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Zeno's A rrow, Divisible Infinitesimals, and

Chrysippus

MICHAEL J. WHITE

In a recent interesting discussion of Zeno's paradox of the arrow, Jon- athan Lear produces a reconstruction of the paradox which, he argues, the differential 'calculus is impotent to solve.'2 In opposition to the received wisdom (and a good deal of contemporary analysis), Lear maintains that Aristotle's response to the paradox is to the point. I find Lear's arguments for both theses compelling, given the assumptions he makes concerning what the 'modern concepts of the calculus' are. However, in the present paper I suggest that the 'conceptual equipment' supplied by a recent grounding of the differential and integral calculi in non-standard analysis is relevant to resolving the paradox as formulated by Lear. My second thesis is really more a 'tentative suggestion': perhaps some of the concep- tual equipment of the nonstandard foundation of calculus, in particular, the concept of 'nontrivial divisible infinitesimals,' can be extracted from several difficult and much controverted passages pertaining to Chrysippean (or `Stoic') doctrines of time, space, and motion.

PART I

The arrow paradox, according to Lear's formulation, is the following: (1) Anything that is occupying a space just its own size is at rest. (2) A moving arrow, while it is moving, is moving in the present. (3) But in the present, the arrow is occupying a space just its own size. (4) Therefore, in the present the arrow is at rest. (5) Therefore, a moving arrow, while it is moving, is at rest.

It might seem that the differential calculus suggests that the paradox can straightaway be resolved by denying premise (1); for the calculus supplies us with a conception of 'instantaneous velocity' or 'velocity at an instant.' The thrust of Lear's response to this line of argument, as I understand it, is as follows. The notion of instantaneous velocity supplied by calculus is - I hope Lear will forgive the pun - a 'derivative' one: 'the limit of velocities at which an object is moving during successively shorter periods of time which converge on a given instant.'3 This account of instantaneous velocity, then, must assume the existence of motion during a series of

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periods of time of which the given instant is a limit. But, to quote Lear in

persona Zenonis contra Owen,

Zeno would not. be at all happy about our simply helping ourselves to the assumption that there exists a period in which the arrow is moving. 'For surely,' he would say, 'if the arrow is moving at all, there is no time it could be moving other than the present. And yet you have admitted that the arrow is not moving in the present, in the sense that it is not actually traversing any distance in the present. You want to say that the arrow really is moving at the present, in the sense that the present is part of a period of time [read: is the limit of a series of periods of time] in which the arrow does traverse some distance. However, you should have admitted that there is no time the arrow could be moving other than the present. So it is absurd for you to say that the arrow is moving at the present in virtue of its moving in some other time!''

One might, I suppose, regard both the 'standard calculus' ' and Owen's resolution of the paradox, to which 'Zeno' is here responding, as a denial of

premise (2), which seems, as Lear points out, to be an expression of a common ancient doctrine that accorded

'ontological priority' of some sort to the present.5 But a number of ancients - Chrysippus for example - would not have been at all eager to dispute the truth of (2).

If one does not wish to relinquish (2), Lear suggests that the 'first line of attack should be premiss (3).' (3) can be attacked, he further suggests, 'by developing a theory of time in which the present can be conceived as a

period of time...one can then proceed, as Aristotle did not, to give a sense to the notion of an object moving at an instant or at the present instants It is, I think, arguable that the conceptual grounding of the calculus in non-standard analysis motivates this sort of resolution of the paradox. The

key is to identify the present with a period of time rather than some

'temporal point'; but the period of time will be of 'infinitesimal' duration. Then, depending on the interpretation of the phrase 'occupying a space just its own size,' either premise (1) or (3) will prove false. One might, according to the interpretation I shall develop, regard the paradox as a

fallacy of equivocation on this phrase: there is a precise sense that may be

given to the phrase which makes ( I ) true and a precise sense which makes (3) true, but these are different senses.

