The Paradoxes of Zeno 1859723683, 9781859723685

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Faris, J. A. (1996). The paradoxes of Zeno. Aldershot (Hants., England) ; Brookfield (Vt., USA): Avebury.

©).A. Faris 1996

Contents

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior permission of the publisher. Published by Avebury Ashgate Publishing Limited Gower House Croft Road Aldershot Hants GUll 3HR England Ashgate Publishing Company Old Post Road Brookfield Vermont 05036 USA British Library Cataloguing in Publication Data Faris,). A. The paradoxes of Zeno. - (Avebury series in philosophy} l.Zeno, of Elea 2.Paradox 3.Philosophy, Ancient I.Title 182.3 Library of Congress Cataloging-in-Publication Data Library of Congress Catalog Card Number: 96-085193

ISBN 1 85972 368 3

vi

Figures Acknowledgements

vii

1 Introduction 2 The Dichotomy

1 6

3 4 5 6 7

The Achilles The Flying Arrow The atomic theory of space and time The Stadium Conclusion

26 35 49

74 108

Appendices A The paradoxes about plurality B Alternative versions of the Dichotomy C The Flying Arrow: concepts employed in the Preamble and Central Text D Vlastos' s treatment of the Arrow paradox

121 123

Bibliography Index

127 132

Printed and bound by Athenaeum Press, Ltd., Gateshead, Tyne & Wear. v

117 119

Acknowledgements

Figures

Distance for traverse Figure 2.1 Achilles, tortoise and tape Figure 3.1 A shorter than shortest? Figure 5.1 Indivisible divided? Figure5.2 Another shorter than shortest? Figure5.3 Atomic theory slope Figure 5.4 Atomic theory circle Figure 5.5 Atomic theory clock face Figure5.6 Crossing Figure5.7 Cinematographic movement Figure5.8 Inter-boundary space Figure 5.9 Initial position Figure 6.1 Final position Figure 6.2 Position t 1 (Mansfeld) Figure6.3 Sequence of positions Figure 6.4 Non-occurring position Figure 6.5 Alternative initial position Figure6.6 Three positions (Barnes) Figure 6.7 Final position (Bicknell) Figure 6.8 Space units and bodies passed Figure 6.9 Figure6.10 Successive positions Figure 6.11 Gaye figure 1 Figure 6.12 Gaye figure 2 Figure6.13 Gaye figure 3 Figure 6.14 Final position?

vi

10 28 55 55 55 57 57 57 60 62 64 78 79 82 85 85 87 88 90 94 96 97 97 98 100

My interest in Zeno was initiated by Peter Nidditch who afterwards became professor at Sheffield, where he died while still in his post. The Dichotomy in particular was one of many topics discussed between us when we were colleagues in Queen's University. I remember him gratefully and with affection. I am grateful to my wife Mary for encouragement at all times, and particularly for comment and advice in relation to the present book. Professor David Evans kindly read an earlier version, and I thank him for his pertinent and helpful comments. I am indebted to many people for useful comments and references and for help of various kinds: in particular to Professor Jonathan Gorman and Mr Robert Hall for advice concerning Chapters 3 and 5 respectively, and to Mr Michael Smallman and other members of staff of the Queen's University Library for bibliographical and general assistance. An earlier version of Chapters 2 and 3 was published in the Irish Philosaphical Journal, volume 3, 1986, and I thank the editor, Professor Bernard Cullen, for permission to make use of the ritaterial here. I am grateful to Professor Cullen also for copious help and advice that he has given more recently in connexion with publication and printing. The exacting final preparation of the copy for the camera has been expertly accomplished by Mr Ivan Strahan; my thanks go to him and to my son George who computerized the diagrams. In addition, I am indebted for useful technical assistance to Fiona McKinley, Textflow Services Ltd, and Mr Julian Strahan.

