Aristotle S Diagnosis Of Atomism

Aristotle's Diagnosis of Atomism
Pieter SjoerdHasper
I Introduction
Aristotle wasnot anatomist.Inseveral passagesscattered over hisextant
works he makes this very clear.In deCaelo ÉÐ 4, for example,he remarks:
It is necessarythat those who posit atomic bodies are in oppositionto
the mathematical sciences and destroy many of the reputable beliefs
and theperceptualphenomena (ðïëëÜ ôùí åíäüîùí êáéôùí öáéíïìÝíùí
êáôÜ ôçí áúóèçóéí), aboutwhich there has been a discussion before in
the booksabout time and motion.1
With regard to the contradiction with the mathematical sciences, one
should think ofarguments like those offered in the small treatise deLinas
Insecabilibus, which appears in the corpus Aristotelicum, but which is
usually taken not to be written by Aristotle himself. In this treatise it is
shown that the assumption that there are indivisible lines is in conflict
with 'the things proved and laid down in mathematics',2
among them
the firmly established conclusion that there are incommensurable lines.3
The reference to the 'discussion in the books about time and motion'
showingthe conflict between atomismand 'many ofthe reputablebeliefs
and the perceptualphenomena', mustbe to the sixth book of the Physics,
1 303a2(M
2 969b29-31; cf. 970al8-20.
3 969b33-70a5
APEIRON a journal for ancient philosophy and science
0003-6390/2006/3902 121-156 $36.00©Academic Printing & Publishing

122 Pieter Sjoerd Hasper
where Aristotle puts forward several arguments for the conclusion that
magnitudes, like lines, times and motions, do not consist of indivisibles.
It isnot immediately clear, however, what 'the reputable beliefs and the
perceptual phenomena' are which he has in mind. It could be that one
of the perceptual phenomena he is thinking ofisthe existence of different
proportionalities between the distance travelled over by a moving object
and the time required, that is, for faster and slower objects, on the basis
of which he refutes the idea that there are indivisible distances and
times.4
It could alsobe that with 'the reputable beliefs' he is referringto,
among others, the idea that in order to compose something, the compos-
ingelements must liesideby side touching and thus have their bounda-
ries together without the elements coinciding — as it is impossible to
distinguish within something indivisible between the boundary and the
interior, indivisibles can only makeup something if this reputablebelief
is repudiated.5
As Aristotle also elsewhere refers to the same argument,6
the latter point seems secure, but since there is no further evidence, the
former point is also plausible.
What is clear, however, about all of the arguments against atomism
which can be related to this passage from de Caelo, is that they target a
very strong version of atomism, namely, that the atoms are indivisible
ina mathematicalor conceptualkind ofway.Thisisobvious with respect
to the conflict with the mathematical sciences. But also the requirement
that oneshould distinguish within touching entitiesbetween aboundary
and the interior, and themachineryofproportionalities used torepresent
faster and slower objectsboth concern a kind of divisibility which goes
beyond the merely physical divisibility of bodies separable by a gap.
That Aristotleuses these arguments may be partly explainable by refer-
ence to the popularity of such strong versions of atomism in the Acad-
4 See for examplePhysica VI2,232b20-3b32.
5 See the very first argument of Physica VI1.
6 In lie Caelo ÐÉ 1, 299al-ll Aristotle seems to distinguish between a conflict with
mathematics and theabsurdity following from atomismthat 'it isnot necessary that
a part of a line be a line (ïõê áíÜãêç ôï ôçò ãñáììÞò ìÝñïò ãñáììÞí åßíáé)', about
which, he says, 'there has been earlier an inquiry in the arguments about motion,
viz. that there are no indivisible lengths.' Wemay suppose that with regard to this
absurdity the reference is to the very first argument of Physica VI1, as it is there
established, on the grounds of the reputable belief mentioned, that a line does not
consist of points, but is rather ever-divisible into lines.

Aristotle's Diagnosis of Atomism 123
emy.7
Suchexplanations, however, will not suffice, since Aristotle clearly
assumes that his arguments apply to every kind of atomism, also to
versions which distinguish between the physicalindivisibilityofatoms
and the mathematicaldivisibility of the same atoms. Forexample, in his
long discussion in de Generatione et Comiptione 12 of Democritus'proof
for the existence of atomshe makes at one moment the remark:
However, also for those who posit [indivisiblebodies and magnitudes]
no less impossible things follow;there has been an inquiry aboutthem
in other [works].9
He can onlybe referringto the same arguments as in the passage quoted
from de Caelo. Yet he counts Democritus' proof among the arguments
which are 'appropriate and physical'.9
Moreover, Democritus cannot
have been an atomist of the mathematical kind, since his atoms varyin
size and shape.10
Sothere is here something in need of explanation: how
could Aristotle have thought that his arguments against atomism as set
out above are also effective against the much weaker atomism of a
physical kind, which seems to have been propounded by Democritus?
This what might be termed a failure to distinguish, as far as the issue
of atomism is concerned, between physical and mathematical divisibil-
ity, cannot be explained by reference to some lack of awareness on
Aristotle's part of the difference between physical and mathematical
division. Thereare enough contextsin which Aristotle isclearly thinking
of a division brought about without a physical separation coming into
being: the division is then actualized by thought, as he puts it.11
Whatis
more, Aristotle employs in some passages technical terms whose use
implies such a difference. InPhysica VIE 8he talks about using (÷ñçóèáí)
7 As appears notably from the arguments in favour of atomism presented at the
beginning of de Lineis Insecabilibus.
8 316bl6-18
9 316al3-14
10 That Democritus' argument for the existenceof atoms as presented by Aristotlein
GCI 2 does not entail mathematical atomism, is a point I have argued for in The
Foundations of Presocratic Atomism' ['Foundations'], Oxford Studies in Ancient
Philosophy 17 (1999) 1-14.
11 SeeMetaphysica è 9,1051a21-31; cf. deAnima ÐÉ6,430blO-14.

a point on a line as theend ofa motionas wellas thebeginning of another
motion, when the moving object stops at that point, 'just as if one would
think it'.12
Obviously the line itself is not being divided by the moving
object stopping there or by the mind thinking it as divided, but ismerely
treated in that way.13
That Aristotle is thus able to distinguish between
the two kinds of division, while still holding that mathematical divisi-
bility implies physical divisibility, actually only serves to add to the
puzzle. Apparently the ability to be mathematically divided entails the
ability to be physically divided, even though the two abilities are very
different logically, that is, in terms of their actualization.
It seems, therefore, that Aristotle is not so much failing to distinguish
between physical and mathematical divisibility, but rather is refusing to
do so, at least with regard to atomism. This also appears from the only
passage where he comments explicitly on the issue, from deCaelo IE 1:
Also the impossibilities concerning physical bodies which those who
posit atomiclines arebound toassert (oca... ðåñßôùí öõóéêþí óùìÜôùí
áäýíáôá óõìâáßíåé ëÝãåéí ôïéò ðïéïàóé ôáò Üôüìïõò ãñáììÜò), we shall
now study a little. For the impossibilities which follow in the case of
[atomic lines] will also follow for physical [bodies] (ôá ... Ýð* åêåßíùí
áäýíáôá óõìâáßíïíôá êáé ôïéò öõóéêïÀò áêïëïõèÞóåé), while those
[which follow] for the latter [will] not all [follow] in the case of the
former, because the former, the mathematical [bodies], are said on the
basis of abstraction (äéá ôï ôá ... åî áöáéñÝóåùò ëÝãåóèáé, ôá ìáèçìá-
ôéêÜ), while the physical ones [aresaid] on the basis of addition (åê
ðñïóèÝóåùò).14
12 262b5-7
13 In his discussion of Democritus' argument for the existence of atoms, Aristotle
remarks that 'it is dear that [amagnitude]divides into separable (åéò ÷ùñéóôÜ) and
into eversmallermagnitudes and into magnitudes coming apart and separated (åéò
áðÝ÷ïíôá êáé êå÷ùñéóìÝíá)' (316b28-9), thus implying that there is a distinction
between adivision intomerely separableparts and oneintoseparated parts.Though
I think that Aristotle is adopting here a distinction drawn by the atomists, who
believe that it is possible that two atoms touch while remaining divided (see my
'Foundations' 8), it may be that even in the present context for Aristotle this
distinction coincides with that between mathematicaland physical divisions.
14 299all-17

Anstotle's Diagnosis of Atomism 125
Aristotle assumes that the number of features and properties involved
in physics and in our conception of physical bodies is larger than the
number involvedin our conception ofthem as mathematical bodies, and
asserts thatwhatin mathematicsis impossible to be the case aboutthese
bodies, is also impossible to be the case about them in physics. Whatis
now remarkable is that he thinks that since in mathematics it is impos-
sible that a body consists of atoms, it is also in physics impossible that it
consists of atoms. It is as if there is one property of being everywhere
divisible (i.e., not consisting of atoms) which is the same in both physics
and mathematics (and can be mathematicallydemonstrated to be the
case about bodies), even though there is a difference between the ways
in which these divisions are brought about.
In this article Iwant to suggest an explanationofAristotle's refusal to
draw a distinction between physical and mathematical divisibility, de-
spite his awareness of the distinction between physical and mathemati-
cal division. This explanation will come in two stages. In the first, long
stage, I shall propose a detailed interpretation of the only text in which
Aristotle is not so much trying to refute atomism from the outside, as it
were,but rather aims to statewhat theidea behind the atomistic position
is: the discussion of Democritus' argument for the existence of atoms in
de Generatione etCorruptione 12. It will appear in the second, shorter stage
that the diagnosis ofatomism presented in that discussion isrelevant for
our problem and coheres with some further remarks made by Aristotle
about atomism.I shall finish by sketching what I think is the underlying
conflict between Aristotle and the atomists.
2 InterpretativeDifficulties
In the second chapter of the first book of his de Generatione et Corruptione
Aristotle presents an argument for the existence of atoms which he
ascribes to Democritus.Wemay summarize this argument as consisting
of two parts. The first part amounts in fact to only one observation, that
the object of discussion, M, is divisible:
(D) M is divisiblesomewhere.
The other part starts with a supposition to be reduced to absurdity:
(1) M is divisibleeverywhere.
That is:

