionsconcerning cohesion and division are for Leibniz intimately boundup with questions about continuity. Indeed, the labyrinth of thecontinuum he frequently mentions is conceived to embrace a hostof physical as well as mathematical concerns. As the seventh itemcited in his prospectus for an encyclopaedic project to be entitled.De Rerum Arcanis (“On the Secrets of Things”), Leibniz lists:

Labyrinthus posterior, ôr on the Composition of the continuum, on time, place,motion, atoms, the indivisible and the infinite. (1676. A VI,3,527)

Cohesion does not appear explicitly on the list, but it is certainlylurking in the background as he cites “motion” and “atoms.”To set as the subject for a single line of inquiry that sort of

amalgam of topics is typical of early modern (and classical)studies in “natural philosophy.” By contrast, the study of continuitytoday falls under the jurisdiction of topology, which is essentiallyconcerned with the structure and properties of various abstractspaces and only incidentally, if at all, with the structure of concretephysical magnitudes and phenomena. But a number of what onemight call “proto-topological” ideas do crop up in natural philo-sophy, ideas that also provide the intuitive underlay for the founda-tions of contemporary topology. Perhaps the most important ofthese is the idea of a “natural whole” that has no “natural joints”or “seams.” And this idea lies at the bottom of the concept ofconnectedness – a concept that is a key feature of all standarddefinitions of a “continuum” in classical contemporary point-settopology.14 Using the framework of point-set topology, I shall setout a few basic topological ideas to help us get quite clear on theconcept of connectedness, and then show how that concept tiestogether with Leibniz’s c. 1671 account of cohesion.15The abstract spaces topology investigates are, on the standard

conception, represents as sets of arbitrary elements we can call“points,” and thus the sets are called “point sets.” Although general

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Figure 1.

topology can be constructed in a number of ways taking differentconcepts as elementary, the basic concept most often used to doso is that of an open set, and my brief sketch will observe thisconvention.16 To a point set or space S there belongs a select familyof subsets that govern its topological structure: these are its opensets. A few simple axioms decide which subsets of S can be its opensets. Basically they tell us that the family of open sets is closed underthe set-theoretical operations of arbitrary union and finite intersec-tion, and that the whole space itself and the empty set both count asopen sets.17In its strict and perfectly general form delimited by those

axioms, however, the concept open set doesn’t automatically yield acomfortable heuristic for thinking about topological properties likeconnectedness. Since the range of point spaces is extremely diverse,the axioms for open sets cover an unimaginably rich spectrum ofcases, and there is no way to “visualize” (or otherwise present easilyto the mind) what it is, in general, for a subset of a space to be anopen set. But it is not hard to get something of a grip on the ideaof an open set if we restrict our attention to more familiar, moreeasily visualized spaces and consider a paradigm case. Imagine atwo-dimensional Euclidean space, like a flat plane, with a circleinscribed in it (See Figure 1).18


Figure 2.

The circle defines three families of points in the plane. First, thereare the points that lie on the circle itself. Second, there are thosefalling strictly within the disk-shaped region enclosed by the circle(but not actually lying on the circle). And third there are those pointsfalling strictly outside of the circle. Now, our paradigm case of anopen set is given by the second family of points: the set of thosepoints falling strictly within the circle. Those points make up anopen disk in the plane, a diskminus the boundary circle that enclosesit. The open disk is the paradigm open set for a two dimensionalspace. For a three dimensional space, the paradigm open set willbe an open sphere – a sphere minus its boundary surface. In a onedimensional space, the paradigm will be an open linear interval, afinite line segment minus its two end-points.In contrast to this, the disk plus its boundary circle is the

paradigm case of a closed set (and likewise the closed sphere andthe closed linear interval are paradigms for their spaces).Let’s sometimes be willing to ignore the fine distinction between

open and closed sets, and in those times just speak of our disk asD (see Figure 2). But when we need to be careful about it, we shalldistinguish the open disk enclosed by the circle, D-minus (for “diskminus the circle”), from the closed disk made up of all the interiorpoints plus the boundary circle,D-plus. And we shall distinguish thespace containing all and only those points that lie strictly outside thecircle as the exterior.With the idea of an open set in hand, we can now use it to define

an array of other topological concepts, most importantly the conceptof a limit point of a region of a given space. We identify a region with

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a subset R of the points in S, and a limit point is defined as follows: apoint p that is an element of the space S is a limit point of a subset Rof that space if and only if every open set containing p also containsat least one other point p# distinct from p (p# $= p) such that p# is anelement of R.19Applying this definition to our example of the circle in the plane,

we get the following results. For any point p lying strictly outside thecircle, in the exterior space, there is always an open disk, containingp, small enough not to spill over onto the circle or into the interiorspace of the disk it encloses. Thus, it is not true that in every open setcontaining a point p in the exterior, there is a distinct point p# thatbelongs to the open diskD-minus – nor is that true even of the closeddisk D-plus. Therefore, no point lying strictly outside of the circle,in the exterior space, is a limit point of our open disk D-minus. Noris any such point in the exterior space a limit point of the closed diskD-plus.But, for any point p lying on the circle itself, it is true that every

open disk that contains p will always spill over into the open diskD-minus and thus contain some further point p# distinct from p, suchthat p# is an element of D-minus. And so we see that every point onthe circle is indeed a limit point of the open disk,D-minus. Likewise,every point on the circle is also a limit point of the closed disk, D-plus. Notice too that every point on the circle is a limit point of theexterior space.Now, finally, with the concept of a limit point in hand, we can

easily define connectedness and head back towards the concept ofcontinuity. A set such as our space S or any subset of it is connectedif and only if it admits of no separation – that is, if and only if itcannot be partitioned into two nonempty, disjoint subsets neitherof which contains any limit points of the other.20 Intuitively put,any way of dividing a connected space into two discrete parts willalways “chop” or “tear” some limit points off one of those parts anddeposit them in other.Suppose we were to take an infinitely sharp Exacto knife and try

to cut the disk D out of the plane. However carefully and smoothlywe cut, we shall face the decision of what to do with those pointslying on the circle itself. If we cut into the plane in such a way thatleaves the circle attached to the disk, we shall succeed in removing


