A Note on Zeno's ArrowAuthor(s): Gregory VlastosSource: Phronesis, Vol. 11, No. 1 (1966), pp. 3-18Published by: BRILLStable URL: http://www.jstor.org/stable/4181773 .Accessed: 08/01/2011 21:01

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at .http://www.jstor.org/action/showPublisher?publisherCode=bap. .

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

BRILL is collaborating with JSTOR to digitize, preserve and extend access to Phronesis.

http://www.jstor.org

A note on Zeno's Arrow.! GREGORY VLASTOS

[1] For if, he says,

[a] everything is always at rest when it is at a place equal to itself (6r=v 1f xa'r& 'r aov),2

[b] and the moving object is always [sc. at a place equal to itself]3 in the "now" (Iatv 8' acxE so yep6F,evov &v 'r7 v5v), then the arrow in motion is mo- tionless. (Aristotle, Phys. 239B5-7; for the text see Ross 657-58).

1 I shall deal only with the main topics, and with these far from exhaustively. For more thorough treatments of the subject the reader may consult works listed in the bibliography at the end of this article (to which reference is made in text and notes by the name of author only). For more extensive references to the scholarly literature see M. Untersteiner, 142 ff. Readers who may be familiar with the account of the Arrow in the chapter on Zeno I contributed to Philosophic Classics, Vol. I, edited by W. Kaufmann (Englewood Cliffs, N.J., 1961), 27ff. at 40-41, are hereby advised that the present Note is meant to supersede it completely. That chapter had been prepared on short notice to fill an urgent pedagogical need and, as I explained at the time (27, n. 1), presented "purely provisional results of work-in-progress." The present Note incorporates results I have reached in a more thorough study of the Arrow made possible by a grant for research on Zeno from the National Science Foundation, to which I wish to express my thanks. 2 I follow the usual translation of this peculiar phrase (cf. e.g. Burnet, "when it occupies a space equal to itself.") I assume that if Zeno had used this ex- pression, his expansion of it would have been xawr Tov laov kaotcu 'r6nov, since r6noq would have been the only word he is likely to have used in this connection: cf. '67rov &XXicramv in Parmenides, frag. 8, 41 (Diels-Kranz). However, Zeno is more likely to have written &v 'cp law kavr& 'r6nc for the context suggests strongly that the construction with xawr& is Aristotelian. Aristotle starts talking of a mobile being xaoX tL as far back as 239A25, using the phrase again at 30, 34, 35 and (twice) at 239B3. Uninterested in conserving the mention of 'r670oq in his summary of the puzzle (t67roq plays no role in the analysis of its reasoning presupposed by his refutation), it would be natural for him to change &v T' rfa kmuvr into xxra. 'r taov in conformity with his 6 uses of xacrci 't in the preceding 15 lines. 3 For the expansion cf. the commentators (Simplicius, Philoponus, Themistius: Lee # 30 to # 34. Lee puts the expansion into the text (he writes: &v 'rij vUV

xm'd& r6 taov), but on frail MS. authority.

3

[2] Zeno argues thus:

[a] A moving thing moves either [A] in the place in which it is or [B] in the place in which it is not.

[b] But it moves neither [A] in the place in which it is, nor [B] in the place in which it is not.

Therefore, nothing moves. (Epiphanius, Adv. Haer. 3, 11; H. Diels, Doxo- graphi Graeci, 590).

A briefer version of [2],' practically identical with its part [b], is ascribed to Zeno by Diogenes Laertius (Vitae Philos. 9, 72) in a passage in which he appears to be following a good source: all

citations in 71-73 are reliable and most of them are letter-perfect. Of the two versions this is the one likely to be the closer to the original: the longer one in Epiphanius looks like an expansion of this one to make it "compl[y] with the rules and conventions of post-Aristotelian syllogisms" (Frankel 7). In any case, the ascription of [2b] to Zeno has the backing of both Diogenes and Epiphanius, and I know of no good reason to doubt it. It is true that Sextus, reporting on three separate occasions (Pyrrh. Hyp. 2, 245 and 3, 71; Adv. Math. 10, 86-89) the use of [2] (with variations) by Diodorus Cronus,& never mentions its Zenonian authorship. But neither does he say that Diodorus was its originator; he seems to imply the very opposite, as Frankel (7, n. 20) has observed, by introducing the argument in one passage with the remark that Diodorus '6v 7rpLpopij'nx6v uvep'rq X6oyov esr s'r '

xtveLaOoE DC (A dv. Math. 10, 87; cf. Frankel 7, n. 20).6 Sextus' failure to cite Diodorus as the inventor of this argument could hardly count against its authenticity: he is as silent concerning the authorship of the other argument whose use by Diodorus he reports in the same connection7 - that if time consists of indivisibles "has moved" will be true of things of which "is moving" was never true - which we know as one of Aristotle's cleverest and most characteristic creations (Phys. 231 B 20ff.).

