Stoic Gunk

22
© Koninklijke Brill NV, Leiden, 2006 Phronesis LI/2 Also available online www.brill.nl Accepted October 2005 1 This sense of the term, though not the concept, is due to David Lewis: see Lewis 1991. Stoic Gunk DANIEL NOLAN ABSTRACT The surviving sources on the Stoic theory of division reveal that the Stoics, par- ticularly Chrysippus, believed that bodies, places and times were such that all of their parts themselves had proper parts. That is, bodies, places and times were composed of gunk. This realisation helps solve some long-standing puzzles about the Stoic theory of mixture and the Stoic attitude to the present. This paper seeks to address two of the central questions to be answered in any account of the physical and metaphysical doctrines of the Stoics. The rst is the question of what attitude the Stoics took towards wholes and parts; and the second, the related question about the divisibility of objects, which is closely connected to the question of what parts objects in fact possess. To my mind modern commentators have misunderstood the Stoics, particularly Chrysippus, on these central issues, or at the very least overlooked an important option. They have thus overlooked an inter- pretation of the Stoics which closely ts the surviving fragments of Stoic writings on the topic: the possibility that the Stoics embraced a non-atomic mereology, and postulated that material objects were such that each of their parts themselves had further parts, without any nite limit. In other words, to use the jargon of contemporary mereology, Stoics postulated the existence of “gunk”: 1 objects which are not composed of mereological indivisibles (mereological atoms), but which are such that all of their parts themselves have proper parts. This paper is divided into three sections, each related to one of its goals. The rst is to explain what gunk is. The second is to examine a range of sources about the Stoic theory of parts and wholes, to show why these sources support my interpretation, and to explain how this interpretation helps with puzzles about mixture and the present. The third is to discuss what seems to be the biggest problem for my interpretation: some testi- monia about the Stoic attitude to innity which are apparently in tension with my claim.

Transcript of Stoic Gunk

Page 1: Stoic Gunk

copy Koninklijke Brill NV Leiden 2006 Phronesis LI2Also available online ndash wwwbrillnl

Accepted October 20051 This sense of the term though not the concept is due to David Lewis see Lewis

1991

Stoic Gunk

DANIEL NOLAN

ABSTRACT

The surviving sources on the Stoic theory of division reveal that the Stoics par-ticularly Chrysippus believed that bodies places and times were such that all oftheir parts themselves had proper parts That is bodies places and times werecomposed of gunk This realisation helps solve some long-standing puzzles aboutthe Stoic theory of mixture and the Stoic attitude to the present

This paper seeks to address two of the central questions to be answeredin any account of the physical and metaphysical doctrines of the StoicsThe first is the question of what attitude the Stoics took towards wholesand parts and the second the related question about the divisibility ofobjects which is closely connected to the question of what parts objectsin fact possess To my mind modern commentators have misunderstoodthe Stoics particularly Chrysippus on these central issues or at the veryleast overlooked an important option They have thus overlooked an inter-pretation of the Stoics which closely fits the surviving fragments of Stoicwritings on the topic the possibility that the Stoics embraced a non-atomicmereology and postulated that material objects were such that each oftheir parts themselves had further parts without any finite limit In otherwords to use the jargon of contemporary mereology Stoics postulated theexistence of ldquogunkrdquo1 objects which are not composed of mereologicalindivisibles (mereological atoms) but which are such that all of their partsthemselves have proper parts

This paper is divided into three sections each related to one of its goalsThe first is to explain what gunk is The second is to examine a range ofsources about the Stoic theory of parts and wholes to show why thesesources support my interpretation and to explain how this interpretationhelps with puzzles about mixture and the present The third is to discusswhat seems to be the biggest problem for my interpretation some testi-monia about the Stoic attitude to infinity which are apparently in tensionwith my claim

Phronesis 512_f3_162-183I 42506 557 PM Page 162

STOIC GUNK 163

2 For an introduction to contemporary mereology the reader may wish to consultSimons 1987

1 Atomless Gunk

I will begin with some notes about terminology The word ldquopartrdquo used asa technical term in contemporary mereology (the theory of parts and wholes)2

is used in a somewhat broader sense than in its non-technical uses in thissense for example the ldquopartsrdquo of a given object A include A itself inanalogy to the usage in set theory according to which each set is one ofits own subsets Some mereologies postulate for convenience a ldquonull partrdquoa part which is a part of every object I will not do so here but in systemswith a null part this is also something the mereologist counts among theparts A proper part of an object is a part which is not the object itselfnor the null part (This is analogous to a ldquoproper subsetrdquo in set theorywhere a proper subset of a set S is any subset of S which is not identicalwith S nor identical with the null set) It is the expression ldquoproper partrdquowhich probably corresponds most closely with our ordinary expressionldquopartrdquo but nonetheless I shall be using ldquopartrdquo in its extended sense toavoid confusion and I shall say ldquoproper partrdquo when I mean a part otherthan the object itself

A mereological atom is an object that has no proper parts (This use ofldquoatomrdquo coincides more with the original meaning of ldquoatomrdquo as an indi-visible (or ldquouncuttablerdquo) object and I stress that this usage of ldquoatomrdquo isnot the use of that term in modern physics or chemistry for famously wecan ldquosplit the atomrdquo of physics and chemistry) To assert the existence ofgunk is to assert that there is an object such that all of its parts have properparts it follows from this that each of its parts can themselves be furtherdivided without end and without at any stage (finite or infinite) reachinga bedrock of indivisible minimal parts (that is the object is not made upof atoms) Alfred North Whiteheadrsquos view of space was like this forexample at the fundamental level there were only regions of space notpoints but there was no limit to how small a region of space might be

The concept of gunk is unfamiliar to many but it is not very abstruseldquoall of its parts have partsrdquo is a natural if sloppy way of characterisingwhat it is for a thing to be gunk (It comes to the same thing as the tech-nical definition only if ldquopartrdquo is taken in the sense of ldquoproper partrdquo andit is further assumed that the object has some parts in this sense) It istempting to say that what makes something a piece of gunk is its beinginfinitely divisible but this is not quite right Gunk is infinitely divisible

Phronesis 512_f3_162-183I 42506 557 PM Page 163

164 DANIEL NOLAN

3 Some readers may be wondering where in this taxonomy of infinite divisibility Iwould include divisibility into infinitesimal magnitudes especially as Michael White(1982 1992) has suggested this as a model for Stoic infinite divisibility (Or rather in1992 he has suggested a ldquofuzzifiedrdquo version of this as our model) If every infinitesimalbody has another infinitesimal part smaller than it and there are none of minimal sizethen it turns out to be a gunky view (albeit one with extra features that strike me asunneccessary) If it is indeterminate whether there is a least size of indeterminate bod-ies which might well be the view in White 1992 then it will be indeterminate whetherthis can be fitted with a gunky framework

but there are ways of being infinitely divisible besides the way gunk isOne way to be infinitely divisible for example is the way a geometricalline-segment is thought to be divisible in a geometry based on pointsevery line segment can be divided into smaller line segments but at theend-point of division there is an infinity of extension-less points whichcannot themselves be divided further Another similar way to be infinitelydivisible might be to be an object which was infinitely large one might beable to take away parts of some fixed finite (non-zero non-infinitesimal)size ad infinitum without having to come to a halt even if those finiteparts themselves are not further divisible A third way if it is coherentis an Aristotelian way in which the potential for infinitely many divisionsmay exist even in an object which has no actual proper parts at all (Thisis what some people have in mind when they speak of an object havinga ldquopotential infinityrdquo of parts but perhaps not what they all mean) So while many commentators have held that the Stoics wished to be committed to infinite divisibility the further claim that the sort of in-finite divisibility they had in mind was gunky is one I am interested indefending3

2 What the Stoics Said

21 Parts Wholes and Infinite Division

It is not an entirely easy matter to determine what the Stoics had to sayabout parts and wholes little of the Stoicsrsquo metaphysical writings havesurvived and it can be hard to pierce the veil of ancient testimonia to findwhat the Stoics in fact argued as opposed to how their opponents or otherreporters thought they argued I shall be discussing some of the fragmentswhich purport to be direct quotations of what the Stoics said as well asthe ancient testimonia I shall first discuss two passages which seem to

Phronesis 512_f3_162-183I 42506 557 PM Page 164

STOIC GUNK 165

4 References of the form ldquoLSrdquo are to Long and Sedley 1987 and the LS translationsare theirs Sometimes as in this case I will not reproduce the entire LS selection LS50C is an extract from Plutarchrsquos On Common Conceptions 1078E-1081A

5 Unless by ldquonumbering the partsrdquo one means assigning a finite number to them(eg a counting number) if that is what is meant it is a case where the parts cannotbe numbered

me to bear directly on the question of what parts Stoics thought materialobjects had Then I shall examine how the hypothesis that the Stoics werecommitted to gunk can shed light on the Stoic theories of mixture and oftime in a way which resolves some of the perplexities of ancient and con-temporary critics I shall be focussing on Chrysippusrsquo theory in what followssince his physical theories seem to be both one of the most fully workedout and best attested to in Stoic thought

The first passage is from Plutarch quoting verbatim from Chrysippusand is strong evidence that Chrysippus at least rejected ldquoultimate partsrdquowhere these are presumably parts which do not themselves have (proper)parts (LS 50C)4