D-plus. But notice that in doing so we shall be slicing off limit pointsbelonging to the exterior space; for the points on the circle are limitpoints of the exterior as well as limit points of D (this holds equallyfor D-plus and D-minus, both of which include the points on thecircle as limit points). The disk D and the exterior space around itshare the circle as a common boundary. Of course this means thatif we should cut into the plane in such a way that leaves all of thepoints on the circle attached to the exterior space, we’ll thereby becutting limit points off the disk D; what we’ll then detach is D-minus, a disk denuded of its limit points. (And we could cut intothe plane in such a way that distributes the limit points in greateror lesser portions between the disk D and the exterior, thus leavingsome of the boundary attached to each piece – but thereby strippingoff some limit points of each one as well.)So, no matter how deftly we impose a cut to separate the whole

space into disjoint parts, some limit points of one part will alwayswind up inside the other part. It is in exactly this sense that aconnected space such as our plane is a seamless whole that has no“natural joints” at which it can be divided. Rather, any cut into aconnected space must be arbitrary and, indeed, disordering.

2.3 Cohesion, Connectedness, and Aristotle’s Dictum:The View of 1671

With the perspective provided by our brief foray into contemporarytopology, we can now throw a little fresh light on Leibniz’s accountof cohesion and continuity. As I mentioned before, on the earlyconatus physics cohesion is a common motion plus something extra– conatus or an endeavor. Here is the significance of adding conatusinto the mix: the parts of cohesive bodies not only move with acommon motion, but also they strive or endeavor to be in the sameplace. And as those parts press together at their boundaries, theystart to interpenetrate very slightly (cf. G VII,572–4 (July 1670),VI,2,263–76 (Winter 1670–1)). At their locus of contact they cometo share an “extremum,” that is, a limit or boundary, and thus cometo overlap – even if only momentarily and at an inassignably smallplace. Two spheres might cohere by striving to cross one another’sboundaries and coinciding at a single limit point; or two flat bodiesmight cohere and thus come to share a single limiting planar surface.

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In his letter to Hobbes of July 1670, Leibniz writes aboutcohesion:

I should think that the endeavor [conatus] of the parts towards each other, ôrthe motion through which they press upon each other, would itself suffice toexplain the cohesion of bodies. For bodies which press upon each other endeavorto penetrate each other. The endeavor is beginning; the penetration is the union.So they are beginning to unite. But when bodies begin to unite, their surfacesor boundaries [initia vel termini] are one. Bodies whose boundaries [termini] areone, ôr ta eschata hen, are according to Aristotle’s definition not only contiguousbut continuous, and truly one body, movable in one motion. (G VII,573)

That account of cohesion as involving an overlap or sharing ofextrema is significant for Leibniz’s thinking about the issue ofcontinuity. For as he notes in a very similar tone in his TheoriaMotus Abstracti (“Theory of Abstract Motion”, 1670–1), written thefollowing winter,

things whose extrema are one, hôn ta eschata hen, are continuous ôr cohering,by Aristotle’s definition too, since if two things are in one place, one cannot beimpelled without the other. (A VI,2,266.)

And, once again, this view which we are finding expressed in the1670 letter to Hobbes and in the TMA is recapped with especialclarity in the 1672 piece “On the Cohesiveness of Bodies,” whereLeibniz writes,

it is clear that any conatus whatsoever already begins to have an effect [iamincipere efficere], even if the effect is smaller than any assignable. Hence itfollows that whatever endeavors to move into another’s place already at its bound-ary [extremo] begins to exist in the other’s place, i.e. their boundaries [extrema]are one ôr [sive] penetrate each other, and consequently one cannot be impelledwithout the other. And consequently these bodies are continuous. (A VI,3,96)

A richly metaphysical upshot of the new conatus-based account ofcohesion is being articulated in these passages: what interpenetrat-ing parts of matter establish in the physical world is not just cohesionand contiguity, but continuity.Is this a tenable conclusion on Leibniz’s part? We can now see

that it is. For “those things whose boundaries are one” would form awhole that satisfies a condition intuitively similar to connectedness.In order to separate those parts from the whole they form, it seemsthat any cut we should impose would “tear” their common boundary


off one part and leave it attached only to the other. If cohesion indeedrequires numerically distinct bodies to share a common boundary,then it can be said to generate continuity in the physical world,in a clear proto-topological sense of the term ‘continuity’. Andconsequently it seems that bodies distinguished by their individualmotions could, nonetheless, still be parts of a continuum, preciselyby being connected in this way.There is a point worth emphasizing about this Leibnizian account

of cohesion that ascribes connectedness to cohesive bodies. Properlyunderstood, Leibniz’s view should be that connectedness superveneson cohesion – and not that connectedness explains or causes it.Cohering parts of matter come to be continuous because they areinterpenetrating at their boundaries, and as those boundaries coin-cide they “become one.” But the shared boundary at the locus ofcontact between cohering parts of a cohesive body is not a “glue”that holds those parts together; it is the result of cohesion, nota source of it. While this point should be clear enough from theaccount, it is not hard to get the order of explanation running back-wards and treat the cohesiveness of a body as due to its partsbeing joined by common boundaries; and indeed we can sometimesfind Leibniz himself reversing the order, as for example in the lastpassage quoted above where near the end he says, “their boundariesare one ôr penetrate each other, and consequently one cannot beimpelled without the other.” This is an error, if only a slip. In thevery next sentence Leibniz sets the account right again, drawing outthe metaphysical consequence from the physical theory of cohesion:“And consequently these bodies are continuous.”