4 Printed as frag. 4 in Diels-Kranz, with no defense of the editorial decision. 5 Also in Pyrrh. Hyp. 2, 242, without ascription to Diodorus or to anyone else. S Just before, at 86, Sextus had given a fuller account of the reasoning by which Diodorus had supported [2bA] and [2bB]. It would be hard to believe that, when he goes on to cite the whole of [2] in the next paragraph, he should refer to it in the above terms (note specially the force of auvepc.r), if he had thought of [2] as a Diodorean invention. 7 Adv. Math. 10, 48 and 85; it is connected closely with [2b] in the latter passage.

4

Once we decide that on the evidence [2b] was Zeno's (the sense, if not the exact wording), we should face the question: Could it really have been a self-contained composition, independent of [1], as the editors have generally assumed? Suppose it were. What would we then make of Aristotle's remark that "there are four arguments by Zeno that give trouble to those who try to refute them" (239B 9-11)? On the hypothesis, [2b] would have been a fifth. Why then did Aristotle ignore it? Because he thought it a silly puzzle, not worth solving, or too easily solved? This would be hard to square with the fact that Diodorus, connoisseur of fine arguments, thought so well of it. Nor is it likely on aesthetic grounds that a construction trenching so closely on the Arrow should have appeared in Zeno's book as a separate puzzle. And there is a third, still stronger, reason for rejecting the hypothesis: As [2b] now stands it makes at [A] a challenging claim - that a thing cannot move 'V 4) ar.L rO'6rca - without putting up the slightest defense for it. To realize how badly it does need defense one need only recall that it would be quite consistent with general usage in Zeno's time (and for a long time after) to think of the r64os in which a thing is as the room or region within which it stands or moves - i.e., as its locale, rather than its precise location.8 Hence to be told out of the blue that a thing cannot move "in the place in which it is" would only provoke the retort, 'And why not? Why cannot the dog move in the kennel, the man in the courtyard, the ship in the bay?' If we take another look at [1 a] with this in mind, its aptness as an answer to just this objection leaps to the eye: Its strategy is to cut down the thing's "place" to a space fitting so tightly the thing's own dimensions (just "equal" to its own bulk) as to leave it no room in which to move. Thus [ a] locks neatly into [2b A] in point of logic, while if it belonged to a separate argument we would have to suppose that Zeno had built into the original of [2] some other backing for [2bA] - and what would that be? In three of the passagesg in which [2] turns up in Sextus the backing for it is just, "for if it is in it [sc. the place in which it is], it rests there." This would be very lame - a patent begging of the question - unless here again the equivalent of [ a] were being pre- supposed. And that this is in fact the case appears from a fourth passage in Sextus (A dv. Math. 10, 86), where the equivalent of [la]

8 Many examples of the wider usage in Liddell & Scott, s.v. r67rog. 9 Pyrrh. Hyp. 3, 71 and Adv. Math. 10, 87; also in Pyrrh. Hyp. 2, 242 (cf. n. 4 above); but not in Pyrrh. Hyp. 2, 245, where the argument is tailored to fit the joke.

5

is given as Diodorus' reason for [2bA]: "and that is why [the body] moves neither in it [i.e. the place which contains it] - for it fills this up, while it needs a larger place in which to move -, nor etc." The fact that the cited argument is loaded with the special assumptions of Diodorean physics (the body is an indivisible which, if it could move, would have to move through a space composed of indivisibles) in no way affects the point at issue, i.e. that Diodorus too, when filling out the argument in [2bA] to make it fully convincing, resorted to precisely the same reasoning as is provided, more tersely, at [1 a] in the Aristotelian sum- mary of the Arrow. This being the case, and taking also into account the first two reasons above against the hypothesis that [la] and [2bA] belonged to separate puzzles, we have good warrant for rejecting it in favor of the assumption - on which the rest of this Note will proceed - that both [1 a] and [2 b] belonged to the Zenonian original of the Arrow. Their subsequent separation in Diogenes Laertius and Epiphanius (or their sources) could be accounted for easily: in the former, a truncated version of the Zenonian original; in the latter an expansion of that fragment.