(2) It is possible that M is divided everywhere.
However, in the situation that M is divided everywhere, there are no
parts with size left. But that means that an object with size consists of
parts without size —which is absurd. So (1)is not true:
(3) M is not divisible everywhere.
This second part of the argument leaves open the possibility that M is
indivisible. However,because of (D)that possibility is sealed off. There-
fore:
(C) M consists of atoms.
As has been pointed out by several scholars, this argument is invalid, as
it trades on an ambiguity of the term 'divisible everywhere'.15
If this
means 'everywhere possibly divided', the step from (3) to (C) is legiti-
mate, since only something which consists of atoms is not everywhere
possible divided. However, the step from (1)to (2)is then not allowed,
as the statementthatsomething iseverywherepossibly divided does not
imply that it ispossible that it is divided everywhere.On the other hand,
if 'divisible everywhere' means 'possibly divided everywhere', there is
no problem with the step from (1)to (2), but then the step from (3)to(C)
becomes unacceptable, as the statement that something is not possibly
divided everywhere doesnot ruleout that itmaybeeverywherepossibly
divided.
As far as it goes, there is nothing wrong with either this summaryof
the argument or the objection raised against it. Things becomeproblem-
is J. Barnes, ThePresocratic Philosophers (2nd ed.; London and New York: Routledge,
1982), 403-4; C.J.F. Williams, trans, and comm., Aristotle's De Generatione et Cor-
ruptione [DGC] (Oxford: Clarendon Press, 1982), 75; W. Charlton, 'Aristotle's
Potential Infinites', in L. Judson, ed., Aristotle's Physics. A Collection of Essays
(Oxford: Clarendon Press, 1991) 129-49, at 135; M.J. White, The Continuous and the
Discrete. Ancient Physical Theories from a Contemporary Perspective [Continuous] (Ox-
ford: Clarendon Press, 1992), 201-2; cf. R. Sorabji, Time, Creation and the Continuum.
Theories in Antiquity and the Early Middle Ages [TCC] (London: Duckworth, 1983),
340-1. It is probably for the same reason that C.C.W. Taylor, 'Anaxagoras and the
Atomists', in: idem, ed., Routledge History of Philosophy I From the Beginning to Plato
(London and New York: Routledge, 1997) 208-43, at 221, calls the argument un-
sound.

atic, however, when one tries to make sense of Aristotle's handling of
the argument. Let me quote for the sake of reference the whole passage:
(1) [I]f it is impossible that magnitudes consist of contactsor points (åî
Üöþí Þ óôéãìþí), it is necessary that there are indivisible bodies and
magnitudes. However, also for those who posit them no lesser impos-
sibilities follow. Therehasbeen an inquiry intothem elsewhere. Butwe
must try to solve [these points] —that iswhy we must state the puzzle
again from the beginning.
(2) There is nothing absurd about every perceptible body being divis-
ible at any position (êáè' üôéïàí óçìåÀïí) as well as being indivisible.
For the one will belong to it potentially, while the other will belong to
it in actuality.
(3) But it would seem to be impossible to be potentially divisible
everywhere at the same time (to ä" åÀíáé áìá ðüíôç äéáéñåôüí äõíÜìåé
áäýíáôïí äüîåéåí áí åßíáé). Forif it were possible, it could also happen
(not so that at the same time it is both, indivisible and divided, in
actuality, but [that it is] divided at any point (äéçñçìÝíïí êáè' üôéïàí
óçìåÀïí)).Therewill thenbenothing left, and thebodywillhavepassed
away into something incorporeal, and would come to be again either
from points or from nothing at all (Þôïé åê óôéãìþí Þ üëùò åî ïýäåíüò).
And how is that possible?
(4) <Further, if I, after having divided it, put together a stick or some-
thing else,itisagain equaland one.Thenthisisobviously the caseeven
if Icut the stick at whatever point (êáè' üôéïàí óçìåÀïí). Therefore, it is
potentially in a state of division everywhere (ðÜíôç áñÜ äéÞñçôáé
äõíÜìåé).16
What then is there apart from the division? Foreven if there
is some property, how is it dissolved into these and does it come to be
from these? Or how are these separated?>17
16 For a defence of this translation, see below, section 3.2.
17 I thus follow D. Sedley, 'On Generation and Corruption 1.2' ['GC 1.2'], in F. de Haas
and J. Mansfeld, eds., Aristotle: On Generation and Corruption, Book I. Symposium
Aristotelicum (Oxford: Clarendon Press, 2004) 65-89, at 75, who proposes to insert
here the lines 316b9-14, whichseem out of placein their original context.Accepting
this proposal will not affect my argument in any significant way,but it does make
one point I am going to make below more immediatelyrelevant.

(5) However, it is clear that it divides into separable and into ever
smaller magnitudes and into magnitudes coining apart and separated.
(6) Neither, then, may one dividing in successive stages bring aboutan
infinite process of breaking, nor is it possible for the magnitudes to be
divided at every point (êáôÜ ðÜíóçìåúïí) at the same time (for it is not
possible), but [only] up to a limit (Üëëá ìÝ÷ñé ôïõ). It is necessary,
therefore, that there are invisible atomic magnitudes in it,especially if,
that is, coining to be and passing awayare to occur by segregation and
aggregation (Üëëùòôå êáé åÀðåñ åóôáéãÝíåóéò êáé öèïñÜÞ ìåí äéáêñßóåé
Þ äå óõãêñßóåé).
(7) This, then, is the argument which appears to necessitate that there
are atomic magnitudes. Letus state, however, that it commits a hidden
fallacy, and [say] in what way this is hidden. For since there is no point
contiguous with a point, there is a sense in which being divisible
everywhere belongs to magnitudes, but also a sense in which it does
not (317a2 Ýðå! ãáñ ïõê åóôß a3 óôéãìÞ óôéãìÞò Ý÷ïìÝíç, ôï ðÜíôç åßíáé
äéáéñåôüí åóôß ìåí a4 ùòõðÜñ÷åéôïéò ìåãÝèåóéí, åóôßä' ùòïõ). However,
it seemsthat, when [being everywheredivisible]hasbeenposited, there
is a point both anywhere and everywhere, so that a magnitude must
be divided into nothing (äïêåÀ ä', ßßôáí ôïàôï a5 ôåèÞ, êáé üðçïàí êáé
ðÜíôç óôéãìÞí åßíáé, þóô' Üíáãêáúïí a6 åßíáé äéáéñåèÞíáé ôï ìÝãåèïò åéò
ìçäÝí). For there is a point everywhere, so that it either consists of
contacts or of points (ðÜíôç ãáñ åßíáé a7 óôéãìÞí, þóôå Þ åî Üöþí Þ åê
óôéãìþí åßíáé). There is, though a sense in which [being divisible]
belongs everywhere, because there is one [point] anywhere, and all
[points] are like each (ôï ä' åóôßí ùò a8 õðÜñ÷åé ðÜíôç, üôé ìßá üðçïàí
åóôß êáé ðáóáé ùò åêÜóôç); but there is no more than one [anywhere]
(for they are not successive), so that [it is] not [divisible] everywhere
(a9 ðëåßïõò äå ìéáò ïõê åßóßí (åöåîÞò ãáñ ïõê åßóßí), þóô' ïõ ðÜíçé).For
if it is divisible in between, itwill alsobe divisible at acontiguous point
(alO åé ãáñ êáôÜ ìÝóïí äéáéñåôüí, êáé êáô' Ý÷ïìÝíçí óôéãìÞí åóôáé all
äéáéñåôüí). But that is not possible, for aposition is not contiguouswith
a position or a point with a point, but that is division or composition
(<ïýê åóôß äÝ,>18
ïõ ãáñ åóôßí Ý÷üìåíïí óçìåúïí óçìåßïõ a!2 Þ óôéãìÞ
óôéãìÞò, ôïàôï ä' åóôß äéáßñåóéò Þ óõíèåóéò).1
'
18 Sedley, 'GC 1.2', 78and note 26,proposes to do without this insertion by reading a