Chrysippus says that when asked if we have parts and how many and of whatand how many parts they consist we will operate a distinction With regard tothe inexact question we will reply that we consist of head trunk and limbs ndash forthat was all that the problem put to us amounted to But if they extend their ques-tioning to the ultimate parts we must not he says in reply concede any suchthings but must say neither of what parts we consist nor likewise of how manyeither infinite or finite I have I think quoted his actual words so that you maysee how far he conserved the common conceptions urging us to think of eachbody as consisting neither of certain parts nor of some number of them eitherinfinite or finite

I am glad that Plutarch went to the trouble of quoting Chrysippus verba-tim since I reject Plutarchrsquos gloss on the passage quoted In rejecting ulti-mate parts either finite or infinite in number Chrysippus need not beldquourging us to think of each body as consisting neither of certain parts norof some number of them either finite or infiniterdquo For despite whatPlutarch might think one can reject ldquoultimate partsrdquo without denying thatan object has parts at all Nor need one say that one cannot number theparts5 there are in fact at least infinitely many parts in any piece of gunk(at least continuum-many indeed which is not the smallest grade of infinity)though whether Chrysippus would have realised that this is a consequenceof his acceptance of real parts of objects together with the denial of ulti-mate parts is another matter which I will take up in Section 3 Howevercertainly the only natural way of reading what Chrysippus as quoted by

Phronesis 512_f3_162-183I 42506 557 PM Page 165

166 DANIEL NOLAN

6 The Greeks did not include a number zero among the numbers7 Sextus mainly has in mind the Epicureans to judge from Against the Physicists

ii 141-144 unless the defenders of indivisible places and bodies who were attacked

Plutarch is saying is that while objects have parts they do not have ulti-mate parts either finite or infinite Plutarch goes on to complain thatChrysippus must be saying that the (ultimate) parts are neither finite norinfinite and so Chrysippus must be attempting some impossible via mediabetween finitude and infinitude But Plutarch has missed the point Chrysippusdenies the existence of ultimate parts and instead claims that matter whiledivisible is such that every division is itself capable of further division(itself has proper parts) Chrysippus need not find some third alternativeto number the ultimate parts6 since Chrysippus simply denies the exis-tence of ultimate parts

The next source discussed comes from the characterisation of the Stoicview by Sextus Empiricus in LS 50F (Against the Physicists ii 121-126139-142) Sextus follows his characterisation and critique of a certain viewof infinite divisibility by saying ldquoThis then was the appropriate reply tothose who say that bodies places and times are divided to infinity namelythe Stoicsrdquo

At the beginning of the discussion that this is drawn from he had char-acterised three approaches and labelled one of them as the view that bod-ies places and times are ldquodivided to infinityrdquo

Next every motion involves three things namely bodies places and times ndash bod-ies to do the moving places for the movement to happen in and times for themovement to take So it is either with these all being divided into infinitely manyplaces times and bodies that motion happens or with them all terminating at apartless and minimal magnitude or with some of them divided to infinity whileothers terminate at a partless and minimal magnitude Taking them in orderlet us start our argument with the first school of thought according to which allare divided to infinity

The discussion then continues with Sextus offering paradoxes of motionagainst those who hold that body place and time are ldquodivided to infinityrdquoTwo things are evident from the above passage however The first is thatthe view that bodies places and times are ldquodivided to infinityrdquo is distin-guished from views according to which division terminates ldquoat a partlessand minimal magnituderdquo Perhaps by this latter expression Sextus intendedonly the atomists who held that the process of division (of bodies at least)terminated in finitely many parts each with a positive finite magnitude7

But the expression is broad enough to encompass theories according to

Phronesis 512_f3_162-183I 42506 557 PM Page 166

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

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STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

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STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

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172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 2: Stoic Gunk

STOIC GUNK 163

2 For an introduction to contemporary mereology the reader may wish to consultSimons 1987

1 Atomless Gunk

I will begin with some notes about terminology The word ldquopartrdquo used asa technical term in contemporary mereology (the theory of parts and wholes)2

is used in a somewhat broader sense than in its non-technical uses in thissense for example the ldquopartsrdquo of a given object A include A itself inanalogy to the usage in set theory according to which each set is one ofits own subsets Some mereologies postulate for convenience a ldquonull partrdquoa part which is a part of every object I will not do so here but in systemswith a null part this is also something the mereologist counts among theparts A proper part of an object is a part which is not the object itselfnor the null part (This is analogous to a ldquoproper subsetrdquo in set theorywhere a proper subset of a set S is any subset of S which is not identicalwith S nor identical with the null set) It is the expression ldquoproper partrdquowhich probably corresponds most closely with our ordinary expressionldquopartrdquo but nonetheless I shall be using ldquopartrdquo in its extended sense toavoid confusion and I shall say ldquoproper partrdquo when I mean a part otherthan the object itself

A mereological atom is an object that has no proper parts (This use ofldquoatomrdquo coincides more with the original meaning of ldquoatomrdquo as an indi-visible (or ldquouncuttablerdquo) object and I stress that this usage of ldquoatomrdquo isnot the use of that term in modern physics or chemistry for famously wecan ldquosplit the atomrdquo of physics and chemistry) To assert the existence ofgunk is to assert that there is an object such that all of its parts have properparts it follows from this that each of its parts can themselves be furtherdivided without end and without at any stage (finite or infinite) reachinga bedrock of indivisible minimal parts (that is the object is not made upof atoms) Alfred North Whiteheadrsquos view of space was like this forexample at the fundamental level there were only regions of space notpoints but there was no limit to how small a region of space might be

The concept of gunk is unfamiliar to many but it is not very abstruseldquoall of its parts have partsrdquo is a natural if sloppy way of characterisingwhat it is for a thing to be gunk (It comes to the same thing as the tech-nical definition only if ldquopartrdquo is taken in the sense of ldquoproper partrdquo andit is further assumed that the object has some parts in this sense) It istempting to say that what makes something a piece of gunk is its beinginfinitely divisible but this is not quite right Gunk is infinitely divisible

Phronesis 512_f3_162-183I 42506 557 PM Page 163

164 DANIEL NOLAN

3 Some readers may be wondering where in this taxonomy of infinite divisibility Iwould include divisibility into infinitesimal magnitudes especially as Michael White(1982 1992) has suggested this as a model for Stoic infinite divisibility (Or rather in1992 he has suggested a ldquofuzzifiedrdquo version of this as our model) If every infinitesimalbody has another infinitesimal part smaller than it and there are none of minimal sizethen it turns out to be a gunky view (albeit one with extra features that strike me asunneccessary) If it is indeterminate whether there is a least size of indeterminate bod-ies which might well be the view in White 1992 then it will be indeterminate whetherthis can be fitted with a gunky framework

but there are ways of being infinitely divisible besides the way gunk isOne way to be infinitely divisible for example is the way a geometricalline-segment is thought to be divisible in a geometry based on pointsevery line segment can be divided into smaller line segments but at theend-point of division there is an infinity of extension-less points whichcannot themselves be divided further Another similar way to be infinitelydivisible might be to be an object which was infinitely large one might beable to take away parts of some fixed finite (non-zero non-infinitesimal)size ad infinitum without having to come to a halt even if those finiteparts themselves are not further divisible A third way if it is coherentis an Aristotelian way in which the potential for infinitely many divisionsmay exist even in an object which has no actual proper parts at all (Thisis what some people have in mind when they speak of an object havinga ldquopotential infinityrdquo of parts but perhaps not what they all mean) So while many commentators have held that the Stoics wished to be committed to infinite divisibility the further claim that the sort of in-finite divisibility they had in mind was gunky is one I am interested indefending3

2 What the Stoics Said

21 Parts Wholes and Infinite Division

It is not an entirely easy matter to determine what the Stoics had to sayabout parts and wholes little of the Stoicsrsquo metaphysical writings havesurvived and it can be hard to pierce the veil of ancient testimonia to findwhat the Stoics in fact argued as opposed to how their opponents or otherreporters thought they argued I shall be discussing some of the fragmentswhich purport to be direct quotations of what the Stoics said as well asthe ancient testimonia I shall first discuss two passages which seem to

Phronesis 512_f3_162-183I 42506 557 PM Page 164

STOIC GUNK 165

4 References of the form ldquoLSrdquo are to Long and Sedley 1987 and the LS translationsare theirs Sometimes as in this case I will not reproduce the entire LS selection LS50C is an extract from Plutarchrsquos On Common Conceptions 1078E-1081A

5 Unless by ldquonumbering the partsrdquo one means assigning a finite number to them(eg a counting number) if that is what is meant it is a case where the parts cannotbe numbered

me to bear directly on the question of what parts Stoics thought materialobjects had Then I shall examine how the hypothesis that the Stoics werecommitted to gunk can shed light on the Stoic theories of mixture and oftime in a way which resolves some of the perplexities of ancient and con-temporary critics I shall be focussing on Chrysippusrsquo theory in what followssince his physical theories seem to be both one of the most fully workedout and best attested to in Stoic thought

The first passage is from Plutarch quoting verbatim from Chrysippusand is strong evidence that Chrysippus at least rejected ldquoultimate partsrdquowhere these are presumably parts which do not themselves have (proper)parts (LS 50C)4

