plato and the instant

Plato and the InstantAuthor(s): Colin Strang and K. W. MillsSource: Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 48 (1974), pp.63-79+81-96Published by: Wiley on behalf of The Aristotelian SocietyStable URL: http://www.jstor.org/stable/4106862 .

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PLATO AND THE INSTANT

Colin Strang and K. W. Mills

I--Colin Strang

I

Plato versus Cratylus

I begin with Plato's treatment of change (kinisis) at Theaetetus 182-3, where he professes to have confounded the flux theorists. Their thesis here is that everything is changing both in respect of place and of quality, all change being deemed to reduce to these two. Plato objects inter alia that if anything is changing in both these ways then no true statement can be made about it; and since, according to the thesis, everything is so changing, it would follow that no true statement could ever be made about anything at all. The argument is presented as a reductio: that is to say, we must ourselves supply the coup de grdce: 'but since there are things about which we can make true statements, there must be things that are not so changing'.

What sort of things are we meant to conclude are not so changing? Well, the whole dialogue to date has been about the physical world, so presumably it is presupposed that some statements about it are true--indeed it has been one of Socrates' main concerns to show some are true and some false. So if some are true, it follows that some physical objects are not always changing in both respects. And we can go further: if we assume, as Socrates and common sense seem to require, that concerning any physical object at any time some statement is true of it, then it would follow that at no time was any physical object changing in the impossible way demanded by the flux men.

It is not necessary to appeal to the Forms here, though of course they may be cited as another kind of thing about which true statements can be made and which are therefore not suf- fering from the fatal Heraclitean flux. (Aristotle makes a similar move, among others, against the flux men at Met. F5, Ioog9a36- 8 and K6, Io63a1o-17). Plato cannot mean to imply here that

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only the Forms are immune, for on that view no intelligible talk could occur outside mathematics and philosophy: he would be saying not simply that only Forms were knowable, but that only Forms could be spoken about-and even in his highest flights of ontological purity he never denied that there could be belief (doxa) about physical objects.

Now if, as I think, Plato never abandoned the view that physical objects were always changing, and if he never denied that there could be true opinion about them, how can he claim, in all apparent seriousness, that this very view entails an absur- dity, namely that no true thing can ever be said about any physical object ?

My solution to this paradox is not novel. But first a closer look at the argument itself:

(I) Any given thing is changing both in respect of place and of quality.

(2) If it were moving only and not also changing in quality, we could say what qualities it had as it moved, e.g. whiteness.

(3) But ex hypothesi it is not moving only, but also changing in quality, so we cannot say that it is white: for the whiteness is also changing, changing into another colour; you cannot catch it because it will not stay put; it is on the way out as you speak.

(4) Hence we cannot make any true colour statement about the thing; and in general, about any such subject any statement is as true as any other.

Now I take it that the same point could be made about a thing's location as is here made about its colour: if it is changing only in quality you can say where it is, but if it is also moving you cannot say where it is. I also take it that qualitative change is meant to cover change of kind, on the principle of Tht. 157b8-c2, where a man or a stone is treated as a collection of sensible qualities.

With one problem already on our hands, that of Plato's apparent inconsistency, we are now faced with another: to locate the teeth of the argument and identify its quarry. Against what not too obviously false flux thesis does the argument have, or appear to have, bite ? What change in a thing precludes description of it? What contrary unchangingness will



PLATO AND THE INSTANT 65

permit description of it ? There are at least two wrong answers to this set of questions, and a third right one which will resolve both our problems.

The first wrong answer is: the thing is changing so fast that you cannot get a true word in. Now one can imagine kaleido- scopic worlds of which this would hold universally. But our world just is not like that: some things in our world are sometimes like that, but most things for most of the time are not.

The second wrong, though more promising, answer is that the world is Protean or chameleon-like, which is a complaint diagnosed in the Timaeus (49). But it cannot be this Protean changeability that is said in the Theaetetus to entail the impossible consequence that nothing can be said about the physical world. In the first place, Tim. 49 nowhere claims that e.g. water is always changing into something else, only that it is always liable to and often does; nor, secondly, does it claim that about the changeable nothing can be truly said-on the contrary, it offers an ontologically healthy way of saying all that needs to be said. (Briefly, and controversially, the Tim. remedy is this: since substances cannot change, water and air and such cannot be substances; so talk that is apparently about substantial change must be construed as talk about the coming to be in space of non-substantial images or appearances.) The Tht. is not in fact here concerned with the problem of how to talk about the (merely) changeable, but with the problem of how to talk correctly about the actually changing.

If this is correct, then the question facing us is this: under what conditions would it indeed hold that the changeable does not defy description whereas the actually changing does ? Well, what thesis, familiar to Plato, about the nature of change would indeed have the consequence that a changing thing (kinoumenon) could not be described ? Evidently, the thesis that a changing thing is changing even at the instant, at every instant throughout the change. Against this thesis Plato could argue, with some plausibility, that since the changing thing has ex hypothesi no one determinate quality at the instant, but a number (perhaps an infinite number) of incompatible qualities, no one thing will be any truer of it than a number of other incompatible things. The nearest one could get to a true statement about it would be the successive denial of qualities ad infinitum (and perhaps this



66 I---COLIN STRANG

is what is meant at I83b4-5 by to 'oud'houtfs' ... apeiron legome- non, viz. " 'not that either' repeated ad infinitum"). Conversely, if the changing thing is not changing at the instant, then some one thing will be true of it at that instant.

Several considerations-apart from the difficulty of finding an alternative-suggest that it is this particular clause in the flux manifesto that Plato is here attacking. Firstly, it is a thesis he has already argued against in the Parmenides (at 152 b-c, as I hope to show), probably with Zeno's Arrow in mind. Secondly, this is just the answer an extreme Heraclitean would have to give if forced to decide between motion and non-motion at the instant. As Plato says of them: 'they are at total war with the static, determined to root it out wherever theyfind it' (I8ob2- 3). Thirdly, Aristotle offers some evidence in Met. F5 that Cratylus held just such a view. Cratylus evidently distinguished between a moderate Heraclitean thesis, which he attributes to Heraclitus himself, namely that you cannot step into the same river twice, and an extreme Heraclitean thesis of his own devis- ing, namely that you cannot step into the same river once. According to Cratylus, his own thesis does, while that of Heraclitus does not, have as a consequence that you should never say anything but only move your finger. Such extreme views, Aristotle says, were held by professed Heracliteans and were based on the belief that 'regarding that which everywhere in every respect is changing nothing can be truly affirmed'. Now whatever Heraclitus himself may have meant by saying, if he ever said it, that you cannot step into the same river twice, Cratylus evidently took him to mean that nothing was the same as itself from one moment to another; for his own dictum only justifies the finger corollary if construed as saying that nothing (moving, of course) is the same as itself even at any moment: and this adds up to saying that the changing thing is changing even at the instant. If this is correct, and Cratylus was saying this sort of thing at the time when Plato was under his influence, then Plato will have been familiar with such doctrines long before he wrote the Parmenides or Theaetetus.