In paragraphs (3), (5)and (6)one can easily recognize the premisses and
steps of the atomistic argument summarized above. What commonly is
also thought to be recognizable, namely in (7), is the charge of trading
on an ambiguity as set out above.20
Indeed does Aristotle distinguish in
(7), the core of his response to the atomistic argument, between two
senses ofbeing divisible everywhere. However, it isexceedingly difficult
to understand the exact nature of the distinction Aristotle draws in (7),
let alone to recognize itas the distinction based on the scope ofthe modal
operator as drawn above. Instead of such a distinction, we have in (7)a
complicated argument showing that it is impossible to be divisible
everywhere in one sense, on the very obscure ground that this would
involve there being two points everywhere.21
Did weneed tobe told that
it is impossible to be divisible everywhere in one sense? And how does
all that show that there is another sense in which it is possible to be
divisible everywhere?
Usually, however, such difficult questions are not even asked; the
problem is circumvented and it is just assumed that Aristotle is merely
pointing out a logical gap in the atomistic argument, along the linesof
question-mark after theprevioussentenceand translating:'...willitalsobe divisible
at an adjacent point?No, because no point is adjacent to a point,...' However, thus
Sedley lets ïõ do double duty, denying both the content of the previous question
and servingasa negation inthe reason for the denial, whichseems awkwardto me,
certainly since an explicit denial of the previous question would be welcome, it not
being an obvious absurdity.
19 316b9-14and 316bl4-7al2
20 For references,see note 15.
21 Sedley, 'GC 1.2', 78and note 25,tries to emend this obscurity away by reading at
317a9 ðëåßïõò äå ìéáò ïõêåßóÀí <ÝöåîÞò> (åöåîÞòãáñ ïõê åúóßí), but the result of this
emendation is an even greater obscurity.For while there is in itself still some sense
in denying that there is more than one point anywhere (dearly üðçïàí has to be
supplied from 317a8 ìßá üðçïàí åóôß —contraSedley, 'GC 1.2', note 25),the denial
that there is not more than one point in succession is completely meaningless. For
how could thereeverbe, just by the meaning of the words, more than one point in
succession, that is, to another point? Moreover, part of the purpose of Sedley's
emendation is to 'make [thefirst clause of317a9]plausiblyinferablefrom the second
clause',but Ifail to seeany plausibilityin the inferencefrom the non-successiveness
of points to the fact of therebeing one, and not morethan one, point in succession.
Finally, Sedley's emendationdestroys the clearly intended relation between 317a8
ìßá üðçïàí åóôß and 317a9ðëåßïõò äå ìéáò ïõê åßóßí.

the objection sketched.22
Apparently, then, it is thought that Aristotle
does not show here that there is a sense in which a magnitude is every-
where divisible, but rather presupposes this sense.23
In this context fre-
quent mention ismadeofAristotle's conception ofan actual whole being
really one and only having potential parts divisible from each other at
potential dividing points,24
as on such a conception it is possible to be
everywhere possibly divided without being possibly divided every-
where (everypotential dividing point canbecome actual, even though it
is impossible that all ofthemtogether become actual).25
In support of this
idea, scholars often refer to the distinction which Aristotle draws in(2)
between potential and actual (in)divisibility. Tomost it seems as if Aris-
totle is drawing here a lexical distinction between two meanings of the
terms Üäéáßñåôïí and äéáéñåôüí, translated here as '(in)divisible', which
in Greek may have a modal as well as a non-modal meaning and thus
may mean '(in)divisible' as well as '(un)divided'.26
It is not the place to
argue the point here, but I would rather interpret this distinction in a
metaphysical way, as the distinction between indivisibility within the
22 F.D. Miller Jr., 'Aristotle against the Atomists', in N. Kretzmann, ed., Infinity and
Continuity in Ancient and Medieval Thought (Ithaca, NY:Cornell UP, 1982) 87-111, at
98; Williams, DGC, 75; Charlton, Totential Infinites', 136; H.H. Joachim, ed. &
comm.,Aristotle:On Coming-to-beand Passing-away (De Generationeet Corruptione)
(Oxford: ClarendonPress, 1922),84;White, Continuous, 201-2;and WJ. Verdenius
and J.H.Waszink, Aristotle on Coming-to-be and Passing-away. Some Comments (2nd
ed.; Leiden: Brill, 1968), 11-14. In the end, Sedley, 'GC 1.2', 79, also arrives at this
view.
23 The only ones to dissent are White,Continuous, 18,and Sedley, 'GC 1.2', 74and 80,
who both deny that Aristotle just assumes that there is a sense of 'divisibility
everywhere' or 'infinite divisibility' which does not entail the possibility of being
divided everywhere or at an infinite number of points. However, as both take
Aristotle merely tobe showingin (7)that acompleted (infinite) division everywhere
isnot possible, theystill have Aristotle presupposing an innocent sense of divisibil-
ity everywhere.
24 This conceptioncan be found in Physica VTH 8, at 262b28-3a3 and 263all-b9.
25 Some version of this account is thought to be behind Aristotle's refutation by
Williams, DGC,72,Miller, 'Aristotle against the Atomists',92-8,and Verdenius and
Waszink, Coming-to-be,13.
26 Thus Williams, DGC,ad locum and 67;cf. Miller, 'Aristotle against the Atomists',
92. The same distinction is drawn in de Anima ÉÐ 6, 430b6-8 and in that context
interpreted in this way by many others. Cf. also Physica VE 5,258a32-b2.

actual situation and divisibility with reference to other possible situ-
ations. In the actualsituation only the whole really exists; the parts into
which this wholeispotentiallydivisible,existinotherpossiblesituations.
However, on either interpretation the distinction seems to express the
very same idea which is supposed to be behind Aristotle's refutationin
(7).27
Thus Aristotle would already in his restatement of the atomistic
argument be pointing forward to his subsequent solution.
It is problematic, however, to interpret the distinction drawn in (2)
between potential and actual divisibility as pointing forward to the later
refutation. First of all, this is not at all how Aristotle presents it.Accord-
ing to him, the restatement is merely a restatement, and the refutation
only takes place in what is explicitly presented as the refutation.28
Secondly, the core of the puzzle is stated by Aristotle in terms derived
from that very distinction: according to him it seems 'impossible that
something is potentially divisible everywhere'.29
That makes it unlikely
that it is this particular distinction, as drawn in (2), which is behind the
later refutation.
To this may be added the fact that in (6) Aristotle mentions the
identification ofgeneration and aggregation as well asofdestruction and
segregation as a premiss to the atomistic argument, or at least as a
consideration which strengthens or even clinches the atomistic argu-
ment.30
The importance of this remark can hardly be overstated. To see
27 Williams, DGC, 72, and HJ. Krämer,Platonismus und hellenistische Philosophie (Ber-
lin: De Gruyter, 1971), 261; cf. Miller, 'Aristotle against the Atomists',92.
28 Somescholars,likeWilliams,DGC, 75-9,especially 75,are prepared tobitethe bullet
in this respect and argue that Aristotle is not doing what he tells us he is doing.
Others even go as far as discarding all references to actuality and potentiality as
interpolations; see Verdenius and Waszink, Coming-to-be, 12-14.
29 Cf. Sedley, 'GC 1.2', 74.
30 Sedley, 'GC 1.2', 77,considers this identification as a part of the conclusionwhich is
neither warrantedby the argument itself nor strengthens the argument, but which
merely represents 'Aristotle's own primary motivefor scrutinizing atomism within
the context ofthe present work.' Ifind it difficult to understand the second sentence
of (6) thus, as if it were saying that therefore there must be atoms, especially ifone
takes an interest in the issue of identifying generationwith aggregation and destruc-
tion with segregation — why did Aristotle not just say so, if this were what he
meant? Moreover, also at 315b20-l Aristotleseems to imply that atomism follows
from the same identification (see note59).

why, we have to return to the accusation levelled above that the argu-
ment for the existence of atomscommits the fallacy of ambiguity. On an
earlier occasion I argued that this accusation falls flat, since Democritus
would deny the meaningfulness of the distinction between two senses
of 'divisible everywhere'.31
He can do so on the basis of his well-known
principle —which may thereforebe called the Atomistic Principle (AP)
— which says that from a unity a plurality cannot come about, nor a
plurality from aunity. Thatis,awhole isnever divisible, since something
can only be divisible if it is in fact already divided and thus a plurality.
Now it followsautomaticallyfrom this Atomistic Principle that the only
way of generation and destruction feasible is by the aggregation and
segregation ofthese primary entities. Aristotle isvery well aware of this,
as appears from a remark he makes in de Caelo ÉÐ 4:
For [Leucippus and Democritus]saythatthere areprimary magnitudes
which are unlimited in number and indivisible in magnitude, and that
from onemanydo notcometobenor from many one, butthatall things
are generated by intertwining and collision of these (êáé ïýôå åî åíüò
ðïëëÜ ãßãíåóèáé ïàôå åê ðïëëþí åí, Üëëá TTJ ôïýôùí óõìðëïêÞ êáé
ðåñéðáëÜîåé ðÜíôá ãåííáóèáé).32
By the use of ïàôå ... ïàôå ... áëëÜ the identification of generation and
intertwining, that is, aggregation, becomes a positive restatement of the
denial inherent in (AP). The point now is that with this identification of
generation and aggregation, Aristotle has thus mentioned a proposition
which can be used by the atomist to ward off the charge of having
committed the fallacy of ambiguity.33
Hence it is all the more unlikely
that the distinction between potential and actual (in)divisibility is al-
ready intimating Aristotle's solution, since then he would be mentioning
31 Seemy 'Foundations'.
32 303a5-8
33 Thus I was too pessimistic when I wrote in my 'Foundations', 13, that 'there is no
direct evidence showing that Democritus ... associated [(AP)] with his argument
for the existence of atoms.' Though Aristotle's mentioning of the identification of
generation and aggregation as at least strengthening (Üëëùò ôåêáé å'ßðåñ at 316b33)
the atomisticargument does not prove that Democritus appealed to it explicitlyin
his argument, it does indicate that Aristotle thought that he used it as a presuppo-
sition.