Chrysippus says that when asked if we have parts and how many and of whatand how many parts they consist we will operate a distinction With regard tothe inexact question we will reply that we consist of head trunk and limbs ndash forthat was all that the problem put to us amounted to But if they extend their ques-tioning to the ultimate parts we must not he says in reply concede any suchthings but must say neither of what parts we consist nor likewise of how manyeither infinite or finite I have I think quoted his actual words so that you maysee how far he conserved the common conceptions urging us to think of eachbody as consisting neither of certain parts nor of some number of them eitherinfinite or finite

I am glad that Plutarch went to the trouble of quoting Chrysippus verba-tim since I reject Plutarchrsquos gloss on the passage quoted In rejecting ulti-mate parts either finite or infinite in number Chrysippus need not beldquourging us to think of each body as consisting neither of certain parts norof some number of them either finite or infiniterdquo For despite whatPlutarch might think one can reject ldquoultimate partsrdquo without denying thatan object has parts at all Nor need one say that one cannot number theparts5 there are in fact at least infinitely many parts in any piece of gunk(at least continuum-many indeed which is not the smallest grade of infinity)though whether Chrysippus would have realised that this is a consequenceof his acceptance of real parts of objects together with the denial of ulti-mate parts is another matter which I will take up in Section 3 Howevercertainly the only natural way of reading what Chrysippus as quoted by

Phronesis 512_f3_162-183I 42506 557 PM Page 165

166 DANIEL NOLAN

6 The Greeks did not include a number zero among the numbers7 Sextus mainly has in mind the Epicureans to judge from Against the Physicists

ii 141-144 unless the defenders of indivisible places and bodies who were attacked

Plutarch is saying is that while objects have parts they do not have ulti-mate parts either finite or infinite Plutarch goes on to complain thatChrysippus must be saying that the (ultimate) parts are neither finite norinfinite and so Chrysippus must be attempting some impossible via mediabetween finitude and infinitude But Plutarch has missed the point Chrysippusdenies the existence of ultimate parts and instead claims that matter whiledivisible is such that every division is itself capable of further division(itself has proper parts) Chrysippus need not find some third alternativeto number the ultimate parts6 since Chrysippus simply denies the exis-tence of ultimate parts

The next source discussed comes from the characterisation of the Stoicview by Sextus Empiricus in LS 50F (Against the Physicists ii 121-126139-142) Sextus follows his characterisation and critique of a certain viewof infinite divisibility by saying ldquoThis then was the appropriate reply tothose who say that bodies places and times are divided to infinity namelythe Stoicsrdquo

At the beginning of the discussion that this is drawn from he had char-acterised three approaches and labelled one of them as the view that bod-ies places and times are ldquodivided to infinityrdquo

Next every motion involves three things namely bodies places and times ndash bod-ies to do the moving places for the movement to happen in and times for themovement to take So it is either with these all being divided into infinitely manyplaces times and bodies that motion happens or with them all terminating at apartless and minimal magnitude or with some of them divided to infinity whileothers terminate at a partless and minimal magnitude Taking them in orderlet us start our argument with the first school of thought according to which allare divided to infinity

The discussion then continues with Sextus offering paradoxes of motionagainst those who hold that body place and time are ldquodivided to infinityrdquoTwo things are evident from the above passage however The first is thatthe view that bodies places and times are ldquodivided to infinityrdquo is distin-guished from views according to which division terminates ldquoat a partlessand minimal magnituderdquo Perhaps by this latter expression Sextus intendedonly the atomists who held that the process of division (of bodies at least)terminated in finitely many parts each with a positive finite magnitude7

But the expression is broad enough to encompass theories according to

Phronesis 512_f3_162-183I 42506 557 PM Page 166

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 3: Stoic Gunk

164 DANIEL NOLAN

3 Some readers may be wondering where in this taxonomy of infinite divisibility Iwould include divisibility into infinitesimal magnitudes especially as Michael White(1982 1992) has suggested this as a model for Stoic infinite divisibility (Or rather in1992 he has suggested a ldquofuzzifiedrdquo version of this as our model) If every infinitesimalbody has another infinitesimal part smaller than it and there are none of minimal sizethen it turns out to be a gunky view (albeit one with extra features that strike me asunneccessary) If it is indeterminate whether there is a least size of indeterminate bod-ies which might well be the view in White 1992 then it will be indeterminate whetherthis can be fitted with a gunky framework

but there are ways of being infinitely divisible besides the way gunk isOne way to be infinitely divisible for example is the way a geometricalline-segment is thought to be divisible in a geometry based on pointsevery line segment can be divided into smaller line segments but at theend-point of division there is an infinity of extension-less points whichcannot themselves be divided further Another similar way to be infinitelydivisible might be to be an object which was infinitely large one might beable to take away parts of some fixed finite (non-zero non-infinitesimal)size ad infinitum without having to come to a halt even if those finiteparts themselves are not further divisible A third way if it is coherentis an Aristotelian way in which the potential for infinitely many divisionsmay exist even in an object which has no actual proper parts at all (Thisis what some people have in mind when they speak of an object havinga ldquopotential infinityrdquo of parts but perhaps not what they all mean) So while many commentators have held that the Stoics wished to be committed to infinite divisibility the further claim that the sort of in-finite divisibility they had in mind was gunky is one I am interested indefending3

2 What the Stoics Said

21 Parts Wholes and Infinite Division

It is not an entirely easy matter to determine what the Stoics had to sayabout parts and wholes little of the Stoicsrsquo metaphysical writings havesurvived and it can be hard to pierce the veil of ancient testimonia to findwhat the Stoics in fact argued as opposed to how their opponents or otherreporters thought they argued I shall be discussing some of the fragmentswhich purport to be direct quotations of what the Stoics said as well asthe ancient testimonia I shall first discuss two passages which seem to

Phronesis 512_f3_162-183I 42506 557 PM Page 164

STOIC GUNK 165

4 References of the form ldquoLSrdquo are to Long and Sedley 1987 and the LS translationsare theirs Sometimes as in this case I will not reproduce the entire LS selection LS50C is an extract from Plutarchrsquos On Common Conceptions 1078E-1081A

5 Unless by ldquonumbering the partsrdquo one means assigning a finite number to them(eg a counting number) if that is what is meant it is a case where the parts cannotbe numbered

me to bear directly on the question of what parts Stoics thought materialobjects had Then I shall examine how the hypothesis that the Stoics werecommitted to gunk can shed light on the Stoic theories of mixture and oftime in a way which resolves some of the perplexities of ancient and con-temporary critics I shall be focussing on Chrysippusrsquo theory in what followssince his physical theories seem to be both one of the most fully workedout and best attested to in Stoic thought

The first passage is from Plutarch quoting verbatim from Chrysippusand is strong evidence that Chrysippus at least rejected ldquoultimate partsrdquowhere these are presumably parts which do not themselves have (proper)parts (LS 50C)4

Chrysippus says that when asked if we have parts and how many and of whatand how many parts they consist we will operate a distinction With regard tothe inexact question we will reply that we consist of head trunk and limbs ndash forthat was all that the problem put to us amounted to But if they extend their ques-tioning to the ultimate parts we must not he says in reply concede any suchthings but must say neither of what parts we consist nor likewise of how manyeither infinite or finite I have I think quoted his actual words so that you maysee how far he conserved the common conceptions urging us to think of eachbody as consisting neither of certain parts nor of some number of them eitherinfinite or finite

I am glad that Plutarch went to the trouble of quoting Chrysippus verba-tim since I reject Plutarchrsquos gloss on the passage quoted In rejecting ulti-mate parts either finite or infinite in number Chrysippus need not beldquourging us to think of each body as consisting neither of certain parts norof some number of them either finite or infiniterdquo For despite whatPlutarch might think one can reject ldquoultimate partsrdquo without denying thatan object has parts at all Nor need one say that one cannot number theparts5 there are in fact at least infinitely many parts in any piece of gunk(at least continuum-many indeed which is not the smallest grade of infinity)though whether Chrysippus would have realised that this is a consequenceof his acceptance of real parts of objects together with the denial of ulti-mate parts is another matter which I will take up in Section 3 Howevercertainly the only natural way of reading what Chrysippus as quoted by

Phronesis 512_f3_162-183I 42506 557 PM Page 165

166 DANIEL NOLAN

6 The Greeks did not include a number zero among the numbers7 Sextus mainly has in mind the Epicureans to judge from Against the Physicists

ii 141-144 unless the defenders of indivisible places and bodies who were attacked

Plutarch is saying is that while objects have parts they do not have ulti-mate parts either finite or infinite Plutarch goes on to complain thatChrysippus must be saying that the (ultimate) parts are neither finite norinfinite and so Chrysippus must be attempting some impossible via mediabetween finitude and infinitude But Plutarch has missed the point Chrysippusdenies the existence of ultimate parts and instead claims that matter whiledivisible is such that every division is itself capable of further division(itself has proper parts) Chrysippus need not find some third alternativeto number the ultimate parts6 since Chrysippus simply denies the exis-tence of ultimate parts

The next source discussed comes from the characterisation of the Stoicview by Sextus Empiricus in LS 50F (Against the Physicists ii 121-126139-142) Sextus follows his characterisation and critique of a certain viewof infinite divisibility by saying ldquoThis then was the appropriate reply tothose who say that bodies places and times are divided to infinity namelythe Stoicsrdquo

At the beginning of the discussion that this is drawn from he had char-acterised three approaches and labelled one of them as the view that bod-ies places and times are ldquodivided to infinityrdquo