vii

1 Introduction

Zeno, the author of the paradoxes which are the subject of this book, lived in Elea, a Greek colony in southern Italy, and was a member of the Eleatic school of philosophy of which the leader was Parmenides (6th-5th century BC). The date of Zeno's birth is uncertain: some evidence suggests that it may have been 515 BC, but the prevalent view is that it was 490 BC. He is commonly referred to as Zeno of Elea, and is to be distinguished from Zeno of Cilium (in Cyprus), the founder of stoicism. A paradox is a proposition or statement that is contrary to received belief or opinion or to common sense; more concisely we may say that it is one that is seemingly absurd. Not just any proposition, however, that is seemingly absurd is normally describable as a paradox. Although a paradox is conirary to received opinion, it is nevertheless a proposition that some people at least think there is some reason to believe, or have a motive for believing. There is normally some sort of argument advanced that appears to lead to the seemingly absurd proposition as conclusion. This is how it is with the paradoxes of Zeno. In each case Zeno put forward an argument leading to a conclusion clearly contrary to received opinion or to common sense. We think of this seemingly absurd proposition as a pa1;adox, but we should probably hesitate to apply the term in the absence of the argument. Such is the basic sense of the term 'paradox': it is a seemingly absurd statement that is supported by some sort of argument. But the term may also be applied to the argument that leads to the seemingly absurd conclusion. That is the sense that people frequently have in mind when they talk about Zeno's paradoxes. They are the arguments that are propounded in support of certain seemingly absurd conclusions. Sometimes, however, what is meant is just the conclusions themselves. People talk about solving the paradoxes, and that is something that has been attempted countless times, though perhaps never successfully. As an 1

appendix to our account of the me~ing of para~ ox it may be worthwh~le t~ spell out what solving a paradox mvolves. It mvolves one of two thmgs.

either (a) showing that the argument leading to the seemingly abs~rd conclusion is fallacious (i.e. either is formally invalid or uses a false premiss), which will mean that the conclusion can safely continue to be regarded as false; or (b) showing that, though the argument is fully valid, the conclusion is to be interpreted in such a way that it is seen after all to be hue. . . Zeno propounded paradoxes about motion and plurahty, m add1bon to at least one or two about place or sensationI. The paradoxes about motwn (o.r

movement: the terms will be used interchangeably) are the best known and 1t is they that are the subject of this book. The book has two main aims. One is to establish and explain what the paradoxes were. The second 1s to attempt to solve them. With regard to the latter there is nothing that need be added here to what has been said in the previous paragraph. With regard to the former, however, some explanatory and other comments of a general kind may be made. Philosophers writing about Zeno's paradoxes a.re not always preeminently concerned about historical exactitude. Somet~es they a:e content in each case to propound a paradoxical argument that IS appropnate to the Zenonian setting (for example, one that is about bodies manoeuvrmg m. a stadium) and not glaringly at variance with the trad1bon. If the result ISm fact identical with Zeno's original paradox, so much the better. If not, well, so long as a philosophically interesting paradox has been constructed, does 1t matter very much? Without any insinuation about how that questwn should be answered the emphasis in the present book will be d1ffe:ent. We will do our best to put before the reader in each case a paradox that IS essenbally the one that Zeno had in mind. This, however, is not altogether an easy or straightforward undertaking. Let me explain briefly why that is ~o. We are very remote from Zeno of Elea. His wntmgs on mob on have not survived. Of his paradoxes about motion one (generally regarded as relatively unimportant) is known to us through references m the wnters Epiphanius (4th century AD) and Diogenes Laertius (3rd century AD), but for those that have become most famous our main, indeed it would generally be said our sole, source of information is Aristotle, who wrote about them m his book, the Physics. In the case of Zeno' s other paradoxes ';e h~ve mdeed some knowledge from sources other than Aristotle. In Plato s dmlogue the Parmenides an argument (to be referred to later) is presented in what purport~ to be a quotation from a book by Zeno2; and it has be~n argued convmcmgly that the 5th century AD writer Proclus in commentmg on th1s passage had access to Zeno' s writings. Simplicius of Cilicia who wrote a commentary on the Physics in the 6th century AD also claimed to have access to a book by Zeno , and his quotations therefrom are also a source independent of Aristotle. These quotations, however, and the passage m the Parmemdes and 2

Pro~lus's comments, all relate only to the arguments about plurality4; for the matn arguments about motion Aristotle is effectively our only source.