in his recapitulation of the argument something which prefigures that
solution and at the same tune something which goes against it. What is
more, the fact that Aristotle with this identification states a principle
which would make the atomistic argument logically invulnerable,
makes it also unlikely that in his subsequent refutation Aristotle is just
pointing out a logical gap in that argument. For merely insisting that
there is a way of being divisible everywhere which does not involve the
actual presence of all possibilities of division would not go to the coreof
the atomist's argument.
So there are in fact three interpretative problems with Aristotle's
recapitulation and subsequent refutation of the atomistic argument.
First, how on earth is it possible to connect Aristotle's obscure remarks
about points not being ordered consecutively and not coinciding in one
spot to the distinction between two senses of 'everywhere divisible'?
Second, how can we give sense to Aristotle's distinction between actual
and potential (in)divisibility insuch awaythat Aristotle'sisnotpointing
forward to his solution? And third, how should we interpret Aristotle's
accusation that the atomist commits a fallacy,based as it is on a distinc-
tion between two senses of 'everywhere divisible' and on some obscure
considerations about points? For a successful interpretation should
make that accusation of fallaciousness pertinent to a refutation of the
core idea ofthe atomist, enshrined in the Atomistic Principle, and, in the
guise of the identification of aggregation and generation, stated by
Aristotle as reinforcing the atomistic argument.
3 The Structure of Divisibility
The interpretative difficulties outlined in the previous section seem to
centre on the issue as towhat the structure ofsomething divisible is.This
is obvious with regard to the status ofthe Atomistic Principle: is it really
the case that something is divisible because it already consists of units
which make it up? It is perhaps not similarly obvious with regard to the
question how tounderstand the distinction between potential andactual
(in)divisibility in such a way that it does not undermine the atomistic
argument, and thus not the Atomistic Principle either, but it remains
clear that this question is thematically related.Only the obscure remarks
about the ordering of points and the impossibility of coinciding points
are not easily connected with the issue ofthe structure ofdivisibles, even
though we have Aristotle's word for it that they are, since he ties them
to a certain conception of divisibility everywhere. Yet,as I shall argue in

this section, they are the key to understanding the whole chapter, pro-
viding as they are Aristotle's ultimateground for rejecting theAtomistic
Principle, the identification ofgeneration and aggregation and the argu-
ment for theexistenceofatoms, and thusthebase for his alternativeview
of the structure of divisibility.
3.1 Two kinds of points
Aristotle's refutation of the atomistic argument is not the only passage
in the corpus Anstotelicum where it is maintained that points are not
ordered consecutively. The principle argument appears in Physica VI1,
where Aristotle, as we already saw above, explains that points cannot
make up a magnitude, because they cannot be ordered consecutively,
since that would require that they have boundaries with which to touch
each other. Some scholars have wondered why Aristotle does not con-
sider a dense ordering relation as a possible relation for points making
up a whole.34
The answer must be that precisely because points are
ordered densely, and not consecutively, they do not make up a whole.
The idea behind this is that the parts into which a whole is divisible and
of which it consists,mustbe independent from each other and contribute
each separately to the whole. If a line were to consist of densely ordered
points, itwould be divisible into one point and the remainder of the line,
sothatthisremainder,which hastobejustasindependent from anything
outside, would not have a limit any more, for lack of a final point —
which would be absurd.35
However, this argument is only of limited help for understanding the
details of Aristotle's refutation of the atomistic argument as provided in
317al-ll. It does show that Aristotle thought that if a magnitude were
to be divisible everywhere in the sense ofpossibly divided everywhere,
it would have to consist of consecutively ordered points. Since that is
impossible, a magnitude cannot be everywhere divisible in that sense.
But it does not explain in any way the further remarks about points in
34 Notably Sorabji, TCC, 369,but also Charlton, Totential Infinites', 136 and D.
Bostock, 'Aristotle on Continuity in Physics VT, in Judson, Aristotle's Physics,
179-212, at 185-6.
35 In 'Zeno Unlimited', Oxford Studies in Ancient Philosophy 30 (2006) 49-85.1attribute
this line of argumenttoZeno.

that refutation. Thereare two obscurities whichhave tobe clarified ifwe
want to understand fully the pointof Aristotle's refutation. The first was
already mentioned above:what does Aristotlemean when he declares
at 317a9 that because points are not consecutive, there is not more than
one point everywhere? Apparentlyhe assumes that if points doexhibit
that ordering, there is more than one point everywhere.But how is it
possible for at least two points to coincide, even if they are ordered
consecutively? Thesecond obscurityconsists in the remarkat 317alO-ll,
which Aristotle even presents as a reason for the previous point, thatif
a magnitude'is divisibleinbetween, it will also be divisibleat acontigu-
ous point.' How isit possible for a magnitudewhich consists of consecu-
tively ordered points tobe divisibleata point?Wouldsuch amagnitude
not ratherbe divisible betweentwo successive points? Themysteryisyet
deeper, since with the phrase 'in between' (êáôÜ ìÝóïí) Aristotle even
refers to such a division between points, as also appears from aparallel
taken from deLineis Insecabilibus: "If points are successive,a line shall not
be cut at either point, but in the middle (áíÜ ìÝóïí).'36
Toshed light on
these dark remarks, we need to go beyond the resources derived from
the argument from Physica VI1 and provide a richer account.
Such a richer account is suggested by again a passage from deLineis
Insecabilibus. There we find the following argument:
(I) Further,if [aline]consists of points, a point will toucha point.When
then from K[lines]ABand CD are drawn, boththe point [B] in AK and
the point [C]in KDwill touchK.Hence [B and Cwill] also [touch] each
other. For a paruess [thing] touches a partless [thing]whole to whole,
so that theywill occupythe same placeand the points touching K [are]
in the same place as each other.37
And if they are in the same [place],
36 972a3-5
37 Most scholars emend the text to read:
ôï ãáñ ÜìåñÝò ôïõ Üìåñïàò üëïí üëïõ åöÜðôåôáé, þóôåôïí áõôüí ÝöÝîåé/ÝöÝîïõóé
ôüðïí ôö Ê, êáé <ôïû Ê> Üðôüìåíáé <á'é> óôéãìáß åí ôö áýôö ôüðö ÜëëÞëáéò.
(two related manuscripts: ÝöÝîïõóé ôüðïí ôö Ê. áú ãáñ Üðôüìåíáé óôéãìáß—all other
manuscripts: ÝöÝîåé ôüðïí ôïõ Ê, êáé Üðôüìåíáéóôéãìáß) —see H.H. Joachim, trans.
&comm., DeLineis Insecabilibus,in: W.D.Ross, ed., The Works of Aristotle VI Opuscula
(Oxford: Clarendon Press, 1913), ad locum; M. Timpanaro Cardini, ed., trans. &
comm., Pseudo-Aristotele-. De Lineis Insecabilibus (Milan: Cisalpino, 1970), ad locum;
D. Harlfinger, Die Textgeschichte der pseudo-aristotelischen Schrift Ðåñß áôüìùí

they also touch [each other], for things which are in the same primary
place38
must touch [each other].
(Ð) And if that is so, a straight [line] will touch a straight [line] at two
points (êáôÜ äýï óôéãìÜò). For the point [B] in AK touches both the
[point] KC* and the other point,40
so that AK touches CD at several
points (êáôÜ ðëåßïõò ... óôéãìÜò).
ãñáììþí. Eine kodikohgisch-kulturgeschichtlicher Beitrag zur Kl rung der berlie-
ferungsverh ltnisse im CorpusAristotelicum (Amsterdam:Hakkert, 1971),329;and M.
Federspiel, 'Notes exegetiques et critiques sur le trait£ pseudo-aristote tien Des
lignes insacables',Rente desEtudes Grecques 94 (1981) 502-13, at 511. Instead I propose
to read:
ôï ãáñ ÜìåñÝò ôïõ Üìåñïûò üëïí üëïõ åöÜðôåôáé, þóôå ôïí áõôüí ÝöÝîåé ôüðïí, ôïõ
Ê 5' áú Üðôüìåíáé óôéãìáÀ åí ôù áýôù ôüðö ÜëëÞëáéò.
I have two reasons for adopting this text. First of all, this is the most economical
emendation, merely changing Éïõ Ê êáé into ôïõ Êä' áú, thus also saving the letters
á'é read by the two divergent manuscripts. The place of the particle may seem
surprising, but in pseudo-Aristotle, Mechanica 23, 855a21-2 there is a parallel.
Secondly, the text which is most commonly adopted contains an imbalance. Forby
reading ôù Êafter ôüðïí, we are obliged to understand the points Â and C to be the
subject with ÝöÝîåé — reason enough to emend that to ÝöÝîïõóé. But since these
points are not mentioned there as the subject, they have to be supplied from the
context, either from the previous sentences or from the next clause. The former
alternative does not work, as then the sentence would go: 'So that [the points
previously mentioned] occupy the same place as K,and, by touching K, the points
are in the same place as each other.' Why mention these points again in the next
clause? The latter alternative does not make for a smooth sentence either: 'So that
[the points] occupy the same place as K, and, by touching K, [...]are in the same
place as each other,' as the postponement of áú óôéãìáß seems awkward.Moreover,
a defect both alternativeshave in common, is that the phrase Tjy touching K' is out
of place, since one would expect T^y occupying the same place as K' or even 'thus',
in order to have a smooth transition from Â and C touching K over Â and C
occupying the same place as Kto Âand C being in the same placeas each other. All
these problems disappear if we take ôá Üìåñç, to be supplied from the previous
clause, tobe the subject with ÝöÝîåé (thus retaining, with most manuscripts, ÝöÝîåé).
Then we canunderstand the sentence up to ôüðïí ('For a partless ... same place') as
a kind of law about partless things in general. The further consequences of this law
are stated then in the next clause in terms of the particular case involving the two
points Â and C touching K —which explains why they are introduced there again,
by way of the phrase áú Üðôüìåíáéóôéãìáß.
38 Following Joachim, De Lineis, ad locum, note I, and Harlfinger, Textgeschichte, 329
(with referenceto two manuscripts) in reading åí ôö áýôù ôüðö ... ðñþôö.