Next every motion involves three things namely bodies places and times ndash bod-ies to do the moving places for the movement to happen in and times for themovement to take So it is either with these all being divided into infinitely manyplaces times and bodies that motion happens or with them all terminating at apartless and minimal magnitude or with some of them divided to infinity whileothers terminate at a partless and minimal magnitude Taking them in orderlet us start our argument with the first school of thought according to which allare divided to infinity

The discussion then continues with Sextus offering paradoxes of motionagainst those who hold that body place and time are ldquodivided to infinityrdquoTwo things are evident from the above passage however The first is thatthe view that bodies places and times are ldquodivided to infinityrdquo is distin-guished from views according to which division terminates ldquoat a partlessand minimal magnituderdquo Perhaps by this latter expression Sextus intendedonly the atomists who held that the process of division (of bodies at least)terminated in finitely many parts each with a positive finite magnitude7

But the expression is broad enough to encompass theories according to

Phronesis 512_f3_162-183I 42506 557 PM Page 166

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 4: Stoic Gunk

STOIC GUNK 165

4 References of the form ldquoLSrdquo are to Long and Sedley 1987 and the LS translationsare theirs Sometimes as in this case I will not reproduce the entire LS selection LS50C is an extract from Plutarchrsquos On Common Conceptions 1078E-1081A

5 Unless by ldquonumbering the partsrdquo one means assigning a finite number to them(eg a counting number) if that is what is meant it is a case where the parts cannotbe numbered

me to bear directly on the question of what parts Stoics thought materialobjects had Then I shall examine how the hypothesis that the Stoics werecommitted to gunk can shed light on the Stoic theories of mixture and oftime in a way which resolves some of the perplexities of ancient and con-temporary critics I shall be focussing on Chrysippusrsquo theory in what followssince his physical theories seem to be both one of the most fully workedout and best attested to in Stoic thought

The first passage is from Plutarch quoting verbatim from Chrysippusand is strong evidence that Chrysippus at least rejected ldquoultimate partsrdquowhere these are presumably parts which do not themselves have (proper)parts (LS 50C)4

Chrysippus says that when asked if we have parts and how many and of whatand how many parts they consist we will operate a distinction With regard tothe inexact question we will reply that we consist of head trunk and limbs ndash forthat was all that the problem put to us amounted to But if they extend their ques-tioning to the ultimate parts we must not he says in reply concede any suchthings but must say neither of what parts we consist nor likewise of how manyeither infinite or finite I have I think quoted his actual words so that you maysee how far he conserved the common conceptions urging us to think of eachbody as consisting neither of certain parts nor of some number of them eitherinfinite or finite

I am glad that Plutarch went to the trouble of quoting Chrysippus verba-tim since I reject Plutarchrsquos gloss on the passage quoted In rejecting ulti-mate parts either finite or infinite in number Chrysippus need not beldquourging us to think of each body as consisting neither of certain parts norof some number of them either finite or infiniterdquo For despite whatPlutarch might think one can reject ldquoultimate partsrdquo without denying thatan object has parts at all Nor need one say that one cannot number theparts5 there are in fact at least infinitely many parts in any piece of gunk(at least continuum-many indeed which is not the smallest grade of infinity)though whether Chrysippus would have realised that this is a consequenceof his acceptance of real parts of objects together with the denial of ulti-mate parts is another matter which I will take up in Section 3 Howevercertainly the only natural way of reading what Chrysippus as quoted by

Phronesis 512_f3_162-183I 42506 557 PM Page 165

166 DANIEL NOLAN

6 The Greeks did not include a number zero among the numbers7 Sextus mainly has in mind the Epicureans to judge from Against the Physicists

ii 141-144 unless the defenders of indivisible places and bodies who were attacked

Plutarch is saying is that while objects have parts they do not have ulti-mate parts either finite or infinite Plutarch goes on to complain thatChrysippus must be saying that the (ultimate) parts are neither finite norinfinite and so Chrysippus must be attempting some impossible via mediabetween finitude and infinitude But Plutarch has missed the point Chrysippusdenies the existence of ultimate parts and instead claims that matter whiledivisible is such that every division is itself capable of further division(itself has proper parts) Chrysippus need not find some third alternativeto number the ultimate parts6 since Chrysippus simply denies the exis-tence of ultimate parts

The next source discussed comes from the characterisation of the Stoicview by Sextus Empiricus in LS 50F (Against the Physicists ii 121-126139-142) Sextus follows his characterisation and critique of a certain viewof infinite divisibility by saying ldquoThis then was the appropriate reply tothose who say that bodies places and times are divided to infinity namelythe Stoicsrdquo

At the beginning of the discussion that this is drawn from he had char-acterised three approaches and labelled one of them as the view that bod-ies places and times are ldquodivided to infinityrdquo

Next every motion involves three things namely bodies places and times ndash bod-ies to do the moving places for the movement to happen in and times for themovement to take So it is either with these all being divided into infinitely manyplaces times and bodies that motion happens or with them all terminating at apartless and minimal magnitude or with some of them divided to infinity whileothers terminate at a partless and minimal magnitude Taking them in orderlet us start our argument with the first school of thought according to which allare divided to infinity

The discussion then continues with Sextus offering paradoxes of motionagainst those who hold that body place and time are ldquodivided to infinityrdquoTwo things are evident from the above passage however The first is thatthe view that bodies places and times are ldquodivided to infinityrdquo is distin-guished from views according to which division terminates ldquoat a partlessand minimal magnituderdquo Perhaps by this latter expression Sextus intendedonly the atomists who held that the process of division (of bodies at least)terminated in finitely many parts each with a positive finite magnitude7

But the expression is broad enough to encompass theories according to

Phronesis 512_f3_162-183I 42506 557 PM Page 166

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 5: Stoic Gunk

166 DANIEL NOLAN

6 The Greeks did not include a number zero among the numbers7 Sextus mainly has in mind the Epicureans to judge from Against the Physicists

ii 141-144 unless the defenders of indivisible places and bodies who were attacked

Plutarch is saying is that while objects have parts they do not have ulti-mate parts either finite or infinite Plutarch goes on to complain thatChrysippus must be saying that the (ultimate) parts are neither finite norinfinite and so Chrysippus must be attempting some impossible via mediabetween finitude and infinitude But Plutarch has missed the point Chrysippusdenies the existence of ultimate parts and instead claims that matter whiledivisible is such that every division is itself capable of further division(itself has proper parts) Chrysippus need not find some third alternativeto number the ultimate parts6 since Chrysippus simply denies the exis-tence of ultimate parts

The next source discussed comes from the characterisation of the Stoicview by Sextus Empiricus in LS 50F (Against the Physicists ii 121-126139-142) Sextus follows his characterisation and critique of a certain viewof infinite divisibility by saying ldquoThis then was the appropriate reply tothose who say that bodies places and times are divided to infinity namelythe Stoicsrdquo

At the beginning of the discussion that this is drawn from he had char-acterised three approaches and labelled one of them as the view that bod-ies places and times are ldquodivided to infinityrdquo

Next every motion involves three things namely bodies places and times ndash bod-ies to do the moving places for the movement to happen in and times for themovement to take So it is either with these all being divided into infinitely manyplaces times and bodies that motion happens or with them all terminating at apartless and minimal magnitude or with some of them divided to infinity whileothers terminate at a partless and minimal magnitude Taking them in orderlet us start our argument with the first school of thought according to which allare divided to infinity

The discussion then continues with Sextus offering paradoxes of motionagainst those who hold that body place and time are ldquodivided to infinityrdquoTwo things are evident from the above passage however The first is thatthe view that bodies places and times are ldquodivided to infinityrdquo is distin-guished from views according to which division terminates ldquoat a partlessand minimal magnituderdquo Perhaps by this latter expression Sextus intendedonly the atomists who held that the process of division (of bodies at least)terminated in finitely many parts each with a positive finite magnitude7

But the expression is broad enough to encompass theories according to

Phronesis 512_f3_162-183I 42506 557 PM Page 166

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 6: Stoic Gunk

STOIC GUNK 167

by Diodorus Cronos (alluded to in ii 143) included people besides those who heldEpicurean doctrines

which bodies places and times are divided infinitely with this divisionldquoterminatingrdquo at infinitely many magnitudes An example of such a viewwould be one that held that space could be divided into points with zeroor infinitesimal magnitudes a view which it can be argued was defendedby Xenocrates among others (and criticised by Aristotle among others)If Sextus can be read as including the latter variety of view then we havehere a case where the Stoic view is distinguished from the view of infinitedivisibility which is more common today according to which for exam-ple space is infinitely divisible not only because there is no minimumsize of regions but also because it can be ultimately divided into pointsSo if we adopt this reading Sextus distinguishes the Stoic view from thetheory of infinite division which holds that division terminates in ldquopart-less and minimal magnitudesrdquo

If the Stoics held that bodies places or times were divided infinitely(as Sextus tells us) but there were no ldquoleast partsrdquo or ldquoultimate partsrdquo (asthe quote from Chrysippus in Plutarch establishes) we have a character-isation of the Stoic view which could virtually only be that of a gunk the-ory Of course the fact that Sextus characterises the Stoic theories in thisway (if indeed he did intend to distinguish them from any view commit-ted to ldquominimal and partless partsrdquo) is no guarantee that the Stoic theorieswere indeed this way it is possible that Sextus may have misunderstoodBut it is certainly strong evidence