2.4 Switching Between Potentiality and Connectedness

All that marks a sharp change from the view Leibniz outlined justa year or two earlier in the 1669 letter to Thomasius, according towhich cohering but distinct parts would still be strictly discontinu-ous and cohesively united only by a common motion. Thus in thatrespect, the divergence of the 1671 TMA account from the earlierone of 1669 entails a change of mind – or, at any rate, a change ofcommitments – on a key point. Whereas in 1669 Leibniz held that“in order to have a variety of boundaries arising in matter a discon-tinuity of the parts is necessary” (A VI,2,435), on the 1671 TMA

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account this claim about discontinuity must be seen as false. Rather,a variety of boundaries can arise in a mass of matter whose parts arestrictly continuous with one another. Does this mean that a varietyof distinct actual parts can, if cohering, form a true continuum?Apparently that is what Leibniz means to defend, and indeed theTMA takes as its first fundamentum praedemonstrabile the followingclaim:Parts are actually assigned in the continuum. (A VI,2,264; italics in original)

The conatus physics of 1671 now stands vividly in contrast with theearlier view of 1669, especially regarding the analysis of continuityand the structure of matter. Whereas in 1669 matter is discontinuousand the continuum is defined as lacking an assignment into parts,now in 1671 parts are assigned in a continuum and cohesive materialbodies are continuous. But Leibniz’s allegiance to the thesis thatthe continuum has actually assigned parts is very short-lived, as hisenthusiasm for that “first principle” dries up along with the conatusphysics of the TMA and its specific claim that cohering bodies sharean extremum or boundary. By the autumn of 1676 he shifts back,this time permanently, to the 1669 view that the continuum has noactual parts and that a variety of boundaries cannot arise in matterwithout some discontinuity.Why does Leibniz switch back and forth on those basic principles

concerning matter and continuity? Part of the answer is simply thathe abandons the interpenetration account of cohesion, and so farI have attributed his doing so to the fact that the conatus physics,upon which that account of cohesion is based, falls through whenin March of 1676 Leibniz comes to reject the notion of an infinites-imal quantity. So we can see his retreat from the view that matter iscontinuous as in part a consequence of his views in the philosophyof mathematics evolving in a way that undermines his original basisfor holding cohesive bodies to be connected quantities.And there is another element in his thought that should also be

mentioned as a possible contributing factor in his abandoning theinterpenetration account of cohesion. Sometime between 1672 and1676 Leibniz comes to accept an ancient line of argument for theimpossibility of a “state of change” or “middle state,” i.e., a statewhich could serve as the common boundary between two contrarystates, such as a state of dying that lies between life and death, or


a state of perfect rest that lies between the rise and fall of a stonethat is tossed straight up in the air.21 On the view he comes to adopt,there are no such middle states mediating change from one state toanother but rather change is to be analyzed as a “composite” or “thepoint of contact or aggregate of two opposite states” (A VI,3,541).If this thesis covers not only cases of alleged middle states joiningtwo opposing states that exist one after another but also the caseof boundaries between distinct bodies that exist one adjacent toanother, it yields an argument against the possibility of cohesionof two bodies via their sharing a common boundary. For the allegedcommon boundary would be just such a middle state between theone body and the other, and allowing the existence of this bound-ary would then contradict Leibniz’s denial of middle states. Sincecohering bodies would therefore have to be joined only by contactin which their separate boundaries are touching, the interpenetrationaccount of cohesion would be disqualified; and the view of materialbodies as continuous (i.e. connected) quantities that rests on it wouldagain be left ungrounded.22 So we can also see Leibniz’s shift awayfrom the c. 1671 thesis that matter is continuous as due in part to hisacceptance of an argument against middle states.While both of those developments in Leibniz’s thought can be

seen as factors that contribute to his abandoning the interpene-tration account of cohesion, and thus can offer us some rationaleto explain why he shifts back and forth between opposing viewsabout the structure of matter (whether it’s discrete or continuous)and the nature of the continuum (whether it’s indeterminate or actu-ally assigned with parts), in fact I think his wavering here is a clueto something still more philosophically interesting and instructive.And this brings us to the interpretive point highlighted in the titleof this paper. I suggest that his changes of mind – especially aboutwhether the continuum has actually assigned parts – signal the pres-ence of two subtly different concepts at work in Leibniz’s thinking,either one of which might be expressed by the term ‘continuity’.On the one hand, there is the concept of connectedness, or at anyrate a similar proto-topological concept that finds expression inAristotle’s dictum, “Continuous things are those whose boundariesare one.” On the other, there is the more metaphysically-orientedconcept of continuity as potentiality, where this is opposed to actu-

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ality, and of the continuum as something indefinite or indeterminate,indifferently assignable into parts but not actually so assigned.As we have seen, the connectedness of an extensum is compat-

ible with (although it does not entail) its containing a variety ofactually assigned boundaries and parts, under the proviso that thoseparts are not separable in the sense that any meaningful cut into aconnected extensum will tear limit points off one part and depositthem in the other. Obviously, however, the potentiality concept ofcontinuity is not compatible with the idea that a continuum mightcontain such a variety of actual parts and boundaries; for on anaccount of continuity as potentiality the presence of actual bound-aries in the whole will always require discontinuity of the parts. Ibelieve that Leibniz’s apparently dramatic changes of mind concern-ing the character of the continuum mark a pair of shifts betweenthose two concepts of continuity or the continuous. Leibniz initiallythinks of the continuous primarily from the point of view of thepotentiality concept that prevents the continuum from containingactually assigned parts and boundaries. He then (c. 1670–1) shifts tothinking of it primarily from the point of view of a proto-topologicalconnectedness concept that allows for a continuum to have actuallyassigned parts and boundaries. And by mid-1676 he shifts back tothe potentiality concept once again.A further small consideration can help to cement our account of

Leibniz’s early double-switch on the nature of continuity. There isa sense in which the connectedness concept naturally aligns with aninterpenetration account of cohesion. If cohering bodies interpene-trate and thus share a common boundary, then, as we have seen, inorder for those bodies to be separated that boundary will apparentlyneed to be “shorn off” at least one of them. Notice how the languageand imagery most natural for discussing the concept of connected-ness are loaded with allusions to the concept of cohesion – as indeedare the topological terms ‘connected’ and ‘separable’ themselves.Interestingly, the samewas true in Aristotle’s time. Consider what hesays about the proto-topological concept of continuity in the criticalpassage from the Physics:

The contiguous is what is in succession and touching. . . . The continuous [tosuneches] is a species of the contiguous. I call things ‘continuous’ when theboundaries of each at which they are touching become one and the same and, as


the name implies, they are “stuck together” [sunechêtai]. But this is not possibleif the extrema are two. It is clear from this definition that continuity belongs tothose things from which a unity naturally arises in virtue of their being joinedtogether [kata tên sunapsin]. (Physics, 227a6, 227a10–15)

I suspect the fact that the connectedness concept tends on itsown to gravitate towards the idea of cohesion may well inclinea thoughtful defender of the interpenetration account of cohesiontowards precisely that concept of continuity. If that’s right, then itis not hard to guess why Leibniz switches in his thinking aboutcontinuity to the proto-topological connectedness concept duringthe very time when he favors the conatus physics and the interpen-etration account of cohesion. That account and the connectednessconcept are, as it were, made for each other. Once that account ofcohesion is abandoned, however, the intuitive pressure to think ofcontinuity as connectedness, rather than as potentiality, is relieved.And in the wake of abandoning that account of cohesion, Leibnizindeed switches back to the view he is originally and more deeplydisposed to have of continuity, namely, that of the potentialityconcept.