It is well known that applications of the calculus developed much more

quickly than its conceptual foundation.7 Indeed, the '8, e-method' was first

rigorously developed in a way which maintains the 'limit concept in its central place' by Weierstrass in the mid-nineteenth century.8 It is also known that, during the first several centuries of the development of the calculus, there was considerable talk of 'infinitesimals' - as well as criticism of this talk. The criticism was inevitable: within the context of

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standard real analysis, if a non-negative infinitesimal is conceived of as a

non-negative real number smaller than any positive real number, there can be only one non-negative infinitesimal, zero. Then, however, a formula for the derivative of the function flx) at xo (assuming that it has one), namely the formula one obtains from (f(xo + Ax) - f(x))/Ax when an in- finitesimal is substituted for the Lx, will be meaningless: dividing by zero is not defined.

The late mathematician Abraham Robinson used the methods of model theory, a branch of mathematics that deals with the properties of 'semantic structures' used to interpret formal languages, in order to restore in- finitesimals to grace.9 Intuitively, we think of the propositions of real analysis as being 'about' the real numbers. In logistic jargon, this amounts to interpreting these propositions in an 'intended model' which has as its basic or 'urelements' the real numbers. Then, (first-order) properties of the reals can be thought of ('extensionally') as sets of these elements, two-place relations among the reals as sets of ordered pairs of these elements, etc. What Robinson did was to effect an

'embedding" of the reals (the urele- ments of the intended model for real analysis) into another, non-standard model, a model which is 'larger' in the sense that it contains elements other than the 'embedded reals.' Such a non-standard model defines what has come to be called a field of 'hyperreal' numbers.l The field of hyperreal numbers is constructed in such a way that it is, in many ways, similar to the field of standard real numbers. More specifically, any property of the reals that is 'first-order' (i.e., that can be defined without quantification over sets of real numbers) is matched by an 'exactly analogous' property in the field of hyperreals. However, properties of the reals that must be defined in terms of quantification over sets or ordered sets of reals (or sets of sets of these, etc.) will be 'matched' by properties of the hyperreals which are 'weaker' in a sense I shall not here try to explain. (Actually, even the 'sense' of some first-order properties of reals is not, intuitively, preserved in its hyperreal 'analogue'; see, for example, the discussion of the Archimedean property in note 25.)

I shall give a relevant example. The set of real numbers is ordered by the less-than-or-equal-to relation

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The set of hyperreals contains infinitesimals or numbers that are 'infinitely small' in the sense that these numbers are not equal to zero; yet each of them is less than any of the 'standard' real numbers all of which are 'embedded' in the field of hyperreals.

In order to continue with the discussion, we must introduce the relation on the hyperreals. When a and b are hyperreal numbers, a -- b is read 'a

is infinitely close to b' or `the difference between a and b is infinitesimal.' A theorem proved by Robinson establishes that when a is any finite hyperreal number, there exists an unique standard 'embedded' real number r, called the 'standard part' of a, to which a is infinitely close.ll When the standard part of a hyperreal a is zero, then, that a is an infinitesimal. It should be pointed out that none of the infinitesimals are 'indivisibles.' The fact there is no smallest positive real number straightforwardly 'transfers' to the hyperreals: there is no smallest positive hyperreal number either.

The non-standard grounding of calculus in the field of hyperreals provides an alternative for those who are inclined to the view that the 8, E-method's limit interpretation of some derivatives is, at best, a useful fiction; for example, 'instantaneous velocity' implies, according to the non-standard conception, neither movement during a temporal point (which is incoherent) nor 'movement,' in a 'derivative' sense, at a temporal point (which might plausibly be viewed as a sort of fiction possessing pragmatic value); rather, instantaneous velocity pertains to movement in the 'fullblooded' sense, distance traversed in a period of time (the distance and time being, of course, infinitesimal).

To re turn to Lear's formulation of the arrow, let us give Zeno his premise (2) and..4'lt us identify the present with an (any) infinitesimal interval of time. Additionally, let us suppose that the size of the arrow is some quantity that can, given an appropriate measure function, be associated with the

positive finite hyperreal number r. Now, in premises (1) and (3) the phrase 'occupying a space just its own size' can be taken to mean either 'occupying a space r' such that r' = r', or 'occupying a space r' such that Y r,' which I shall refer to (for reasons that will later become evident) as the 'is-equal-to' and the

'is-not-unequal-to' interpretations of the phrase. Let us do what Aristotle was unwilling to do and grant the truth of

premise (I) for the 'is-equal-to' interpretation. It does, I belive - pace Aristotle - have some intuitive plausibility on this reading. However, given the