Accounts of these arguments , and discussions of them, are indeed to be fo~d m three comparatively ancient writers (who may be referred to as 'the anctent commentators'): Simplicius himself, Themistius who lived in the 4th

century AD and wrote a paraphrase of the Physics, and Johannes Philoponus who was roughly contemporary with Simplicius and wrote a commentary; but these accounts are all fundamentally derivative from Aristotle. It will be seen later that there are at least two cases in which commentators have found

reason for .suggesting that Aristotle's version does not adequately represent the Zenoman ongmal; however that may be, in general the nearest we can get to determining what the Zenonian original was is by determining what Aristotle thought it was. We are remote from Aristotle also, as indeed from all classical authors. The manuscripts of the Physics upon which modem texts are based are copies wh1ch are beheved to have been made at various times from the tenth century to the fourteenth, and probably no one has any idea how many mtervenmg stages, that IS to say how many copyings, each with its own quota ?f misunderstandings, misreadings, haplography, dittography, lapsus calamt

a~d

other errors, have occurred in the transmission of any one

manuscnpt from ~ original in the fourth century BC. Despite all hazards a very h1gh proportwn of the text of the Physics is regarded as established beyond reasonable doubt. Inevitably, however, there are cruces here and there, and unfortunately some of these occur in the parts of the Physics on wh1ch our knowledge of Zeno' s paradoxes about motion is based. In addition to ~uch textual problems there are of course ordinary problems of translatiOn such as are liable to arise in the interpretation of any passage m a fore1gn lan.guage. In pursuance of our policy of getting as close as po~s1ble to Zeno 1t wdl be necessary to look from time to time at linguistic quesbons of both kmds that have been encountered by editors of Aristotle's text and other scholars. Our aim will be to explain and discuss these questions in a way that will be intelligible and interesting to the general reader. Attention will be restricted to problems whose resolution has, or may be thought to have, a bearing on the essential logical character of Zeno's arguments. The many linguistic questions which fall outside that category will in general be disregarded. Aristotle talks about Zeno's paradoxes in several places. The most comprehensive passage is in Book VI, chapter 9, of the Physics, 239b5-240a18. Anstotle says (239b9-11) 'There are four arguments of Zeno's about motion which give trouble to those who seek to refute them', and he gives a short statement of each of the four together with a comment of his own. We shall deal with the arguments in the order which Aristotle assigns to them in this passage (and which he seems to regard as standard). He refers to the first 3

argument as the Dichotomy (239b22) and to the second as the Achilles (239 b14); we shall follow him in using these names, and for the third and fourth arguments shall use respectively titles now commonly accepted, namely the Flying Arrow and the Stadium. The Dichotomy and the Achilles will be the subjects of Chapters 2 and 3 respectively. The Flying Arrow will be dealt with in Chapter 4 and in part of Chapter 5, and the Stadium in Chapter 6. The object of the paradoxes and some other questions of a general kind will be considered in Chapter 7. One of the general questions discussed in Chapter 7 will be what Zeno's object was in propounding the paradoxes, and reasons will be given for thinking that the paradoxes were put forward in support of the doctrines of Parmenides, and in particular that those about motion were put forward in support of the Parmenidean doctrine that reality is motionless, i.e. that there is no such thing as movement. This, like the other doctrine of Parmenides that there is no such thing as plurality, is so utterly in conflict with common sense that it may well be asked, What is the point in considering arguments in its support? Let us consider this primarily in connexion with the Dichotomy and Achilles arguments. These two arguments appear to go along so smoothly in accordance with ordinary and seemingly sound principles of reasoning from premisses that appear to be true to their ridiculous conclusions that it is natural to feel compelled to ask what is wrong, and a satisfactory answer, if obtained, to that question being in itself sufficient reward, surely no further justification for pursuing the enquiry should be required. There is, however, a not negligible by-product, namely that in examining these arguments we find ourselves led into interesting and important questions about infinity, space and (to some extent) time. With regard to the other two paradoxes the situation is less clear, since the arguments do not carry us along in the same apparently straightforward way, and there are difficult problems of interpretation. Here too, however, there is a worthwhile by-product though with a difference: again we find ourselves dealing with interesting problems about space and time; the difference is that here (at least according to the view that I take) these arise out of consideration less directly of the paradoxical arguments themselves than of the theories that some commentators have propounded about them.

2 3

4

Plato Parmenides 127-128. By Joh':' M Dillon, Introduction to Proclus' Commentary on Plato's Parmemdes (translated by Glenn R Morrow and John M D'll ) 1 on , pages xxxlviii-xliii. d. A Some of the quotations referred to will be found m' A togeth ·th d · · ppen lX b er Wl . a 1scuss10n of the main characteristics of the paradoxes a out plurality as compared with those about motion.