Nor would it be hard to account, on the same assumption, for Aristotle's quite different version of the Arrow. We need only suppose that he recognized in [1 a] and [1 b] the significant part of the argument, and scornfully ignored the rest: witness his drastic abbreviation of the Race Course in the same passage (239 B 11-13).10 That he should have kept [1 a] is understandable: on any reconstruction of the puzzle, this is the heart of its reasoning. As for [1 b], all of this too might well have figured in the original as the sequel to [2bA] duly backed by [1 a] (see the conjectural reconstitution of the argument below), except for its mention of the viv at the end.1' This is one of Aristotle's favorite technical terms. He used it commonly as a name for the durationless instant;12 but occasionally, in controversial contexts, he also allowed

o ? Cf. this with the considerably fuller, though still abbreviated, account of the same argument at 263 A 5-6. 11 Cf. Calogero 131-38. la How much of an Aristotelian innovation it is one can see by comparing Platonic usage. For Plato the "now" remains an interval; he uses 'z viv as short for 6 v5v Xp6voq (Parm. 152 B5). The closest he comes to Aristotle's instantaneous "now" is in something he calls the kocv(v "this queer thing situated between motion and rest, not itself in any time [Cornford, Comparing Nic. Eth. 1174B8, 'it occupies no time at all,' i.e. has no temporal stretch], while to it and from it the moving changes to a state of rest and the resting to a state of motion"

6

himself to use it to denote the atomic quantum of duration, integral multiples of which would make up all larger temporal intervals, if time were discontinuous.13 Since neither of these two uses of viv have any known precedent,'4 it would be most unsafe to assume that Zeno had anticipated one of them across a gap of a hundred years or more. Its presence here is explicable as an Aristotelian plant: by sticking it into his account of the puzzle Aristotle makes it all the easier for his readers to feel the appositeness of his refutation, which centers in the claim that "it [Zeno's argument] assumes that time is composed of 'nows': if this were not granted, the argument would not be valid."'5 If it did not have this function, it is doubtful that Aristotle would have found a place for it in his ultra-compressed account of the puzzle, even if it had figured in the Zenonian original. How expendable it is becomes apparent in a number of the summaries of the argument in the Aristotelian commentators: though undoubtedly dependent on our present Aristotelian passage, they drop the "now", probably not even aware that they have done so, and certainly not expecting that their

(Parm. 157 D6-E 3). Thus Plato's k(aqtpsvr is a limit of temporal extension, itself extensionless, and one might wonder if this, in all but the name, is the Aristo- telian "now." But there are great differences: Plato does not explicate in rigorous terms the concept he has in mind, does not elucidate it as the temporal analogue of the geometrical point (so fundamental for Aristotle's analysis of the "now" Phys. 231 B 6 ff. et passim), does not specify its formal properties (especially the crucial one, of non-consecutive succession, so clearly identified by Aristotle for the "now", 218 A 18 et passim). Aristotle is at pains to distinguish his V5V from the & xEpvij, remarking (perhaps with implied opposition to Plato's use of the term) that the latter refers to what happens in an imperceptibly small [stretch of] time. Phys. 222B15. Cf. Owen 2, 101 ff. 18 For this second usage (united with the first by the fact that the durationless "now" is also indivisible, but never confused with it by Aristotle to my know- ledge) see e.g. the argument against the thesis that time and motion (no less than spatial extension) "are composed of indivisibles," 231B18ff.: at its conclusion he represents the refuted thesis as holding that time "is composed of 'nows' which are indivisible" (232A19). 14 There is no hint of the latter in Plato (Parm. 152 Aff.; 155 D) and certainly none of the former (cf. n. 12 above). "I 239B31-33; cf. B8-9. That "time is not composed of indivisible 'nows"' (loc. cit.) is the very foundation of Aristotle's theory of the temporal continuum. He announces it (218A8) early in the second paragraph of his essay on time (Phys. IV, 217B29ff.) and reintroduces it in the first chapter of Physics VI: cf. n. 13 above.

7

readers would miss it.16 It becomes quite superfluous to the sense and to the force of the reasoning when [1 a] and [lb] (the latter with modifications) are spliced into [2b], with a transitional sentence added to smooth out the reasoning:

The arrow could not move in the place in which it is not (= [2 b B]).17

But neither could it move in the place in which it is (= [2bA]): For this is a place equal to itself [supplied]; And everything is always at rest when it is in a place equal to itself (= [1 a]). But the flying arrow is always in the place in which it is ([1 b] with modifi-

cations). Therefore, the flying arrow is always at rest.

Pending a fuller, or better grounded, utilization of our textual data, it is reasonable to assume that some such argument was the origi- nal of the Arrow.18

II

Why should Zeno have thought [1 a] true? There are two possible answers, depending on how we read its "when" (Cf. Black 128 and 144-46). Is Zeno saying (I) that everything is at rest for any temporal interval during which it is "at a

place equal to itself", or

(II) that everything is at rest for any (durationless) instant"I in which it is "at a place equal to itself" ?

