6

Dans les corps il n'y apoint deßgureparfaite:

L e i b n i z o n T i m e , Change , a n d C o r p o r e a l

Substance

S A M U E L L E V E Y

I .

T h e Cartesian theory o fcorporea i substance is grounded i n the simple

and beautiful idea that b o d y is to be understood completely i n terms

o f extens ion—that is, that the attribute o f extension constitutes the

w h o l e nature or essence o f corporeal substance. Descartes advertises

this thesis i n a famous passage f r o m the Principia Philosophiae:1

Each substance has one principal property that constitutes its nature and essence, and to which aU its other properties are referred. Thus extension in length, breadth, and depth constitutes the nature o f corporeal substance . . . Everything eke that can be attributed to body presupposes extension, and is merely a mode ofan extended thing. (Prinapks i . 53; A T viiia. 25)2

T h e modes o f the extended t h i n g that Descartes goes o n to

m e n t i o n are shape and motion, w h i c h he says are 'uninteUigible' apart

1 This 'georaetrical' conception of body is evident through many of Descartes's earHer writings, as wefl, for instance, in Rule 12 o{Ruksfor the Direction of the Mind (c.1628), where 'shape, extension and motion' are akeady being promoted as the 'simple natures' essential to body (cf. A T x. 419), and Rule 14, where Descartes, after declaring 'we should be delighted to come upon a reader favorably disposed towards arithmeric and geometry',moves immediately to a discussion of the geometrical nature of extension in which 'body' and 'that which is extended' are rapidly identified with 'extension' itsetf (cf. A T x. 441 ff.). For a good discussion ofDescartes on the nature ofbody, see Daniel Garber, Descartes' Metaphysical Physics (Chicago: University of Chicago Press, 1992), ch. 3, esp. 6 3 ^ , 7 S - 9 3 - See ako Margaret Wikon, Descartes QLondon: Routledge, 1978), 8 3 - 8 , 166-9.

2 Translations of Descartes foUow CSM 1; translations of Leibniz foUow AG, L, and, especiaUy, Richard Arthur's translation volume, LOC. I have sometimes made modifications without comment. Inaccuracies are my own.

Leibniz on Time, Change, and Corporeal Substance i 4 7

f r o m extension. Extension itseff is to be 'distinctly recognized as

constituting the nature' o f body, and indeed o f corporeal substance,

for ' i n this way w e w ü l have a very clear and distinct understanding'

o f b o d y (Principles i . 63; A T vi i ia . 30—1). Thus the centerpiece o f

the Cartesian metaphysics o f body is the idea 'that the nature o f

corporeal substance consists simply i n its being something extended'

(Principles i i . 19; A T vi i ia . 51). T h e underlying hope is, ofcourse, that,

w i t h the principal component o f the theory o f corporeal substance

simply being the theory ofextension, our understanding o f t h e nature

o f body w i U finaUy be guided b y the concepts arising i n the most

perfectly conceived and perfecth/ understood disciphne i n aU o f h u m a n

k n o w l e d g e — n a m e l y , the science ofgeometry .

Critics inc luding Leibniz have been quick to p o i n t out that this

'geometrical' conception o f body, for aU its dar i ty and precision,

cannot denver aU. the concepts that w i U be demanded by a cor­

rect physics o f bodies (notable among them, inertia and antitypy)

and, therefore, that the Cartesian theory o f corporeal substance is

at best incomplete as a total account. 3 B u t i t is Leibniz alone w h o

criticizes the Cartesian account for relying o n concepts that are not clearly understood and that in themselves cannot be true o f the nature

3 To pick a few famous examples: Locke on what he caUs 'soHdity': 'Sobdity is so inseparable an idea from body that upon that depends its fiUing ofspace, its contact, impuke and communicadon of motion upon impuke,' where, by contrast, 'extension includes no sohdity, nor resistance to the motion ofbody, as body does'( Essay II. xiii. 11, 12); and Leibniz on antitypy, speaking here through the character Phuarete in his 1711 dialogue 'Conversation of Philarete and Ariste': 'Philosophers who are not Cartesians do not agree that extension is sufficient to form body; they ako require something eke that the ancients caU antitypy, that is, that which renders a body impenetrable to another' (GP vi. 580); Leibniz on inertia: 'I discovered that this, so to speak, inertia ofbodies cannot be deduced from the initiaUy assumed notion ofmatter and motion, where matter is understood as that which is extended or fiUs space, and motion is understood as change of space or pfoce' (A V I . iv. 1980). And, finaUy, Leibniz in the Discours de Metaphysique, article 18: 'Force is something different from size, shape, and motion and it can be judged from that that not aU that is conceived in body consists uniquely in extension and its modifications, as our modems are persuaded' (A V I . iv. 1559). For some recent discussions, see Garber, 'Leibniz and the Foundations of Physics: The Middle Years', in Kathleen OkruhHk and James Robert Brown (eds.), The Natural Philosophy of Leibniz @3ordrecht: D. Reidel, r985), 2 7 - 1 3 0 , at 30, 78 fF.; Robert Sleigh, Leibniz and ArnauU: A Commentary on their Correspondence [Leibniz and Amauld] fNew Haven: Yale University Press, 1990), 166 fF.; Bernard WiUiams, Descartes: The Project of Pure Enquiry ^London: Penguin Books, 1978), 228 fF.; and R . S. WooU10use, Descartes, Spinoza, Leibniz: The Concept of Substance in Seventeenth Century Metaphysics @London: Routledge, 1993), 94—6, 102—15.

148 Samuel Levey

o f corporeal substance; indeed, as he puts i t i n Art ic le 12 o f the

Discours de metaphysique ( i 6 8 6 ) , 'the notions involved i n extension

contain something imaginary and cannot constitute the substance o f

a body ' (A V I . i v . 1545). W h a t was supposed t o be the irreproach­

able geometry o f extension i n fact presents an inherenth/ flawed

basis for understanding the nature o f actual bodies; its fundamental

concepts are n o t so transparent after aU. This b o l d strand o f Le ib­

niz's critique appears i n several texts f r o m the 1680s i n w h i c h he

advances various forms o f an argument concerning bodüy shape to

reach the conclusion that the Cartesian attribute o f extension is n o t

something that apphes to actual bodies as they are absolutely o r

i n themselves, b u t rather i t is on ly something ' i m a g i n a r y ' — t h a t is,

something that is n o t i n things outside o f us, b u t only appears to

be (cf. A V I . iv . 70). Consider t w o such passages from this period,

the first from a piece w r i t t e n i n 1683 and t i t led ' M i r a de natura

substantia corporea' ( 'Wonders Concerning the Nature o f Corporeal

Substance'):

Even th0ugl1 extension and motion are more distinctly understood than other quahties, since aU the rest have to be explained using them, i t must in fact stiU be acknowledged that neither extension nor motion can be understood distincdy by us at aU. This is because, on the one hand, we are always embroUed i n the difficulties concerning the composition of the continuum and the infinite, and, on the other, because there are i n fact no precise shapes [certaefigurae] i n the nature ofthings, and consequendy no precise motions [certi motus]. A n d just as color and sound are phenomena, rather than true attributes of things containing a certain absolute nature without relation to us, so too are extension and motion. (A V I . iv. 1465)

T h e second is from the 1686 piece 'Specimen i n v e n t o r u m de a d m i -

randis naturae generahs arcanis' ( 'A Specimen o f Discoveries o f the

Admirable Secrets o f N a t u r e i n General'):

hideed, even though this may seem paradoxical, i t must be reatized that the notion o f extension is not as transparent as is commonly beHeved. For from the fact that no body is so very smaU that i t is not actuaUy divided into parts excited by different motions, i t foUows that no detemunate shape can be assigned [assignari] to any body, nor is an exacdy [exactum] straight line, or circle, or any other assignable shape of any body, found in the nature of things . . . Thus shape involves something imaginary, and no other sword

Leibniz on Time, Change, and Corporeal Substance 149

can sever the knots that we tie for ourselves by a poor understanding of the composition ofthe continuum. (A V I . iv. 1622)

I f t h i s argument concerning extension, shape, and the c o n t i n u u m has

attracted only sparse attention over the years from Leibniz's c o m ­

mentators, 4 its basic points are nonetheless famihar from many o f h i s

wr i t ings . 5 T o offer just the simplest oudine, Leibniz's argument is

c u m u k t i v e l y estabhshed across these three points, each being used as

a p k t f o r m to reach the next:

(I) There are n o precise shapes i n actual bodies, because o f the

subdivision o f t h e c o n t i n u u m t o in f in i ty .