(IE) Further also the [circumference] of a circle will touch a straight
[line] at several points (êáôÜ ðëåßù). For both the [point] in the circle
and the [point] in the straight [line] touch the intercontact (ôçò óõíáöÞò)
as well as each other.41
Here,just as in deGeneratione et Corruptione 12, there is talk of there being
'two' or 'several' points at which something happens to two entities,
there division, here touch. Here, however, we have an argument justify-
ing such talk, an argument which suggests an underlying picture. If we
are able to reconstruct this picture, we might try it out on the argument
of de Generatione et Corruptione 12, to see whether it fits there.
As I showed above,in the argument of de Generatione et Corruptione I
2,wehaveaplaceoftouch, as distinguished from two contiguous points,
referred to by the phrase êáôÜ ìÝóïí. In the present argument we have
the same situation: Ê is the place where lines ABand CD touch; it is not
a full member of the successively ordered series of points. This appears
first of all from the switch from AB and CD to AK and KD in (I).If K
were a full member ofAK, then 'thepoint in AK' would be a misleading
way of referring to the last point B of AB, as clearly it must. This is
confirmed by the parallel case in (ÐÉ) of a circumference and a straight
line, where in each of them there is a point meeting at 'an intercontact'.
Thus we should represent the situation as:
K
B C D
39 Reading here for the time being, with Joachim, De Lineis, ad locum, and Harlfinger,
Textgeschichte 330, ôçò ÊÃwhere most manuscripts have ôç ÊÃ. Other proposed
emendations are <ôçò Ýí> ôç ÊÃ(Ï. Apelt, ed., Anstotelis quaeferuntur ...de lineis
insecabilibus [Leipzig· Teubner, 1888],ad locum,and Hayduck) and <ôçò Ýí> ôç ÊÄ
(Timpanaro Cardini, De Lineis, ad locum).Later on, however, I shall proposemy
own emendation.
40 Reading, with most manuscripts, ôçòåôÝñáò instead of Apelt's åôÝñáò, which until
Harlfinger, Textgeschichte,399, was acceptedbyeveryone.
41 971b5-18

Having said that, however, I must add straight away that matters are
rather complicated. First of all, at one stage in (I)the argument crucially
treats Kas being just the same kind of indivisible as the points making
up the lines. Moreover, it is not immediately clear how (Ð)is related to
(I): is the other, second, point referred to in (Ð)different from K and C
or not?From (Ð) as it stands it may appear that Kand C count as one, so
that 'the other point' cannot be either of them, but then two difficulties
arise. The first is that there is no argument on offer that there is such a
different point besides 'KC' which point B in AK could touch, even
though the phrase 'the other point' seems to imply that we have already
encountered it. The second is that in (ÐÃ) the intercontact, which is
analogous to K,and the point in the straight line, analogous to C, seem
tobe enough to count two points at which the circumferencetouches the
straight line.42
In order to solve these two difficulties we should take (EH)
as given and try to accommodate (Ð) in order to fit its scheme. This
scheme seems to be as follows:
(a) Line÷touches line y at intercontact ñ — implied in (ÐÉ).
(â) The point a in x43
touches ñ —see (ÐÉ).
(ã) The point a in ÷touches the point bin y — see (ÉÐ).
(ä) Therefore ÷touchesyboth at ñ and at b— cf.the conclusion of (ÉÐ).
Applied to the exampleof (I) we get:
(áú) AB = AK touches CD = KD at K — starting point of (I).
(âé) The point Bin AK touches K— assumed in (I).
(ãé) The point Â in AK touches the point C in KD— argued for in(I).
(äé) Therefore AKtouches CDboth at Ê and at C.
42 These two problems have been noticed before, by Timpanaro Cardini, De Lineis,
99-100. The basic outline of my solution is also the same as hers.
43 That is, the final constitutive point in line÷(whichby hypothesis consists ofpoints),
and thus the one 'closest to' p.

Now in (II) we do read that AK touches CDat two places, but as the text
stands there is no reference to Kand C separately in (Ð), eventhough we
do have separatestatementswith regard to K — cf. (â,) — and C —cf.
(ã,) — in (I). Therefore I propose to emend the text and to read in(II):
Þ ãáñ åí ôç AK óôéãìÞ êáé ôçò Ãêáé ôçò åôÝñáò Üðôåôáé óôéãìÞò.44
For the point [Â]in AK touches both the [point] C and the other point
[seil. K].
Thus the picture is that AB touches CD both at the first 'real' point C
contiguous with the last point B of AB and at the intercontact K between
AB and CD. Byphrasing it in this way, I have made it already clearthat
this picture fits the argument of deGeneratione et Corruptions 12.The only
adjustment we have to make is to read 'x is divisible from y at p' where
in de Lineis Insecabilibus we have 'x touches y at p'. Moreover, we have
seen how it is argued in deLineis Insecabilibus that both Band C as well
as K are in the same place, so that there are more points than one
everywhere.
Sothe distinctive feature ofthe picture presupposed by the argument
of de Lineis Insecabilibus which enables us to make sense of Aristotle's
remarks in his refutation of the atomistic argument is that a line made
up of points consists of two kinds of points, both of which are succes-
sively ordered. This composition can be represented as follows:
CICICICICICICIC
where a 'C' represents a constitutive point and a T an intercontact. Inde
Lineis Insecabilibus it is argued that every constitutive point coincides
with its neighbouring intercontact, so that there are two points every-
where — just as Aristotle claims in de Generatione et Corruptione would
be the case if a line were to be divisible everywhere in the pernicious
sense — and a line is both divisible at each intercontact, between the
constitutive points, and divisible at the nextconstitutive point, precisely
44 ThusIemendthephraseôçÊÃof themajority ofthe manuscriptstoôçòÃ. Timpanaro
Cardini, DeLineis, ad locum, has proposed anotheremendationin order to secure
thesameinterpretation,by reading<ôçò Ýí> ôçÊÄ. Not only ismy emendation more
economical, but it alsoexplains why the dative ôç (in the manuscripts ôç) appeared:
ò was somehowread as Ê and added to Ã.

because the intercontact coincides with that constitutive point. In this
way we can also understand why Aristotle saysinhis refutation that the
fact that if a magnitude is divisible in between, it is also divisible at the
next point, is explanatorilyrelated to the fact that there is morethan one
point everywhere, as is indicated by the ãáñ at317alO.
What is more, with this picture in mind we can also get abetter grasp
of the refutation in deGeneratione et Corruptione 12 as a whole,because it
also allows us to understand some other clauses. On Aristotle's own
model, being divisible everywhere means having a point anywhere,
where a point is a possible place of division and touch, by which
magnitudes are then marked off from each other. If being divisible
everywhere is understood, however, as the possibility ofbeing divided
everywhere, there is, as Aristotlesays, not only a point anywhere, in the
sense that it is possible to be divided there, but also everywhere, in the
sense that is possible to be divided at all of them.But if that is the case,
we face a problem. On the one hand, we assume that if a magnitude is
divided everywhere, there are some ultimate building blocks without
size into which the magnitude is divided. Thus the magnitude would
consist ofnothing but constitutive points. On the other hand, all ofthese
constitutive points must be divided from each other.Tothatpurpose we
need points as places at which the magnitude is divided. A magnitude
with a point everywhere would then be a magnitude divided at allof
them, so that it consists of nothing but intercontacts. Thiswould explain
the exclusivedisjunction at 317a7 that if there is a point everywhere, a
magnitude 'either consists of contactsor of points': the points which are
everywhere cannot fulfil both functions,being an intercontact as well as
being an ultimate constituent.45
This problem, though, is not developed by Aristotle. Instead he
continues the argument with a reformulation of the distinction between
45 Similar disjunctions appear at 316b4 and b!5. The specificuse of áöÞ as the point or
place of contact(rather than the fact of contact) seems to be confined to GC1, and
it is striking that all cases appear in discussions of atomists' views. This is obvious
in GCI 2, 316b4, b!5, 317a7, and I 8, 315b31, and very plausible in GCI 8, 316bl2
and 19,327al2. As thephrasing in that last passage ('[a body] can be segregated at
the points of contacts (êáôÜ ôÜò ÜöÜò), as some say') suggests that it concerns
technical vocabularyadopted by the atomists, it seems likely thatby using the same
terminology in GC12, Aristotleexploits their way of talking in his discussion of the
argument for the existence of atoms.

a healthy and a pernicious sense of divisibility everywhere. Again, if
there is one point, only in the sense of a (possible) place of division
anywhere, there is no problem. However, if it is to be possible that a
magnitude is divided everywhere, there must be both an intercontact
and a constitutive point everywhere. For if two constitutive points are
divisible at the intercontactbetween them, they are,by the argumentof
de Lineis Insecabilibus, also divisible at one of the two constitutivepoints,
which is contiguous to the other and coincident with the intercontact.
This whole construction is impossible, Aristotle declares up to three
times in the refutation,46
because points are not successively ordered.
Since at 317a9 this impossibility of successively ordered points is used
to rule out there being two points anywhere, he must be referring to the
intercontact and the constitutive point as the two points which would be
successive. This is in line with the argument of de Lineis Insecabilibus,
where the intercontactis treated asjust another indivisible, belonging to
the same group as the constitutive points. However, after that he seems
to switch, now referring tothe two kinds ofpoints separately as the points
which cannot be successive. As Iexplained earlier, 'in between' refers to
an intercontact, while 'at the contiguous point' indicates the constitutive
point which is contiguous to the constitutive point 'at the other side'of
the intercontact. Similarly, Aristotle distinguishes at 317all-12 between
óçìåßá and óôéãìáß,neither of which are successive. This distinction,
which seems more than a verbal variation, is best explained as the
distinction between intercontactsand constitutive points. Moreover, this
would also explain the distinction between division and composition
drawn at 317al2,which seems tocorrespond to that between óçìåßá and
óôéãìáß: points involved in division are óçìåßá and intercontacts,
whereas points from which a magnitude is thought to be composed are
constitutive points, óôéãìáß.47
Thereisnothing wrong with such a switch,
however, since the contiguity between an intercontact and a constitutive
46 And he repeatsit in GC12,317al5-16.
47 As far as I know, there is no passagein Aristotle wherehe talks about composition
from óçìåßá; he does call points óçìåßá, but only if they are (possible)places of
division and touch (see here at 316b20 and 25).If he wants to refer specifically to
composition from points of touch and division, he switches to Üöáß, as we see in
317a7.FortherestitshouldbenotedthatAristotleusesóôéãìáßbothfor intercontacts
and for constitutive points, just as he here implies that there are two óôéãìáß
everywhere, one intercontact and one constitutive point.