16 Of the three summaries of the Arrow in Simplicius, the longer one (Phys. 1011, 19ff. = Lee # 31) includes the "nows", but the shorter ones (1015, 19ff. =

Lee # 30; 1034, 4ff. = Lee # 32) do not. Philoponus (Phys. 816, 30ff. =

Lee # 33) conserves the "now". Themistius does not in Phys. 199, 4ff. = Lee # 34 or in 200, 29ff. 7 It is entirely possible that a reason may have been given for this proposition, such as the one in the versions of [2] which Sextus ascribes to Diodorus in Pyrrh. Hyp. 3, 71 and 3, 89: "for it can neither do nor suffer anything where it is not." IL A trivially different version, sticking more closely to Aristotle's [1 b] at the price of a slight roughness in the transition from [2 bA] to [1 a], would repeat the first two premises as above and then proceed as follows:

For everything is always at rest when it is in a place equal to itself (-= [1 a]). And the flying arrow is always in a place equal to itself (= [I b] without I vu3v). Therefore, the flying arrow is always at rest.

19 Hereafter I shall always mean 'durationless instant' when I say 'instant.'

8

Could (I) be the right reading? Its linguistic plausibility is so great that the second sense of 6Tov might not even occur to one here,20 as it apparently did not to Aristotle: he seems to have taken it for granted that the 68av in [la] would refer to a stretch of time.20a But there is a strong objection to this reading: it would require us to debit Zeno with a gratuitous fallacy. For only if we know that an object is in the same place for some stretch of time, would we be entitled to infer that it is at rest during that stretch. And that is just what we do not know in the case of the arrow. To all appearance it is never in the same place in the course of its flight during any sizeable period. Is there then any reason to think that it would be in the same place during smaller periods? On the face of it, none whatever: there is no more reason a priori why the flying arrow should be in the same place for a billionth of a second, than for a whole second. So if Zeno wanted us to believe the contrary, he would have to give us his reason. Otherwise, he would be begging the very question to be proved - that the arrow, moving by hypothesis, is in fact resting - and to beg it for smaller periods would not make the offense to logic the smaller.2' Now on the present reconstruction of

20 Cf. the behaviour of Liddell & Scott. Elucidating the "when, at the time when" sense of 6'e (&8av = 6're &v), the authors say that it is used in the indicative with imperfect or aorist "to denote single events or actions in past time," with present "of a thing always happening or now going on," etc. They make no provision here or subsequently in any of their statements or examples for sense (II) above, i.e. for the use of 85e to refer not to an event or a period but to an instantaneous limit of an event or period, e.g. to the start or finish of a race. This rarer, but perfectly authentic, sense of 6Te has evidently not occurred to the original authors or later editors of the dictionary. '0a I think we may infer from the context (239 B 1-4) that the "indivisible nows" which Aristotle believes (ibid. 8-9; 31-33) are being "assumed" by Zeno's argument are atomic stretches, not extensionless instants (i.e. that he is using the v5v in the second of the two senses in which he employs the term: cf. n. 13 above). For since he says that there is neither motion nor rest in a "now" in 239B1-4, where he is using "now" in the sense of instant, the assumption that time consists of instants would have warranted in his view the conclusion that the flying arrow is neither moving nor resting. But he says (ibid. 30-32) that the assumption warrants the conclusion that the arrow is resting. So unless he is being very careless, he must be thinking of the "nows" of the supposed assump- tion not as instants, but as atomic durations. 21 I am not implying that the hypothesis is that the arrow is moving over every interval, however small. It would be quite legitimate to hold that the hypothesis is, as such, non-committal as to motion over micro-intervals and does not exclude a priori the possibility that motion might be discontinuous after all. But to assert (I) Zeno would have to go much further than profess agnosticism as to

9

Zeno's argument there is absolutely nothing - not one word - to say, imply, or even suggest, that he was offering us some reason for believing that the flying arrow stays put for tiny intervals. Hence the only possible way of bringing any such premise into the argument would be to assume that this is something which Zeno's reader already believes, so that Zeno does not have to argue for it, or even mention it: he can just take it for granted.