(lT) T h e notions ofshape, m o t i o n , and extension are not i n things

outside us b u t rather are only imaginary, hke those o f color,

heat, and sound.

( I I I ) Shape, m o t i o n , and extension are n o t quahties that can consti­

tute substance.

As is evident from the passages above, Leibniz w i U typicaUy add as

weU that extension and m o t i o n are n o t as distincdy understood as

is c o m m o n l y beheved and suggest that aU this arises from 'a p o o r

understanding' ofdi f f icult ies o f t h e composit ion o f t h e c o n t i n u u m .

U n d e r the Cartesian attribute o f extension, Leibniz seems to say,

corporeal substance can neither be nor be distincdy conceived. A n d

what he often does say is that, i f there were n o t h i n g but m o t i o n

and extension i n t h e m , bodies w o u l d n o t be substances but only

phenomena 'uke rainbows and m o c k suns' (A V I . i v . 1648; cf. A V I .

i v . 1464—5). Some ofLeibniz ' s most subde philosophy is involved i n

assembhng the reasoning that leads from (I) to (II) to ( I I I ) i n this hne o f

argument about precise shapes and imaginary notions; as a w h o l e and

i n each o f its steps i t merits close study. B u t the focus o f the present

chapter w i U fäU only o n its first po int , Proposition (I ) , that there are no

* But see Sleigh, Leibniz and Amauld, n o fF.; Robert Adams, Leibniz: Deierminist, Tkeist, Idealist [Leibniz] 0Sfew York: Oxford University Press, 1994), 229 fF.; Phüip Beeley, 'Math­ematics and Nature in Leibniz's Early Metaphysics' ['Mathematics and Nature'], in Stuart Brown (ed.), The Young Leibniz and his Philosophy (1646—76) flDordrecht: Kluwer, 1999), 123-45, M 129-32; and see RichardArthur's 'hitroduction' in L O C . I discuss this argument in 'Leibniz on Precise Shapes and the Corporeal World', in Donald Rutherford andJ. A. Cover (eds.), Leibniz: Nature and Freedom ^>JewYork: OxfordUniversity Press, 2005), 6 9 - 9 4 .

5 Cf. A V I . iv. 1545, 1622, 1648; GP ii. 77, 9 8 - 9 .

i 5 0 Samuel Levey

precise shapes i n things, because o f t h e subdivision o f t h e c o n t i n u u m

to i n f i n i t y , and i n particular o n j u s t one ofLeibniz ' s arguments for i t .

2 .

So, n o w : w h y , and i n what sense, are there n o precise shapes i n bodies?

A n d what has this t o do w i t h the subdivision o f the c o n t i n u u m to

infinity?

Leibniz has t w o central arguments for the c k i m that there are n o

precise shapes i n bodies, and, as advertised, each rests o n considerations

about the inf in i te subdivision o f t h e c o n t i n u u m . O n e o f those t w o is

reaLly the primary argument, showing up i n aU o f t h e m a i n texts, and i t

concerns the division ofbodies i n t o parts. B y contrast the other is o n l y

a secondary argument; i t concerns the structure o f time and change and

occurs just once, i n a short and somewhat puzzhng text from i 6 8 6 ,

'Dans les corps i l n ' y a p o i n t de figure parfaite' or 'There is N o Perfect

Shape at aU i n Bodies.' I t is principaUy this document ^iereafter: ' N o

Perfect Shape') and the secondary argument i t contains, concerning

t i m e and change, that w i U be under scrutiny i n what foUows, t h o u g h

there w i U be several occasions o n w h i c h w e shaU appeal to a significant

earUer w r i t i n g that abo addresses the nature o f t i m e and change i n

order to flesh o u t the argument o f ' N o Perfect Shape.' A n d to begjn

the discussion, a sketch o f the pr imary argument concerning the

divis ion ofbodies i n t o parts w i U first be i n order.

I n the pr imary argument for Proposit ion (Г), Leibniz typicaUy

contends that, since the universe o f m a t t e r is i n essence 'a vessel fiUed

Qj/eno] w i t h h q u i d ' , the motions o fany b o d y w i U be 'propagated' across

the w o r l d to make at least some impact o n every other body (A V I .

i v . 1646—7). A n d thus every b o d y is constandy buffeted b y an i n f i n i t y

o f i m p u k e s transmitted t h r o u g h the p l e n u m and raining d o w n o n i t

from aU sides. T h e result o f t h i s ongoing exposure to aU the motions

i n the universe is that 'there is n o b o d y so smaU that i t is not actuaUy

subdivided' (A V I . iv . 1647), and indeed every p a r t i c u k r body w i U be

actuaUy subdivided i n t o an i n f i n i t y o fd is t inct parts. 6 B u t n o particular

6 The argument evidently supposes that bodies cannot undergo elastic deformation, in the sense that they cannot change shape without being actuaUy divided into distinct parts; this is a steady underlying feature ofDescartes's own view ofbodies (cf. Pnndpks ii. 34—5,

Leibniz on Time, Change, and Corporeal Substance 151

geometrical shape can exactly describe the inf initely complex actual

structure ofsuch inf inite ly divided bodies, ' for none can be appropriate

[satisfacere] for an i n f i n i t y ofimpressions' (A V I . i v . 1648). T h e shapes

embedded i n traditional g e o m e t r y — e v e n i n its rekt ive ly sophisticated

early m o d e r n f o r m studied and i n part developed by Descartes and

Leibniz (inter alios)7—can offer only approximate and necessarily

inexact or imprecise representations o f actual bodies. 8 That is the

sense i n w h i c h 'there are no precise shapes i n bodies': i t is not that

bodies are themselves indeterminate or imprecise w i t h respect to

shape, b u t rather that geometrical shapes are imprecise w i t h respect to

actual bodies.

' N o Perfect Shape' opens w i t h a version o f t h i s argument. Leibniz

considers, as a simphfied case o f a b o d y w i t h a precise geometrical

shape, a straight hne, and he argues that no such t h i n g actuaRy exists

i n nature:

There is no precise and ftxed [arrestoe] shape i n bodies because of the actual

division ofits parts to infiruty.

A T viiia. 59—60) and one that Leibniz appears to accept. See my 'Leibniz on Mathematics and the ActuaUy Irrfinite Division of Matter' ['Leibniz on Mathematics'], Phihsophical Review, 107 (1998), 4 9 - 9 6 , at 5 1 - 2 .

7 For some discussion, see Emily Groshok, Cartesian Method and the Problem of Reduction [Cartesian Method] (Oxford: Clarendon Press, 1991), esp. chs. 1 and 2; and see Paulo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century [Phihsophy ofMatkematia] @^JewYork: OxfordUniversityPress, 1996), ch. 3.

8 As wiU be suggested below, the account of those Leibnizian actual bodies appears to describe them as fractal in structure, and, if so, then it is in fact true that they cannot be exacdy described in terms oftraditional geometrical shapes. See my 'The Interval ofMotion in Leibniz's Pacidius Phihdethi pnterval ofMotion]' Nous 37 (2003), 371-416. There are further strands in Leibniz's thought about why precise shapes are imaginary that involve the activity of the imagination in (1) phstering over non-uniformities in things that faU below the threshokl of conscious perception (and are detected only by 'petite perceptions') to produce the appearance of uniform precise shapes in nature, and in (2) imagining 'fictive' or ideal hmit entities ofmathematical constructions, such as imagjning a circle as a polygon with infinitely many sides—whose sides 'appear only indistincdy' and then, by a further extrapolation, disappear altogether to form a shape with no sides at aU (e.g., Leibniz writes: 'For entities of this kind, i.e., polygons whose sides do not appear distinctly, are made apparent to us by the imagination, whence there arises in us afterwards the suspicion of an entity having no sides. But what if that image does not represent any polygons at aU? Then the image presented to the mind is a perfect circle', A V L iii. 499). A close examination ofLeibniz's nuanced attitudes toward the hmits ofmathematical constructions would be demanded by a fiJl account. My thanks to Richard Arthur for his very helpful suggestions on this point at the Florence Leibniz Workshop in November 2000 and in private correspondence.