II

Zeno's Arrow

Motion at the instant is the obvious Heraclitean solution to Zeno's Arrow paradox. I have already suggested that Plato has this paradox in mind when at Parm. I52b-c he denies motion at the 'now': let us see if we can uncover Plato's own solution to it.

A version of the paradox that will serve my purpose (effec- tively, Aristotle's at Physics 239b5-7) runs like this:

(PI) At any now the moving arrow is at a place equal to itself;

(P2) that which at a now is at a place equal to itself is non- moving;

(C) hence, the moving arrow is at rest throughout the duration of its motion.

The extreme Heraclitean simply denies PI, though giving emphatic assent to P2. Aristotle's solution (Phys. 239b8-9) was to say that the conclusion did not follow without the support of a further premiss representing one of Zeno's hidden presupposi- tions, namely

(P3) a duration is a sum of indivisible nows, but that this additional premiss (construing indivisible nows as durationless instants) was simply false.

Now I am not so much concerned with what Zeno himself meant by his argument as with the challenge later thinkers read into it. Vlastos ('A Note on Zeno's Arrow', Phronesis XI. I,pp. I I-15) may well be right in thinking that Zeno argued fallaciously from 'non-moving' to 'at rest', failing to see that at a now the arrow is neither moving nor at rest since it takes time to stay put as well as to move. But even if we correct this, to us, simple error by putting 'non-moving' for 'at rest' in the conclusion, the argument can still be troubling, especially for those who remain dissatisfied with Aristotle's solution (namely, that since a duration is not a sum of durationless instants, motion over a duration is quite compatible with non-motion at every instant within it). For granted that the arrow is merely non-moving, and not resting, at each now, one may still be puzzled as to how it ever gets any moving done. One may, of



68 I-COLIN STRANG

course, not be puzzled, and many are not: they are content to think of motion as simply constituted by the fact of being at different places at different times. Aristotle was one such, and Russell was another. But Plato, on my reading of him, was not so easily satisfied: 'it cannot make a move without making a move', to adapt a crucial dictum of his from Parm. I56c7, where he says oude min metaballei aneu tou metaballein ('and it certainly does not make a transition without making a transition' [my italics]).

Accordingly, I want to argue that for Plato the conclusion did not follow even with the addition of P3; for if the indivisible nows are taken to be atomic durations and not durationless instants, then P3 is true but motion is still possible. You may even, if you wish, say that for each atomic duration the thing is at rest (though it might be preferable to say 'non-moving'); but whether or not the thing moves in the course of, say, a hundred atomic durations depends on whether or not it is at rest (or non-moving) in the same place for each of them: if at different places for each of them, then certainly in motion over the composite duration. So when, and how (comes the insistent question) does the moving get done? But now there is an answer: not in any time, but between the atomic nows.

To attribute a view of this kind to Plato requires some temer- ity, seeing that despite Aristotle's attribution to him of some theory about atomic lines no one has yet been tempted to see any echo of this in the Parmenides passages I now want to exam- ine. Translations of both of them are offered in the Appendix.

III

The Anatomy of Becoming: Parmenides 15ie3-I52e3

The ostensible thesis of this passage is that the One both becomes and is both older and younger than itself; but when stripped of its schematic uniform it is seen to be about change or process or becoming (genesis) in general, as Plato takes the trouble to tell us when at 152c6-d2 he announces his conclusion simply in terms of 'anything that is becoming' (pin to gignomenon) and then repeats it in terms of 'the One in the course of becoming older'.




That it becomes older is quickly established: I52a3 The time it partakes of is travelling time; so if it

keeps pace with time it is always becoming older than itself.

Time is evidently thought of here as a pace-setter: if and only if a thing ceases to exist does it fall behind and so stop growing older. The temporal item that does the travelling I take to be the present, or the now, an item conceived of as always being one and the same, travelling from past to future but nevertheless always remaining between them, as the reference to 'the now time which is between the was and the will be' (b3-4) shows. In this sense the now is always the same. But in another sense it is always different, for the path travelled by the one moving now comprehends a multiplicity of stationary nows with which the moving now from time to time, or from now to now, coincides. In this sense the now is always different. Only some such distinction as this, akin to Aristotle's in Physics IV. I I, between the travelling now that is always the same and the stationary nows that are always different, will save Plato from inconsistency. For he says at d8 that 'the now is always present to the One throughout its existence; for whenever it exists it always exists now'; yet elsewhere he speaks of the One as encountering the now (cI) and being caught by the now (c2-3)-but one can hardly encounter what is always with one. What is always with one, or with the One, is the travelling now with which it keeps pace; what it encounters are the stationary nows standing shoulder to shoulder along the temporal highway.

A thing's being older, as distinct from its becoming older, is a more difficult notion to put across. Plato says:

152b2 It is older when, in the course of becoming (older), it is level with the now time which is between the was and the will be.

I take this to mean: at any now the thing is older when, in the course of travelling along the nows in company with the moving now, it comes level with or encounters that now. That is, the now time referred to is whichever of the many nows is being encountered by the thing and the moving now. To be so encountered by the moving now and its retinue of existents is the one great moment, indeed the only moment, in the life of a stationary



70 I-COLIN STRANG

now: for then and only then is it, as the moving now always is, between past and future.

Plato goes on: b4 For of course it will not, as it travels from the once to

the after, skip the now (hyperbisetai to nun). (For hyperbainein in this sense see Rep. 528d9; in the repeat at c7 the word used is parelthein.) The point being made here is straightforward: since the thing keeps pace with the moving now, and the moving now coincides in turn with each of the many nows, the thing cannot skip any of them.

b6 When it encounters the now it then desists from

becoming older; and it then is already older and is not becoming older.

The essential point here is that becoming (genesis) does not occur at the now; but of course to desist from becoming (at the now) is not to pause awhile. What the thing enjoys at the now is an absence of both motion and rest: for clearly a thing which keeps pace with time never stops growing older, any more than time itself stops.

The next two sentences explain why the thing cannot be in process at the now:

c2 For if it were (sc. when it encounters the now) moving onward (profon), it would never be caught by the now.

c3 For that which is moving onward is such as to be in contact with both the now and the after, letting go of the now and catching hold of the after, becoming (or, being in process, gignomenon) between the two, the now and the after.

The last sentence is crucial to my thesis. It is Plato's very precise account of what it is for a thing to be profon moving onward. What he is denying is not that a thing can move but that it can move onward at the now. When it is moving onward it neither is at the now nor is at the after, nor is it moving onward either at the now or at the after, nor is it between the two: it is moving onward, is in process, between the two.

Plato then generalises the conclusion of b6: c6 But if it is necessary that anything that is becoming

should not bypass the now, then when it is level with




it (the now) it always desists from becoming and then is whatever it may have been becoming.

Since at the now it is not becoming (gignomenon) but is whatever it was becoming, we can say, pace the flux men, what it is, where it is and what it is like: it is thus and not otherwise. The passage ends with the sentence already quoted:

d8 Furthermore the now is always present to the One throughout its existence: for whenever it exists it always exists now.

And this conveniently ensures that about a changing thing something true can always be said. So much for the Heracliteans.