3. THE METAPHYSICALLY PROBLEMATIC STRUCTURE OFLEIBNIZIAN MATTER

3.1 The Actuality of Matter

The two concepts of continuity I have tried to distinguish inLeibniz’s thought – potentiality and connectedness – are not mutu-ally exclusive. It is possible to hold a continuum to be a connectedspace in which parts and boundaries are merely potential or inde-terminate, indifferently assignable but not actually assigned. And Ithink such a two-aspect account is in fact probably nearest Leibniz’sown most considered view of the continuum. Recall, for example,how both of our two continuity concepts are visible in the 1669letter to Thomasius. First, continuous quantity is said to be inter-minatum or unbounded: it is not cut into parts, and boundaries(termini) are not actually assigned in it. But second, Aristotle’sdictum about continuity is also invoked: continuous things are thosewhose boundaries are one, that is, they’re connected.

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I suspect also that it is the potentiality concept that really runsmost deeply in Leibniz’s thought about the continuum. During thatepisode in the 1670s when he reverses his view about the poten-tiality of the continuum, Leibniz is being brought on the strengthof an interpenetration account of the cohesion of matter to allowthe connectedness concept full sway, so to speak, in his thinkingabout continuity. It holds full sway even to the point of being able tooverride the potentiality concept’s requirement that a continuum beonly indifferently divisible into parts rather than have parts actuallyassigned in it.How can this have happened? How can a passing account of

cohesion have tempted Leibniz to suspend the rules stemming fromwhat I claim is his deepest current of thought about continuity? Theanswer, I think, is that that account of cohesion engages a line ofLeibniz’s thinking that runs more deeply than any of his views ofcontinuity: it engages his commitment to the actuality of matter. Atall points Leibniz holds that matter is “actual” in the sense of thatterm which contrasts with being merely potential: matter is fullydeterminate or definite, ontologically speaking; all the details of itsstructure are in some important sense filled-in or actually assigned.This view is clear already in a passage at A VI,2,435 (from the

1669 letter to Thomasius), quoted above, where matter is said tohave parts and boundaries “actually assigned” in it. And likewise inthe later metaphysics it shines out across Leibniz’s writings as hecomes to call material bodies “actuals” precisely to underline theircontrast with continua, which are merely indeterminate, potential,and express only how it is possible for divisions to be made inthe extended world “without having to bother with divisions actu-ally made” (G IV,491). We have also considered already a handfulof later passages that detail this commitment, passages in whichLeibniz writes that unlike in the continuum, “in actuals there isnothing indefinite” (1706. G II,282), or that “in real things, namelyin bodies, the parts are not indefinite . . . but are actually assigned acertain way” (1704. G II,268). He also writes in 1705 to PrincessSophie that “the mass of bodies is actually divided in a determinatemanner, and nothing in it is precisely continuous,” and then carriesthe point ahead quite vividly a few lines later:


There are, therefore, always actual divisions and variations in the masses of exist-ing bodies, to whatever degree of smallness one might go. It is the imperfectionof our senses that makes us conceive physical things as mathematical beings inwhich there is some indeterminacy. (1705. G VII,562–3)

There is a bounty of such examples, and even where the texts arenot so explicit in emphasizing how the actuality of matter’s internalstructure of parts and boundaries sets it apart from the indeter-minacy of a continuum, the basic idea that matter is actually divided– indeed, actually infinitely divided – and not just divisible intoparts is a resounding and pervasive theme in the theory of matter.“Created things are actually infinite,” Leibniz writes in a fragmentdated sometime between 1677 and 1685, “For any body whateveris actually divided into several parts” (VE 1129); and similarly: “Isuppose that every body is actually divided into several parts, whichare also bodies” (VE 1127). In a letter to Burcher de Volder hestates unequivocally, “In truth, matter is not continuous but discrete,and actually infinitely divided” (G II,278). And – once again in the31 October 1705 letter to Sophie – the actually infinite divisionof matter is enlisted directly in illuminating the contrast betweenthe determinacy of the structure of matter and the indeterminacyessential to continuity:

The better to conceive the actual division of matter to infinity, and the exclu-sion that there is of all exact and indeterminate continuity, we must consider thatGod has already produced there as much order and variety as it was possible tointroduce there until now, and thus nothing has remained indeterminate there. (GVII,562f.)

It might not be overstating the case to say that the actuality of matteris just about Leibniz’s deepest commitment of all in metaphysics (or,at any rate, a commitment than which none is deeper).Keeping the actuality of matter in mind, it becomes easier to

see how Leibniz can come, even temporarily, to reject the require-ments laid down by the potentiality concept of continuity. We mightreconstruct the path of his thought summarily as follows: (1) Onthe strength of the interpenetration account of cohesion, cohesivematerial bodies are connected, for their parts share common bound-aries; (2) by Aristotle’s dictum (which the new account of cohesionhas now raised to salience) what is connected is continuous; and so,cohesive material bodies are continuous; (3) but matter is actual:

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its parts and boundaries are actually assigned; hence, (4) there areactually assigned parts and boundaries even in a continuum.Yet as I mentioned before, the interpenetration account of

cohesion is fleeting, and once it has gone the connectedness conceptno longer holds full sway in Leibniz’s thought. By autumn of 1676,at the latest, the potentiality concept reëmerges in what is to be itspermanent place at the center of his analysis of continuity.