152 Samuel Levey

Let there be, for example, a straight Hne ABC. I say that i t is not exact [qu'ette n'estpas exacte]. For w i t h each part ofthe universe sympathizing wi th aU the others, i t is necessarily the case that, i f the point A tends along the straight iine AB, the point B should have a tendency i n another direction. For w i t h each part A striving to carry wi th i t aU the others, but particukrly the nearest B, the direction o f B wiU be composed of that o fA, together w i t h some others; and it is not at aU possible that B, which is indefinitely close to A, should be exposed to the whole universe in precisely the same way as A , i n such a way that A B composes one whole that has no subdivision. (A V I . iv. 1613)

There is a httle diff icuky w i t h this passage, as Leibniz seems to shift

between talking about A and B as 'points' i n the hne and taUting

o f t h e m as 'parts' o f the hne (not an equivalence that can be safely

assumed and moreover one that his official theory, earfy and iate,

exphcitry denies: points are never t o be parts ofHnes but onh/ bounds

o f t h e m ; cf. A V I . i v . 1648; GP i v . 491; A V I . v i . 152; GP i i . 370,

520, etc.). B u t the gist o f the argument is clear enough. Since the

different parts o f any hne w i U be exposed differendy to the motions

o f the w h o l e universe, n o Une, however smaU, could be u n i f o r m

and undivided, as the 'exact' straight hnes o f tradit ional geometry

are imagined to be. Actual h n e s — s u c h as the edges ofbodies or the

paths carved out b y the vertices o f m o v i n g objects—must instead be

div ided i n t o distinct parts that i n fact deviate from one another i n their

directions, however subdy, and thus at best can onh/ appear to f o r m

a single exacdy straight hne (cf. A V I . i i . 255). Generahzing this result

from hnes to bodies, w e find that n o actual b o d y could be u n i f o r m

and undiv ided i n any part, even i fbodies m i g h t sometimes offer the

appearance o f b e i n g undiv ided wholes. T h e constant flux o f t h e finer

sub-parts i n fact always k y s d o w n an inf ini te subdivision o f t h e w h o l e

body, and, once again, this makes the assignment o f a 'precise and

fixed shape' impossible.

A t this p o i n t i n ' N o Perfect Shape' Leibniz anticipates a rejoinder to

this first hne ofreasoning. ( A n d what w e are caUing Leibniz's secondary argument against the existence ofprecise shapes i n things graduaUy

comes i n t o pIay, as discussion turns to the topics o f t i m e and change.)

T h e rejoinder itseH" is left unstated, b u t i t apparently goes something

hke this. Suppose a given hne or b o d y is taken as i t is just i n a single

Leibniz on Time, Change, and Corporeal Substance 153

instant, leaving aside the changes i t undergoes i n the surrounding

interval, and focus only o n its momentary shape. This momentary

shape m i g h t be precise. For i f one disregards the differing tendencies

t o w a r d m o t i o n i n the distinct parts and considers the shape o f t h e hne

or the body w i t h its parts ' f ixed' or 'frozen' intact at a single moment ,

the subdivisions that are about t o give way i n k t e r moments might

not present an obstacle t o a precise geometrical description o f i t at this

one instant. Perhaps the hne is exacdy straight i f t a k e n i n a single and,

so to speak, 'frozen' m o m e n t .

I n a f e w rather compressed remarks Leibniz offers a reply, or perhaps

t w o rephes, to this ' f rozen-moment ' suggestion. This is the first reply

to the d a i m that a momentary precise shape m i g h t be assigned to the

body i n a frozen moment :

I t is true that one could always draw an imaginary Hne each instant, but this Hne wiU endure i n the same parts only for this instant, because each part has a motion different from every other, since i t expresses the whole universe in a different way. Thus there is no body that has any shape for a definite time, however smaU i t might be. N o w I beheve that what is only in a moment [n'est que dans un moment] has no existence \existence\, since i t begins and ends at the same time. (A V I . iv. 1613)

I f t h e hne reaUy is going to be held exacdy straight for some t ime,

i t can be so only for an instant: stopping the course o f changes i n

the hne so that a single precise and fixed shape m i g h t be assigned to

i t narrows the w i n d o w o f t ime i n w h i c h that shape can be assigned

d o w n t o a single m o m e n t . B u t n o w , since the aUegedly precise shape

can exist only i n a m o m e n t , w e find that there is n o such shape

after aU. For, as Leibniz says, 'what is only i n a m o m e n t has n o

existence'.

Y e t Leibniz's defense for the crucial premise o f this argument is

rather hard t o make out , despite its simple statement. W h y should i t

be that 'what is only i n a m o m e n t has n o existence'? I t is perhaps

being assumed here that whatever 'has existence' must have b o t h a

beginning and an end, and that these must be opposite states o f i t . I f

so, the hypothesis o f something that 'is only i n a m o m e n t ' and thus

'begins and ends at the same t i m e ' w o u l d result i n contradiction and

is thereby reduced to absurdity w i t h Uttle trouble.

i 5 4 Samuel Levey

Maybe such an easy redudio is what Leibniz has i n m i n d there, 9

b u t w e shaU n o t pursue i t . 1 0 For i n the very next sentence Leibniz's

discussion appears to t u r n t o w a r d a quite distinct reply to the possibihty

o f a precise momentary shape. A n d this n o w cuts to the heart o f the

secondary argument for Proposition ( I ) — t h e argument that concerns

the nature o f t i m e and change. Leibniz writes: ' I have proved elsewhere

that there is n o middle m o m e n t , or m o m e n t o f change, b u t only the

bst m o m e n t o f the preceding state and the first m o m e n t o f the

f o l l o w i n g state. B u t this supposes an enduring state' (A V I . i v . 1613).

T h e three points being made i n that passage should be clearly

distinguished, since they w i U have to be considered separately i n a

short w h i l e .

(1) There is n o middle m o m e n t or m o m e n t ofchange.

(2) Change i n tact involves only the kst m o m e n t o f t h e preceding state and the first m o m e n t o f t h e foUowing state.

(3) This anah/sis ofchange supposes an enduring state.

W h e n Leibniz says that he has proved aU this 'ekewhere', he is

referring to his 1676 dialogue Pacidius Philalethi, i n w h i c h he presents

his first philosophy o f m o t i o n and takes o n the paradoxes o f the

composit ion o f t h e c o n t i n u u m . 1 1 B y l o o k i n g back to the Paddius w e

shaU be able to understand the grounds for (1), (2) and (3), and this w i U

9 We should not say that this is deünitery not what Leibniz has in mind. In a text that the Akademie editots date to the winter 0 f 1 6 8 2 - 8 3 , 'An Corpora Sint Mera Phaenomena', Leibniz offers a similar Hne of argument concerning the composition of a cube from two touching triangles. If the triangles were to touch orJy for a single moment, he argues, this would not be enough for a cube to exist, for 'ifthey are in that situation ofthe cube only for a moment. . . it would foUow that the cube would arise and perish simultaneously' (A VI. iv. 1464—5). Here 'arise' [nasci] and 'perish' [mierire] do appear to be opposite states; and there is nothing comparable in that text to the discussion ofmiddle moments that foUows in the present one. So perhaps Leibniz has two Hnes ofargument: the too^riefreA<cftO, and the more subde one explored next.

1 0 For some discussion ofthis passage, and ofthe general Hne ofargument in 'No Perfect Shape', see Adams, Leibniz, 231—2. Adams describes the premise that what is only in a moment has no existence as 'a large and dubious assumption'. See ako Sleigh, Leibniz and Amauld, 211. Marc Bobro has ako offered an instructive discussion of that premise in his paper 'Leibniz on Instantaneous Perceptions: A Momentary Lapse ofReason?', deHvered at the Eastern Divkion meeting of the American Philosophical Association on December 30, 1999, in Boston.