Another piece of the puzzle provided by Sextus is evidence that theStoics took these parts into which bodies are supposed to be infinitelydivided to be real parts and that it was not just that there were unlimitedpossibilities for creating parts through acts of division (which is all thatAristotelian ldquoinfinite divisionrdquo is often taken to be) Sextus talks as if theStoics are committed to bodies being ldquodivided to infinityrdquo and not merelydivisible to infinity In the Greek of AP ii 121 as well as saying that thefirst option involves patildentvn toEcirctvn efiw eacutepecurrenrouw temnomdegnvn (ii 121) orefiw ecircpeiron tdegmnetai (ii 123) or efiw ecircpeiron tdegmnesyai (ii 131) (every-thing is infinitely divisible or ldquodivisible ad infinitumrdquo in Buryrsquos 1936translation) Sextus also characterises the division of body as efiw ecircpeirasasympmata (ii 121) ldquointo infinite bodiesrdquo or as Bury 1936 translates it ldquointoan infinite number of bodiesrdquo

There is more substantial evidence from Sextus that the Stoics did notintend their account of infinite division to be merely about a potential

Phronesis 512_f3_162-183I 42506 557 PM Page 167

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 7: Stoic Gunk

168 DANIEL NOLAN

8 This interpretation of Sextusrsquo account of the Stoics runs counter to Dirk Baltzlyrsquos1998 interpretation of Against the Physicists i 352 Baltzly suggests that Sextus hasthe Stoics in mind when he attributes to unnamed ldquodogmatistsrdquo the view that properparts of bodies are ldquosomehow in usrdquo ie mind-dependent I reject Baltzlyrsquos interpre-tation since the views Sextus explicitly attributed to the Stoics seem to be quite dif-ferent from the views of the unnamed dogmatists and as Baltzly himself points outthe views attributed to the unnamed dogmatists are in conflict with things we knowfrom other sources that Chrysippus wanted to maintain such as his response to theScepticrsquos ldquogrowing argumentrdquo

infinity as for instance Aristotle and other Peripatetics treated divisionEvidence suggesting that Sextus took the Stoics to be committed to theactuality of the parts which make up the infinite division can be found in the argument against the Stoics in LS 50F (Against the Physicists ii139-42 in Bury 1936) Sextus in his argument against the position heidentifies as the Stoic position argues that if a body can be divided with-out end there will be no ldquofirst partrdquo to begin movement For this argumentto touch the Stoic position the Stoics would presumably have to believethe body has the division in actuality and not merely in potentiality forsaying there is no first part is different from saying that it is merely pos-sible that there be no first part Sextus seems to be treating Stoic divisionas not merely potential Interestingly Sextus does not include this ldquono firstpartrdquo argument in the battery of similar arguments he offers againstStratorsquos Peripatetic account of infinitely divisible bodies in the same dis-cussion (Against the Physicists ii 155-167 in Bury 1936) this omissionwould be explained if Sextus took the Peripatetics to be committed onlyto potential parts while the Stoics had a stronger commitment to the actu-ality of a bodyrsquos parts I do not want to say that Sextusrsquo argument aboutthe possibility of movement succeeds against the Stoics of course merelythat it presupposes that the Stoics would have accepted that the parts ofthe body are already there whenever it begins to move The mere fact thatSextus offers the argument is evidence that Sextus interpreted the Stoicsas committed to actual parts and the fact that he does not offer it againstStrato is evidence that Sextus perceived a contrast here between the twoviews of division and parthood8

It seems that Sextusrsquo evidence plus the quote from Chrysippus we oweto Plutarch together establish that the Stoics treated bodies places andtimes as gunk actually divided into infinite bodies though without any ultimate parts As well as evidence from ancients such as Plutarchand Sextus Empiricus however the thesis that Stoics believed in gunksolves two longstanding puzzles in the interpretation of Stoic physics the

Phronesis 512_f3_162-183I 42506 557 PM Page 168

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 8: Stoic Gunk

STOIC GUNK 169

9 Here and elsewhere when I use the word ldquosubstancerdquo the reader should be care-ful not to read an Aristotelian conception of ldquosubstancerdquo into what I say (ThoughAlexander when he uses the word ldquosubstancerdquo in reporting Stoic views may or maynot have intended Aristotelian connotations)

problem of the Stoic theory of ldquomixturerdquo and the Stoic theory of the rela-tion between the present and time

22 Blending

Alexander in his ldquoOn Mixturerdquo offers a description of Chrysippusrsquo three-fold account of types of mixture (Alexander On Mixture 216 14-218 6LS 48C) First there is ldquojuxtapositionrdquo (paraydegsiw) when two or more sub-stances9 are ldquoput together in the same place and juxtaposed with oneanother lsquoby joiningrsquo as he says while they each preserve their own sub-stance and quality at their surface contact in such a juxtaposition asoccurs one may say with beans and grains of wheat when they are placedside by siderdquo (LS 48C) Then there is ldquofusionrdquo or ldquothrough-and-throughfusionrdquo (sugxEcircsiw) when ldquothe substances themselves and their intrinsicqualities are destroyed together as he says happens in the case of med-ical drugs when the things mixed together undergo mutual destruction andanother body is generated out of themrdquo (LS 48C)

So far so good But interpreters have more trouble with the third sortwhich I will follow LS in calling ldquoblendingrdquo which occurs when

certain substances and their qualities are mutually extended through and throughwith the original substances and their qualities being preserved in such a mix-ture for the capacity to be separated again from one another is a peculiarity ofblended substances and this occurs only if they preserve their own natures in themixture He [Chrysippus] believes that such a coextension of blended bodiesoccurs when they pass through one another so that no part among them fails toparticipate in everything contained in such a blended mixture otherwise the resultwould no longer be blending but juxtaposition (LS 48C)

This krccedilsiw (krasis) or mcurrenjiw (mixis) is no minor detail in Stoic physicsfor ldquoblendingrdquo is a pervasive relation in Stoic thought It applies to exam-ples like water and wine (LS 48D) and ldquoblendingrdquo is the relation pneumabears to other material substances It is the relation that fire bears to hotbodies light to illuminated air and in virtue of the role of pneuma it isthe relation which souls bear to the bodies of animals and humans (AlexanderOn Mixture 217-218 on p 119 of Todd 1976) Apparently such blendingsometimes produces a mixture which is of greater volume than any of its

Phronesis 512_f3_162-183I 42506 557 PM Page 169

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 9: Stoic Gunk

170 DANIEL NOLAN

10 Again the primary ldquoStoicrdquo I am attributing this view of blending to is Chry-sippus Alexander (On Mixture 2164) tells us that some later Stoics like Sosigeneshold more Aristotelian accounts of blending but dismisses these as theories which areldquoin many respects inconsistentrdquo adopting some Aristotelian doctrines while not sufficientlyrepudiating the traditional Stoic theory

individual ingredients and at other times the volume of the blend is thesame as that of one of its ingredients (An example of the former wouldbe the mixture of water and wine while an example of the latter is lightin air or perhaps the pneuma in other bodies)

Contemporary commentators have not been very happy with Stoic ldquoblend-ingrdquo10 Long (1974 p 160) calls the Stoic theory of blending ldquoingeniousbut untenablerdquo R Todd while accepting the Stoics had a distinctive the-ory of mixture when it came to pneuma holds that the Stoics could nothave intended to apply this theory of mixture literally to more mundanecases such as the mixture of water and wine incense in air heat in ironetc since this would be ldquoinherently paradoxicalrdquo (Todd 1976 p 46) andGould (1970 p 112) says of the Stoic theory of mixture that ldquothe criticsagain and again harp on its paradoxical character and have no difficultyin branding it logically unsoundrdquo though Gould says little about why Isthis because modern commentators take what Alexander attributes toChrysippus to be incoherent

There are certainly readings on which the theory of ldquoblendingrdquo set outby Alexander is incoherent If the original substances survive blending(they are ldquopreservedrdquo) then surely they will be parts of the blend andpresumably parts of the original substances will be parts of the blend aswell So Chrysippus would be in trouble if ldquono partrdquo of the blend ldquofailsto participate in everything contained in such a blended mixturerdquo unlessthe parts blended change their parts which is at least odd However thereare other ways to take the story of blending presented An account can begiven whereby blending is different from ldquojuxtapositionrdquo and ldquofusionrdquo andin which every part of the blend which fills a region occupied by the mix-ture contains parts of all of the original substances mixed Furthermoreaccording to this account for any magnitude equal or less to the totalmagnitude of the blend there is a part of the blend which is of that sizeand has as parts some of the parts of each of the original substances whichare blended This would give us a theory according to which the originalsubstances are ldquoextended through and throughrdquo in a certain sense This isa sense in which no matter how small a sample of the blend is taken thatsample will contain parts of all of the original substances blended and in

Phronesis 512_f3_162-183I 42506 557 PM Page 170

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 10: Stoic Gunk

STOIC GUNK 171

11 Another less ambitious claim that might be salvaged is that there could be amixture which had a division into a privileged set of parts M such that each memberof M had parts of each of the original mixed substances in it and the members of Mwere further such that each member of M had proper parts that were also among themembers of M No matter how ldquosmallrdquo then there would always be some parts ofthe mixture that contained parts of each of the blended substances This claim wouldbe cleaner in that it would not invoke any notions of location or magnitude (theldquosmaller thanrdquo relation invoked in the use of ldquosmallrdquo is just the relation of ldquobeing aproper part ofrdquo which does not involve notions of magnitude) ndash but it is harder tointerpret the reports of the Stoic claims that all parts of the mixture contain parts ofall of the mixed substances as implying commitment only to this purely mereologicalclaim

which original substances can be ldquospread outrdquo so that for example a dropof wine could come to have parts across the entire ocean if it is blendedwith the ocean (LS 48B) without supposing there is any unblendedresidue of the original substances anywhere11