Under the influence of Paul Tannery (249ff.) a number of distin- guished scholars have made just this assumption. Supposing (T, i) -

"T" in honour of the father of this hypothesis - that Zeno's arguments were directed against Pythagoreans whom they supposed (T, ii) to hold a remarkable doctrine, called "number-atomism" by Cornford, these scholars have also supposed (T, iii) that Zeno's opponents believed that time, no less than matter and empty space, was made up of indivisible quanta. Elsewhere22 I have argued against (T, i) and (T, ii). This is not the time to resume that argument. But it might be just as well to remind the reader that that protest (preceded by such fundamental work as that of Heidel, 21ff., Calogero 115ff. and van der Waerden 151ff.) has been sustained in several later contri- butions (including Owen 1, 211ff.; Booth 90ff.; Burkert 37ff., 264ff.; Untersteiner 197ff.) However, (T, i) and (T, ii) were at least put forward on the basis of presumptive textual evidence. In this respect they are in a totally different category from (T, iii), for which not a single item of positive textual evidence has ever been offered. Aristotle's last-cited remark does not constitute such evidence. For neither here (i.e. 239B8-9 and 31-33) nor anywhere else does Aristotle say, or even suggest, that this or any other argument of Zeno's was directed against Pythagorean philosophers.23 Nor does Aristotle tell us here that Zeno said (or claimed, maintained, etc.) that time is composed of "in- divisible nows." His remark can be read perfectly well as only tracking down the assumption which, in Aristotle's own judgment, was logically entailed by Zeno's argument, and would have to be added to its premises, to validate the conclusion.24 If I were to say, 'In arguing that

motion or rest over micro-intervals; he would have to assert rest; and this would be begging the question. 22Gnomon 25 (1953), 29-35 at 31ff.; Philos. Review 68 (1959), 531-35 at 532ff. 23 Nor is any such thing said, or suggested, by Eudemus or by any other ancient authority. 84 This, I trust, is all Lee (78) means in saying that Aristotle here "points out... the necessary presupposition of the argument."

10

P entails Q, you are assuming the truth of R,' I would not be implying, and might not even wish to suggest, that you already believe R. All I could be, strictly, understood to imply is that, since 'P entails Q' entails R, you cannot maintain the former unless you are also prepared to stomach the latter. Therefore, if one wished to cite Aristotle as a witness of Zeno's profession of the discontinuity of time, one would have to produce other evidence tending to show that Aristotle wished us to understand his remark as ascribing such a doctrine to Zeno. Such evidence does not exist. That the quantization of time was espoused by Zeno himself or by his Pythagorean contemporaries thus remains a pure conjecture,25 and a most implausible one: so abstruse a speculation as the replacement of the temporal continuum by an atomic conception of temporal passage could not have been seriously entertained, let alone professed, until well after the much more concrete hypothesis of the atomic constitution of matter had become thoroughly assimilated by the philosophical imagination, i.e. well after Zeno's time.26 So we can be confident that if Zeno had expected his readers to concur with (I), he could not have presumed on their doing so because of their antecedent philosophical commitments; he would have had to produce an argument for (I), if that is what he was asserting by means of [la]. Since there is no trace of such an argument, we have good reason to discount this first reading of the "when."

(II), on the other hand, allows a viable, and very simple, explanation of the fact that Zeno thought [1 a] true, and so plainly true that he felt no need to argue the point or even so much as mention it as his reason for [ a]; If we think of the arrow as occupying a given position for a time of zero duration, it will be obvious enough that it cannot be moving just then: it will have no time in which to move.27 To derive [la] we will then require only the following additional premise:

H. If the arrow is not moving when it is "at a place equal to itself," it must be at rest at that place.

26 For the "suggestion" that this was "a Pythagorean formulation, arising out of their point-atom theory" see Lee 105-06. 26 For this reason among others I would reject the interpretation of the fourth Zenonian paradox of motion as an argument against temporal indivisibles. Cf. p. 43, n. 31, of my essay cited in n. 1 above; and note that this interpretation of the paradox is unheard of in antiquity, and goes flatly against the one which all of our ancient authorities take to be the obvious sense of the paradox. 27 Cf. Black 133-34, the first four paragraphs of his exposition of what he calls "a modem version of the paradox."

11

This hypothetical is the crucial tacit premise of the puzzle. If this were granted, [1 a] would certainly follow, and the success of the whole argument would be assured28. How then would H have struck Zeno? Would it have looked to him a hazardous inference in need of argu- mentative support? I shall try to convince the reader - if he does need to be convinced - that, on the contrary, it would have seemed to him trivially true.