1 1 In his 'hitroduction' to L O C , Arthur ako notes this connection between the argument of 'No Perfect Shape' and the Pacidius.

Leibniz on Time, Change, and Corporeal Substance 15 5

t h r o w the entire secondary argument for Proposition (I) into m u c h

sharper rehef.

3-

Early o n i n the Pacidius, Leibniz offers a detailed discussion o f the

nature o f c h a n g e (see especiaUy A V I . i i i . 535—41), and i t is there

that he gives his p r o o f t h a t there is n o middle m o m e n t i n change. I n

this discussion Leibniz handles a few different examples ofchange, the

most v i v i d being that o f d e a t h — o r what the interlocutors caU the 'act

o f d y i n g ' . T h e dialogue t a k e s up the question whether the act o f d y i n g

could occur i n a single m o m e n t o f change that transacts the passage

f r o m hfe i n t o death, where hfe and death are held to be opposite

states. Leibniz's argument against middle moments is straightforward,

and i n hght o f i t one m i g h t say that i t is the 'logic ofchange' itserfthat

precludes middle moments. T h e argument runs across many hnes o f

the dialogue and cuhninates i n this exchange between Pacidius, w h o

prays the role o f Socrates, and Charinus, the principal interlocutor:

PAcroras. Isn't death a change? C H A M N u s . Undoubtedly. P A c m n J S . I understand i t to be the act ofdying.

CHAHrNUS. And so do I .

P A c m r o s . Is someone who is dying ahve? C H A M N u s . That's a puzzHng question. P A C T D i u s . Or is someone who is dying dead? C H A R t t i u s . This I see to be impossible. For to be dead means for one's death

to be past. P A c m r o s . I f death is past for the dead, then i t wiU be i n the future for the

hving, just as someone who is being born is neither about to be bom nor already born.

CHAtuNUS. So i t seems. P A c m i u s . Therefore i t is not the case that someone who is dying is ahve. C H A M N u s . Agreed.

P A c m r o s . So someone who is dying is neither dead nor aHve. CHAPJNUS. I concede this.

P A c m r o s . But you seem to have conceded something absurd. CHAPJNUS. I do not see the absurdity yet. P A c m r o s . Doesn't Hfe consist i n some particular state? CHAPJNUS. Undoubtedly.

Samuel Levey

P A c m r o s . This state either exists or i t doesn't exist. C H A R n v u s . There is no third alternative. PACroras. We say that anything i n which this state does not exist is kcking

hfe. C H A P j N u s . Yes.

PAcroros. Isn't the moment ofdeath the moment at which someone begins to k c k Hfe?

CHAPJNUS. W h y not? P A c r o r o s . Or is i t the moment at which he ceases to have Hfe? CHAPJNUS. Precisely.

P A C r o r u s . I am asking whether Hfe is absent or present at that moment. CHAPJNUS. I see the difficuky, for there is no reason why I should say one

rather than the other. P A c r o n J S . Therefore you must either say neither or both. CHAPJNUS. But you have blocked this way out for me. For I see weU enough

that a given state is necessarily present or absent, and cannot at the same time be both present and absent, nor neither present nor absent.

PAcroros. What, then? CHAPJNUS. Yes, what?, otherwise I am stuck. (A V I . i i i . 535)

Charinus finds himserfstuck, for the hypothesis that 'the act o f d y i n g '

occurs i n a single middle m o m e n t results i n a tri lernma. I f the act o f

dy ing i t se l fwere to occur i n a middle m o m e n t , either (i) i t w o u l d

have to be a combinat ion o f t w o opposite states, hfe and death, w h i c h

is contradictory, or (ii) i t w o u l d have to be a state that is one neither

o f hfe n o r o f death, w h i c h v ioktes the principle o f the excluded

t h i r d (Leibniz writes: Tertium nullum est), or, finaUy, (i i i) that middle

m o m e n t w o u l d have to be either the ks t m o m e n t o f h f e or the first

m o m e n t o f death, but there could no reason w h y i t should be one

rather than the other.

I n the face o f t h i s perplexity, the interlocutors conclude that there

is n o momentary state o f change o r middle m o m e n t . T h e logjc o f

change, so t o speak, rules i t out . T h e occurrence ofchange itseU"must

therefore instead always take p k c e over some stretch o f t i m e .

PAcroros. W e concluded that a state ofchange is impossible. C H A M N u s . Yes, we did, i f the moment of change is assumed to be the

moment ofa mediate or common state. P A C I D I U S . But don't things change? C H A M N U S . W h o would deny it?

Leibniz on Time, Change, and Corporeal Substance 157

P A C r o r a s . Then change is something. CHARLNUS. O f course. P A c r o r u s . Something other than what we have shown to be impossible,

namely a momentaneous state. CHAPJNUS. Yes. P A C l D m s . Then does a state ofchange require some stretch oftime? CHAMNus. So i t would seem. (A V I . i i i . 538)

T h e reahty o f change, i t wovdd seem, is n o t open for negotiation,

even i n the face o f the present difficulties. I f t h a t is a smaU concession

to c o m m o n sense ( ' W h o w o u l d deny it?'), stih, i t is one that continues

to pose a chaUenge to Leibniz's inquiry . W h a t could change be?

T h e interlocutors t u r n their attention to the nature ofchange and i n

p a r t i c u k r to this 'stretch o f t i m e ' that is required b y a state ofchange.

I n the first major step, Charinus proposes a n e w theory ofchange that

identifies change itserfas an aggregate o(tu>o moments:

CHAMNus. I think I have finaUy found a way out. For I would say that it [change] is composed from both [the last moment of the eariier state and the first moment ofthe later state], and that, although i t is usuaUy caUed momentaneous, i t i n fact contains two moments . . .

P A c r o r u s . Y o u have spoken correcdy, and consistendy wi th what you said above, so that I have no objection to this opinion ofyours. (A V I . i i i . 541)

Thus, o n the anarysis o f change that emerges i n the Pacidius, a state o f

change actuaUy consists o f a n aggregate o f t w o immediately neighbor­

i n g moments containing t w o opposite states—'the last m o m e n t o f t h e

earher state and the first m o m e n t o f t h e k t e r state'. For instance, the

act o f d y i n g w i U be an aggregate that consists o f t h e ks t m o m e n t ofhfe

and the first m o m e n t o f death. A n d thus change minimaUy requires a

'stretch o f t i m e ' that includes t w o moments rather thanjust one.

I t is b y this p o i n t clear that the discussion ofchange i n the Pacidius provides the account Leibniz draws u p o n i n ' N o Perfect Shape' w h e n ,

i n the k t t e r , he reports that there is 'no middle moment , b u t only the

bst m o m e n t o f t h e preceding state and the first m o m e n t o f t h e foUow-

i n g state' (A V I . iv . 1613). T h e logic ofchange imphes (1) that there is

n o middle m o m e n t or m o m e n t ofchange. T h e n e w theory ofchange

imphes (2) that change is an aggregate o f t w o neighboring moments

containing t w o opposite states. N o t i c e , though, that those facts about

the logic o f change and the n e w theory o f change do not yet entail

i 5 8 Samuel Levey

Leibniz's t h i r d c laim from ' N o Perfect Shape'—namely, (3) that this

analysis o f c h a n g e 'supposes an enduring state'. For those t w o facts

about change stiU perfectly weU aUow that change m i g h t be an aggre­

gate o f n o t h i n g more than t w o moments containing t w o opposite m o ­

mentary states. N 0 enduring state has yet been supposed. T o see w h y

the analysis ofchange additionaUy requires enduring states—and thus

to l o o k stiU m o r e deeply i n t o the nature o f t h e 'stretch o f t i m e ' required

for change to o c c u r — w e need t o t u r n from the discussion ofchange

to the metaphysics o f t i m e , and i n part icukr to a strand o f t h o u g h t

about the nature o f t h e c o n t i n u u m , once again i n Paddius Philalethi.