point implies and isimplied by the contiguityoftwo intercontacts or two
constitutive points.
3.2 The atomisticargument refuted
It may be that with the help of the argument from deLineis Insecabilibus
we have been able to provide an explanation of the obscure remarks
about points in Aristotle's ultimate refutation of the atomisticargument,
as well as of some other elements of that refutation. However, that does
not mean that we now have an answer as to what the point of these
remarks is.Ostensibly they are meant to underpin Aristotle's claim that
being everywhere divisible in one sense cannot be a feature of a magni-
tude. But since it seems to follow immediately anyway that if a magni-
tude is everywhere divisible in that pernicious sense, it is made up of
consecutively ordered points (just as in Physica VI 1), one may still
wonder what the relevance is of these remarks about there then being
more than onepoint everywhere and themagnitude beingbothdivisible
in between and at the next point. In order to provide an answer to this
crucial question, we have to return to the two other interpretative
difficulties we had with Aristotle's recapitulation of the atomistic argu-
ment.
The first problem with this recapitulation was to explain Aristotle's
distinction between potential and actual (in)divisibility in such a way
that it doesnot point forward to the subsequent refutation, orevenimply
something which is inconsistent with the AtomisticPrinciple.It is pos-
sible toprovidesuch an explanationby considering how Aristotle thinks
about the relation between actuality and potentiality. Apotentiality has
to be understood in terms of its concomitant actuality: if a pile ofbricks
has the potentiality tobe a house, thispotentiality isunderstood in terms
of the actuality that they in fact make up a house —a very trivial point.
Now according to one way of applying this scheme to an object which
is potentially divisible at a certain place, we get the following picture:
such an object consists of two parts which at the moment make up one
whole, but which can be separated from each other. This view of the
object can be represented as follows:

According to this conception, the parts are lying ready to be separated
out; this can be done by merely destroying the container of the whole.48
That Aristotledoes use the term 'potential divisibility' in such a context,
also appears from his application of it to numbers, when he calls a
countable multitude 'that which is divisible potentially into non-con-
tinuous [parts].'49
In the case of a number, the parts are clearly present
in the actual whole in a somehow determinate way, rather than forming
one undifferentiated stretch.50
That such a conception ofpotential divisi-
bility as applicable to numbers is relevant in the contextof an argument
for the existence of atoms, is confirmed by Aristotle's remark in deCaelo
ÉÐ 4 that what the atomists in fact do when they state their Atomistic
Principle and identify generation and aggregation, is that they 'in away
produce all things which are as numbers and from numbers (ôñüðïíôßíá
... ðÜíôáôá ïíôÜ ðïéïàóéí áñéèìïýò êáéåî áñéèìþí)'.51
Now such an understanding of potential divisibility, in terms of the
possibly separated parts which only need to be brought to actuality by
an act of division, does not lead to problems as long as one considers
each possibility of division distributively.Toeach separate possibility of
division justtwo potential parts correspond. However, ifone applies the
same conception of potential divisibility to all possibilities of division
collectively, absurdities follow. For according to that conception, one
needs to posit, corresponding to all these potential divisions together,
potential parts which are there to be actualized by these divisions.The
only parts which canfulfil this function are sizeless, since any other part
is not a part which canbe actualized by a division everywhere.
48 Thereis an analogous Aristotelian use of the term 'matter7
, e.g. in Metaphysica Æ10,
1035al7-21, wherethe halves ofline arereferredto asthe matterofthat line.Inother
passages, likePhysica ÐÉ7,207a35-bl, 'matter' refers to the undifferentiated aspect
of a divisible body.
49 Metaphysica Ä 13,1020alO-ll
50 Cf.also Aristotle's general description of ðïóüí in Metaphysica Ä13,1020a7-8: 'what
is divisibleintothings present in it ofwhichboth or each is ofsuch anature as to be
one thingand a this something (ôï äéáéñåôüí åéò ÝíõðÜñ÷ïíôá ùí ÝêÜôåñïí Þåêáóôïí
åí ôé êáé ôüäå ôé ðÝöõêåí åßíáé)' — one should compare ÝíõðÜñ÷ïíôá here with
ÝíõðÜñ÷åéí at 316b32.Thereason why Aristotleapplies thus this whatonemay call
structured conception of quantity also to magnitude, is that he wants to thinkof
magnitude primarily in terms of being measurable(1020a9-10).
51 303a8-9

Thus with this what one may call structured conception of potential
divisibility we have not only identified a way of understanding the
distinction between potential and actual divisibility which does not
preclude the puzzle constituted by the atomistic argument being stated,
but also the very idea which is the source of that puzzle. Moreover, it is
also an idea which can be expressed in terms of defining generation and
passing-away as aggregation and segregation respectively. The whole
which consists of two potential parts put together is destroyed by the
segregation of its parts; it comes into being again by the converse
aggregation of the parts. Each separate division and unification thus
amounts to nothing more than moving the given parts around in a
certain way. Already before a division, the whole has an internal struc-
ture consisting in the parts being aggregated; with the division merely
the relation of forming an aggregated whole disappears, but the parts
do not change for themselves.
There is also textual evidence that it is this conception of potential
divisibility which is targeted by Aristotle. It can be found in the lines
316b9-14, which I, following Sedley, have transposed to be inserted
before 316b28.52
The argument there runs as follows:
(a) A stick having being actually divided at some point, putting
together the two halves will result in the same stick asbefore.
(b) This is true of every point on the stick.
(c) Therefore the stick ðÜíôô| äéÞñçôáé äõíÜìåé.
(d) Thereforethere is, absurdly, nothing apart from divisions.
In order that (d) follows from (c), we need to understand the unique
phrase ðÜíôç äéÞñçôáéäõíÜìåéas denoting the potential state of division
everywhere and to think of this potential state as somehow already
present now.53
Interpreted thus, (c)is also easily inferable from (a) and
52 See the full translation of the passagein section2.
53 Sedley, 'GC12', 75-6, wants to understand the phrase äéÞñçôáé äõíÜìåé in termsof
a virtual or conceptual division having been carried out by someone, which is
brought about when one has 'run througha purely mentalprocedure of registering
divisions within some magnitude'.Thereare several reasons why such an interpre-
tation isratherunfortunate.One isthatit isimpossibletorun through such amental

(b) together. For (a) presents division and putting together again as a
completely neutraloperation after which the stick is exactlythe same as
before. Thus (a) invitesus to conceiveof the one stickas already consist-
ing of two halves divided potentially from each other at some point—
that is, to adopt the conception of divisibility in terms of parts already
lying ready to be separated and thus actualized. Since, as (b) says, (a)
applies to every point on the stick, it follows(c)that the stick is already,
be it potentially, in a state of being divided everywhere, so that (d)
readily follows. Thus we have here the very argument given above
applying the structured conception of potential divisibility to all possi-
bilities of division collectively.
Soit should be this conception ofpotential divisibility as involvingan
internal structure which Aristotle needs to refute. It is here that my
interpretation of the refutationbecomes important. For if one compares
the picture suggested by the argument taken from deLineis Insecabilibus,
according to which a line consists of successively ordered points, with
constitutive points and intercontacts alternating, with the picture repre-
senting this conception of potential divisibility, then one may see how
the former picture is to be derived from the latter. For according to the
conception ofpotential divisibility as involving an internal structure, the
divisible object consists of constitutive parts touching at intercontacts.
Now if one retains this conception in the case of something which is
everywhere potentially divisible, then one has to assume that such a
magnitude still consists of intercontacts and constitutive parts. Since in
the case of something everywhere divisible the only constitutive parts
available are points, one gets the picture presupposed by the argument
from de Lineis Insecabilibus: the places of division are retained as inter-
contacts, but the parts shrink to constitutive points.
procedure, the mind being finite. More importantly, one would not want the
conclusion (d) to be dependent on a process bringing about the state of affairs
described by (c)and (d), as they state something about the stick itself and are part
of an argument purporting to show that the stick itself is not everywhere divisible,
but consists of atoms.Thirdly, as Sedley himself notices (76),'even within a single
[physical] atom such "potential" divisions could be arguably be registered in
thought', so that the argument thus interpreted would lead straight away to the
very problematic thesis of mathematicalatomism. Finally, in the argument (c) is
derived from (a), which is not concerned with conceptual division at all, but is
explicitly couched in terms of actualseparation and putting together.