Let me begin by pointing out that even today most people (perhaps even some readers of this journal!) would think it so. Here is one example from a distinguished modern philosopher: "If a flying arrow occupies at each point of time a determinate point of space, its motion becomes nothing but a sum of rests" (James 157) - presumably because occupying a point of space at a point of time entails the antecedent of H and therefore its consequent. Evidently it has not occurred to James that H involves a substantial inference - and an invalid one.29 Yet James had at his disposal tools of analysis by means of which he could easily have satisfied himself that H, so far from being tautologously true, is certainly false. Its antecedent is indeed true: the arrow does not move when (i.e. in the instant of zero duration at which) it occupies a space equal to its own bulk. But its consequent is false (in the broader sense in which "false" covers senseless statements no less than significant falsehoods): to say that the arrow is at rest for an instant is, strictly speaking, senseless. This can be established,

S for example, by means of the familiar v formula (v, velocity; s,

distance; t, time). Since a body at rest has zero velocity and covers no distance, the values required for v and s to represent the state of rest will be zero. On the hypothesis that the body is at rest in an instant,

0 the value of t will also be zero, and we will then get v = , i.e. arithmeti-

28 It might be thought that [1 a] could be granted and the conclusion escaped by arguing that even if at rest in each instant the arrow need not be at rest during the whole of its flight. This would be a mistake. If we conceded sense and truth to the statement that the arrow is at rest in each instant of its flight, we would be admitting that the whole period of the flight can be accounted for in terms of rest, which would be only another way of saying that the arrow is at rest throughout the whole period. Cf. Owen 1, 216-17. '9 A good, clear statement of its error in Chappell 203: "What is at rest is motionless, but we cannot infer that what is motionless is at rest without the added premise that it is motionless through a period of time, or at more than one moment [= instant]: for a single moment there is neither motion nor rest."

12

cal nonsense. The only way to get the required v = 0 is to assign a value greater than zero to t, i.e. to represent the body as being at rest during some temporal interval, however short.30

In Physics VI (234A32-B7; 239A11-17) Aristotle reached an equivalent result without benefit of algebra by a conceptual analysis of the "now". He demonstrated that, since "resting consists in being in the same [place] in some [interval ofl time" (239A26), it follows that

(1) in any "now" there can be neither motion nor rest: it is only true to say (2) that [any body, whether moving or resting] is not moving while it is over against something [i.e. while having a determinate position] in a "now"; but (3) it would not be the case that [any body, whether moving or resting] could be over against a stationary body in a [period of] time: for if the latter were the case [i.e. if a body had a determinate position for some temporal interval, however small], then a moving body would be at rest (239 B 1-4).

Here at last we do get the denial of H we have been looking for: It is implied in (2) that the antecedent of H is true (the arrow would not be moving in the "now") and in (1) that its consequent is false (the arrow would not be at rest in the "now"). But this comes only at the high point of an intensive exploration of temporal concepts, begun in the Academy,31 and continued with rare diligence and penetration in the Physics. And even so, for all the brilliant advance in insight this treatise represents, it does not take Aristotle far enough to enable him to understand clearly the fact that the antecedent of H, though certainly true, is true not for physical, but semantic, reasons :32

The sense in which the arrow is not moving in any instant is vastly different from that in which the Rock of Gibraltar is not moving in any day, hour, or second. To say that the Rock is moving in some period would be merely false. To say that the arrow is moving in any instant

80 A perfectly good sense may nonetheless be given to 'instantaneous rest' as a limit: cf. the explanation of 'instantaneous velocity' in Section III below. 31 Cf. Owen 2, 92ff. 32 Some readers may think I am underestimating Aristotle's insight at this point. They may retort that since Aristotle has come to see that a body is neither moving nor resting in an instant (while knowing all too well that a body must be either moving or resting during an interval), he must have grasped the semantic difference in the two types of assertions. I am not questioning that Aristotle has some understanding of the difference. All I am claiming is that it is not extensive enough to enable him to (i) state formally the crucial point (in some such way as I proceed to do above or in some logically equivalent way) and, therefore (ii) see its full implications for questions of immediate concern to him. The second deficiency will be apparent in n. 35 below.

13

would be (strictly speaking) senseless:33 it is non-moving and non- resting in the same way in which e.g. a point is non-straight and non- curved, non-convex and non-concave - the predicates are not falsely applicable, but inapplicable. If this is not understood, the antecedent of H will itself seem fully as paradoxical as its consequent, and will provoke the question, 'But if the arrow is not moving in any given instant of its flight, when and how does it manage to move?' To answer this question one must expose the confusion lurking behind the expec- tation that, if the arrow is to move at all, it must move in the instant. One must point out that to ask, 'How can the arrow be moving during an interval when it is non-moving in every instant contained by that interval?' would be like asking, 'How can the arc be curved when none of its points are curved?': motion (or rest) apply to what happens not in individual instants but in intervals (or ordered sets of instants), as curvature is a property not of individual points but of lines and surfaces (or ordered sets of points). Only when this has been understood will one be in a position to see that the arrow's non-moving in any given instant is absolutely irrelevant to its moving or resting during some interval containing the given instant,34 and that conversely the arrow's moving (or resting) during a given interval allows no inference whatever that it is moving (or resting) in any instant contained by that interval.35