A c c o r d i n g t o Leibniz, the c o n t i n u u m o f t i m e is divided up i n t o

intervals, and at each p o i n t o f d i v i s i o n there are moments assigned i n t o

the c o n t i n u u m (A V I . i i i . 552—3); i n fact, since every interval is itserf

inf in i te ly subdivided i n t o smaUer intervals, every interval contains an

i n f i n i t y o f m o m e n t s (A V I . i i i . 564—5). B u t the metaphysical nature o f

the relation o f t h e indiv idual moments to the c o n t i n u u m o f t i m e is o f

paramount importance. I n particular, i t is crucial t o Leibniz's account

o f t h e c o n t i n u u m that moments always be either ends or beginnings

o f extended intervak o f t i m e — t h e termini o f in terva ls—and not

independent elements o f t ime. For, i f moments are aUowed to be

independent elements o f t i m e , then the c o n t i n u u m o f t i m e itseh°

w f f l be an aggregate o f n o t h i n g but moments, finaUy dissolving i n t o

i n d i v i d u a l moments as i f i t were just so many grains ofsand. A n d this

w i U raise the paradoxes o f t h e composit ion o f t h e c o n t i n u u m .

C H A M N u s . Supposing we concede to you that space is an aggregate ofnothing but points and time an aggregate of nothing but moments, what do you fear so much from this?

P A C i D i u s . I f you admit this, you wiU be swamped by the whole stream of difficulties that stem from the composition ofthe continuum, and that are dignified by the famous name ofthe labyrinth.

C H A M N u s . This preface is capable ofstriking terror into someone even from afar! (A V I . iv. 548)

That is where the trouble reaUy starts, ofcourse, though i t is beyond the

scope o f t h i s chapter to pursue Leibniz's inquiry into that k b y r i n t h . 1 2

1 2 For some discussion of that inquiry, see Arthur, 'RusseU's Conundrum: On the Relation of Leibniz's Monads to the Continuum', in James Robert Brown and Jürgen

Leibniz on Time, Change, and Corporea Substance 159

Let us instead m o v e straight t o its conclusion. T h e Pacidius' final

resolution o f the difficulties o f the composit ion o f the c o n t i n u u m ,

as apphed t o t i m e , requires that t i m e be inf inite ly subdivided i n t o

intervak, but that there be n o moments i n t i m e apart f r o m the

beginnings and ends o f those intervals. This is h o w Leibniz describes

the divis ion o f t h e c o n t i n u u m (and this description covers space, t i m e ,

m o t i o n , and matter).

Accordingfy the division ofthe continuum must be considered to be not Eke the division ofsand into grains, but Hke that ofa sheet ofpaper or tunic into folds. A n d so, although there occur some folds smaUer than others mfinite in number, a body is never thereby dissolved into points or [seu] niinima . . . I t is just as i f we suppose a tunic to be scored wi th folds multipUed to infinity i n such a way that there is no fold so smaU that i t is not subdivided by a new fold . . . A n d the tunic cannot be said to be resolved аП the way down into points; instead, although some folds are smaUer than others to infinity, bodies are always extended and points never become parts, but ahvays remain mere extrema. (A V I . i i i . 555)

Just as the tunic is d iv ided into folds w i t h m folds ad infinitum, but is not

thereby resolved i n t o a mere coUection o f p o i n t s , the c o n t i n u u m o f

t i m e is divided i n t o intervak w i t h i n intervak ad infinitum, but i t is never

resolved i n t o a dust ofsingle moments. M o m e n t s always remain mere

extrema and are never free-standing independent elements o f t i m e .

I n the face o f t h e fact that independent moments are impossible,

and that there is ' o n i y the last m o m e n t o f the preceding state and

the first m o m e n t o f the foUowing state', the only coherent idea o f a

state that exists at a m o m e n t is the idea o f t h e 'kst m o m e n t ' or 'first

m o m e n t ' of some enduring state. M o m e n t s are themselves always

ends or beginnings ofintervals. Likewise, momentary states are always

ends or beginnings o f enduring states. This is w h y i n ' N o Perfect

Shape' Leibniz says, o f h i s o w n analysis ofchange, that i t 'supposes an

enduring state' ( A V I . iv . 1613); i n fact, since i t holds change to be

an aggregate o f t w o moments containing opposite states, that analysis

Mittektrass (eds.), An Intimate RAation fDordrecht: KIuwer, 1989), 171-201, as wdl as the 'Introduction' to L O C ; and Michael White, 'The Foundations of the Calculus and the Conceptual Analysis of Motion: The Case of the Early Leibniz', Pacific Philosophical Quarterfy, 73 (1992), 281—313. See also my 'Leibniz on Mathematics', 'Interval ofMotion', and 'Leibniz's Constructivism and Infinitely Folded Matter', in R_occo Gennaro and Charles Huenemann (eds.), New Essays on the Rationalists 0Sfew York: Oxford, i999), 134-62.

i 6 o Samuel Levey

supposes two enduring states and two extended intervak o f t i m e , the

one preceding and the one foUowing.

So the logic o f change precludes change from taking p k c e i n

m i d d l e moments; the n e w theory o f change demands that change is

an aggregate o f t w o moments containing t w o opposite states; and n o w

the metaphysics o f t i m e demands that those t w o moments containing

t w o opposite states are themselves always end moments o f i n t e r v a k o f

time and that they contain the momentary ends o f e n d u r i n g states.

W i t h this total account o f t i m e and change i n hand, w e can at

bst see exactly w h y the ' frozen-moment ' suggestion for assigning a

precise shape to a b o d y at a m o m e n t goes w r o n g . T h e strategy b e h i n d

the ' frozen-moment ' suggestion was to f i n d a stable p o i n t i n the flux

o f changing parts at w h i c h a shape could be assigned to a b o d y — a

'precise and f ixed shape'. T h e p o i n t o fs inghng out a given m o m e n t i n

time was intended to h o l d the shape o f t h e b o d y ' f ixed' (arrestie). T h e

p o i n t ofconsidering the body as having a shape that exists only i n the

m o m e n t , and thus o f disregarding the tendencies o f its parts t o move

i n different directions, was to ehminate from v i e w the subdivisions

that w i U break open i n foUowing moments, thereby 'freezing' the parts

intact so that the w h o l e b o d y m i g h t prove more u n i f o r m and tractable

to being exactly described b y a precise shape o f t r a d i t i o n a l geometry.

T h e strategy for assigning a precise shape t o a body at a m o m e n t

that imagines a frozen m o m e n t i n t i m e runs afoul b o t h o f t h e ' logical '

and o f t h e metaphysical requirements o n change, fbr i t has t o suppose

that a precise shape w i U be a non-endur ing momentary state that

exists i n an independent middle m o m e n t . B u t there can be n o frozen

m o m e n t o f this sort, since aU moments are essentiaUy beginnings or

ends o f i n t e r v a k i n w h i c h m o t i o n occurs. A n d there is n o momentary

state i n things apart from the end o f one enduring state and the

beginning o f t h e next. M o m e n t a r y states presuppose enduring states,

just as moments themselves presuppose intervak o f t i m e . T h e task o f

assigning a shape to a b o d y at a m o m e n t therefore runs back i n t o the

original diff iculty o f assigning a shape t o a m o v i n g body, where this

shape is never a merely momentary state o f i t b u t w i U always be the

beginning or end o f a n enduring state. I f t h e r e are to be precke shapes

i n things, they must be enduring states o f t h e m .

This result n o w leads i n t o a p r o f o u n d consequence o f Leibniz's

account, one that significandy raises the phUosophical stakes i n his

Leibniz on Time, Change, and Corporeal Substance 161

crit ique o f the Cartesian theory o f corporeal substance. For faffing

under Leibniz's attack w i U be not ordy the Cartesian notions ofshape

and extension b u t ako the idea o f a n enduring state.