'is-not-unequal-to' interpretation, it must be false; for, ex hypothesi, the arrow moving in To vv is moving; but if T6 Vi5V is of infinitesimal duration, then the distance r' traversed in the interval by the arrow must be r

(not-unequal-to the length of the arrow). For premise (3), if we assume that

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the present is of (nontrivial)12 infinitesimal duration, then (3) is false according to the 'is-equal-to' interpretation. But for the 'is-not-unequal-to' interpretation, it must be regarded as true: in any infinitesimal time in- terval the arrow will traverse some r' such that Y r. So, according to this reconstruction, the non-standard conceptual apparatus for the differential calculus does suggest a resolution of the paradox: the paradox involves the

fallacy of equivocation, with the phrase 'occupying a space just its own size' cast as the culprit

PART II

In the remainder of this paper, I wish to entertain the hypothesis that Chrysippus toyed with the concept of 'divisible infinitesimals'; I shall examine several rather puzzling passages concerning Chrysippean or 'Stoic' doctrine found in Sextus Empiricus and Plutarch in the light of this hypothesis. My aim is to determine if the hypothesis is of any use in making some sense of these passages. A. The cone, 'that which is greater without exceeding,' and the distinction

between things that are 'not equal' and things that are 'unequal'(Plu- tarch, comm not 1079e-1080d).

This passage pertains to Chrysippus' response to a paradox due to Demo- critus concerning the cone. If a (right circular) cone should be cut parallel to the base, 'what ought one think about the surfaces of the orifices (Tv 61volalli Tas Twv Tpr?p,&Twv do they turn equal or inequal

&VLOOU? yiyvowlvs)?'13 Democritus may well be thinking of the cone as 'hollow': hence, when it is sliced parallel to the base, the cutting yields two 'holes' bounded by circles.14 According to Democritus, if the circles are unequal they will make the 'cone uneven, giving it many step- like notches and asperities.' But if they are equal, 'the orifices will be equal and the cone, being constituted of circles that are equal and not unequal, will have the appearance of a cylinder, which is absurd.'15

This looks like an illustration of the Epicurean complaint that geometers do not deal with the world 'as it really is.' One of the standard illustrations used to introduce the concept of integration in elementary calculus texts is the calculation of the volume of a right circular cone by ascertaining the limit of the summed volumes of 'circumscribed' stacks of n cylinders of equal height h as h approaches 0 and n is 'indefinitely increased.' Suppose that one were to assume a 'realistic' view of this procedure and were actually to conceive of the cone as constituted of an infinite number of such cylinders. Question: is the circumference of one of these cylinders (the

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circle that is the 'surface of the orifice' at the top of the bottom segment obtained when a plane is passed through the cone) equal or unequal to the circumference of a cylinder 'on top of it in the stack' (the circle that is the surface of the orifice at the bottom of the top segment of the cone)? 16 If the answer is

'unequal,' one does obtain a 'stairsteps' of cylinders that cannot be identified with the cone; if the answer is 'equal,' then one obtains one big cylinder, which also obviously cannot be identified with the cone.l7

Chrysippus' response is to declare that the surfaces (the circles which, on this interpretation are equivalent to the circumferences of the cylinders) are neither equal nor unequal. Despite great controversy concerning this whole passage, I am confident that this answer is intimately connected with (i) the doctrine, attributed by Plutarch to Chrysippus, that something can 'be greater without exceeding' (TV vziiov xav J.1T1 uiTEpE\ov) and (ii) the doctrine Plutarch appears to ascribe to Chrysippus concerning the non- equivalence of 'not being equal' zlvi roov) and 'being unequal'

More controversial is the interpretation I should like to give to these various distinctions: 'to be greater and exceed' = 'to be unequal' = 'to differ by a finite amount and *)'; 'to be greater without exceeding' = 'to not be equal and to not be unequal' = 'to differ by a (non-zero) infinitesimal amount but Although I shall not defend this inter- pretation in detail against its competitors, it does seem to me somewhat more plausible than the rather similar interpretation of Sambursky, who reads 'to be greater and to exceed' as I do but is forced to interpret 'to be greater without exceeding' not as specifying a relation between two

quantities but as characterizing an infinite series of relations b=a + E, where e is, in effect, a variable having as values an infinite series converging to zero (which entails that either a or b must vary, as well).19 B. The 'Stoic ' account of motion xaT 6povv J.1EpLO'TOV 6iGaTqw (Sextus,