0 88 As much so as would the - result for v in the formula above, and for parallel reasons. 84 And, therefore, that the disjunction in [2a] is not exhaustive: it stops short of a third possibility which (stated with pedantic completeness) would run: "or [C] from the place, Po, in which it is at a given instant, io, to a place, Pi, in whicb it is not at io, during an interval which contains both io and the later instant, il, at which it is at pl." a6 While certainly allowing the inference that it is moving (or resting) at any instant contained by that interval. This - and the implied distinction between motion (or rest) in (or, for) an instant (which is senseless, as has just been explained) and motion (or rest) at an instant (to which sense can and must be given: cf. Section III below) - Aristotle totally failed to grasp. For positive evidence of this failure see Phys. 236 A 15ff., where he is dealing with a period of rest, CA, immediately followed by a period of change, AD, with A as the boundary instant common to the two periods. He remarks that if something were at rest throughout CA, it would follow that xxl &v 'r& A ipeLe, line 18. This inference is, of course, correct. But it appears to be in flagrant contradiction with Aristotle's formal doctrine that 'pcrLeZv is impossible &v -C vwov (234 A 32-34; and cf. the citation from 239B in the text above). To get around the apparent inconsistency Aristotle would have to understand how vastly different is the

14

We may now return to Zeno. It should now be clear that to see the falsehood of H, or even to suspect it, he would have needed to possess a clear-cut understanding of the instant/interval distinction and to bring this to bear on H in spite of the fact that it does not mention "in- stants"), thereby realizing that its evident truth for intervals is not the slightest reason why it should be true for instants. Zeno's ability to meet this condition may be gauged from the fact that, in all pro- bability, he did not even have a term for "instant" and could only get at this concept indirectly by thinking of what would happen to the time of the flight "when" the arrow was "at a place equal to itself," i.e. by thinking of duration cut to zero as the distance traversed was cut to zero. Is it surprising if in such circumstances he should have thought of the arrow's being at rest in such a time as being no more than verbally different from its not moving in it, and therefore felt as certain of the former as he was entitled to feel of the latter?36

III

We saw above what William James made of the Arrow. Here is one of Bertrand Russell's glosses on it:

Philosophers often tell us that when a body is in motion, it changes its position within the instant. To this view Zeno long ago made the fatal retort that every body always is where it is; but a retort so simple and so brief was not of the kind to which philosophers are accustomed to give weight, and they have continued down to our own day to repeat the same phrases which roused the Eleatic's destructive ardour. It was only recently that it became possible to explain motion in detail in accordance with Zeno's platitude, and in opposition to the philosophers' paradox. 2, 1582.

Here Zeno is given credit for having grasped the following truth: If time-specifications are made not in terms of temporal intervals, but in terms of instants, then it is possible to say that a moving body

sense of iv in 236A18 (where Av -T& A = "at instant A") from e.g. Iv 'r v5v Ogv 7ri9oxe... QpqLiv, 234A34 (where &v = "in" or "within" [i.e. for] a given instant.) 86 It should be evident on inspection that in the Zenonian argument as recon- structed above (end of Section I) "not moving" and "resting" are being used as logically equivalent expressions: so e.g. at the third step where the arguer says "cannot move" though his inference, if valid, would have entitled him to say "must be at rest." But I am not entering this as evidence for the above con- clusion, since the extent to which the relevant texts have preserved Zeno's original wording is unknown.

15

is at one and only one place at any given instant just as unambigu- ously as we can say this of a body which is at rest. This insight liberates the philosopher from the idea of an instantaneous state of motion37 - an idea which, though (strictly) nonsense, is not obvious nonsense. For we do say such things as 'the ship is now moving at the rate of ten miles per hour,' meaning by "now" 'at this instant,' and our dynamics cannot dispense with the notion of "instantaneous velocity." How so, if the instantaneous state of motion makes no sense? How could motion at an instant make sense, if motion in (or for) an instant does not? It was only with the greatest labour and after many false moves that it was found possible to give a satisfactory answer to this question - "to explain motion in detail in accordance with Zeno's platitude" - by showing that 'velocity at instant, i' can be understood to mean no more than the limit of average velocities over intervals approaching zero and always containing i, where "approaching zero" can be defined without covert appeal to an instantaneous state of change (or to its mathematical twin, the infinitesimal) by employing only variables quantified over intervals of finite length: the set of inter- vals containing i approaches zero if, and only if, for any preassigned, arbitrarily small, interval e, there is always a member of the set such that the difference between its length and zero is less than e.