The way this can be done requires that the original substances and thesubstance formed by blending are entirely composed of gunk If the sub-stances have divisions of minimum size and the minimal parts are to sur-vive the blending then we will be unable to avoid juxtaposition ratherthan blending since the atoms of each original substance will not containany of the other blended substances Furthermore if minimal parts sur-vive the blending and if regions the size of a single atom can only con-tain one atom at a time regions the size of a single atom will not containa blend but only a sample of one of the original substances Likewiseeven if the substances are made up of infinitely many point-occupiers thefundamental point-occupiers will continue to be of their unblended sub-stance and unless points are simultaneously occupied by more than onepoint-occupier the putative blend will remain in some sense a juxta-position of heterogenous point-occupiers occupying their zero-magnitudeor infinitesimal magnitude areas (the ldquopointsrdquo)

However if each of the substances to be blended has no ultimate partsthe blend itself can contain all of these parts without there having to beany continuous region in the blend which is wholly occupied by part ofonly one of the blended substances For there can be a division of theblend such that no matter how many stages of division are carried outall of the so-far divided proper parts of the blend contain proper parts ofboth of the blended substances (This can be easily generalised for blendsof more than one substance but for convenience I shall assume that thereare only two substances to be blended) Furthermore one can insist that

Phronesis 512_f3_162-183I 42506 557 PM Page 171

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 11: Stoic Gunk

172 DANIEL NOLAN

12 I am here assuming in line with many typical mereological theories that anobject can have parts which do not fill a continuous region Whether Chrysippus wouldfollow me in this is another matter but I am concerned at the moment to explore anoption

any continuous region of the blend is wholly occupied by a piece of theblend which has parts of the original blended substances among its ownparts Thus any ldquosamplerdquo of the blend thought of as a spatially contin-uous piece of the blend will ldquoparticipate in everything contained in sucha blended mixturerdquo that is it will have a part of each of the substancescontained in the blend

Of course there is a sense of ldquosamplerdquo where we can acquire a samplewhich contains only one of the original substances for one way of tak-ing a sample is to remove one of the blended substances (as water is sup-posed to be removable from a waterwine blend by the application of anoily sponge (LS 48D)) Since the substances are ldquopreservedrdquo I take it thatthe substances and their parts would have to be literally parts of such agunky blend it is hard to see how one could consistently deny them thestatus of parts of the blend while holding that the substances as a wholeare preserved The parts of these substances however do not whollyoccupy any region within the blend since any region in which they arelocated also contains proper parts of the other blended substance Neitheris it a case of juxtaposition with the region of the blend being composedof regions wholly occupied by parts of one of the blending substancesadjacent to regions wholly occupied with parts of the other blending sub-stance each region within the blend is wholly occupied only by a part ofthe blend which is not a part of either of the original substances12

However every continuous region in which the blend is found containsparts of each of the blended substances on this model so the blendedsubstances are indeed ldquoextended through and throughrdquo None of the con-tinuous subregions is entirely filled with a part of one of the original sub-stances of course but this does not mean that parts of the substances arenot ldquoextended throughrdquo such regions

23 Where Are the Blended Substances While They are Blended

The question of whether this is a case of ldquotwo bodies in the same placerdquoor ldquotwo objects at the same positionrdquo and if so whether that is a serious(or even fatal) problem for the Stoics is one which arises at this point ndashand objections to the Stoic theory of mixture as being committed to two

Phronesis 512_f3_162-183I 42506 557 PM Page 172

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 12: Stoic Gunk

STOIC GUNK 173

13 See further Sorabji 1988 chs 5-614 Other interesting questions such as how the question of the possibility of two

bodies (or other objects) in one spot relates to ancient conceptions of places and therelations of ldquoplacesrdquo to bodies are ones I shall put aside here

bodies in the same place are made by both ancient and contemporary commentators13 There seem to me to be at least two interesting questionshere Firstly does this gunky account of mixture by itself commit its pro-ponent to holding that two bodies can be in the same place Secondly ifa defender of this gunky account of mixture maintains that it is a case of two bodies in the same place is this a problem for this theory of mixture14

As for the first question it seems to me that a believer in gunky bod-ies has several options when it comes to saying what it is for a body tobe in a place (or at a position or a location ndash Aristotelians will see animportant distinction to be drawn here but I do not think any such dis-tinction will be important for current purposes) Take it for granted thatthe mixture is in a certain place and that certain other objects are partsof that mixture what follows about their place Deductively not verymuch Models are possible where an object is ascribed a location withoutany of its proper parts being ascribed any location at all Normally wethink that when an object occupies a region some at least of its properparts occupy subregions my foot for example occupies part of the spacethat I as a whole take up But even for objects conceived of as ultimatelybeing made up of atoms there is a question about whether every properpart has a location especially once we allow proper parts made up ofparts which are spatially scattered Where is my left kidney plus my rightfoot You could say that it occupies a disconnected region ndash one part ofthe region located with my kidney one part with my foot Or you couldsay that it occupies a connected region which includes those two sub-regions ndash maybe those two subregions connected with a line or with a thincylindrical region One might even want to say that disconnected parts areonly found at quite large regions ndash maybe the object composed of my twohands (if we believe in such an object) can be found no smaller locationthan the region occupied by my upper body or some other larger partwhich contains the two hands as proper parts Finally you might not wantto say that such disconnected objects have a location at all ndash perhaps theonly objects which have locations are spatially connected objects Someanswers here are better than others no doubt and there may well be one

Phronesis 512_f3_162-183I 42506 557 PM Page 173

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 13: Stoic Gunk

174 DANIEL NOLAN

correct answer to these questions ndash but at least the question can be raisedand apparently raised intelligibly

The question becomes more acute when we are dealing with objectswhich do not resolve into mereological atoms If material bodies are notcomposed of mereological atoms then plausibly some objects O will besuch that when they exactly occupy a region R each continuous (non-zerosized) subregion of that region will be exactly occupied by a part of Oand only a part of O Such Os are relatively well-behaved There mayalso be objects Q ndash even parts of well-behaved objects such as the Os ndashsuch that whenever we are tempted to say they occupy a region we canfind some other object Qrsquo which has no parts in common with Q whichit is also tempting to say occupies that region (In the case at hand Q andQrsquo together constitute a well-behaved object O and each of the objectswhich exactly occupies one of the subregions of the region O exactlyoccupies contains parts of both Q and Qrsquo) Are we forced to say anythingin particular about where such objects as Q and Qrsquo are located

It is hard to see that we are without further principles to constrain us ndashand in addition this is a case where we might expect much less guidancefrom our ordinary ways of assigning objects to positions We could findlocations for Q and Qrsquo easily enough ndash we might locate Q at the small-est continuous region which contains an object which exactly occupies itand which is an object which contains Q as a part for example in whichcase Q and Qrsquo may even turn out to both take up exactly the same regionas O Or we might assign locations to only some of the objects out therendash for instance for well-behaved objects such as O but none for strangeobjects like Q and Qrsquo If we take this second option Q and Qrsquo will notbe located anywhere at all (though we may still say that they are in aloose sense since they will retain an important connection to the placewhere O is) If we do this then we are not forced to say the mixed sub-stances are in the same place ndash the mixture is in a specific location trueenough but while they remain mixed the components are not in a placeat all (at least in the strict sense)

I am not sure how best to choose between these two options and otheroptions which might suggest themselves for understanding the blending ofgunk Maybe we are dealing with physical (or metaphysical) questionswhich might be resolved differently in different possible gunky worldswith different spaces or space-times (or things very much like spaces andtimes) Or maybe our linguistic practices fix somehow what the correctway is to describe the position of gunky bodies of different sorts It seemsto me however that a defender of gunky mixture might hold either option

Phronesis 512_f3_162-183I 42506 557 PM Page 174

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 14: Stoic Gunk

STOIC GUNK 175

15 It may also be that a gunky blend is the best model for Anaxagorasrsquo theory ofmixture (see Anaxagoras Diels-Kranz frs 6 11-12) I do not know whether we couldattribute such a worked-out conception of matter to Anaxagoras on the basis of thesurviving evidence however in particular I am inclined to suspect that if Anaxagorashad explicitly proposed a worked-out gunky conception of division we would haveheard more about it from Aristotle

(or some third option) without being convicted of obvious inconsistencyThe option seems open to accept the above account of mixture withoutbeing committed to there being two mixed objects in the same place asthe mixture at the same time