PH 3.76-80 and M 10.123-142). The latter passage discusses a doctrine concerning motion, attributed at the end of the passage (M 10.142) to 'those of the Stoa' and, at the beginning of the passage, to those of the opinion that 'all things are divided "to infinity"

The doctrine is that

a moving body 'effects the crossing' of a continuous, divisible interval in one and the same time and does not occupy the first part of the interval with its first part, and secondly, in order, the second part; but it passes through the whole divisible interval at one time completely

The discussion in PH is more succinct, and perhaps clearer. Sextus asserts that those who hold bodies, places, and times to be divisible to infinity

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cannot accept the possibility of the progressive sort of motion because it will be impossible 'to discover anything that is first whence the thing said to move will move.'21 One is reminded, of course, of the version of Zeno's Stadium paradox in which the runner can never begin.

To those who hold a doctrine of motion &8pows through a divisible interval, Sextus gives three choices. The spatial interval over which such motion is possible can be (I) limitless, (II) 'precisely bounded' (qrp6s &xp?(3wav ?repvwpvo?EVOV), or (III) 'small, but not precisely bounded' ([LLxp6v

ov ITP63 &xpv(3wav 8i T7EpL

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A finite number of repeated additions of a to itself, in other words, will yield a sum greater than any given positive real number; non-ceasing repeated addition will eventually surpass any finite quantity.

For Robinson's non-standard ordered field of 'hyperreals', this axiom

fails when a is any infinitesimal and b is any finite, non-infinitesimal number: for any natural number n, n repeated additions of any in- finitesimal to itself yields an infinitesimal. So the force of the sorites is avoided if the 'small but not precisely bounded' interval is an infinitesimal one. Repeated additions of such an infinitesimal 'hair's breadth' to itself will always yield an infinitesimal and never a finite, non-infinitesimal sum that would give rise to the paradox derived from Option (II), namely equivalent velocities for all moving objects. Motion over the 'small but not precisely bounded' interval, identified with any infinitesimal interval will occur &Op6w-;, in the sense of 'in a time interval t such that t 0.' But if we consider (an imaginary point in) a body traversing an infinitesimal distance E in an infinitesimal time 8 and a second body traversing the also in- finitesimal distance 2E in the same time 8, the velocity of the second will be twice that of the first (2 - /6 is equal to 2e/8) even though both motions take place 'all at once' (i.e., 6 0). C. The divisible present alone 'obtains' (im&PXELv), while the past and future

'subsist' but, of the present, part is future and part is past (Plutarch, comm not 1081c-1982a)

The Stoics are characterized by Plutarch as 'not admitting a least time nor wishing the now to be without parts' (iX6tXLaTov xp6vov &'TTO?d'TTOUOL vq6k To vv ETVat t Aristotle, in fact, distinguishes a use of 'now-locutions' according to which such locutions refer to a finite in- terval of time.27 The extent of the 'now' or

'present' referred to by such a use of a now-locution is heavily context-dependent. But, according to the characterization by Owen of such uses of now-locutions, it is in principle possible to find, for any given use characterizing a given stretch of time as

'present,' another use of a now-locution characterizing a smaller 'sub- stretch' of time as present, i.e., a use that implies that part of the original time interval is

'past' and part is 'future.' It is this model of 'now' or 'the

present,' which Owen terms the 'retrenchable present,' that he finds in the combination of Chrysippus' interval-conception of the present and another doctrine attributed to Chrysippus by Plutarch:28 'but in the third, fourth, and fifth books pertaining to Parts, he [Chrysippus] affirms that of present (lvzaTqx6Tos) time, some is going to be while some has already been.'29

There is yet another doctrine ascribed by Plutarch to Chrysippus that

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generates difficulties, however: Chrysippus says 'in the work concerning the Void and in certain others that the past and future parts of time do not obtain but subsist while only the present (To vaTqxds) obtains.'3 Plutarch is quick to point out that Chrysippus' doc- trine that the 'divisible present' is actually 'partly future and partly past' seems to nullify this distinction between 'obtain' and 'subsist' with respect to time. It appears that the two doctrines are not consistent. Several ways of

attempting to deal with the apparent inconsistency are available. The simplest, I suppose, is to admit that there is an inconsistency, assuming that since the passages giving rise to it apparently come from different parts of Chrysippus' corpus, he either changed his mind concerning at least one of the doctrines or did not recognize the difficulty. Another approach is to conceive of the distinction between 'obtain' and 'subsist,' applied to time, as relative not merely to the time the assessment is made but also to the particular 'scope' one attaches, in that context, to the retrenchable-present locution one is using.