We may be grateful to Russell for helping us see how important and

87 I.e. of a motion which is accomplished in, or within, an instant (cf. the first sentence in the citation from Russell). 38 Russell's most frequently cited gloss on Zeno - 1, 347 ff.: the historical fable and further leg-pulling of the "philosophers" - is a brilliant piece of writing and would be wholly delightful if it were not confusing for those who are not fully up to Russell's tricks (see e.g. what poor James made of it, 186, n. 1). Here Russell tells us that Zeno (and Weierstrass following unknowingly in Zeno's footsteps) have proved "that we live in an unchanging world and that the arrow, at every moment [ = instant] of its flight, is truly at rest" - this on the strength of the fact that Zeno is supposed to have proved that there is no instaneous state of change (= "Zeno's platitude" above), as though this entailed the vastly different propositions that there are instantaneous states of rest and that the moving arrow is always in one of these! For still another Russellian analysis of the Arrow see 4, 805: here Zeno's denial of motion (suppressed in the citation from 2, 1582 in the text above) is acknowledged and pegged on Zeno's assumption that there can be no motion unless there are instantaneous states of motion. Another assumption is imputed to Zeno in the still different analysis of the Arrow in 3, 179: "the plausibility of the argument seems to depend upon supposing that there are consecutive instants." There seem to be almost as many Zenos in Russell as there are Russells.

16

how true is that insight he has called, with pugnacious understatement, "Zeno's platitude." With our own analysis of the texts behind us, we need not be disturbed by the accompanying historical fable.38 If the foregoing interpretation is correct, Zeno has indeed seen that the arrow does not move in a given instant. But he could only have had a faint glimmering of what this means, else he would not have jumped to the conclusion that it must be resting at that instant and during all intervals containing that instant and that all bodies must be resting during all intervals containing all instants. Recognizing Zeno's mistake, we need not belittle his achievement. Zeno's paradox is not a bad first move in the direction of "Zeno's platitude."39

Princeton University

89 I am deeply indebted to Professor Carl B. Boyer, of the department of mathematics of Brooklyn College, for a criticism which has prompted some revisions in the penultimate paragraph of Section II and in note 35. I am also indebted to Professor Gunther Patzig for some useful suggestions.

Bibliography

Black, M. Problems of Analysis (Ithaca, 1954). Booth, N.B. "Were Zeno's arguments directed against the Pythagoreans?",

Phronesis 2 (1957), 90-103. Burkert, W. Weisheit und Wissenschaft (Nuirnberg, 1962). Calogero, G. Studi Sull' Eleatismo (Rome, 1932). Chappell, V. C., "Time and Zeno's Arrow," Journal of Philosophy 59 (1962), 197 ff. Diels, H. and Kranz, W. Fragmente der Vorsokratiker, 6th edition, Berlin, 1952. Frankel, H. " Zeno of Elea's Attacks on Plurality," American Journal of Philology

63 (1942), 1ff. and 193ff. (Transinto German in Wege und Former Fruh- griechischen Denkens [Munich, 1955], 198ff.)

Heidel, W.A. "The Pythagoreans and Greek Mathematics," American Journal of Philology 61 (1940), 1 ff.

James, William, Some Problems of Philosophy (New York, 1911). Lee, H. D.P. Zeno of Elea (Cambridge 1936). Owen, G.E.L. (1) "Zeno and the Mathematicians," Proc. Arist. Society 58

(1957-58), 199-222. Owen, G.E.L. (2) "TtO6vmL '8 Lptv6.Levx", Aristote et les ProblUmes de Mdthode

(papers presented at the Symposium Aristotelicum held at Louvain in 1960) (Louvain, 1961), 83-103.

Tannery, P. Pour l'Histoire de la Science Hellgne (Paris, 1877). Ross, W. D. Aristotle's Physics (Oxford, 1936). Russell, B. (1) Principles of Mathematics (London, 1903). - (2) "Mathematics and the Metaphysicians," in The World of Mathematics,

ed. by J. R. Newman (New York, 1956): the date of Russell's essay not given.

17

- (3) Our Knowledge of the External World (London, 1914). - (4) History of Western Philosophy (New York, 1945).

Untersteiner, M. Zenone: testimonianze e frammenti (Florence, 1963). van der Waerden, B. L. "Zenon und die Grundlagenkrise der griechischen

Mathematik," Mathematische Annalen 1172 (1940), 141 ff.

18