3.2 Trouble in the Account: Minima and Unconnected Matter

With the potentiality concept thus reinstated, the actuality of matterbecomes decisive in settling the issue about the relationship betweenmatter and continuity. Matter is not continuous. For when continuityis thought of primarily from the point of view of the potentialityconcept, it will follow that matter – as something actual, deter-minate, assigned, &c. – simply has to be seen as a discrete quantityand not a continuous one. And as we have seen that is in fact theview that Leibniz defends over the next thirty or more years of hislife.23But the connectedness concept has not been altogether banished

from Leibniz’s philosophy. On the contrary, a look at a passage froma less familiar (and somewhat less metaphysical) writing of the lateryears turns up a remarkably exact statement of connectedness in thefollowing definition of the continuum:

A continuum is a whole any two of whose co-integrating24 parts (ôr parts whichtaken together coincide with the whole) have something in common, and indeed ifthey are not redundant ôr have no part in common – that is [sive], if the aggregateof the magnitude of the aggregated parts is equal to the whole – then they have atleast some one boundary [terminum] in common. (c. 1686–92? GM VII,284)

So the idea of connectedness does still remain alive in his thought;it has simply slipped back out of the spotlight. And from itsbackground position it manages to leave its imprint forever uponLeibniz’s metaphysics of matter. For when he composes his accountof matter as a discrete quantity rather than a continuous one, whatresults is a theory set in counterpoint to both of the two concepts ofcontinuity we have explored. On his view, not only is matter discreteand not continuous in the sense that it is actual rather than potential,but also matter is not connected. Its parts are always discontinuousthings in Aristotle’s sense: they are those whose boundaries are two.


Each and every part of matter has “its own separate boundaries,”and thus no body made up of them, however cohesive, can truly beconnected.Yet consider how this will be equally true of any mass of matter

you like, for the parts of any mass of matter are themselves alwaysjust more bits of matter, and are therefore discrete rather thancontinuous quantities. Thus having still finer parts of their ownwhose boundaries are two and actually separate, these bits of matterare likewise not connected; and neither are the finer parts of thebits, nor the finer parts of the finer parts, and so on, ad infinitum.Matter as a discrete quantity is fully actual and actually unconnectedeverywhere.That consequence is disastrous for Leibniz’s metaphysics of

matter. It entails, among other thing, that matter ultimately resolvesinto a powder, so to speak, of isolated points; or if not points,exactly, then at least isolated simple indivisible particles – whatLeibniz in various places in the early writings calls ‘minima’. Thisis a deep problem because on his view minima can be only modesof extended things and never parts of them nor things in their ownright; and he is often at pains to declare that a resolution of matterinto isolated minima is impossible. Indeed, as I have argued at lengthelsewhere, much of the design his theory of matter is controlledspecifically by the task of avoiding even the possibility that anypart of matter, however small, should turn out to be minimal.25Matter’s being actually unconnected everywhere, however, shattersthat commitment by requiring the existence of isolated minima orpoint-like particles of matter.

3.3 A Leibnizian Argument for the Commitment to Isolated Minima

It is not difficult to secure a fairly tight argument to demonstrate thisresolution of matter into minima – or at any rate, the existence of atleast some isolated minima (infinitely many!) in matter – and in factwe can comfortably follow the lead of young Leibniz himself. For in1670 he hit upon an intriguing strategy for showing the existence ofinfinitesimal particles in a continuum, and only a slight modificationof that strategy is needed to produce a demonstration of the existenceof isolated minima in a body of matter that is actually unconnectedeverywhere.

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Figure 3.

Here is Leibniz:

Let there be a line ab, to be traversed by some motion. Since some beginning ofmotion is intelligible in that line, so also will be a beginning of the line traversedby the beginning of the motion. Let this beginning of the line be ac. But it isevident that dc can be subtracted from it without subtracting the beginning. And ifad is believed to be the beginning, from it ed can be subtracted without subtractingthe beginning, and so on ad infinitum. For even if my hand is unable and my soulunwilling to pursue the division to infinity, it can nevertheless be understood atonce that everything that can be subtracted without subtracting the beginning doesnot involve the beginning. And since the substraction can be done to infinity (forthe continuum, as others have demonstrated, is divisible to infinity), it followsthat the beginning of the line, i.e. that which is traversed in the beginning of themotion, is infinitely small. (1672–3, A VI,3,98–9)

Leibniz’s tactic of inverting the pattern of Zeno’s celebrated dicho-tomy paradox works to dazzling effect, and it can be transposed toan argument for the existence of an isolated extremum. Though theargument will only be sketched here, it will, I think, be sketched inenough detail to indicate how a fully rigorous statement would run.Relying on Leibniz’s diagram above, suppose the line ab is

unconnected everywhere, i.e. that ab itself and every part of abcan be exhaustively decomposed into two parts neither of whichincludes any limit points of the other. We can show that theextremum a (the left-hand endpoint) is an isolated element of ab– that is, we can show that the extremum a can be separated fromthe rest of ab. Since ab is unconnected everywhere, the segment cbcan be separated from any segment that includes a – indeed, froma metaphysically rigorous point of view cb is already actually soseparated, and thus we shall say that a is isolated from cd. Likewise,the segment db can be separated from any segment that includesa, so a is isolated from db. The same holds for eb, and indeed forany proper right-hand segment of ab you like, for ab is unconnectedeverywhere. The extremum a is therefore isolated from every properright-hand segment (every proper right-hand part) of ab, for “evenif my hand is unable and my soul unwilling to pursue the separation


to infinity, it can nevertheless be understood at once that everythingthat can be separated from a segment that includes the extremum ais isolated from a.” But every segment of ab overlaps some properright-hand segment or other of ab; ab itself is nothing more thanthe sum of its proper right-hand segments, plus the extremum a– the extremum a is the only element of ab that does not overlapany proper right-hand segment of ab. Since a is isolated from eachof those right-hand segments, a can be separated from the sum ofthem all, i.e. a can be separated from the rest of ab. Indeed, froma metaphysically rigorous point of view, it is already actually soseparated. Hence the extremum a is an isolated element of ab.Obviously the argument can be applied equally to any point of