However, on the assumption that the distinction is intended as an 'ontological' (although, obviously, 'time-relative ontological') distinction, there is yet another approach to the problem of consistency posed by Plutarch. Although there may be now-locutions that refer to a finite in- terval of time, these 'retrench', according to this interpretation of Chrysippus, not to a 'real point' but to an 'indefinite' infinitesimal but divisible interval of time. It is this 'indefinite' infinitesimal interval that

'actually obtains' surrounding some 'incorporeal' (i.e., `conceptual')31 temporal point that might be imagined to divide the future and the past. Yet since this now is an interval and not a point, other 'imaginary points' may be selected within the interval (at various 'infinitesimal distances' from the initially selected point); and each of these must be in either the future-direction or the past-direction from the initially selected point. In other words, according to this way of construing the dilemma, the apparent inconsistency results from an ambiguity of 'past' and 'future': the now is a time interval of indefinite infinitesimal length; relative to an imaginary point selected within this interval, any finite interval 'to the front or to the rear' involves the sort of future or past time that 'only subsists and does not obtain' while any infinitesimal interval 'to the front or rear' really 'obtains.' Of course, the selection of the initial imaginary point in the infinitesimal interval is purely arbitrary since it is the indefinite interval and not any point in it that is 'present.' Consequently the distinction between past and future within the interval is, in a sense, arbitrary; what Chrysippus means to emphasize in saying that 'TO-D iVE(IT'qx6Tos yp6vov T8 wkv zlv1 To

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Si is that the present is an interval rather than a point.32 D. Counterevidence? (Diogenes Laertius, 7.150 and Plutarch, comm not

1070a-b) The former passage, from the text as 'emended' by von Amim, reads as follows:

Hence, they further claim that the cutting [i.e., division] proceeds to infinity. Which cutting, Chrysippus says, is infinite but not 'to an infinite'; for it is not the case that there is some infinite to which the cutting proceeds. Rather, it is inexhaustible.33

The latter passage constitutes an account of Chrysippus' view on the matter of `our parts.' He advises drawing a distinction: employing a `gross' sense of 'part,' we are composed of head, etc. as 'parts.' But, 'if they advance their questioning to the ultimate parts (Ta 9CFXCtTOt he says, 'it is not proper to respond with any answer of this sort; but one must say neither from what sort [of minimal parts] one is constituted nor, likewise, of how many, whether of a infinite or of a limited number'.34

Sambursky cites the former passage as evidence of Chrysippus' care in 'avoiding any expression which could be interpreted as reference to an actual infinitely small quantity-which he rejected as did the Peripatetics.'36 Todd also interprets the former passage as an attempt to avoid a 'terminated infinite series':

That is, Chrysippus would have reasoned that the sum of an infinite number of magnitudes (the product of a terminated infinite division) would be infinitely large, and in the light of this principle formulated his concept of an interminable infinite series. 36

Todd also mentions the latter passage as denying that 'body is composed of an infinite number of bodies.'37

I believe that both passages do indeed deny that infinite division 'ter- minates' in a 'ultimate' quantity or in a largest number of these quantities. But such a denial does not entail denial of the existence of an 'actually infinitely small' quantity or an 'actually infinitely large' number of these quantities. The point of the passage from De communibus notitiis is

evidently simply that, since body is infinitely divisible, there can be no least quantities to serve as the 'ultimate constituents' of our bodies; so, a fortiori, one cannot say how many 'ultimate parts' there are or of what sort they are. The passage from Diogenes Laertius can be read in the same way. If the 'iaTl Tt' of the key, unemended clause, 'Ob yap EOTL w a?revpov, eis 6 -YCVFTOLL ?