4 ·

Once i t has been announced i n ' N o Perfect Shape' that Leibniz's

analysis o f t i m e and change 'supposes an enduring state', the topic o f

imprecision comes i n t o p k y . Leibniz writes:

N o w aU enduring states are vague [vagues], and there is nothing precise about them. For example, one can say that a body wiU not leave some pkce greater than itseU' for a definite [certaine] time, but there is no place where i t may endure that is precise or equal to the body. One can thus conclude that there is no moving thing ofa definite shape. (A V I . iv. 1613-14)

Leibniz's conclusion that 'there is no m o v i n g t h i n g o f a definite shape'

can be achieved i n more than one way from the premise that enduring

states are n o t precise. T o see just what argument is being advanced,

that premise itserfneeds to be examined. W h y is i t that 'aU enduring

states are vague, and there is n o t h i n g precise about them'?

Leibniz exphcitiy caUs enduring states 'vague', and, given his expert­

ise w i t h the Sorites, 1 3 i t may weU be that he means t o d a i m that the

very idea o f a n enduring state is subject to this ckssical paradox. Y e t i t

is difficult to see h o w that charge ofvagueness could be substantiated

i n this case. W h a t is i t about an enduring state demands that i t be

vague? Leibniz's o w n example does httle to help. I f t h e r e is n o place

where a body may endure that is 'precise or equal t o the body ' , the

p r o b l e m seems to be that bodies and the piaces that they occupy are

imprecisely f i t ted t o one another. I f 'pLace' is taken to be a Cartesian

n o t i o n , then perhaps a phce cannot be 'precise and equal to the

body ' for the same reason as b e f o r e — n a m e l y , that Leibnizian bodies

cannot be exacdy described by the traditional geometry assumed by

the Cartesian account. Even i f t h i s is not the reason, however, the

idea o f an imprecision i n fit between prace and body seems to rely

o n the idea that there is something imprecise about the n o t i o n o f an

1 3 Leibniz develops perhaps the most sophisticated discussions and uses ofthe Sorites in early modem philosophy; cf. A VI. iii. 539 fF., A VI. iv. 69—70, A VI. vi. 302, 321.

l 6 2 Samuel Levey

enduring body, presumably an imprecision concerning shape. W h a t

else could i t be? B u t i f t h a t is the idea h e r e — i f t h e sub-argument for

the vagueness o f e n d u r i n g states presupposes that bodies are imprecise

i n shape—then the overaU argument o f ' N o Perfect Shape' moves i n a

circle, since the imprecision o f e n d u r i n g states is actuaUy and exphcidy

being used i n that piece as a premise for the thesis that bodies have

no precise shape. T o find support for the c la im that aU enduring states

are vague, one therefore cannot appeal to the imprecision o f shape i n

bodies. O t h e r reasons w i U need t o be tendered.

A natural idea m i g h t be that enduring states are vague because they

are n o t precisely bounded i n t i m e . Perhaps i t is unclear just w h e n

they begin and end. B u t i t is far f r o m obvious that Leibniz holds the

relevant sort o f u n c k r i t y to infect the durat ion o f e n d u r i n g states. W e

could p u t the question this way: fbr every possible enduring state,

must there be moments i n t ime that are 'borderhne cases' o f m o m e n t s

at w h i c h the enduring state obtains? I t is hard to see w h y this should

be true. Certainly for some enduring states there could be moments i n

t ime that are borderhne cases o f m o m e n t s at w h i c h the state obtains.

For example, suppose I have been p o o r for a whUe, b u t m y poverty is

graduaUy being removed b y the successive addi t ion o f single pennies

whUe w r i t i n g this chapter; i t may then be that this very m o m e n t is

on ly a borderhne case o f a m o m e n t at w h i c h m y enduring state o f

poverty stiU obtains. M y enduring state o f p o v e r t y may thus n o t be

precisely bounded i n t ime. B u t the vagueness o f the enduring state

i n that example is clearly due n o t to its be ing an enduring state b u t

rather to its being a straightforward case o f vagueness tout court. I t is

n o t obvious that an enduring state can be vague simply because i t

is an enduring state. E n d u r i n g states that are vague seem as t h o u g h

they w i U always be vague for other reasons, and i t cannot simply be

presumed that those other reasons w i U be operative i n every case o f

an enduring state.

Thus i t seems that Leibniz has yet to produce pbusible credentiak

for his d a i m that aU enduring states are vague. StiU, he does have sohd

grounds fbr saying that 'there is n o t h i n g precise about t h e m ' . N o t aU

imprecision need be due to vagueness. T h e sohd grounds he has for

h o l d i n g that there is n o t h i n g precise about enduring states are to be

f o u n d i n his v i e w o f t h e nature o f t i m e . So again w e r e t u m to ideas

arising i n the Pacidius.

Leibnizon Time, Change, and Corporeal Substance 163

T h e fundamental diff iculty for enduring states is that n o interval

o f time can be assigned d u r i n g w h i c h an enduring state can persist

the same and u n i f o r m throughout. Leibniz is quite exphcit that the

m o t i o n o f a body through any interval w i U suffer this inf ini te division;

indeed, i t is the very centerpiece o f his theory o f m o t i o n and his

account o f t h e structure o f t h e c o n t i n u u m itseUi

C H A M N U S . Then what i f w e say that the motion ofa moving thing is actuaUy divided into an infinity o f other motions, each different from the other, and that i t does not persist the same and uniform for any stretch oftime?

P A C i D i u s . Absolutely right, and you yourself see that this is the only thing left for us to say. But it is abo consistent wi th reason, for there is no body which is not acted upon by those around it at every single moment. (A V I . i i i . 564-5)

Just as a body's state o f m o t i o n is never 'the same and u n i f o r m for

any stretch o f t i m e ' , its shape undergoes constant change as weU. A n y

stretch o f time d u r i n g w h i c h one shape m i g h t be assigned as 'the'

shape o f a given body is i n fact divided into smaUer subintervals o f t i m e

d u r i n g w h i c h the body actuaUy takes o n a number o f d i s t i n c t shapes.

Likewise for each o f the smaUer subintervak, and so o n ad infinitum. Thus the shape o f the body does n o t persist the same and u n i f o r m

t h r o u g h any stretch o f t i m e , but rather dur ing every extended interval

o f t i m e the b o d y goes through an m f i n i t y o f d i s t i n c t transformations.

This again goes r ight t o the heart o f Leibniz's analysis o f the

composit ion o f the c o n t i n u u m i n w h i c h he tries to speU out a

m i d d l e path between describing the c o n t i n u u m o f t i m e as 'purely

continuous' and describing i t as 'purely discrete'. T h e c o n t i n u u m o f

t i m e is never undivided and u n i f o r m hke a geometrical hne, n o r is

i t ever divided aU the way into moments hke the division o f sand

i n t o grains. Instead, the divided c o n t i n u u m o f time turns out to

have a 'scahng' structure o f intervab w i t h i n intervab hke 'a tunic

scored w i t h folds mult iphed to i n f i n i t y ' (cf. A V I . i i i . 555), never

b o t t o m i n g out into a mere coUection o f moments. This account

o f the c o n t i n u u m o f time, space, matter, and m o t i o n describes a

structure that can be recognized from a contemporary perspective t o

be fractal—or something very closely akin to fractal structure. Fractak

are, broadly speaking, geometric objects that have structure o n aU

scales o f magnification, and, although the m o d e m science o f fractal

i 6 4 Samuel Levey

mathematics came together rargely i n the pioneering w o r k o f B e n o i t

Mandelbrot i n the k t t e r h a l f o f t h e twent ie th century, the 'pre-history'

offractal mathematics can be traced weU back i n t o the mathematics o f

the eighteenth century and indeed i n t o Leibniz's seventeenth-century

wri t ings . Leibniz's o w n mathematical wri t ings are shot t h r o u g h w i t h

insights that anticipate fractal mathematics b o t h i n brge impressionistic

gestures and i n some fine technical details. 1 4 W h a t is most remarkable

about the connection w i t h fractal structure here is h o w perfecdy i t fiUs

out Leibniz's critique o f t h e Cartesian theory o f c o r p o r e a l substance.