And so much for Zeno: when Zeno tries to argue from a thing's always being something (somewhere) to the impossibility of its becoming (moving), Plato shows just how a thing can, throughout a stretch of becoming, be something, so by implica- tion defeating the paradox. To put it another way: the proposition that whenever it exists it exists now, together with the proposition that at the now it is not in process, might be thought to threaten the Zenonian conclusion that throughout its existence it never changes; but Plato, if I am right, shows that the process (genesis) takes place, gets done, between the (atomic) nows or, more strictly, between the now and the after. (This case of betweenness must be sharply distinguished from that of the 'now time' that is said at b3-4 to be between the was and the will be.)

As we have seen, Plato's account of what it is for a thing to be in process requires (I) that there should be some temporal item between the now and the after, (2) that this item should be in contact with both and (3) that the thing should, when in this intermediate position but not elsewhere, be in process. The temporal item in question, which in my first passage is only embryonic, is in my second born, named and exhibited.

IV

The Birth of the Instant: Parmenides I55e4-157b5

This second passage I shall treat more briefly. Its topic is instantaneous change, for which Plato here reserves the term metaboli, and which I shall call 'switching'. A switch is what



72 I-COLIN STRANG

occurs between a state (stasis) and a process (kinisis), and vice versa. Plato is not concerned with processes themselves, nor with the condition of a thing at a now during a process, nor with how it gets its moving done: these were the topics of my last passage.

i56aI-b8 establishes the existence of pairs of contrary states

and pairs of contrary processes between them, called in I57a2 staseis and kiniseis respectively. The rest of the argument I present in synoptic form, repetitions omitted. Metaballein is translated as 'switch' throughout. c3 (I) The proposition that the thing was first at rest and

then in motion entails that it switched.

cI (2) However, there is no time for the thing to switch in, since

c6 (a) at all times it is either in motion or at rest; and c9, e6 (b) when it switches it is neither in motion nor at

rest nor (therefore) in any time, since d4 (c) when switching from one condition to the other

it is not still in the one (nor is it yet in the other). C7 (3) Nevertheless it certainly does not switch without

switching [or perhaps: does not make a switch without making a switch].

c8 (4) When, then, does it switch? dI, e5 5) (a) It switches in the sudden (cf. en hoi, d2); dI, d6 (b) the sudden is a weird sort of thing (atopon, dI,

d7); d6 (c) it squats between motion and rest; d6 (d) it is not in any time; d3, d6 (e) from it and to it the moving thing switches to

rest, and the resting thing to motion. Cornford's reading of the last sentence seems to me right: 'This means that if a thing passes, say, from motion to rest, it is in motion up to (eis) the moment of transition, and at restfrom (ek) that moment. This is substituted for the description above (d2) of the instant as the time at which the transition occurs' (Plato and Parmenides, p. 20I, n. I).

Plato then applies these findings to all the cases he has listed in a4-ci. At e7 he spells out one case in detail: being and not being are two states (staseis), while coming to be and passing away are the two processes (kiniseis) between them; a switch




(metaboli) occurs at the juncture between a state and a process. In I57a4 ff. the other cases are presented, somewhat ellip- tically, as changes between two states; but the word there used for change is, quite properly, ienai (go) and not metaballein: what he is referring to, in this elliptical way, are the switches that occur at the terminal points of a process between two states.

V

The Atomic Now

Both passages are offered by Plato as serious analyses. Neither has any proprietary connection with Unity or the One, the nominal topic of the dialogue; allthatis said applies indifferently to anything capable of change. The first passage is inserted into the schema of the dialogue with minimal and transparent formality, and it says nothing of importance that is proprietary to the species of change it uses as an example, namely growing older. The second passage is outside the schema altogether. If both are seriously meant, then it is reasonable to take them as jointly offering a consistent picture of time and change. In any case, the canons of interpretation require that this be assumed until the attempt to read a consistent story into them has failed.

The 'sudden' of the second passage is clearly what we mean by a durationless instant. It is contrasted with chronos, which is equally clearly time qua duration. Then to say that the sudden is not in any time is to say that it is no part of any duration. The now is nowhere mentioned in this second passage, though the first passage operates extensively with both time (chronos) and the now, once using the expression 'the now time'. So what is a now? My account of the first passage has to some extent already anticipated my answer, so I must now justify it.

There are two things which strongly suggest that a now is a duration, and a third which proves that it is an atomic one. The first is the use of 'the now time', just cited. The second is the fact that the now is treated as an old familiar, whereas the sudden is exhibited, after a laborious birth, as a weird novelty. The third, and most telling, clue is the mention in the first passage (i52c3) of a temporal item between the now and the



74 I-COLIN STRANG

after, an item extracted and glanced at in Section III and now to be examined more closely.

This item, which I shall call X, could not, on any account of the now, be a duration: for any duration coming after a now of whatever kind would necessarily be part of the after, whereas X is said to be between the now and the after. So if X is not a duration but a durationless instant, the now cannot itself also be a durationless instant; for the durationless instant X would then have to be between a durationless now and the period it limits, namely the after: which is impossible.

If this is correct, then the now is a duration; and it is then not difficult to see that it must be an atomic one. For if it is divisible it must be able, contrary to the hypothesis, to contain motion: for if motion is possible but cannot occur within an atomic duration, then it must occur within a divisible duration. Hence, unless some arbitrary veto is imposed, it must be able to occur in any divisible duration. But ex hypothesi it cannot occur at the now. Ergo the now is indivisible.

If this bit of argumentation is thought to be more precise than the text will bear (though I do not think it is), consider a different approach. The interpretation I am urging can be represented graphically in a way that fits the text like a glove. Think of a graph whose horizontal axis represents discrete time and whose vertical axis represents discrete space. Motion will be represented on it as a step-like progression. The successive treads of the stairway will be successive atomic durations, and the risers successive atomic places. A square on the graph will represent an atomic object at an atomic place at an atomic time (though it will be convenient to represent an atomic object on the graph by a thick line along the base of a square, as if it were the edge of the object).

Now consider two successive treads and the riser between them: the riser will represent the boundary between two atomic durations and also the transition or switch between two atomic places. Mark the riser with an arrow pointing upwards, indi- cating direction of motion. Then the posture of the arrowed riser will fully satisfy the description Plato gives (I) of an object at the temporal item (X) between the now and the after and (2) of the sudden. For (I) the object in question is, as I52c3-6 describes it, in contact with both the now and the after, letting




go of the now and catching hold of the after and in process between the two. And (2) if we take the first tread to be the last atomic duration of a period of rest, and the second tread to be the first of a period of motion, then the riser between them does indeed squat between motion and rest and (assuming Cornford's interpretation) the resting thing does indeed switch to motion 'from it and to it' (I56d6-e3).