Whether or not the option is open to say that the ingredients of a blendstrictly speaking lack a location the sources seem to indicate that Chrysippusand the Stoics did not avail themselves of this option The ancient sourcesagree that Chrysippus held that the blending substances ldquocoextendrdquo (Dio-genes Laertius in LS 48A) or ldquomutually coextend through and throughrdquo(eacutentiparektecurrennesyai diEacute ˘lvn) according to Alexanderrsquos On Mixturequoted in LS 48C One gets the impression reading the sources that thepneuma is meant to be everywhere (albeit in different concentrations)rather than for example nowhere in particular It should be noted how-ever that the problem of multiple bodies being found in each otherrsquos loca-tions (at least in the sense that any region that contains within it a partof one will contain within it a part of the other) need not arise just becauseof the Stoicsrsquo theory of mixture Gunky division all by itself even if it ishomogeneous may allow that we can find two objects A and B with noparts in common that go together to make up a piece of gunk C such thatfor any part of C that exhaustively occupies a region it contains piecesof each of the objects A and B Perhaps this strikes us as bizarre (thoughwhether it is deeply bizarre or merely unfamiliar is a further question)and perhaps it can be argued that this does some violence to our conceptof location or at the very least to our ldquocommon conceptionsrdquo aboutobjects in space Such a construction may not have examples in our sim-plest models of gunk such as the open regions of Euclidean space Butallowing that there are such objects A B and C is a distinctive option thatfalls prey to no obvious incoherence or metaphysical intractability

If Chrysippus treated blendable substances as gunk he would have hada consistent story to tell about ldquoblendingrdquo not available to his atomistrivals nor rivals if any who accepted infinite divisibility into infinite ulti-mate (proper-partless) parts15 Whether Chrysippus managed to articulatethis theory of mixing in an entirely consistent manner however is another

Phronesis 512_f3_162-183I 42506 557 PM Page 175

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 15: Stoic Gunk

176 DANIEL NOLAN

16 White 1986 also defends an interpretation of Stoic mixture in which the volumeof the blend need not be equal to the sum of the initial volumes of the elements

matter Alexanderrsquos report suggests that Chrysippus incoherently insistedthat the blend had no parts which were parts of only one of the originalsubstances I am not sure whether the best course is to construeChrysippus-according-to-Alexander charitably for example by assumingthat by ldquopartrdquo Chrysippus meant ldquocontinuous partrdquo or to assume Chrysip-pus fell into an understandable incoherence or to suspect that Chrysippuswas careful enough but Alexanderrsquos gloss is imperfect

One interesting thing about this gunky construal of blending is that noconclusions about the volume of the blend follow simply from the assump-tion that a blend is created such that for one infinite division (for examplea division into parts wholly occupying regions of non-zero magnitude)every one of those parts of the blend contain parts of the blended sub-stances So this story of blending is consistent with the blend being nolarger than one of the blended substances (as in presumably the case ofa human body and the associated blended soul) but also with the blendbeing larger than either of the blended substances (as for example whenwater is added to wine and the volume of the blend is the sum of the vol-umes of the substances to be blended) It is also consistent with more implau-sible theories of volume for instance that the sea could be blended intoa cup of wine or a drop of wine added to water could produce a volumeinfinitely larger Since these latter results are consistent with but do notfollow from Chrysippusrsquo account of blending the ancient objections tothe Stoic account which merely ridicule these latter conclusions are mis-guided For there is nothing to stop the Stoics specifying the subsequentvolume of blendings on a case by case basis or by sorting blendings intoseveral categories16

Furthermore there is nothing to stop the Stoics from saying whatChrysippus apparently said that a quantity of substance could be spreadthrough a much greater area by blending with another than it could on itsown For example a drop of wine could do so by blending with the entiresea (LS 48B) or gold supposedly does so by blending with certain drugs(LS 48C) Plutarch denigrates this as ldquoabsurdrdquo by using the example ofa dissolved leg as filling large volumes of the sea (LS 48E) but ratherthan ldquostamping with derision on their absurditiesrdquo Plutarchrsquos example leg(which he gets from the Academic Arcesilaus) seems only to show thatPlutarch is a bad judge of what is absurd

Phronesis 512_f3_162-183I 42506 557 PM Page 176

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 16: Stoic Gunk

STOIC GUNK 177

17 Compare Whitehead on spatial points every region which includes a ldquopointrdquo inhis sense will have a subregion which does not include that point Indeed Samburskynotices this ldquostriking close kinshiprdquo between Chrysippusrsquo view of time and that ofWhitehead (Sambursky 1959 p 106) though without drawing any conclusions about

So far I have been concentrating primarily on the Stoicsrsquo theory of divi-sion of body rather than of place or time Stobaeus holds that ldquoChrysippussaid that bodies are divided to infinity and likewise things comparable tobodies such as surface line place void and timerdquo (LS 50A) While I willlater be suspicious of Stobaeusrsquo report of Chrysippus it is independentlyplausible that the incorporeals which are analogous to bodies will betreated similarly in Chrysippusrsquo physics Certainly Chrysippusrsquo views asso far outlined are consistent with space being made up of gunky regions(as in Whitehead) rather than of points Assuming that Chrysippus took agunky attitude to time however solves another standing difficulty of Stoicinterpretation and it is to this I shall now turn

24 Time

Chrysippus seems to have denied that any time is strictly speaking pre-sent (see Stobaeus at LS 51B ldquoHe [Chrysippus] says most clearly that notime is entirely presentrdquo) Nevertheless apparently some bodies exist inthe present and act in the present (some attributes are said to ldquobelongrdquoldquoholdrdquo(Iacutepatilderxein) (51B) and some lekta are presently true (see Sextus EmpiricusLS 51H though Sextus may be assuming this as fact rather than as some-thing the Stoics would be committed to)) What could be going on

If ldquonowrdquo or the present was supposed to lack temporal magnitude orwas supposed to be thought of as some sort of limit between the past andthe future (as Aristotle did in Physics IV13 222a10) Chrysippus wouldunderstandably be reluctant to admit a part of time corresponding to iteven among the incorporeals If his conception of time was that it wasmade up of sub-intervals in the fashion of gunk there would be manyintervals of time which in some sense include the present However nonecould be entirely present since for any region of time it will have asmaller subregion which is entirely in the past or entirely in the future(and surely something which has a non-present part is not entirely pre-sent) While smaller and smaller regions can be specified each of whichruns up to or through the present there will be no region correspondingentirely to the present since the present appears to be instantaneous butevery interval of time has some finite magnitude17 Since there are not

Phronesis 512_f3_162-183I 42506 557 PM Page 177

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 17: Stoic Gunk

178 DANIEL NOLAN

whether the Stoics may have shared Whiteheadrsquos ldquogunkyrdquo conception of processes andpresumably time

18 I hope to have avoided many of the other issues surrounding Chrysippusrsquo ontol-ogy of time whether it exists or is a mere something and other questions about itsontological status There are also questions about the Stoic theory of limits egwhether limits are or are not nothings and this is obviously relevant to the questionof the status of the present thought of as a limit of the past and future The claimwhich I do take the Stoics (or at least Chrysippus) to be making is that there is nopart of time which is completely present and it is this claim which is illuminated bythe realisation that the Stoics had a gunky conception of the relation of parts to wholes

19 Todd 1973 suggests an emendation ldquofor division to infinity is inconceivable(eacutekatatildelhptow)rdquo but I think this emendation is unlikely given how similar the unamendedpassage in Stobaeus is to Diogenes Laertius and because of the evidence from Sextusthat the Stoics accepted division into infinitely many bodies (and so presumably didnot find it inconceivable) In any case Toddrsquos emended passage would pose a verysimilar challenge to my interpretation of Chrysippusrsquo view on infinite division sinceit leaves intact the suggestion that Chrysippus allowed that division was in some senseinfinite while keeping the suggestion that there is no infinity produced in division

temporal points for a gunk-loving Stoic but only intervals divided intosmaller and smaller intervals without limit no region can be entirely ldquopre-sentrdquo Chrysippus says what we would expect him to say given a con-ception of time as gunk made up of intervals on the assumption thatChrysippus identified ldquotimesrdquo with these intervals of time18

3 A Puzzle about Infinity

The interpretation of Chrysippus in particular and the Stoics in generalas holding that bodies and perhaps place and time as well are gunk facesone main obstacle Gunk has an infinite number of parts (continuum-many in fact) But Chrysippus seems to have denied that bodies containan infinite number of parts Perhaps most explicit is Stobaeus ldquoButalthough these are divided to infinity a body does not consist of infinitelymany bodies and the same applies to surface line and placerdquo (LS 50A)though Diogenes Laertius can be read as indicating something similarldquoDivision is to infinity or lsquoinfinitersquo according to Chrysippus (for there isnot some infinity which the division reaches it is just unceasing)rdquo (LS50B)19 Finally Plutarch who I have already mentioned characterisesChrysippus as rejecting the claim that objects have infinitely many partsldquoyou may see how he conserved the common conceptions urging us tothink of each body as consisting neither of certain parts nor of some num-ber of them either infinite or finiterdquo (LS 50C)

Phronesis 512_f3_162-183I 42506 557 PM Page 178

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 18: Stoic Gunk

STOIC GUNK 179

Fortunately we can dismiss Plutarchrsquos charge for Plutarch charac-terises this ldquourgingrdquo as being what Chrysippus is doing in the passagePlutarch quoted above (see p 165) And it is clear that in that passageat least Chrysippus was not urging us to ldquothink of each body as consist-ing neither of certain parts nor of some number of them either infinite orfiniterdquo but was instead urging us to think of each body as consisting nei-ther of certain ultimate parts nor some number of them either infinite orfinite Chrysippus is exactly right to urge us in this fashion since Chry-sippus rejects the existence of ultimate parts especially those which sup-posedly make up bodies Perhaps Stobaeus makes the same mistakemisreading a point about ultimate parts as a point about parts in generalFurthermore the passage quoted from Diogenes Laertius can be read notas rejecting infinite division but instead as rejecting infinite division whichsomehow reaches a certain infinite stage in the way that infinite divisioninto indivisible points or indivisible point-occupants would The infinitedivision of gunk does not result in such a ldquofinal stagerdquo of course