I f t h e actual bodies are fractaüy divided, then i t is indeed impossible

for any tradit ional geometrical shape to describe them, ^ v e n the

simplest cases o f fractal structures, such as the K o c h curve , 1 5 can be

described only as hmits o f m f i n i t e series o f t radi t iona l shapes; and such

constructions, t h o u g h something Leibniz could perhaps gHmpse, faU

b e y o n d the reach o f t h e Cartesian geometry ofextension and shape.)

Again, the detaik underly ing this account are beyond the compass o f

the present chapter. B u t the basic hne o f t h o u g h t that time is 'fractany

divided' i n t o intervals w i t h i n intervals w i U iUuminate the grounds for

Leibniz's key premise that 'there is n o t h i n g precise' about enduring

states.

Because o f t h e fractal structure o f t i m e , the very idea o f a n enduring

state turns out to be an imprecise n o t i o n , not because i t is a vague

n o t i o n , b u t because o f i t s inexactness as a representation o f t h e nature

o f persistence t h r o u g h t i m e . A body w o u l d have an enduring shape,

for example, only i f there could be some assignable stretch o f time

t h r o u g h w h i c h its shape persists the same and u n i f o r m . B u t i n fact,

t h r o u g h every such interval, any 'precise and fixed' shape can offer

onh/ an approximate guide t o the actual shape o f the constandy

ttaraforming body. N o t i c e that this w i U h o l d n o matter whether

the purported 'precise and f ixed' shape o f a b o d y is supposed to be

fractal or 'merely' geometrical. A n y interval o f t i m e t h r o u g h w h i c h a

" See B. B. MandeRjrot, The Fractal Geometry of Nature [Fractal Geometry] ^Jew York: W. H . Freeman and Company, 1982), 405 fF, 412—13, 419 ('The number and variety of premonitory thrusts is overwhehning'). See ako section 6 ofmy 'Interval ofMotion'.

1 5 FMge von Koch, 'Sur une courbe continue sans tangente, obtenue par une construction geometrique äementaire', Arkiv för Matematik, 1 (1904), 681-704. See ako Heinz-Otto Peitgen, Harmut Jürgens, and Dietmar Saupe, Chaos and Fractah: New Frontiers of Science ^Jew York: Springer Veriag, 1992), 89; and Mandelbrot, Fractal Geometry, 42 fF.

Leibniz on Time, Change, and Corporeal Substance 165

single shape is said to be the enduring shape o f the body is actuaUy

divided up i n t o several subintervaLs d u r i n g w h i c h the body takes o n

a number o f different shapes. Likewise for other supposed enduring

states. T h e y one and aU involve something imaginary i n the same way.

Thus the very n o t i o n o f a n enduring state is only an imprecise n o t i o n

rekt ive to the actual course ofchanges i n things through t ime, for the

actual states of things can never be described w i t h perfect accuracy by

the n o t i o n o f an enduring state. T h e n o t i o n o f a state that endures

t h r o u g h some interval can offer ordy an approximation, always less

than perfecdy correct, o f w h a t actuaUy happens over the course o f a n y

stretch o f t i m e . A n d so, i n this way, the n o t i o n o f a n enduring state is

an inexact n o t i o n .

FinaUy Leibniz's main thesis from ' N o Perfect Shape' can be

defended and what w e have been caUing his 'secondary argument'

from the nature o f t i m e and change for Proposition ( I ) — t h a t there

are n o precise shapes i n things, because o f the subdivision o f the

c o n t i n u u m to i n f m i t y — c a n be brought fuUy to hght. There can

be ' n o precise and f ixed shape i n things' and ' n o m o v i n g t h i n g o f

a definite shape', because bocUly shape has to be understood as an

enduring state. B u t , w h e n held up against the reahty o f constant flux

i n the material p l e n u m and the fractal division o f t h e c o n t i n u u m o f

t i m e itserf, the n o t i o n o f a n enduring state proves to be onh/ an inexact

one. This means that there is n o t h i n g that can be accurately described

as having an enduring state; and hence there is n o t h i n g that can be

accurately described as having a p a r t i c u k r shape.

O n this reading o f his argument, Leibniz's c k i m that there are

'no precise and fixed shapes i n things' is vindicated, as is his c k i m

that there is n o t h i n g precise about enduring states. T h e theories o f

t i m e and change that are the legacy o f Pacidius Philalethi make sense

o f his argument i n 'There Is N 0 Perfect Shape at aU i n Bodies',

and give i t sohd grounds (or grounds that Leibniz could reasonably

suppose to be sohd, t h o u g h the 'fractal theory' he holds is i n fact

n o t w i t h o u t difficulties o f its o w n 1 6 ) . O n this reading o f Leibniz,

however, his specific assertion that enduring states are 'vague', i f t h a t

is meant t o invoke the Sorites, w i U be contravened i n favor o f the

claim that enduring states are inexact; and thus the present account

1 6 See my 'The hiterval ofMotion', and footnote 18 below.

l 6 6 Samuel Levey

o f h i s argument against the existence ofprecise shapes i n things may

be a t o u c h revisionary w i t h respect t o what Leibniz actuaUy says.

(At any rate, the c laim that enduring states are 'vague' has been

interpretively ckr i f ied here i n a way that Leibniz himseM" does n o t

c k r i f y i t . )

StiU, his final example i n ' N o Perfect Shape' abo seems to accord

best w i t h the claim o f inexactness rather than vagueness. Just after

concluding that there can be n o m o v i n g t h i n g o f a definite shape,

Leibniz writes:

For example, i t is impossible for there to be found i n nature a perfect sphere that would compose a moving body i n such a way that this sphere could be moved through the least space. I n a рДе ofstones, one could easih/ conceive an imaginary sphere that passes through аД these stones, but one could never find any body whose surface would be precisely spherical. (A V I . iv. 1613-14)

W h a t is the force o f saying that one could never find any body

whose surface is precisely spherical? N o t h i n g i n Leibniz's metaphysics

o f matter could underwrite the c k i m that a b o d y m i g h t have only

an mdeterminate and vague surface boundary. I t is one o f h i s most

important and most frequendy asserted metaphysical doctrines that

actual bodies and the material p l e n u m they f o r m are everywhere fuUy

determinate (cf. A V I . i i i . 495; GP i v . 567; G P i i . 268, 2 7 8 - 9 , 282;

G v i i . 562—3, etc.). Thus, w h e n he says that one could never find

any body w i t h a precisely spherical surface, this is n o t because bodies

are ordy vaguely bounded, but because the actual surface o f a n y given

actual body is deterrninately not spherical. T o describe an actual b o d y

as having a spherical surface can give only an approximate account o f

its actual surface—approximate, b u t n o t exact. A n d hkewise for aU

other traditional geometrical descriptions.

Let us at last sum up Leibniz's secondary argument, from the nature

o f t i m e and change, for the c laim that there are n o precise shapes i n

bodies, because o f the division o f the c o n t i n u u m to i n f i n i t y . Bodies

i n the p l e n u m are undergoing constant change, b u t the analysis o f

change demands that there be n o middle m o m e n t or momentary state

o f change; thus change always occurs over a stretch o f t i m e , and

i n fact i t consists i n an aggregate o f t w o moments containing t w o

opposite states. Since the c o n t i n u u m o f t i m e cannot be composed

Leibniz on Time, Change, and Corporeal Substance 167

o f isolated moments, there can be no moments i n t i m e apart from

the beginnings or ends o f e x t e n d e d intervak o f t i m e . Hence n o state

that is only momentary or exists only for a m o m e n t can be assigned

to a body, since that w o u l d require the existence o f an isokted

m o m e n t that is n o t the beginning or end o f an extended interval.

A n d so, i n part icukr , n o precise shape that is only momentary can

be assigned to a body; hence, a precise shape can be assigned to a

b o d y ordy i f an enduring state can be assigned to i t . Because o f the

fractal divis ion o f t ime, however, the very idea o f an enduring state

turns o u t to be only inexact and imprecise, and therefore the very

idea o f a shape that can be assigned to a body is only an inexact

and imprecise n o t i o n that does n o t apply to any actual body as i t is

absolutely or i n itseU7. Thus there are n o precise shapes at аИ i n actual

bodies.