Since we know from Physica VI 1 that this consecutive ordering of
indivisibles is impossible, Aristotle thus reduces to absurdity the con-
ception ofadivision as a mereactualization ofalready present parts. One
might object to this conclusion,by pointing out that it only reduces to
absurdity this conception in combination with the hypothesis that a
magnitude is everywhere divisible; logically speaking Aristotle could
have rejected this hypothesis, just as the atomist does. However, we do
not need to assume that it is Aristotle's purpose in de Generatione et
Corruptione 12 to refute atomism as such. As he makes clear a numberof
times throughout the chapter, there are independent grounds for deny-
ing the existence of atoms. The clearest statement to that effect has
already been quoted in the introduction. Another statement occurs even
after the refutation, at 317al4, thus indicating that it isnot Aristotle's aim
in the refutation to reject atomism as such, for then he would have
referred to that refutation. Rather,assuming that magnitudes are every-
where divisible,hewants toshow howonecanunderstand that they are
everywhere divisible without falling into the absurdities to which the
atomist reduces the assumption of magnitudes being everywhere divis-
ible. As he says at the beginning ofthe recapitulation (in316bl4-19),it is
this puzzle he wants to solve. He does so by rejecting on the basis of an
argument the underlying premiss that a divisible whole has an internal
structure of already present parts being aggregated. Since this idea can
also be expressed by defining generation and passing-away in termsof
aggregation and segregation respectively, he consequently also rejects
this way of defining generation and passing-away.
That it is the specific purpose of Aristotle's restatement and sub-
sequent refutation of the atomistic argument to argue for a rejection of
the conception which he supposes to underlie that argument, also ap-
pears from the way in which he proceeds after the refutation. He says
there:
Hence thereissegregation and aggregation,but neither into atomsnor
from atoms (for there are many impossibilities), nor in such a way that
everywherea division comesabout(for ifapoint werecontiguous with
a point, that would have been the case),but there is [segregation] into
small and smaller [parts], and aggregation from smaller [parts]. But
unqualified and completegeneration (ÞáðëÞ êïß ôåëåßá ãÝíåóéò) is not
defined by aggregation and segregation, assomeclaim,[whoalso claim
that] change in what holds together (åí ôù óõíÝ÷åé) is alteration. That
is just where everything goes wrong (Üëëá ôïàô' åóôßí åí ö óöÜëëåôáé
ðÜíôá).Forthereisgeneration without qualification, aswellasdestruc-

tion,notby aggregation and segregation,but whensomething changes
fromthis to this asa whole (ïôáí ìåôáâÜëëô) Ýêôïàäååéòôüäå üëïí). They,
however, think that every such changeis an alteration; but it is differ-
ent. For in what underlies there is something corresponding to the
account (êáôÜ ôïí ëüãïí) and something corresponding to the matter
(êáôÜ ôçí ûëçí). When, then, there is a change in these, there will be
generation or destruction. On the other hand, when there is a change
in the affections, that is, accidentally, therewill be an alteration.
... Now this much has been settled, that it is impossible that gen-
eration is aggregation, as some claim it to be.54
The ultimate source of the trouble, Aristotle states here once again —
'that is just where everything goes wrong' —, is that aggregation and
segregation define generationand destruction. Asthis definitionaliden-
tification comestogether with the conception of potential divisibility as
involving an internal structure of latent parts lying ready to be actual-
ized, it is this conception which has to be rejected.
The alternative conception of something divisible, which we would
expect Aristotle tosubstitute for theonecausing somuch trouble, should
be one envisaging that a divisible objecthas only one point anywhere at
which it is divisible, albeit without an internal structure ofparts; rather,
the divisible object consists of a homogeneous stretch that may be
divided and thus individuated in an unlimited number of ways, but
which in itself is without any individuation. This is a conception Aris-
totle defendselsewhere, for example in Metaphysica Â5:
[A]ny shape whatsoever is similarly present (åíåóôéí) in the solid
[body] or none [is]. Hence, ifnot even Hermes [ispresent] in the stone,
neither [will] half the cube [be present] in the cube as something
determinate (ïûôùò ùò ÜöùñéóìÝíïí). Therefore the surface [wiU] not
[be present] either, for if any [surface] whatsoever were present, also
the one which determines the half (ÞÜöïñßæïõóá ôï Þìéóõ)would have
existed. The same account also applies in the case of a line, a point and
a unit.55
54 GC12,317al7-27; 30-1
55 1002a20-5

And this isprecisely the conception we seeAristotlepropounding in the
passage from deGenerations etCorruptione 12,where he ishappy to admit
that there is segregation and aggregation, but denies that the generation
is complete yet when something holding together is formed;56
to have a
complete generation there needs to be a change 'from this to this as a
whole', so that 'in what underlies', that is, the subject from which the
generation occurs, 'there is a change in ... something corresponding to
the account and [in] something corresponding to the matter.' Applied to
such typicalcases ofgeneration by division like thatofastatue of Hermes
out of a block of stone, or of destruction by division like that of a whole
divided it into two halves,57
we may think of the change in the aspect
corresponding to the account as the replacement of one individuating
form with two new individuating forms,and of the change in the aspect
corresponding to the matter as the disappearance of one homogeneous
stretch of matter and the arrival of two other stretches of matter, both as
individuated by the one or the two forms respectively. Similarly, we
might view the division of a whole into two parts as the change in the
whole as identified by its unindividuated matter from the this of being
one to the this of being two.58
56 Here óõíå÷Þò is used in the sense of havingno gaps in it, not in the stronger senses
of being divisible everywhere or of having a real unity. This weaker sense we
encounter also elsewhere in Aristotle, e.g. Physica ÉÐ 4, 203a22, VI1, 231bl6 and
Metaphysica Ä26,1023b32-4.
57 Thoughin the contextAristotle also seems tobe concerned withsavingtransmuta-
tions of elements as cases of generation, as appears from his remark at 317a27-30
that assegregated things become smaller, theybecomemore susceptible ofdestruc-
tion, that is, of transmutation into a different element.
58 Cf. Physica HI 1, 201a3-5. Lest the reader worry about Aristotle's statement at the
beginning ofthe second chapter of GCI,where he says at 315b24-8 that for the issue
whether generation should be defined by aggregation it is basic and makes a huge
difference whether there are atoms, I think we can interpret these qualificationson
a secondary-order level: the investigation whether the argument for the existence
of atoms is sound is indeed basic and does indeed make a huge difference to our
understanding of the issue. This interpretation is even the only option available if
one agrees with what I say in the next note aboutlines 315b20-2.

4 From Mathematical Divisibility to
Physical Divisibility
Where does this account of Aristotle's diagnosis of the atomistic argu-
ment leave us with respect to the relation which Aristotle presupposes
between mathematicaland physical divisibility?Though it is clear that
during his discussion of the atomistic argument Aristotle is thinking
primarily in terms of physical divisibility, his diagnosis applies equally
to mathematicalatomism.For whathehas achieved isthat hehas shown,
one, that the argument summarized at the beginning of section 2 de-
pends crucially on the assumption of a principle like the Atomistic
Principle, and, two, that such an assumption is inconsistent with a
magnitude being divisible everywhere. Since the argument does not in
any way presuppose a specific kind of divisibility, this result also per-
tains to the case of mathematical divisibility.
It is this general nature ofAristotle's diagnosis which, I want to argue,
provides the basis of a justification of Aristotle's disregard for the
difference between physical and mathematical divisibility in so far as it
concerns the issue of atomism.In deGenerationeet Corruptione 12 Aristotle
refers a number of times to the impossibilities consequent upon
atomism.59
As I already said in the introduction, he must be thinking of
the arguments found in Physica VI and in de Lineis Insecabilibus, all of
which are concerned with mathematical or conceptual (in)divisibility.
Whether he actually thought they were immediately sufficient to refute
all kinds of atomism or not, the least we can say is that he has thus
sufficient grounds for rejecting mathematical atomism. Thereforehe has
also enough reason to reject in the realm ofmathematics a principle like
59 In addition to the two passages mentioned earlier, there is also such a reference to
be found in 315b33. Presumablyeven the 'many impossibilities' which at 315b20-l
are said to follow from the identification of generationand aggregation, refer to the
same problems with atomism (cf. Williams, DGC, 64), as this identification is
considered by Aristotle to be a premiss clinching the argument for the existenceof
atoms. I would say that this also appears from the context: 'If, on the one hand,
generation isaggregation,manyimpossibilities follow. On the otherhand, thereare
other arguments, forceful and not easy to dissolve (áíáãêáóôéêïß êáé ïõê åýðïñïé
äéáëýåéí) that it cannot be otherwise' (315b20-2). I cannot think of any other
argument than the atomistic reduction to absurdity of the possibility of division
everywhere (which even withoutthe explicit supposition that generationis aggre-
gation is forceful enough) — cf. Sedley, 'GC 1.2', 84.

(AP), together with the idea that adivisible whole isstructured into parts
lying ready to be separated out. Soas a general argument the argument
for the existence of atoms as reported in de Generatione et Comtptione 12
is not sound, because the underlying view ofthe structure of something
divisible is not generally true.
As a consequence of this lack of generality with regard to the truthof
the underlying principle, the onus is on those who do want to maintain
its correctness in some field or another. Why should we adopt such a
principle in the area of physics, if we cannot accept it in the area of
mathematics? The physical atomist owes Aristotle at least some justifi-
cation.
Now one may think that there is one available to the atomist, one
pointing to the different nature of physical and mathematical divisions.
Whereas a mathematical division is brought about by the mind and
consists in the marking off by some boundary of parts within a whole,
sothat atleast insome sensethewhole remains intact, aphysical division
actually destroys the whole. An object which is physically divisible can
therefore not be completely firm and must to some extent fail to resist
attempts at deformation or dissolution. So in order to account for the
relative firmness and solidity which at least some objects exhibit, the
thought mightbe,weneed completelyfirm and solid parts and a limited
number of places where a separation might be brought about — the
fewer places the harder the object. It is such a physical argument which
Epicurus seems to be hinting at in his letter to Herodotus:
Further, among bodies, some are aggregates, and some are those from
which aggregates have been made. These are atomic and unchange-
able, lest they are all to be destroyed into not being (åß'ðåñ ìç ìÝëëåé
ðÜíôá åéò ôï ìç äí öèáñçóåóèáé) and ifthey are to survive, being strong,
in the dissolutions of aggregates, as they are full by nature and are not
such that they will be dissolved at any place or in any way (áëë'
éó÷ýïíôá ýðïìÝíåéí åí ôáÀò äéáëýóåóé ôùí óõãêñßóåùí, ðëÞñçôçí öýóéí
ïíôÜ, ïõê Ý÷ïíôá ïðÞ Þ üðùò äéáëõèÞóåôáé). Hence the principles of
bodies must be atomic natures.60
60 Ad Herodotum 40-1