Another option to consider is that perhaps Chrysippus only believed ina potential infinity of division that there is no finite limit to the numberof parts that could be created through a process of dividing but it is notthe case that all of those parts are already in existence This is the viewattributed to him by many prominent contemporary authors includingLong and Sedley (1987 p 303) and JB Gould (1970 p 116) LongSedley and Gould all take Diogenes Laertiusrsquo claim above (LS 50B) tobe evidence for this view and it would fit with Stobaeusrsquo report

I reject this reading for four main reasons The first two are pieces oftextual evidence I have already mentioned Sextus Empiricus draws a con-trast between Stoic division and Peripatetic division or at least Stratorsquos(see p 168) and offers an objection to the Stoics but not the Peripateticsthat seems to require that the parts of a body all be in existence If Sextushas understood the Stoics rightly then they believed in actual parts incontrast to Strato The second consideration is that the Stoic theory ofmixture makes sense on the reading where the parts all exist but it isharder to make sense of if no mixed body actually has an infinity of partsIndeed Long and Sedley draw attention to this tension between the Stoictheory of mixture and the potential parts reading (Long and Sedley 1987p 304) I suggest the lesson of this tension is that the Stoics did not con-ceive of their infinite division of bodies as always merely potential ratherthan that Chrysippus and others made a relatively simple blunder

The third reason is that there is positive evidence that the Stoicsbelieved that a not-yet-divided object was already composed of infinite

Phronesis 512_f3_162-183I 42506 557 PM Page 179

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 19: Stoic Gunk

180 DANIEL NOLAN

20 Todd also cites Plotinusrsquo Enneads VI 1 26 as evidence that ldquoIn general Stoicismrejected potential beingrdquo (1973 p 23 n 14) but I do not quite see how the relevantpassage of Plotinus is to be read in this way with equal justice it could be read asa complaint by Plotinus that the Stoics treated all being as potential I do not wish tohazard any specific interpretation of Plotinus VI126 but I doubt that a defender ofthe claim that the Stoics embraced only potential infinities and potential parts need bevery troubled by it

parts Plutarch De Comm Not 1079a says ldquoStoics believe that mandoes not consist of more parts than manrsquos finger nor the world than manFor division pulverizes to infinity and among infinities there is no moreor lessrdquo (LS 50C) First this suggests that men and their fingers alreadyhave parts unless we are to suppose that the cosmos does not have partseither (which is certainly in tension with other things the Stoics want tosay about things in the world being parts of the cosmos as well as in tension with common sense) Second there are more than finitely manyparts of each for on the assumption that the parts of a manrsquos finger arepart of that man the only obvious way that there could be no more partsin the finger than there were in the man would be if there was no finitelimit to the number of parts of either I suppose we might doubt thatPlutarch has correctly understood the Stoics here as well but the moststraightforward understanding of this passage is that Plutarch is reportingthat Stoics believed in infinitely many parts of such objects as fingersmen and the cosmos and were prepared to say so (Sambursky 1959 p 97interprets the passage in this way and his translation of Plutarch 1079aon p 141 suggests this reading even more strongly than the Long andSedley translation)

My fourth reason to hold that the Chrysippus did not maintain a doc-trine of merely potential division is that the Stoics do not seem in gen-eral to have used potential being as opposed to actual being in theirmetaphysics in contrast to Aristotle and the Peripatetics Apart from theevidence about parthood and the infinitude of division I know of no viewattributed to the Stoics which suggests that they would have adopted thisPeripatetic ontological status (The claim that there is no such survivingattribution is highly falsifiable and I would welcome counter-examples ifthere are any) Todd (1973 p 23 n 14 and 1976 p 57-58 n 148) alsomakes the claim that this device ldquois not explicitly employed by the Stoicsrdquo(1976 p 58 n 148)20 If appeals to potentiality and potential being are otherwise absent from Stoic doctrine then we would need good reason tosuppose they should be read into the Stoics here The ldquopotential infinityrdquointerpretation of the Stoics should be rejected

Phronesis 512_f3_162-183I 42506 557 PM Page 180

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 20: Stoic Gunk

STOIC GUNK 181

21 Recall that Plutarch LS 50C also suggests that ldquoStoics believe that man doesnot consist of more parts than his finger nor the world than man For division pul-verizes bodies to infinity and among infinities there is no more or lessrdquo

So my opinion is that we should not change our view of the Stoic con-ception of division Most likely Stobaeus is mistaken about Chrysippusrsquoview but even if he is not I think my thesis about what Chrysippus andthe Stoics believed about division may survive If I am right about theStoic conception of division then it follows from Chrysippusrsquo view thatthere are an infinite number of existing parts of any body (or place ortime) But I do not need to claim that Chrysippus would have realisedthis Believing that objects are without smallest parts and divided intoparts without ceasing but failing to believe that there were an infinitenumber of such parts is a natural mistake for someone like Chrysippusto make Consider the picture Chrysippus is working with we have abody which is made of sub-bodies which are themselves made of sub-bodies and so on without limit It is tempting to suppose that there is nocompleted totality of these bodies but only that for any finite number ofsub-divisions there are more sub-divisions than that Without any com-plete collection of these sub-bodies then it is plausible that there is noway of appropriately measuring their quantity This tempting suppositionwould be a mistake but not one that even mathematicians would havebeen able to diagnose properly before Georg Cantorrsquos work on infinity inthe late 19th century Just as we do not suppose that Aristotle was famil-iar with the works of Weirstrauss on limits we should not suppose Chrysippushad available to him the insights of Cantor Neither the supposition thatChrysippus realised that his account committed him to an actual infinityof bodies nor the supposition that Chrysippus failed to realise this arecompletely compelling In the absence of further information about theposition of Chrysippus in particular or the Stoics in general this questionmay even be unable to be settled I am inclined on balance to think thatthe Chrysippus did realise that his view was committed to an infinite num-ber of bodies and did hold that there were infinitely many parts of eachbody21 but in doing this I am discounting Stobaeusrsquo report in favour ofother apparently more reliable sources like Sextus Empiricus

Phronesis 512_f3_162-183I 42506 557 PM Page 181

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 21: Stoic Gunk

182 DANIEL NOLAN

22 Thanks to Dirk Baltzly Sarah Broadie Jon Hesk Tom Holden Calvin NormoreDavid Sedley an anonymous referee for this journal and audiences at the 2001Australasian Association of Philosophy meeting 2001 Creighton Club meeting and atthe University of St Andrews for helpful discussions of the material in this paper

Conclusion

If I am right Chrysippus articulated a theory of division which was impor-tantly different from those of his Peripatetic and Atomist rivals and whichwas deployed in important areas of Stoic physics such as the issue of thenature of mixture (so important in explaining the connection between pneumaand the passive matter it acted upon) and the nature of the present Thereare other issues which may also be illuminated by this interpretation ofthe Stoics particularly their attitudes to questions concerning motion andquestions concerning geometry However since these bring up somewhatindependent and also independently thorny questions of interpretation con-cerning the Stoic theories of limits magnitude mathematical objects andso on I have not addressed them here or mentioned them only in pass-ing As with any interpretation of Stoic physics we are faced with frag-mentary evidence largely filtered through hostile sources so it may be thatno interpretation can hope to be conclusively demonstrated At the veryleast however the ldquogunkrdquo option deserves to take its place in the rangeof alternatives to be considered when we try to understand the Stoic posi-tion in Hellenistic physical and metaphysical debates22

Department of PhilosophyUniversity of St Andrews

References

Baltzly D 1998 ldquoWho are the Mysterious Dogmatists of Adversus Mathematicos ix352rdquo Ancient Philosophy 18 145-170

Bury RG 1936 Sextus Empiricus III (Against the Physicists and Against theEthicists) Loeb Classical Library Harvard University Press Cambridge MA

Gould Josiah B 1970 The Philosophy of Chrysippus EJ Brill LeidenLewis David 1991 Parts of Classes Basil Blackwell OxfordLong AA 1974 Hellenistic Philosophy Charles Scribnerrsquos Sons New YorkLong AA and Sedley DN 1987 The Hellenistic Philosophers 2 vols Cambridge

University Press CambridgeSambursky S 1959 Physics of the Stoics Routledge amp Kegan Paul LondonSimons P 1987 Parts A Study in Ontology Oxford University Press OxfordSorabji R 1988 Matter Space and Motion Duckworth London

Phronesis 512_f3_162-183I 42506 557 PM Page 182

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183

Page 22: Stoic Gunk

STOIC GUNK 183

Todd R 1973 ldquoChrysippus on Infinite Divisibilityrdquo Apeiron 71 21-29mdashmdash 1976 Alexander of Aphrodisias on Stoic Physics EJ Brill LeidenWhite MJ 1982 ldquoZenorsquos Arrow Divisible Infinitesimals and Chrysippusrdquo Phronesis

273 239-254mdashmdash 1986 ldquoCan Unequal Quantities of Stuffs Be Totally Blendedrdquo History of Philosophy

Quarterly 34 379-389mdashmdash 1992 The Continuous and the Discrete Clarendon Press OxfordWilliamson T 1994 Vagueness Routledge London

Phronesis 512_f3_162-183I 42506 557 PM Page 183