So i t is against the backdrop o f h i s 1676 m q u i r y into the nature

o f t i m e , change, and shape that Leibniz finds the Cartesian theory

o f b o d y to rest o n a 'poor understanding o f the composit ion o f

the c o n t i n u u m ' . T h e Cartesian theory had defined bodies i n terms

o f shape, b u t n o w , w i t h a newly enhghtened understanding o f the

c o n t i n u u m , i t appears that the very n o t i o n ofshape cannot accurately

describe anything as i t is absolutely or i n itserf, and thus the n o t i o n

o f b o d y that i t defines can be only ' i m a g i n a r y ' — t h a t is, one ofthose

notions 'that are n o t i n things outside o f us, b u t whose essence i t

is t o appear to us' (A V I . iv . 70). T h e Cartesian theory o f body

cannot offer an account o f the true nature o f actual bodies as they

are i n themselves; and thus i t coUapses as a metaphysics o f corporeal

substance.

5-

I f L e i b n i z is r ight that there are n o precise shapes i n bodies, and that

the n o t i o n o f shape that can be assigned t o bodies is ordy imprecise

and imaginary, then the Cartesian theory o f corporeal substance

stands refuted—as w o u l d any theory that rests so heavily o n the

'geometrical ' concepts o f extension and shape. 1 7 B u t i t seems that

1 7 My thanks to Dan Garber, whose discussion helped me to clarify this section considerably.

i 6 8 Samuel Levey

Leibniz's considerations about the n o t i o n o f an enduring state w i U

have far m o r e p r o f o u n d consequences than just the defeat o f the

Cartesian conception o f b o d y . I t is important here t o consider the

difference between the primary argument, w h i c h says that there are

n o precise shapes i n things because assignable shapes are geometrical

w h i l e bodies are fractaUy divided, and the secondary argument, w h i c h

says that shape must be an enduring state b u t that t i m e is fractaUy

divided.

If , as the pr imary argument suggests, the problem w i t h the n o t i o n

o f shape is that actual bodies are fractal i n f o r m w b i l e assignable

shapes are ordy geometrical, then w e could at least i n principle have

a theory ofbodies that substitutes the fractal account o f b o d U y shape

for the o l d geometrical one, even i f this means that the shapes o f

actual bodies are unassignable b y us and that w e can never even i n

principle grasp the true f o r m o f a n y p a r t i c u k r b o d y . 1 8 T h e resulting

account thus w o u l d n o t achieve the h i g h hopes o f the Cartesian

project t o found the metaphysics o f b o d y o n concepts that are clearly

and chstinctiy understood. 1 9 Y e t those w o u l d have been take hopes

firom the start. Leibniz stresses i n his criticisms o f Descartes that

the k t t e r faik to consider the difficulties o f the k b y r i n t h o f the

c o n t i n u u m , and so fails to appreciate the difficulties w i t h the very

1 8 Presumably Leibniz would seek to reconcile this deviation of the real shapes of things from the precise assignments that our geometry can offer by suggesting that the error of our assignments can always be made 'smaUer than any gjven error', even if they can never be perfectly accurate in describing (fractal) reaUty. The 'approach' or 'approximation' of mathematics to reakty is a staple feature ofhis accounts from his early writings right through the rest of his career. As he writes in De organo sive arte magna cogitandi (c. 1679): 'Even if no straight Hnes or circles can be assigned in nature, it is nonetheless sufficient that shapes can be assigned that differ firom straight fines and circles so Htde that the error is less than any given error—which is sufficient in order to demonstrate certainty as weU as usage' (A VI. iv. 159). See Beeley, 'Mathematics and Nature', 131. See ako Arthur's 'Introduction' to LOC. Leibniz's view appears to be divided by two competing intuitions: the first, that actual curves wiU 'converge towards continuity', the second, that actual curves are 'fractaUy divided'. Leibniz apparendy wants to have it both ways, but those two intuitions wiU be difficult to reconcUe.

1 9 StiU, it is worth noting that, as early modern mathematics began to pursue ideas beyond the compass of Cartesian geometry, it was indeed exploring a dark and paradoxical subject; Descartes was not totaUy unjustified in drawing the official boundaries ofgeometry at algebraic curves—the only curves that he was confident admitted of'precise and exact measurement'—and withholding from the nascent infinitary methods the same daim to securing a distinct understanding of reaHty that he attributed to geometry. See Mancosu, Philosophy of Mathematics, 72 fF, and 141 - 5 ; and Groshok, Cartesian Method, 42 ff.

Leibniz on Time, Change, and Corporeal Substance 169

n o t i o n o f extension. 2 0 I f Leibniz is right about the nature o f those

difficulties and what is required to overcome them, then the Cartesian

geometrical conception o f b o d y ofFers only an iUusion oftransparency

to the m i n d , whereas the true understanding o f t h e nature o f b o d y is

fär more difficult to reach, and indeed i t cannot be achieved w i t h o u t

negotiating a w h o l e maze ofparadoxes concealed behind the thought

that mathematics w i U be a guide to a t r u l y sohd metaphysics.

I f Leibniz's pr imary argument against the existence o f precise

shapes i n things poses a sharply focused chaUenge to the geometrical

conception o f b o d y , and offers a visionary n e w account to replace i t ,

his secondary argument from t i m e and change is quite another matter.

T h e charge that the n o t i o n o f an enduring state is only imaginary,

because time itserf is fractaUy divided, raises a m u c h more serious

difficulty, and one that is hardly unique to the theory ofDescartes.

Perhaps the idea o f fractal structure m i g h t coherendy repkce the

traditional n o t i o n o f geometrical shape and y ie ld a n e w account o f

body; b u t what n o t i o n is going to repkce the idea o f an enduring

state? I t is not at aU clear that there is a paraUel repkcement to be

made here. N 0 new conception o f an enduring state seems to be

f o r t h c o m i n g to take over where the o l d one faUed. I t is quite clear

o n Leibniz's theory o f the c o n t i n u u m that n o state can be merely

momentary; n o w i t seems that n o state can endure t h r o u g h t ime

either. Y e t what t h i r d alternative could there be?

I t appears that the consequences ofthis secondary strand ofLeibniz 's

critique o f the Cartesian theory o f corporeal substance are very far-

reaching indeed. N o t only w i U no body that is defined i n terms o f

magnitude, shape, and m o t i o n be t r u l y substantial, b u t also n o t h i n g

that is supposed to have e i t h e r m o m e n t a r y or enduring states w i U

ever be more than imaginary. Reahty itseH" must somehow shp these

bonds o f t i m e and change altogether. Perhaps our grasp o f t h e nature

o f t i m e and change is mdistinct and rests, as Leibniz suggests, o n a

p o o r understanding o f t h e composit ion o f t h e c o n t i n u u m . B u t , even

w i t h his n e w theory o f the c o n t i n u u m i n hand, i t is n o t clear that

2 0 Descartes was not the only predecessor to suffer this criticism from Leibniz. In the Pacidius, just after announcing the necessity ofsolving the difficulties ofthe labyrinth ofthe composition ofthe continuum, Leibniz writes: 'Neither Aristotle nor Gahlei nor Descartes was able to avoid this knot, although one ofthem pretended it didn't exist, one abandoned it as hopeless, and the other broke offhis discussion' (A VI. iii. 547).

i 7 0 Samuel Levey

any coherent account o f s u c h a reahty b e h i n d the appearances m i g h t

be ofFered—nor that w e could understand such an 'enhghtened'

theory, even i f one were actuaUy proposed. For, w e m i g h t begin to

suspect that, apart firom t i m e and change, reahty can neither be n o r be

distincdy conceived. 2 1

Dartmouth College

2 1 ш addition to those already mentioned in the notes, I want to offer my thanks to the participants ofthe 2000 Horence International Leibniz Workshop for very helpful discussion ofthe topics I treat here; to the Florence Centre for the History and Philosophy ofScience and the Dipartmento di Filosophia deU'Universita di Firenze for their sponsorship; and to Alessandro Pagnini and, most especiaUy, to Massimo Mugnai for making it aU possible.