VI

Physics versus Geometry

If this is correct, then Plato will be one of those who, according to Aristotle, gave in to Zeno's dichotomy (Phys. 187ai-3; cf. De Lin. Insec. 968a18-23) by postulating indivisible lines to account for the possibility of motion in physical space. Now it would be good Platonic doctrine to maintain that a perfect circle can, but motion cannot, exist in geometrical space (which is continuous), whereas motion can, but a perfect circle cannot, exist in physical space (which is discrete). The author of the Seventh Letter (if not Plato himself, then a good Platonist) says of physical circles that they are full of the contrary of cir- cularity: tou gar eutheos ephaptetai pantii (343a7), which I take to be making the same point as Protagoras made against the geometers (Met. 997b35-a4), namely that a circular thing always makes extended contact with a straight-edge (cf. Plut. Quaest. Plat. v. 2-4; i oo4A-C). Vlastos (p. 14) compares the twin theses that there is no motion at the instant and no curvature at the point. It would be consistent in Plato to give similar accounts of both: motion occurs at the juncture between two atomic durations, and curvature at the junction between two atomic lines (i.e. there is a discrete change of direction at the juncture, so that a finite number of them will add up to a complete revolu- tion). Plutarch (loc. cit.) drew the conclusion that for Plato the paths travelled by physical bodies were never circles proper but always straight lines.

Now Plato can hardly have been deaf to the geometer's proof of incommensurable lines, nor blind to the fact that atomic lines into incommensurable lines won't go. But a non-Platonist geometer who supposed that his theorems were precisely applicable



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to things in this world would fail to appreciate Plato's distinction between geometrical lines, squares and circles and physical ones. The author of 'On Indivisible Lines' (which I take to be the first draft of a Ph.D. thesis) never tires of instancing cases where the doctrine of atomic lines conflicts with the theorems of geometry. He is full of 'geometricians dogmas', of a kind I can imagine Plato expressing impatience with. His impatience with some geometricians' hobby horse about points is recorded by Aristotle (Met. 992a20), who says that in this connection Plato postulated indivisible lines.

What Aristotle supposed Plato to mean by this remains ob- scure. But he seems to have thought that the Platonic thesis could easily be defeated by pointing to the fact that even atomic lines must have points as termini. This fact could hardly have escaped Plato. Indeed, by my account, the termini of atomic magnitudes played a leading r61le in his picture of the spatio- temporal world. So if I am right about Plato, Aristotle has gravely misunderstood him. Yet I would prefer to believe this of Aristotle than to believe of Plato that he thought there could be atomic lines without termini. Yet again, if Plato repeatedly made his point (loc. cit.), how can Aristotle have failed to see it? Did Aristotle never voice his simple objection and get Plato's reply ? Or is he just reporting hearsay? These questions I gladly leave unanswered, both for lack of time and lack of an answer.

APPENDIX Parmenides 151e3-152e3

I51e3 The One partakes of time. And if it partakes of time it both is and becomes (i) both younger and older than itself and the others, and also (ii) neither younger nor older than itself or the others.

e6 Existence belongs to it if it is one. Now to exist is simply to partake of being along with present time, just as "was" is sharing in being along with past time and "will be" with future time. Therefore it partakes of time, if of existence.

I52a3 The time it partakes of is travelling time; so if it keeps pace with time it always becomes older than itself [and therefore also younger than itself: a5-b2].




b2 It is older when, in the course of becoming older, it is level with the now time, which is between the was and the will be. For of course it will not, as it travels from the once to the after, skip the now. When it encounters the now it then desists from becoming older; and it then is already older and is not becoming older. For if it were [sc. when it encountered the now] moving onward it would never be caught by the now. For that which is moving onward is such as to be in contact with both the now and the after, letting go of the now and catching hold of the after, becoming [or, being in process] between the two, the now and the after. But if it is necessary that anything that is becoming should not bypass the now, then when it is level with it [the now] it always desists from becoming and then is whatever it may have been becoming.

d2 And so the One, when in the course of becoming [older] it encounters the now, desists from becoming and is then older [and also younger than itself: d4-8].

d8 Furthermore the now is always present to the One throughout its existence; for whenever it exists it always exists now.

e2 Therefore the One always is and becomes older and younger than itself.

Parmenides I55e4-I57b3

155e4 If the One is as we have described it, then since it is one and many and neither one nor many and since it partakes of time, it must, qua being one, at some time partake of being and, qua not being one, at some time not partake of being. Now when it is partaking it cannot at the same time not partake, or when it is not partaking partake. So it is at different times that it partakes and does not partake; only in this way could it both partake and not partake of the same thing.

I56ai So there is also a time when it is acquiring being and when it is losing it. How else could it sometimes have and sometimes not have the same thing unless it sometimes acquired and lost it?



78 I--COLIN STRANG

a4 Now to acquire being is what we call becoming [or, coming to be], and to lose it ceasing to be [or, passing away]. So the One, in acquiring and losing being, comes to be and passes away.

bI Further to its being one and many and its coming to be and ceasing to be [sc. one and many]: when it is becoming one it is ceasing to be many, and when becoming many ceasing to be one. And while becoming one and many it must be collecting and dispersing; and when it is becoming like and unlike it must assimilating and dissimilating; and when becoming larger and smaller and equal, increasing, diminishing and equalis- ing.

ci When the moving thing comes to a stop and when the resting thing switches to moving, that of course cannot be in any time. That a thing once at rest should later be moving, or a once moving thing later be at rest without its switching is an impossible state of affairs. But there is no time in which it can be at once neither in motion nor at rest. On the other hand it does not switch without switching. Well, when does it switch? For it is not switching either while at rest or while in motion or when in time.

dI Whatever [sc. temporal item] it might be in when it switches is a puzzle. Is not this puzzling thing the sudden? For the sudden seems to mean something like: that from which a thing switches to either [i.e. motion or rest]. For it does not switch from rest while still at rest, nor from motion while still in motion; but this puzzling entity, the sudden, squats between motion and rest, and is not in any time, and to it and from it the moving thing switches to rest and the resting thing to motion.

e3 Thus the One, if it rests and moves, must switch from one to the other (otherwise it could not do both); and when it switches, it switches suddenly; and when it switches it is not in any time nor is it then in motion or at rest.

e7 The same holds for the other switches too: when it switches from being to passing away or from not being




to coming to be, it is then between certain rests and motions [i.e. between a state and a process of some kind] and it then neither is nor is not nor is coming to be nor passing away.

157a4 By the same reasoning, when it goes from one to many and from many to one [sc. when it is at the terminal points of the kinisis between these two staseis] it is neither one nor many nor dispersing nor collecting. [And similarly for like and unlike and for large, small and equal: a6-b3].



PLATO AND THE INSTANT

Colin Strang and K. W. Mills

II-K. W. Mills

In the first part of this paper I shall consider the texts which Strang has cited in support of his thesis that Plato had an atomic conception of time. In the second part I shall make some comments on the logic of Zeno's arrow-paradox.

Part I : Plato on the 'now' and the 'sudden'.

(A) Parmenides 151e-152e. As Strang indicates, the following sentences (i52b-c) are

especially important for his case: 'When it encounters the now it then desists from be-

coming older; and it then is already older and is not becoming older. For if it were (sc. when it encounters the now) moving onward it would never be caught by the now. For that which is moving onward is such as to be in contact with both the now and the after, letting go of the now and catching hold of the after, becoming (or, being in process) between the two, the now and the after.'