division that can be assigned in ab, and according to Leibniz therewill be actually infinitely many such points of division in ab – justas there will be in any quantity of matter. Consequently his accountis committed to the existence of infinitely many isolated minimaexisting in any quantity of matter.I suggested a little while ago that matter, as an unconnected

quantity, would actually be divided into finer and finer parts ad infin-itum. That claim should be qualified somewhat. The actual divisionof matter into an infinity of ever finer parts only follows frommatter’s being unconnected if we additionally suppose that no partof matter can be perfectly simple in the sense of being minimal ortotally without parts. (A minimal part would be trivially connected,since there is no way to partition it into two nonempty disjoint partsthat have no limits in common.) Now, as I said, it is in fact anelement of Leibniz’s view that there are no minimal or partless partsof matter. So he will in fact be committed to the consequence that allof matter – every single bit of it however small – is actually dividedeverywhere into an infinity of still smaller parts. And with this sortof infinite division in place, some further problems for his meta-physics of matter are going to arise. For example, by Leibniz’s ownlights, infinite division everywhere should have as a consequencethe proposition that there isn’t any matter.26 Also, under yet otherLeibnizian principles about parts and wholes, infinite division every-where will entail that matter is subject to a vicious regress of partswithin parts ad infinitum.27 But as those “infinity problems” arenot immediate consequences of matter’s being unconnected every-

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where, I shall be content at this point simply to raise them ascharges and rely on previous discussions to fill them out. For theconcerns of the present paper focus narrowly around the concepts ofcontinuity and the way Leibniz’s account of the structure of matteris immediately responsive to those concepts.With the difficulties facing the Leibnizian account of matter

before us, it is interesting to note that Leibniz was not forced intoa commitment to the proposition that matter is unconnected every-where. He commits himself to it by reacting to the connectednessconcept when he formulates the conditions that a part of matter mustsatisfy in order to be discrete and not continuous. But in doing so heoverreacts. His real basis for holding that matter is discrete and notcontinuous is that whereas continuity requires potentiality, matter is“fully actual.” But the actuality of matter carries no special prohibi-tion against matter’s being connected. Or at least it’s not just obviousthat a connected magnitude must somehow fail to be fully actual.Once we have this idea clearly in mind, a new possibility emerges.Why not take matter to be a fully actual connected quantity? Thatis, why not propose that matter has a variety of parts and boundariesactually assigned in it, and that the neighboring parts of cohesivebodies share a common boundary at their locus of contact?On a proposal of this sort matter would still come out as discrete

and not continuous, and the difficulty about matter resolving intoa powder of isolated minima would arise because one would notbe committed to matter’s being unconnected everywhere. WhenLeibniz demands that matter as a discrete quantity must not beconnected, he is essentially treating connectedness as a sufficientcondition for continuity. But once the potentiality concept is backin place as the focal point in his analysis of continuity – as it isafter 1676 – there is no longer any compelling reason to think thatconnectedness must be sufficient for continuity. By holding matterto be unconnected as well as fully actual, Leibniz effectively closesoff what should have been a wide-open possible view of its structure.But alas, even this “possible view” of the structure of matter maynot truly be open to Leibniz, for quite a different reason: an actu-ally connected quantity whose contiguous parts share their commonboundaries would run into conflict with Leibniz’s denial of middlestates – if, that is, his denial of middle states in fact extends widely


enough to include a denial of shared boundaries. Perhaps it does notextend so widely; but just as likely his various commitments at thispoint simply cannot be brought into perfect balance.In the end I suspect Leibniz is never truly or completely aware

of the presence of two different continuity concepts in his thought,and thus I suspect that he simply sees his denial of the connec-tedness of matter as part and parcel with his denial that matteris a continuous quantity. Had he sorted out properly those twoconcepts of continuity, perhaps Leibniz would have achieved a moresatisfactory theory of the structure of matter. Having sorted themout ourselves, I think we can begin to see with more clarity justwhat that theory actually is, and more importantly, just what Leib-niz’s thoughts actually are on the distinction between matter andcontinuity.28

NOTES

1 The open territory here remains very large. In this paper I shall leave aside(among the many related topics in Leibniz) many of the nuances of Leibniz’slater thought concerning continuity and continuous orderings, his views about theconstruction of points, lines, and planes, his account of a link between continu-ity and possibility, etc. For a good discussion of some aspects of Leibniz’s lateraccounts of continuity not taken up in the present paper, see Timothy Crockett,“Continuity in Leibniz’s Mature Metaphysics,” also in this issue.2 I am responsible throughout for translations of Leibniz and Descartes, butin translating Descartes I have consulted Cottingham, Stoothoff, and Murdoch,eds., The Philosophical Writings of Descartes Vol. 1 (Cambridge: CambridgeUniversity Press, 1985), and in translating Leibniz, Leroy Loemker, ed., Philo-sophical Papers and Letters (Dordrecht: Kluwer Academic Publishers, 1969)and R. Ariew and D. Garber, eds., Philosophical Essays (Indianapolis: HackettPublishing Co., 1989). Also, for Leibniz’s writings in A VI,3, I have consultedmanuscripts of Richard Arthur’s forthcoming translation volume, Leibniz andthe Labyrinth of the Continuum: Leibniz’s Writings on the Continuum 1672–1686, to be published in the Yale Leibniz Series. I abbreviate the primary textsthus: A = Berlin Academy Edition, Samtliche Schriften und Briefe: Philosoph-ische Schriften, Series VI, Vols. 1–3, 6 (Berlin: Akademie-Verlag. 1923–80);AT = Adam and Tannery, eds., Oeuvres de Descartes, revised edition (Paris:Vrin/C.N.R.S., 1964–76); C = Couturat, ed., Opuscules et Fragments inédits deLeibniz (Paris: Alcan, 1903); G = Gerhardt, ed., Die Philosophischen Schriften,Vols. 1–7 (Berlin: Weidmannsche Buchhandlung, 1875–90); GM = Gerhardt,ed., Mathematische Schriften von Gottfried Wilhelm Leibniz, Vols. 1–7 (Berlin:A. Asher; Halle: H.W. Schmidt, 1849–63). References to AT, G and GM are