&xotr6tX'qXT63 on,' is given 'wide scope', it expresses a pro- position that the defender of divisible infinitesimals would surely accept: it is not the case that there is some aireLpor (infinitely small quantity or

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infinitely large number of those quantities) in which division terminates; division can always proceed beyond any such quantity.

There is, so far as I am aware, no independent evidence that Chrysippus would have accepted the doctrine which Todd believes to lie behind his denial of the 'actual infinite', namely that an infinite number of mag- nitudes of some given size would yield an infinitely large sum. As Alexander of Aphrodisias implies in his De mixtione, the Stoic doctrine of 'total blending' (xpiais 6Xuv) seems ultimately to be conflated with 'justaposition' if 'infinite divisibility' if given the 'potential, 6-z method' interpretation. 38 Of course, Alexander additionally argues that the alternative, 'actual infinite' interpretation also yields difficulties for the Stoic doctrine. Although the passage is less than pellucid, his criticisms seem to be two in number: (i) 'what is constituted from an infinite number of parts each having some magnitude and extent (wlyz86s TL xai 8L6tcrTaCFLV i-X6VTWV) is infinite i.e., infinitely large in extent)'; (ii) the idea of infinitely small but still divisible magnitudes would yield 'still further infinite bodies' (Xzlu av zlvi awp,aTa (ii) might be in- terpreted - as Todd apparently interprets it4 - as implying 'different sizes' of infinitely small and infinitely large quantities. The defender of divisible infinitesimals need have no fear of this consequence. (i), however, is another matter. Given a collection of magnitudes all of some fixed size E, we can conclude from Archimedes' axiom that, for any finite magnitude 8 such that 8 > E, there is a natural number n such that e added to itself n times exceeds 8. This principle holds true regardless of how small we make e, provided that it is some finite magnitude. So we might conclude that any E, 'added to itself an infinite number of times,' would yield a sum surpas- sing any finite 8. So, according to this interpretation, the cogency of criticism (i), generously interpreted, relies on Archimedes' axiom. E. Did Chrysippus accept Archimedes' axiom ? I very much doubt whether this question can be apodictically answered one way or the other. The point of the argumentation in Part II of this paper has been to suggest that the hypothesis that he did not accept the axiom for a class of 'divisible infinitesimals' is not an absurd conjecture. In fact, the hypothesis does, I think, enable us to make sense of a number of reports of Chrysippean doctrine, reports for which it is difficult to find a different, unified explanation. Perhaps some small additional support for the 'divis- ible infinitesimal' hypothesis can be drived from the similarity of ter- minology between Plutarch's reports of Chrysippus' strategem for dealing with the 'constitution' of the pyramid and cone and the statement of his 'axiom' by Archimedes, who was an almost exact contemporary of Chrysippus.41

'

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Plutarch, comm not 1079d:

Archimedes, De Sphaera et Cylindro, I. Post. V:

F. Conclusion I believe that it is, then, a real possibility that Chrysippus postulated the existence of divisible infinitesimals. If he did, however, the status of these infinitesimals remains rather obscure. I have thus far adopted a rather 'ontological' conception of these quantities. In particular, in order for Chrysippus successfully to avoid several criticisms of his doctrines of time, space, and motion, it must be the case that Archimedes' axiom does not hold for these quantities.

It is conceivable, however, that the basis of Chrysippus' apparent dis- tinction between

'being greater but not exceeding' and 'being greater and exceeding' was epistemological. One quantity is greater than and exceeds another if there is a discriminable difference between them. Suppose, however, that there is not a discriminable difference between two quant- ities. Given Chrysippus' doctrine of infinite divisibility, we probably should be judging rashly were we to decide that the two quantities are precisely equal (assuming that we can give some legitimate sense to 'precise equality'). Hence, the idea of an 'indiscriminable difference' between two quantities. It is far from clear, however, that it is possible to 'map' such indiscriminable differences onto divisible infinitesimals of the sort found amongst Robinson's hyperreals. Perhaps the key question is whether it is sensible to deny Archimedes' axiom for 'in discriminable differences': is it, in other words, sensible to maintain that any finite sum of indiscriminable differences is itself an indiscriminable difference?43 If Chrysippus' dis- tinction between being greater but not exceeding and being greater and exceeding, his distinction between not being either equal or unequal and being both not equal and unequal, etc. is to be identified with d distinction between indiscriminable and discriminable differences and the answer to

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the preceding question is 'no,' then much of the criticism of Chrysippean doctrines concerning time, space, and motion by Plutarch, Sextus, and Alexander seems warranted.