Strangcomments that this'account of what it is for a thing to be in process requires (I) that there should be some temporal item between the now and the after, (2) that this item should be in contact with both, and (3) that the thing should, when in this intermediate position but not elsewhen, be in process'. Strang argues, on the basis of requirement (I), that this temporal item 'could not, on any account of the now, be a duration: for any duration coming after a now of whatever kind would necessarily be part of the after' (and not between the now and the after). However, this argument takes it for granted that the after is a duration-a duration extending forwards in time from the now. Why not take the alternative view that the after

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is a durationless instant-any instant you like, however close to the now, so long as it is later than the now? It will then be necessary (not merely possible) for a duration to intervene between the now and the after and the notion of a duration's being part of the after will no longer make good sense. Again, Strang's 'most telling' argument for his view that the now itself cannot be durationless is wholly subverted if, as just suggested, we think of the after as a durationless instant, and hence of the intermediate item as a duration; for in this argument he draws his conclusion from the impossibility of a durationless item's being 'between a durationless now and the period it limits, namely the after'. It is thus not hard to interpret the text in a manner which robs Strang's argument of its force.

Strang brings two further arguments in support of the thesis that the now is a duration. He himself treats these as having less weight than the one already considered, and to me they appear quite insubstantial. The first of these lesser arguments was also used by Vlastos, who put the matter as follows ('A note on Zeno's arrow', Phronesis, XI, 1966, p. 6 n. 12): 'For Plato the now remains an interval; he uses to nun ('the now') as short for ho nun chronos ('the now time'), Parm. I52b5'. The argument depends on the contrast between this expression 'the now chronos' and the later description of the so-called sudden as 'not in any chronos' (Parm. I56d). Chronos, claims Strang, is 'clearly time qua duration. Then to say that the sudden is not in any time is to say that it is no part of any duration'. Hence (as it seems to Strang and Vlastos at any rate) to say of the now that it is a time (the now time) is to say nothing else than that it is a duration. But why not say of the durationless instant not only that it is no part of time qua duration, but also that it is a time in the sense that it has--or rather, is-a position in time? The chronos-descriptions of the now and the sudden can be seen as incompatible; but the notion that they must be thus interpreted seems to me indefensible.

The second of Strang's two lesser arguments for the thesis that the now (unlike the sudden) is not durationless, is this: 'The now is treated as an old familiar, whereas the sudden is exhibited, after a laborious birth, as a weird novelty'. But Strang also comments as follows on Parm. I55e-I57b: 'Its topic is instantaneous change, . . . which I shall call 'switching'. A




switching is what occurs between a state (stasis) and a process (kinesis) and vice versa. Plato is not concerned with processes themselves, nor with the condition of a thing at a now during a process, nor with how it gets its moving done: these were the topics of my last passage (151 e- I52e)'. So even if the nows (and not only the suddens) were durationless instants, the suddens would still be no more than a subclass of the nows-a now would belong to this subclass if, and only if, it constituted the boundary between a state and a process, or between a process and a state. The sudden qua sudden would still be distinguish- able from the now qua now; and why should not the sudden qua sudden be a weird novelty even though the now qua now were an old familiar ?

To show that, if the now is a duration, it must be an atomic duration, Strang produces the following argument:

'If it is divisible it must be able, contrary to the hypothesis, to contain motion: for if motion is possible but cannot occur within an atomic duration, then it must occur within a divisible duration. Hence, unless some arbitrary veto is imposed, it must be able to occur in any divisible duration. But ex hypothesi it cannot occur at the now. Ergo the now is indivisible.'

But why should we accept the twice-mentioned hypothesis that motion cannot occur at (be contained in) the now? Plato's reason (i 52c) is that if a thing were moving at the now itself, it would ipsofacto be already proceeding beyond the now and so, even when at the now, would already have gone past it. What Plato here claims is that the now does not have enough temporal room in it to accommodate motion: any movement, however small, takes a thing out of the now. From this it follows that that in which motion can occur must have in it more temporal room than the now does. But is there, in the durationless instant, more temporal room than in the now? Clearly not. But this creates a difficulty for Strang's contention that Plato's answer to the question 'When, and how, does the moving get done?' was 'Not in any time, but between the atomic nows'. For to answer thus is (on Strang's own view of the matter) merely another way of saying: 'At one or other of the durationless instants at which adjacent atomic durations meet'. But how can we reasonably suppose that Plato, having ruled out motion at a



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now on the ground that there is not, in the now, sufficient temporal room for it, went on to affirm that there is motion at something else (the instant) in which there is this same lack of room (or even less room, given that the instant has no duration at all)? It seems that Plato's reasoning excludes not only the possibility of a motion's being contained in an indivisible duration, but also (and this is what Strang fails to take account of) the possibility of its being contained in an indivisible non- duration. The relevant thing is the indivisibility as such.

Commenting on the relation of Plato to Xeno, Strang writes: 'The proposition that whenever a thing exists it exists now, together with the proposition that at the now it is not in process, might be thought to threaten the Zenonian conclusion that throughout its existence it never changes; but Plato, if I am right, shows that the process (genesis) takes place, gets done, between the (atomic) nows'. But Plato does not really show any such thing. On the contrary, as pointed out in my last para- graph, he uses a principle from which it follows that, if the nows are atomic durations meeting at boundaries which have no duration, change (process of any kind) can no more happen between the nows than at them. Interpreting the nows of Plato's argument as atomic durations has thus the effect of con- verting that argument, not into a method of defeating the arrow-paradox, but into a particular presentation of that paradox. If we interpret the nows as durationless instants, such that any two, however close, have between them a divisible interval, we will be entitled to say that change can occur between the nows; and the argument that the now has not enough room in it to contain any change at all, works no less well if the now is durationless that if we take Strang's view of it.

I conclude that there is nothing in this section of the Parmen- ides which conflicts with the supposition that Plato viewed time in much the same way in which Aristotle viewed it. On the other hand, the supposition that Plato held an atomic view of time brings with it the conclusion that the argument of this passage is such as to lay him open (whether he realised it or not) to Zeno's arrow-argument. It is thus more plausible to claim either that Plato had a non-atomic conception of time, or that he had not articulated any clear conception of it, than to attribute to him an atomic conception.




(B) Parmenides I55e-I57b. The most vital part of this second Parmenides passage is the

following: 'When the moving thing comes to a stop and when the

resting thing switches to moving, that of course cannot be in any time ... There is no time in which it can be at once neither in motion nor at rest... Whatever it might be in when it switches is a puzzle. Is not this puzzling thing the sudden ? For the sudden seems to mean some- thinglike: that fromwhich a thing switches to either (i.e. motion or rest). For it does not switch from rest while still at rest, nor from motion while still in motion; but this puzzling entity, the sudden, squats between motion and rest and is not in any time' (i56c-d, with omissions).

The argument here is this: (i) When in a time, a thing is either in motion or at rest. (2) When at that, which we shall call 'the sudden', at which it makes a switch, a thing is neither in motion nor at rest (it has either finished moving but not begun resting, or finished resting but not begun moving). Therefore: (3) When at the sudden, a thing is not in a time.

We must ask: Would this argument make good sense if Plato's conception of time were the one which Strang attributes to him?