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to volume and page numbers; references to C and VE are to page numbers;references to A are to series, volume and page numbers. Also, I follow Arthurin adopting a notational convention (initiated by Edwin Curley) for translatingthe Latin term seu, which (like sive) corresponds to the English term ‘or’ where itimplies equivalence: seu = or in other words, that is to say, etc. In the translationsof the Latin, I mark the occurrences of this “ ‘or’ of equivalence” in Leibniz’s textwith a circumflex, thus: ôr.3 The earliest passage I know of where Leibniz explicitly defends the plenumhypothesis is A, VI,525 dated to March 1676. And there he takes pains to pointout that the hypothesis is not merely presupposed; rather, he argues for it. Butthe view of nature as a plenum seems clearly to be a backdrop for many of hisdiscussions from 1671 to 1672 onward.4 In fact in this letter Leibniz presents two possible ways in which discontinuitymight arise: “first, in such a way that contiguity is at the same time destroyed,when the parts are so pulled apart from each other that a vacuum is left; or insuch a way that contiguity remains” (A VI,2,435). His ensuing discussion makesit clear that it is the latter possibility which keenly interests him, and that it is theone he holds to be how discontinuity is actually introduced into matter.5 Aristotle’s definition occurs at Physics 231a21, 227a10–15, and Metaphysics1069a5–8.6 Perhaps we should say it is a mere contiguum: a contiguum that is not also acontinuum. For on the usual Aristotelian account that Leibniz observes here, thecontinuous is a special case (the limiting case) of the contiguous: the continuousare both “those whose boundaries are together” and “those whose boundaries areone.” See, for example, Aristotle, Physics 227a10f., and Leibniz’s remarks aboutthe continuous and the contiguous at A VI,3,94, and 537.7 It’s hard to capture the subtle imagery at work here in a close translation.Leibniz’s materia turbans evokes a picture of a body of matter as a disorderlycrowd: a multitude of agitating individuals, each with its own distinctive actions,yet nonetheless collectively displaying a sort of unity by participating in a greatercommon motion.8 In a piece whose date is not clearly established (the Akademie editors place itanywhere between 1677–1695), Leibniz writes in much the same vein: “The prin-ciple of cohesion is harmonizing motion [motus conspirans], and that of fluidityis varying motion” (VE 495).9 This is to remain Leibniz’s view, in the slightly evolved form of the thesis thatnothing is absolutely fluid or solid, but rather that all matter is to some degreeor other pliant, and that motion accounts for this. In a piece dated to 1683–6,“On the Existing World,” he writes: “Therefore it must be said that no point canbe assigned in the world which is not set in motion somewhat differently fromany other point however near to it, but, on the other hand, that no point can beassigned which does not have some motion in common with some other givenpoint in the world; under the former head, all bodies are fluid; under the latter, allare cohering. But to the extent that a common or proper motion is more or less


observable, a body is called on solid, or a separate body, or perhaps even a fluid”(VE 420).10 There’s a lot of text here. For an overview, see Hartz, G. and Cover, J.“Space and Time in the Leibnizian Metaphysic,” Noûs 22 (1988): 493–519. Seealso, J.E. McGuire, “ ‘Labyrinthus Continui’: Leibniz on Substance, Activityand Matter,” in Machamer and Turnbull (eds.), Motion and Time, Space andMatter (Columbus: Ohio State University Press, 1976) 290–326; Glenn Hartz,“Leibniz’s Phenomenalisms,” The Philosophical Review 101 (1992): 511–49;Richard Arthur, “Russell’s Conundrum: On the Relation of Leibniz’s Monadsto the Continuum.” In Brown and Mittelstrass (eds.), An Intimate Relation(Dordrecht: Reidel, 1989) 171–201; and Chapter III of my “Matter, Unityand Infinity in Early Leibniz” (Ph.D. dissertation, Syracuse University, 1997).Although the period of Leibniz’s writings that I have in mind here is a vaguelybounded one, spanning more than a decade, I believe the purest expressions ofthe views hat he holds during this time emerge near its end, in the years closelysurrounding 1705. For that reason, and for brevity, I shall refer to the accountsof matter and continuity that Leibniz defends in this period as belonging to “the1705 metaphysics.”11 Elsewhere is “Leibniz’s Constructivism and Infinitely Folded Matter,” forth-coming in Genarro and Huenemann (eds.), New Essays on the Rationalists(Oxford University Press, 1999). Also, in section 1 of that paper the modalizedspin on continuity is discussed briefly.12 More precisely, it’s something that is not a feature of the Cartesian view asLeibniz would have seen it. For in Descartes’ Optics one can find an ideas ofan “action or tendency to move” (which “it is necessary to distinguish from”movement itself) that could naturally be read as a cousin concept to Leibniz’sconatus (cf. AT VI,88). But while Leibniz certainly read Principles 1–2 closely,there is no clear evidence to suggest that he is (in 1670–2, at any rate) aware ofthat particular element of Descartes’ views in the Optics. My thanks to AlisonSimmons for bringing that element of the Optics to my attention.13 In one important sense, however, the conatus physics does stay alive. Thepicture of the material world as invested throughout with centers of motion – thepicture of the conatus world with its principles of action that are in each of theinfinitely many parts of matter and ground their physical properties – this willquietly keep its grip on Leibniz’s thought for some time, and rearise explicitly inhis later efforts to found physics upon a metaphysics of immaterial and “active”first principles: another role that is to be ascribed to the monads. In this way onemight say that the ontology of conati “transforms” into the ontology of monadsin the later years. But as a basic account of physical phenomena – of motion,cohesion, impact, and so on – the early theory of endeavors has run its course bylate spring of 1676. For some (albeit brief) discussion of Leibniz’s rejection ofinfinitesimals see fn. 8 of my “Leibniz on Mathematics and the Actually InfiniteDivision of Matter,” The Philosophical Review, Vol. 107, No. 1 (January 1998).14 Michael White is the first to make this point that the concept of a seamlessor natural whole finds expression in the contemporary topological concept of