Arizona State University

NOTES

1 Jonathan Lear, 'A Note on Zeno's Arrow,' Phronesis 26 ( 1 98 1 ), pp. 91-104. 2Ibid., p. 100. 3 Ibid., p. 99. 4Ibid., p. 95. 5 Ibid., p. 103, note 4. s Ibid., p. 98. ' See, for example, Abraham Robinson, 'The Metaphysics of the Calculus,' in The Philosophy of Mathematics, ed. Jaakko Hintikka (Oxford, 1969), pp. 153-163. 8 Ibid., p. 162. 9 Robinson's principal comprehensive work on non-standard analysis is Abraham Robinson, Non-Standard Analysis, 2nd edition (Amsterdam, 1974). A number of his papers on the topic have been collected in the second volume of his 'selected papers': Selected Papers of Abraham Robinson: Volume 2 Nonstandard Analysis and Philosophy, ed. W. A. J. Luxemburg and S. Komer (New Haven and London, 1979). 10 H. Jerome Keisler, Elementary Calculus (Boston, 1976). This text demonstrates the pedagogical feasibility of Robinson's non-standard grounding of calculus. " Robinson, 'The Metaphysics of the Calculus,' p. 155. 12 Zero is, in Robinson's usage, a 'trivial' infinitesimal. 13 Plutarch, comm not 1070e.

The term 'Tpfip' seems to eventually acquire the technical sense of 'segment' of a figure, in particular, segment of a circle. But in connection with the circle, it apparently was first applied to lunes and sectors of the circle; hence, I suppose, the connection with its etymological sense, which I use in the translation of the term in this passage. See Thomas Heath, A History of Greek Mathematics, Vol. I (Oxford, 1921), pp. 184, 187-189.

Plutarch, comm not 1079e. The translation here is dependent on the Loeb translation of H. Cherniss, Plutarch's Moralia, Vol. 13, Part 2 (Cambridge and London, 1976), p. 821. Cherniss' Loeb edition possesses many valuable notes, although, as will become apparent, I cannot accept all his conclusions concerning the argumentation in this very complex work. ls The picture of a stack of an infinite number of cylinders, each with an infinitesimal height is, in one respect, inaccurate: apparently, not every cylinder in such a 'stack' can have a unique cylinder 'sitting on top of it.'

The assumption here is that each cylinder would possess a height h such that but h = 0. If h = 0, then the 'cone' would turn out to be the circle that is its base. Furthermore, it seems arbitrary to identify the cone with one infinite 'stack' of cylinders each of which is of some infinitesimal height h rather than another 'stack' containing a different infinite number of cylinders each of a different infinitesimal height h'. Perhaps the cone could be

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identified, indifferently, with any such infinite stack of cylinders the 'summation' s of whose volumes is such that s r, where r is taken to be the 'real number' value of the volume of the cone. See Kreisel, Elementary Calculus, p. 235, Theorem 3: 'Given a function f continuous on [a,b] and two positive infinitesimals dx and du, then the definite integrals with respect to dx and du are the same.' (The definite integral of a function f continuous from a to b with respect to b

dx is defined a as the standard part of the infinite

Reimann sum with respect to dx, i.e., ffix)dx = st a b 18 See comm not 1080c. There are problems with the interpretation of the last part of this

passage, discussed by Cherniss. He claims that Plutarch 'misinterprets the first example to mean that they [i.e., Chrysippus and disciples] denied the equivalence of OX vaa and offboa and the second to mean that they denied the equivalence of'L

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in a minimal time t, and all spatial intervals are multiples of d, all temporal intervals multiples of t, it seems that if a body traverses a distance D (= nm 'without stopping', it will do so in a time interval T=nt. But then, the speed DlT = ndlnt = dlt. But dlt was assumed to be constant for all moving bodies. By analogous reasoning, if all bodies traverse some fixed finite distance d 'all at once' (i.e., in some fixed time t), evidently all bodies must move at the speed dlt, if t is some finite interval of time not equal to zero. If t=0, then it appears that all bodies move at the speed d/0, whatever that 'non-ratio' here might signify (the 'literally instantaneous' traversal of any finite distance?). PH 3.80.