Let there be four successive atomic durations: AI ,A2, A3, A4. Let there be three distinct locations: LI, L2, L3. Let there be an object O which in AI occupies LI, in A2 occupies L2, in A3 occupies L3, and in A4 is still at L3. As for the durationless boundaries between the atomic durations, let us give the name 'DI' to that between AI and A2, the name 'D2' to that between A2 and A3, and the name 'D3' to that between A3 and A4-

In the course of explaining the theory of time attributed by him to Plato, Strang writes: 'You may even, if you wish, say that for each atomic duration the thing is at rest (though it might be better to say 'nonmoving'). But whether or not the thing moves in the course of, say, a hundred atomic durations depends on whether or not it is at rest (or non-moving) in the same place for each of them: if at diferent places for each of them, then certainly in motion over the composite duration'. Pre- sumably, then, for any composite period of two atomic durations,



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we shall say that a thing is in motion over that composite period if it is not at the same place in the second constituent atom as in the first. And if the thing is at the same place in the second constituent atom as in the first, we shall say it is at rest. Thus the object, O is in motion over the period consisting of AI and A2, but at rest over the period consisting of A3 and A4. But note the following remark by Plato at Parm. I53c (a sentence omitted from the quotation at the beginning of this section): 'That a thing once at rest should later be moving, or a once moving thing later be at rest without its switching is an impossible state of affairs'. So the object O must at some point switch from its moving (in the period AI-A2) to its resting (in the period A3-A4). But when does this switch take place ? It seems that the answer can only be: At D2.

Now I have already had occasion to mention that the answer, given by Strang on Plato's behalf, to the question 'When, and how, does the moving get done?' is 'Not in any time, but between the atomic nows'. At precisely what point, then, does O do its moving in the period AI-A2 ? The answer can only be: At DI. But what about the period A2-A3 ? Here again we have a composite time over which, if we use Strang's criterion, we have to say that O moves; for O is not at the same place in A3 as in A2. When, then (on Strang's account) does this moving in A2-A3 get done? Clearly, at D2.

Strang's account thus commits Plato to holding both (a) that O makes a switch (from moving to resting) at D2 and (b) that O is in motion at D2. But if O makes a switch at D2 (and so is then at a sudden), yet none the less is in motion at D2, it will not be exceptionlessly true that, when at a sudden, O is neither in motion nor at rest. Consequently, proposition (2) of the argument at I56c-d (see my above formulation of that argument) will no longer be forthcoming. So Plato's proof that, when at the sudden, a thing is not in a time, collapses.

Again, we have already noted that O is in motion over the period A2-A3, but at rest over the period A3-A4. But the period A3-A4 as a whole is later than the period A2-A3 as a whole. So this case too, it seems, comes within the scope of the rule that it is 'an impossible state of affairs' for a once moving thing to be later at rest without its switching. But at what point can the switch from the moving in A2-A3 to the resting in




A3-A4 occur? It cannot occur at D2, for then the motion has not yet run its course. But it cannot occur at D3 either, for then the period of rest has already begun. And to say that the switch is made at A3 is to concede that in this case the thing is in a time at the sudden at which it switches (for A3 is a constituent part of time); but the argument at I56c-d is designed to show that, when at the sudden, a thing is not in a time.

Thus, if we follow Strang, we find ourselves in a seemingly insoluble dilemma. If we take the view that the last atomic duration in a composite period of motion is the immediate pre- decessor of the first atomic duration in the subsequent period of rest, we are faced with the conclusion that the sudden at which the switch occurs is (contrary to Plato's own assumption) something at which a motion occurs-the motion which finally takes the thing to the place at which it then rests. Yet if the final atom in the period of motion is one and the same with the first one in the period of rest, then there is nothing between the two periods (for they overlap); and so (again contrary to Plato's assumption) the expression 'that at which the switch is made from the one to the other' has nothing at all to refer to. I conclude that Plato's comments on the sudden have more weight as evidence against Strang's suggestions than as evidence in support of them.

In his paper 'Tithenai ta phainomena' (Aristote et les probldmes de la mithode (Symposium Aristotelicum, Louvain, 1961), pp. 83-103; reprinted in Aristotle (essays edited by J. M. E. Moravcsik), pp. 167-190) Owen puts the argument at Parm. 156c as follows: 'If there is no time in which a thing can be neither A nor not-A, neither still nor moving, it baffles us to say when it makes the change from the one to the other'. Commenting on the argument, Owen writes: 'It is generally held that Plato's purpose was to show that there can be no period of time during which a thing can be neither A nor not-A, and consequently that the change from one to the other must occur at a point of time ... (But) by the same law of excluded middle not only is there no period but there is no point of time at which a thing can be neither A nor not-A'. However, though sure that Aristotle saw that the argument can be applied as well to instants as to durations, Owen appears to be in some doubt as to whether Plato himself saw this; and the doubt is well-grounded. If Plato's premise that, when in a time, a thing is either in motion or at



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rest, is interpreted as meaning that at any instant, no less than in any period, a thing is either in motion or at rest, his conclusion (viz. that, when at the sudden, a thing is not in a time) will mean that, when at the sudden, a thing is at no instant whatever, as well as in no period; but to take the argument as proving this is to take it as proving that there is never such an occur- rence as being at the sudden-i.e. never any making a switch from moving to resting or vice versa. But Plato himself declares, at I56c, that both (a) that it is impossible for a once resting thing to be later in motion, or a once moving thing to be later at rest, without its switching, and (b) that nothing switches without making a switch. Hence, if there is never any switching at all, it cannot be that anything either rests after previously moving, or moves after previously resting. But can we really suppose that Plato believed that things once resting can never move, nor things in motion ever stop ? Surely we cannot.

I therefore accept the common view (as also does Strang) that what Plato's argument tells us about the sudden is that it is not an extended time (a duration) but an instant. Yet we should note that the various arguments in the 'dialectical exercise' of the Parmenides are exploratory rather than dogmatic: Plato appears to be more concerned here with clearing the ground for, and exciting our interest in, the elucidation of this or that concept or conceptual connexion, than with presenting 'final answers' or 'the real truth'. So it might be the case that, though some given thesis cannot rightly be said to be assumed or proved by the arguments in the text, these arguments were none the less meant to be seen as steps in the direction of that thesis. How- ever, to suggest what might be is one thing; to show that it is so, quite another. My own concern has been to show no more than that certain interpretations by Strang and by Owen are not in good harmony with what is explicit in the Parmenides.

(C) Theaetetus I8Ib-I83b. Strang claims (a) that at Theaetetus I82a ff. Plato advances

the thesis that, if things are changing both in place and in quality, all statements about them are equally true (and equally false), and (b) that we cannot make good sense of this thesis unless we take the changing referred to in its if-clause as a




changing which goes on at each instant, and not as something needing an extended time to happen in. The first claim, I think, is true if suitably interpreted. The second claim seems to me false.