And we should not rule out the possibility that Epicurus has taken over
this argument from Democritus, since there are numerous references to
the impassiveness of the latter's atoms.61
Aristotle, however, is not impressed by such physical observations.
Of course he is aware that it iseasier to divide, say, water than stone, but
he reduces such differences to differences in nature between the stuffs
and their qualities, notably wetness and dryness.62
And he is willing to
say that larger bodies may break easier than smaller,63
but that does not
mean that there are indivisibles, as he remarks in de Generatione et
Corruptione I 8:
Further, it is also absurd that small bodies are indivisible, while large
bodies are not.For as it is (íõí), reasonably, it is larger bodies rather
than small bodies which are broken up, for some dissolve easily, like
the larger ones (ôá ìåí ãáñ äéáëýåôáé ñáäßùò, ïßïí ôá ìåãÜëá), for they
collide into many bodies; but on what ground doesbeing completely
indivisible belong to small bodies rather than large bodies?*4
61 Mostpromising inthis respect aretestimonies wherethe impassiveness ofDemocri-
tus'atoms isexplained by theirhardness (äéá óôåññüôçôá),such as Luria [Democritea
(Leningrad: Nauka, 1970)) fr. 212 =Simplidus, In Physica 81.34H., and Luria fr. 215
= 68Al DK=Diogenes Laertius 9.44.However, certainly Aristotlehas the tendency
to ascribe the áðÜèåéá of the atoms to their unity and continuity (in the strict sense
of not being capableof being separated), notably in GC19,327al and Physica íºÐ
4, 255al3. This would also fit with those testimonies in which their being solid
(íáóôüò) and not participatingin the void (Üìïéñïòôïõ êåíïà) isinvoked to thesame
purpose, as in Luria fr. 212 = Simplidus, In Physica B1.34ff. and In de Caelo 609.18,
Luria fr. 214 = 68 A14 DK = Simplidus, In de Caelo 242.19-20, and Luria fr. 217 =
Aetius 1.3.18. Foras Iargued in my 'Foundations', 9,these qualifications should not
be taken as explanatory of the indivisibility and thus unity of the atoms, but as
constitutiveofit.Therefore theirimpassiveness would againbebased on theirunity,
so that there is no space for an independent physical justification for adopting the
Atomistic Principle.
62 See GC Ð 2 and Meteorologica IV 4.
63 One should distinguish this pointfrom another one madeby Aristotle,that smaller
bodies are easy or easier to destroy (åàöèáñôá or åýöèáñôüôåñá) (GCI 2,317a27-9
and de Caelo ÉÐ 6, 305a6-7), for then Aristotle is thinking of the transmutationof
elements: it is easier to turn a smallquantityof water into air than a largequantity.
64 326a24-9

Not only does Aristotle think that size is merelyproportional, however
vaguely, to breakability, so that there is no reason to assume that there
is a size which corresponds to complete unbreakability, but he also
explains the proportionality by invoking the circumstances, viz. that
larger bodies have a bigger chance of hitting other bodies, rather than
their inner constitution.
It isno wonder thatAristotle refusesthus to countenancethe physical
evidence which the atomist could put forward to justify the acceptance
of (AP) in the realm of physics. For Aristotle's diagnosis goes further
than merely causing a dialectical shift in the burden of proof — it puts
up a real challenge for the atomist. For if one has shown, like Aristotle,
that a mathematically divisible entity does not have a structure serving
as a ground for its divisibility, but rather consists of a structureless,
homogeneous stretch of matter, then it appears impossible to come up
with some feature because of which that same entity is physically
indivisible. We find the idea in an argument which Aristotle ascribes to
'some of the ancients', who must be the Eleatics:
Some of the ancients thought that what is, is by necessity one and
immovable. For the void is not and it would not be possible to move
while a separate void is not, nor again are there many, as what keeps
apart is not. And if someone holds that the whole is not continuous,
but, though divided, <consists of parts which> touch — this does not
differ from sayingthat [thewhole]is many,and not one, and void.For
if it is divisible everywhere, thereis no unit (ïàèÝí åßíáé åí), so that the
whole is not many, but void. If [the whole is divisible] here but not
there, this looks like something contrived. For up to what size and on
what ground (ìÝ÷ñéðüóïõ êáé äéá ôß)is this [part]of the whole like this
and full, and that [part] divided?65
It is easy to recognize here the reductio part of the argument for the
existence of atoms as set out above. The atomistic conclusion that some-
thing isdivisible here but not there, however, isruled out on the ground
of considerations of homogeneity:if we compare two outwardly identi-
cal, homogeneousentities, one which consists of two atomsand another
which is itself an atom, what could ever explain the divisibility of the
65 GC18,325a2-12

Aristotle's Diagnosisof Atomism 153
former entity, given the indivisibility of the latter, or what the indivisi-
bility of the latter, given the divisibility of the former?66
That Aristotleendorses this argument himself,alreadyappeared from
his response to the suggestion that there should be some small indivis-
ible bodies, because smaller bodies do not break so easily. The rhetorical
question he asks there: 'On what ground does being completely indivis-
ible belong to small bodies rather than large bodies?' is identical to the
one asked by the Eleatics: 'Up to what size67
and on what ground is this
part of the whole like this and full, and that part divided?' The same
question appears once again in the next passage:
Further, is there one nature of all these [indivisible] solids, or do some
differ from others, just as some would be fiery, and others earthy in
bulk? Forif there is one nature of all,what is it that separates them?Or
on what ground do they not become one upon touching, just as water
when it comes into contactwith water? For the later [water] does not
differ from the earlier [water] (ïõäÝí ãáñ äéáöÝñåé ôï ýóôåñïí ôïõ
ðñïôÝñïõ). If, on the other hand, they are different, of what kinds are
they? And clearly these one should posit as principles and grounds of
the accidents rather than the shapes.68
As there is no void toseparate the atoms, they form, when touching, one
homogeneous whole if they are to have, as they do, one kind of matter:
just asthe water distributed over several drops, at the earlier stage, does
not differ from the water assembled into one bigger drop, at the later
stage,69
so the matter of the solids does not differ. To what may the
66 For more on the use of the argument from homogeneity by the Eleatics, see my
'Foundations', 4, and S. Makin, Indifference Arguments (Oxford: Blackwell 1993).
67 I take it that ìÝ÷ñé ðüóïõ refers to the size of an atom, so that a body with a larger
size containing that atom is divisible and a part of that atom is indivisible: it is
indivisible up to that size and divisible down to that size (cf.ìÝ÷ñé ôïõ in 316b32).
As the emphasis in the context seems to be on the upward perspective, I have
translated 'up to what size'.
68 326a29-bl
69 This seems to me to be the most plausibleinterpretationof the phrase. Thealterna-
tive is to let ôï ýóôåñïí and ôïõ ðñïôÝñïõ refer to different cases, but what could
those be? The putative atoms touching and water touching? That seems unlikely,
since, firstly, the point about the water will not be acceptableto an atomist, and,

atomist then point in order tojustify his claimthat atomswhen touching
do not become one, but, in accordance with the Atomistic Principle,
remain divided and are therefore physically divisible, that is, separable
only at the points ofcontact?
This is a real challenge, one which cannot be met by simply positing
irreducible dispositions for separation at some places and not at other
places. Since that is equivalent to adopting a structured conception of
divisibility in the realm of physics, and thus to adopting the Atomistic
Principle, it would merely amount to ignoring the challenge. The only
way I can imagine of undermining Aristotle's use of the argument from
homogeneity and his diagnosis ofatomism is to argue that it isbased on
too strict a criterion for ascribing properties to something. As I have
shown on an earlier occasion,70
the atomist may reply to Aristotle by
distinguishing between the geometrical homogeneity ofany entity with-
out gaps and the dispositional homogeneity which is merely exhibited
by indivisible entities. Entities consisting of two touching atoms are not
dispositionally homogeneous, as they are divisible at one place and
indivisible at allother places, even though they are geometrically homo-
geneous. Now the property of being divisible at one place, and thusof
having a structure, canbe ascribed, the atomist should argue, by taking
into account all (possible) states of affairs, both the actual one and the
non-actual ones, concerning the entity, whereas the property of being
geometrically homogeneous can be ascribed to it merely by reference to
one possible situation (which is at least imagined to be the actualone).
Aristotle's refusal to distinguish with regard to the issue of atomism
between mathematicaland physical divisibility would thus come down
to a failure to distinguish between these two criteria for ascribing prop-
erties.
The philosophical question still stands whether one should call this
failure a mistake. But it seems that as long as we have not settled that
extremely difficult matter, we are not in a position to accuse Aristotleof
the mistakeof failingto distinguish between physical and mathematical
secondly and more importantly, one would then expect ôï ðñüôåñïí not to differ
from ôïõ õóôÝñïõ.Moreover, in the context the verbäéáöÝñåé has alreadybeen used,
at 326a30, to refer to a difference in nature between solids.
70 See my 'Foundations', 10-11.

divisibility in his refutation of atomism. Whatin any case remains isthat
Aristotlehas given us aphilosophically very acutediagnosis of atomism.
Faculty of Philosophy
University of Groningen
Oude Boteringestraat 52
9712 GLGroningen
Netherlands
p.s.hasper@rug.nl

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