Abraham Robinson, 'Function Theory on Some Nonarchimedean Fields', in Selected Papers ofAbraham Robinson: Volume 2 Nonstandard Analysis and Philosophy, p. 87. Cf. the distinction between the 'additive' and 'multiplicative' versions in 'The Metaphysics of the Calculus,' p. 158: a 'multiplicative' version of the axiom holds for 'hyperreals': for all hyperreals a, b, if 0 < a < b, then there exists an n e *N such that n . a > b. *N is the hyperreal 'analogue' of the property of being a natural number in real analysis. However, unlike N, *N may contain infinitely large numbers. It is for this reason that the multi- plicative version of the axiom holds for the hyperreals.

Actually, forms of 'Archimedes' axiom' seem to antedate Archimedes and are sometimes referred to as 'Eudoxus' postulate or axiom' (cf. Euclid V, Def. 4). The actual statement of his axiom by Archimedes in De sphaera et cylindro I, Postulatum V (quoted below in Part II, Section E of this paper) has the following import:

for all magnitudes a, b, if0 < a < b, then for any magnitude c 'homogeneous with a and b,' there exists a natural number bn such that (b-a) + (b-a) + ... + (b-a) > c.

n times With respect to this postulate, see E. J. Dijksterhuis, Archimedes (New York, 1957), pp. 146-149. Dijksterhuis concludes his discussion with the comment that, 'to express it in modern terms, he [Archimedes] excludes the existence of actual infinitesimals; the mag- nitudes he is going to discuss are to form Eudoxian systems' (p. 149). zs comm not 1081c.

See Physics 4.13.222a20ff. G. E. L. Owen, 'Aristotle on Time,' in Motion and Time, Space and Matter, ed. P. K.

Machamer and R. G. Turnbull (Columbus, Ohio, 1976), pp. 18-19. 2 comm not 1081 f. 30 Ibid.

It seems that closely connected with the Stoic view that continuous magnitudes such as time and space are not 'constituted' of 'limit entities', such as points or planes without depth (cf. Stoicorum Veterum Fragmenta (SVF) ed. J. von Arnim, Vol. II, Frag. 482 (2.482)), is the idea that such 'entities' are 'incorporeals', i.e., that they exist in intellectu but not in rebus. According to the report of Diogenes Laertius, Posidonius - here as elsewhere a 'Stoic rebel' - dissented from this view, holding such entities to exist not only xaT' iitilvotav but also xaO' {1'ITO'TaOW (7.135). 3z comm not 1081 f. 33 SVF2.482. 34 comm not 1079b-c. 35 Sambursky, p. 94. 31 R. B. Todd, 'Chrysippus on Infinite Divisibility,' Apeiron 7 (1973), p. 22. 37 Ibid., p. 23.

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38 Alexander, mixt 222.6-14. 39 Ibid., 222.15-22. 4o R. B. Todd, Alexander of Aphrodisias on Stoic Physics (Leiden, 1976), pp. 208-209. If there are non-equal (i.e., infinitesimals, the inverses of these will be non-equal infinite 'hyperreals'. And it is necessary that there be such inverses if the hyperreals are to obey the 'inverse law': r, 1 /r = 1. 41 The dates for Chrysippus are 281/77-208/4 B.C. (cf. Diogenes Laertius, 7.184) and for Archimedes c.287-212 B.C. (cf. Dijksterhuis, pp. 9-10).

Plutarch, comm not 1070d, 1080c; Archimedes, De sphaera et cylindro I, Postulatum V (II 8.23-7 Heiberg). 43 Another question is whether there is a least discriminable difference. There is, of course, no least non-infinitesimal finite hyperreal. I am not certain what the correct answer to either of these questions is; but I am inclined to think that our initial intuitions suggest (a) that any indiscriminable difference added to itself `eventually' (i.e., with some finite number of additions) does yield a discriminable difference, and (b) that there is, perhaps relative to the 'discriminator' and the conditions of observation, a least or 'threshold' discriminable difference.