The first appearance in the Theaetetus of the thesis that all things are changing or moving is at 152d-e. The thesis is introduced as part of Plato's exposition of the doctrine that to know is the same as to perceive (a doctrine which has made its own first appearance at 151d-e). But, as one cannot know what is not the case, so (if to know is nothing else than to perceive) one cannot perceive things as other than they are-as they appear to be, so they must really be. And this, Plato comments (I52a-c), is precisely how matters do work out on Protagoras' theory that each man's measure of how things are is himself. For Plato understands Protagoras to mean that the mere fact of a thing's appearing to a man to be of a certain character is sufficient to justify the man's asserting that the thing is of that character. The wind's appearing warm to you (your feeling it as warm) justifies your verdict (viz. 'The wind is a warm one'); but the same wind's appearing cold to me (my feeling it as cold) justifies my contrary verdict (viz. 'The wind is a cold one'). Where, then, shall we look for an unqualified truth (a truth which is truer than what is incompatible with it) about a thing on which opposite verdicts are thus equally justified ? Plato's answer, on Protagoras' behalf, is that there is no unqualified truth- nothing auto kath' hauto (I52d, 153e, 156e, I57a, I82b); and this is where the talk of change and motion comes in. For the answer thus proposed can be put as follows: 'The truth about things changes as we take now one man's standpoint, now another's-no matter where we suppose it to lie, it proves to be no more there than somewhere else'.

So Plato, as I interpret him, employs the flux-formula ('All things are on the move, changing, flowing') as one of his ways of saying that there is no unqualified (i.e. objective) truth (with which, if there were such, what we perceive as truth might conflict). Another way Plato has of putting the point is: We should not try to say of things what they are, but only what they come to be for this man or that in his own experience of them

(i 52d-e, I57a-b). How, then are we to think of this coming-to- be? Is it a physical process of some sort? Plato's answer, on behalf of the flux-men, is 'Yes'; for he offers as theirs the theory



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that there are certain slow motions which, when they interact, emit quicker motions which are the sine qua non both of qualities and of the perceptions of those qualities (I 53d- I54a, 156a- 157c, I82a-b. The motions of this theory must be physical motions if the talk of their quickness or slowness is to make sense. There is, of course, bound to be a certain obscurity so long as this literal use of the term 'motion' is not distinguished from its metaphorical use in talk of non-objectiveness (of not being fixed by any but conflicting criteria which, because of their conflict, give no fix). But though I believe the term to be used in both these ways in the course of Plato's first account of the flux- thesis (I52d-I57c), it cannot be said that he points to the distinction between them at that early stage.

However, I believe that the distinction is made in, and by means of, the later argument at I8Ib-I83b. I cannot here deal with every aspect of this argument; but I shall now paraphrase it in sufficient detail to show how, as it seems to me, Plato makes the distinction. My paraphrase will, I hope, also show how differently I see the argument from the way in which Strang sees it.

(I) 181b-e: 'Since the axiom of the flux-men is that nothing is fixed (objectively determined) at all, they are committed to rejecting qualitative fixity no less than spatial fixity. Their view must be that, as there is nothing which is rather here than there, or more turned in one direction than in any other, so too there is nothing which is more young than old, more soft than hard, more white than black'.

(2) I82a-d: 'It is true that, at an earlier stage, we attributed to the flux-men a story of how both whiteness and the percep- tion of whiteness could arise as the transient effects of certain interactions. But since their theory is one that rejects qualitative fixity, they cannot recognise a being-white which is not also a not-being-white; nor, indeed, a being-F which is not also a not-being-F, let F be any property you like'.

(3) I82d-I83a: 'It follows that, for the flux-men, there is no perceiving which is not a not-perceiving-they cannot, that is, make any distinction between perceiving and not-perceiving. So the message of the flux-men proves to be that the premise 'Knowing is perceiving' warrants the conclusion 'Knowing is not-perceiving'. We began by supposing that the flux-theory




would serve to buttress the proposed equation of knowing with perceiving; but the theory is now seen to have as its consequence that this account of knowledge is no more true than are other incompatible accounts'.

(4) I83a-b: 'It is, indeed, a consequence of the flux-theory that statements generally (not merely those about knowledge) are all of them as true as, and therefore as false as, their own contradictories. Nor does a route which we indicated earlier- that of saying that though there is nothing which a thing is, it may come to be such or so for this man or that-offer the flux-men any escape from this pit of nonsense. If, for example, we say of something that it becomes warm for A but not-warm for B, we can be asked: 'Is it really the case that what the thing becomes for A is warm ? And that what i t becomes for B is not-warm ? To these questions, as to others, the flux-men can answer "Yes" if, and only if, they also answer "No".'

So one of the things which, on my interpretation, emerge (and are meant to emerge) from this argument is that the motions which it is the primary concern of the flux-men to impute to things are of such a sort that no statement about a thing's physical character (and therefore no statement about its physical movings or restings) could ever be true at all. Plato leads us to see, among other things, that the doctrine of the flux-men does not, in the end, permit them to sponsor the bit of physical theory of which, in the beginning, he presented them as the sponsors.

Now if the r61le of the flux-theory in the Theaetetus is anything like what I have claimed it to be, Plato's bringing it in here has nothing at all to do with the analysis of temporal process qua temporal. Rather, he uses the language of flux as a style for presenting the metaphysical doctrine that, since there is no truth apart from each man's private truth, nothing can ever be established as the truth (this being a doctrine we shall have to accept if being known is to be no more than appearing); and Plato's response to this doctrine is to argue that, where there are none but private truths, there are no truths at all which are anymore truths than falsehoods. It is more illuminating, I think, to compare Plato's argument here with modern discussions of whether a private language is possible, than to construe it as concerned with instants and instantaneous processes. 4



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As to the theory proposed by Cratylus, it seems to me that our position on this is the same as it is on Plato's doctrine of indivisible lines-viz. that our evidence is too scanty to permit any confident opinion about it. It does, however, seem reasonable to comment that, if Plato wanted his readers to think of Cratylus (or of any other particular flux-man) in connexion with the argument at I8Ib ff., it was strange that he should choose to assert with such emphasis (I 79e- I8oc) that none of the flux-men ever says anything one can get to grips with.

I conclude that the Theaetetus contains no evidence either that Plato accepted the thesis that moving 'gets done' at instants, or that this theory was of interest to him, or that he had even so much as heard of it.

Part 2: Some remarks on the arrow-paradox.

Consider the following version of the arrow-argument: (I) No time in which a thing cannot be at more than one

place is a time in which a thing can move. (2) Every indivisible time is a time in which a thing cannot be

at more than one place. (3) No indivisible time is a time in which a thing can move.

The premises (I) and (2) have an affinity to those used in Aristotle's version of the argument at Physics VI, 239b 5-7; and the inference from (i) and (2) to (3) is clearly valid. But to stop at (3) is not to have got as far as Zeno hoped to get. For that, we should need to prove this further conclusion.

(5) No divisible time is a time in which a thing can move. For if (3) and (5) are both true, then there is no time at all in which a thing can move; and this, it seems, is the result Zeno wanted.

However, we cannot reach (5) from (3) without the help of some further premise. Let us take Strang's hint (compare his P3) and introduce the following as our further premise:

(4) Every divisible time is a sum of indivisible times But is the step from (3) and (4) to (5) a valid step? And (whether or not it is so) would Plato have thought it so?