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connectedness. For a fine discussion of some related issues in Aristotle’s philo-sophy (including Physics 227a6 and 227a10–15 that I quote below on pages 24–5)and the philosophy of topology, see his “On Continuity: Aristotle Versus Topo-logy?” History and Philosophy of Logic, Vol. 9, No. 1 (1988), 1–12.15 I appeal to point-set topology in order tomotivate in a clear way the concept ofconnectedness; I am not proposing here to model Leibniz’s account of matter andcontinuity in that framework. The latter project would not be a straightforwardone, for (inter alia) the idea of “indistant” contiguous closed regions would notbe consistent with the usual metric topology. That is, the usual way of defining ametric on a topological space won’t allow for two distinct surfaces to be at “nodistance” from one another (though a non-standard approach might still do it).More importantly, however, the ontology of point sets would simply be anathemato Leibniz. The right approach to capturing Leibniz’s account would, I suspect,require a model in combinatorial topology. I discuss this approach in a workin progress, tentatively entitled “Discontinuity and the Structure of Motion inLeibniz’s Metaphysics.”16 Cf. James Munkres, Topology (Englewood Cliffs, NJ: Prentice Hall, 1975);Michael Henle, A Combinatorial Introduction to Topology (New York: Dover,1979); Paul Alexandroff, Elementary Concepts of Topology (New York: Dover,1961).17 % is a set of open subsets of space S.(1) The whole space S and the empty set are elements of %.(2) The union of any number (finite, countable or uncountable) of sets of from %

is in %.(3) The intersection of any finite number of sets from % is in %.18 We shall assume throughout that all the point spaces discussed have the “usualtopology” for Euclidean spaces, namely, the order topology and its self-products.19 In fact, this definition of a limit point is broad enough to include all the interiorpoints of a set as limit points, as well as those that fall on the boundary. For ourpurposes, we shall be considering only those limit points that are also boundarypoints (i.e. those limit points that are not also interior points).20 A separation of a topological space S is more usually defined as a pair ofnonempty, disjoint, open sets S and V such that the union (U & V) = S. Ourdefinition in terms of limit points is equivalent, but brings out more clearly theissues of our immediate concern.21 Leibniz’s fullest expression of the argument for this conclusion, which I won’trehearse here, occurs near the outset of Pacidius Philalethi (cf. A, VI,3,534–41),though when he writes the Pacidius in autumn of 1676 he has been warming upto that line of argument for some time.22 It should be noted that in a passage near the end of the Pacidius’smain discus-sion of middle states there is a remark that poses a prima facie challenge to mysuggestion here that the argument against middle states might be seen as rulingout the existence of common boundaries and with it the related account of cohe-sion. The primary interlocutor, Charinus, while offering an example of a perfectly


round sphere resting on a perfectly flat table, says in passing: “It is clear that thesphere does not cohere with the plane, and that they have no extrema in common,otherwise one would not be able to move without the other” (A VI,3,537). Thiscertainly suggests that the account of cohesion as boundary-sharing is in the air(even if it’s not exactly asserted), and the interlocutors do not go on later to pointout that the argument against middle states defeats that account of cohesion. But inmy view, also to be defended in “Discontinuity and the Structure of Motion,” thisaccount of cohesion, like so many of the ideas mentioned early in the Pacidius, isrehearsed as a feature of Leibniz’s previous thought, and it is not an element ofthe most considered views that are ultimately endorsed in the dialogue; moreover,Leibniz’s final commitments in the Pacidius, and the overall gist of his remarksabout middle states and continuity that occur as the dialogue progresses, do seemto militate against the account of cohesion as boundary-sharing.23 This is not to say that the texts are always unwaveringly clear on this point.As Robert Adams has pointed out, Leibniz can be “quoted on both sides ofthe question whether bodies are continuous or whether they only appear to becontinuous” (Adams, Leibniz: Determinist, Theist, Idealist. (New York: OxfordUniversity Press, 1994), p. 233). Yet I do not believe there is a very seriousworry about Leibniz’s views even in most of the passages (after 1676) in whichhe can be quoted apparently on the other side of the question. In the yearsshortly following 1676 Leibniz will sometimes call matter ‘continuous’, but Ithink it is fairly clear in those contexts that he only means that matter is notinterrupted by void spaces (which accords with a use of ‘continuous’ that hementions occasionally in 1676; cf. A VI,3,542). Adams cites a troubling passageat G IV,394, from 1702, where Leibniz seems to say outright that body [corpus]is continuous. Though continuity itself does seem to be at stake there, it isless than clear to me that in the crucial sentence Leibniz is characterizing thenature of actual material bodies and not, say, an abstract concept of “body” thatsimply does not weigh-in on the question whether actual bodies are ultimatelydiscrete or continuous. Still, even if Leibniz wavers in places, his consideredview about material bodies is abundantly clear and a deeply committed feature ofhis metaphysics.24 The Latin term cointegrantes derives from integro = “to make a whole,” soLeibniz’s subsequent gloss on “co-integrating parts” is in fact nicely capturedalready in the base meaning of his technical term, though that does not comethrough in my translation.25 Seemy “Leibniz onMathematics and the Actually Infinite Division ofMatter,”op. cit.26 The relevant principles are: (1) No infinite aggregate can form a true whole, orbe truly one; and (2) what is not truly one does not truly exist (cf. G II,97,251).Thefirst of those is especially motivated by the link between the concept of an infinitewhole and Galileo’s paradox. I take this up in detail in “Leibniz on Mathematics,”op. cit.27 Leibniz’s own statement of the regress problem – and the argument it yieldsfor the existence of immaterial substantial unities – can be found at, for example,

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G II,261, 267, 296. I discuss the regress problem together with some others(including the “unity problem” mentioned in the previous footnote) in “Leibniz’sConstructivism and Infinitely Folded Matter,” op. cit.28 Earlier versions of this paper were presented at Dartmouth College andat the March 1998 Pacific Division meetings of the American PhilosophicalAssociation. My thanks to both audiences for helpful input. Special thanks alsoto Christie Thomas and Walter Sinnott-Armstrong for providing comments onearlier written drafts, and to John O’Leary-Hawthorne, Jan Cover and TimothyCrockett for helpful discussion.

Department of PhilosophyDartmouth CollegeNew Hampshire, USA

Matter and Two Concepts of Continuity in Leibniz

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