There is an argument in the Theaetetus which has a bearing on this matter. At 201d ff. Plato sketches a theory according to which everything is either an incomposite or else a composite




consisting of incomposites. He frequently calls the incomposites 'letters', and the composites 'syllables'. The theory further affirms that, whereas composites (syllables) are definable and knowable, incomposites (letters) are indefinable and unknowable. But at 2o3c-2o5e Plato sets himself to prove that either the letters are just as knowable as the syllables, or the syllables just as unknowable as the letters. The argument is tortuous, but the following quotations (from the revised Jowett translation) will serve to display its basic skeleton:

(204e) Then as many things as have parts are made up of parts ... But all the parts are admitted to be the all, if we are to regard the entire number as the all ... (2o5a) When a thing has parts, all the parts will be a whole and an all . . . Then, as I was saying before, must not the alternative be that either the syllable is not the letters, and then the letters are not parts of the syllable, or that the syllable will be the same with the letters, and will therefore be equally known with them? ... (2o5b) But if letters are not parts of syllables, can you tell me of any other parts of syllables which are not letters ? ... (2o5d) Then is not the syllable in the same case as the elements or letters, if it has no parts and is one form ?

Of the theses in this argument, the one to notice is the claim that a sum of parts just is all its parts ('All the parts are admitted to be the all'-cf. 205d: 'All the parts are acknowledged to be the same as the whole'). Plato's meaning here, it seems, is that all those things, and only those things, are true of any whole of parts which are also true of every part of that whole. So let us express the claim as follows:

(6) If anything is a sum of parts, then it is itself an F if and only if, all its parts are Fs. Let us now add to this two further premises which, though not in Plato's text, appear unobjectionable:

(a) All the parts of a thing are Fs if, and only if, the thing is a sum of Fs.

(b) If anything is a sum of Fs, then it is a sum of parts. From (6) and (a) we can derive: (c) If anything is a sum of parts, then it is itself an F if, and only if, it is a sum of Fs. From (b) and (c) we can derive; (d) If anything is a sum of Fs, then it is



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itself an F if, and only if, it is a sum of Fs. But (d) warrants the following:

(7) If anything is a sum of Fs, then it is itself an F. We now see more clearlywhat it was for which Plato was groping when he asserted the identity of a sum (whole) of parts with the parts themselves. He was working towards the expression of a particular view about what is involved in being a part of a thing. In effect, he is laying it down that, if the alleged parts of a thing are not homogeneous with that thing, then they are not its parts.

(The examples which, at 204b-d, Plato presents as conform- ing to his whole-of-parts principle can, I think, be seen as con- forming to my (7). For 6 is a sum of numbers and itself a number; the acre (plethron) is a sum of areas and itself an area; the race-course (stadion) is a sum of distances and itself a distance. The last example-the army (stratopedon)-may appear more difficult; for an army may be thought to be a sum of soldiers, and yet is not itself a soldier. But we may also see it as a sum of battle-groups and itself a battle-group.)

The relevance of (7) to the argument in the Theaetetus is plain. Substituting 'incomposite' for the variable 'F', we obtain: If any syllable is a sum of incomposites, then it is itself an incomposite (and therefore unknowable too, if every incomposite is unknowable). It is equally plain that (7) is relevant to the second half of my version of the arrow-argument. Substituting 'indivisible time' for 'F', we obtain:

(8) If anything is a sum of indivisible times, then it is itself an indivisible time. From (4) and (8) we can further derive:

(9) Every divisible time is an indivisible time. From (9) it follows that there is no divisible time; and from this, together with (3), we can infer the Zenonian conclusion that there is no time at all in which a thing can move. We may also note that (3), (9), and (5) constitute a syllogism in Celarent, just as

(I), (2), and (3) do. It seems, then, that if Plato had accepted both (I) and (2),

he would have had no other course (had he wished to avoid the Zenonian Conclusion) than either to reject (4) or to abandon a thesis about wholes and their parts on which he relies in the Theaetetus. Is there, then, any passage in a Platonic




text which might be held to indicate that Plato had doubts as to whether this thesis was in fact correct? The answer to this question is 'Yes'.

The relevant passage is Parmenides 145b-e. Plato is there supposing the One to be a whole of parts, and he brings to bear the principle: 'The One is all its own parts,.. . and the One is also the whole' (I45c, Cornford's translation). He further claims that, though it is clearly the case that every part of the One is contained in the whole of the One, it is equally clearly not the case that the whole of the One is such as to be contained in every part of the One; and, this being so, his principle allows him to deduce that the One is, and is not, contained in the One.

In terms of my (7), the argument comes out as follows. To the theses already introduced, two more are now added:

(Io) The One is a sum of parts of (things contained in) the One.

(11) The One is not itself a part of (a thing contained in) the One. But (7), it seems, permits the introduction of:

(12) If the One is a sum of parts of (things contained in) the One, then the One is itself a part of (a thing contained in) the One. But (Io) and (12) yield the contradictory of (II).

How, then, are we to escape from this inconsistency? There is little temptation to think that (Io) or ( 11) is the culprit: no matter which particular whole of parts we take the One to be, these two theses appear quite acceptable. But if the trouble lies in (I2), its source must apparently be traced back to (7), the generalisation from which (12) was derived.

But it might be suggested that the trouble should be looked for, not in (7) as such, but in the assumption that (12) follows from (7). In making this assumption we take it for granted that 'part of (contained in) the One' is a proper substitution for the variable 'F' in (7). But perhaps we should recognise some such restriction as the following: that no expression 'G' is a proper substitution for this variable if there is anything x such that the statement 'x is a sum of Gs' is entailed by the statement 'x is a sum of parts'. Now if the One is a sum of parts at all, it can only be a sum of such parts as are parts of (contained in) the One. 'rhe expression 'part of (contained in) the One' would



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thus be disallowed as a proper substitution for the variable in (7). (And the expression 'part' would be similarly disallowed.) So if the restriction just proposed were adopted, (12) would no longer be derivable from (7) as one of the cases falling under it; and (7) would thus be rescued from the attempt to overthrow it.

Finally, a point relating to my (I). It may well be thought that, though a thing cannot move in a time in which it is at one place only, there is nothing to prevent its being in motion at such a time. For remarks bearing on this matter, see Owen ('Zeno and the mathematicians', Proc. Aristot. Soc., LVIII, 1957-8, pp. 199-222, especially p. 216 ff.) and Vlastos (op. cit., p. 11 ff.). The way is then open for making the further point that a thing moves in a divisible time so long as it is in motion at each of the indivisibles within that divisible: to suppose it must also be in motion in each indivisible, is a mistake. But if this is right, what does it matter whether or not some of the indivisibles have posi- tive duration (are atomic durations)? For any period over which a thing moves, we may say of it that it is moving at each extended indivisible (at each of those that have a non-zero duration), no less than at each instantaneous indivisible (at each of those that have a zero duration); and if only the indivisibles really are indivisibles, then (whether or not they have a non-zero duration) there will be none of them in which a thing can move. So the thesis that there actually are temporal indivisibles which have each a non-zero duration, may well be thought irrelevant to the arrow-paradox.



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