Notes 1

Produs in his commentary on Plato's Parmenides (Produs 694.25) says that Zeno had forty arguments in all, and Elias (6th century AD commentator on Aristotle and Porphyry) mentions the same figure: Elias, p. 109 (see Diels (1934), 29A15). For titles of works referred to see Bibliography. Different writings by the same author are distinguished either by dates or by numerals in square brackets. 4

5

2 The Dichotomy

It can be seen that that passage assumes acquaintance with the details of the argument. The second passage is more informative. It is at the beginning of

Aristotle's conspectus, mentioned earlier (page 3) of Zeno's four arguments. It is very short and refers back to text A 'There are four arguments of Zeno

about motion', Aristotle says, 'which give trouble to those who try to solve the puzzles they involve'. He continues:

Text B (Physics, 239b11-13) The first is that which says that motion is impossible because the moving body must arrive at the half-way point before it reaches the end ('dia to proteron eis to hemisu dein aphikesthai to pheromenon e pros to telos').

The Dichotomy is so called because it makes us~ of the idea. of an infinite sequence of halves or mid-points that can be identlhed rn any lrne or d1stance by repeated division into two. . . It will be convenient to have before us at the outset four passages rn wh1ch Aristotle states or refers to the Dichotomy argument. They willbe referred to as Texts A, B, c and D. In the Physics the first reference to the D1chotomy 1s rn Book VI, chapter 2. Aristotle has been argu~ng that trme 1sco:'trnu?~~' by which he says he means infinitely divisible ( d1aueton e1s ae1 dm1reta. l: and that if time is continuous magnitude must be too. For they are both d1v1s1ble by the same number of divisions and in the same proportions. Further, 1f one is infinite, 50 is the other, and in the same sense, 1.e. m respect of extrem1hes or in respect of divisions. He goes on:

Text A (Physics 233a21-31) . Hence Zeno's argument makes a false assumption when it asserts that 1t is not possible to traverse an infinite ':'umber of thrngs or to touch an infinite number of things, one by one, rna frn1te time. For there are two senses in which distance and time, and in general any contrnuum may be said to be infinite, namely in respect of division or rn respect of extremities. Accordingly, though a thing cannot in a finite time touch things that are infinite in respect of qu~tity, it can rna frn1t~ trme touch things that are infinite in respect of d1v1s1b1hty. For the hme 1tself 1S infinite in this sense. So it is in an infin1te not a flmte time that 1t 1s possible to traverse what is infinite, and it is by means of parts of tun_e that are infinite not finite in number that 1t 1s poss1ble to touch what 1s infinite 2•

6

Our next passage (which will be given in two parts Cl and C2) occurs in Book VIII of the Physics. Aristotle has been drawing a distinction between points which exist potentially and those which exist actually. If a body which has been moving in one direction stops and reverses, the point at which it does so exists actually but the intermediate points which it passes over in continuous movement in either direction exist potentially only. He proceeds:

Text C1 (Physics, 263a4-11) We may answer similarly those who ask Zeno's question, claiming that the half-distance must always be traversed before the whole, that these half-distances are infinite in number and that an infinity of things cannot be traversed; or those who put the same argument in a different form and claim that as the moving body in the course of its motion reaches each half-way point ('kath' hekaston gignomenon to hemisu') it should be possible to count the preceding half-distance, so that when the whole distance has been traversed an infinite number would have been counted; which is admittedly impossible. Text C2 (Physics, 263a11-b8) In our first discussion of movement we solved thi,fproblem by pointing

out that the time has an infinite number of parts within itself; for there is nothing absurd about traversing an infinite number of things in a time that is infinite in this way. The infinite is to be found equally in the distance and in the time. This solution, however, though sufficient for the person asking the question (which was whether it is possible in a finite time to traverse or count an infinite number of things) is not adequate as a true account of the facts. For if one leaves out the distance and the question of whether an infinite number of things can be traversed in a finite time, and asks about the time itself (for it contains infinite divisions), this solution is inadequate. The true one, which we 7

mentioned recently, must he given. This is that if one divides a

continuous line into halves one uses one point as two: ~n; makes t~e . t a be inning and an end. This is done bot y one w o s~::~f~~d by o~e who bisects a line. Whe':' this division is made ~either the line nor the motion will be cor:hnuous: For con~~~e~:

(or positions). Aristotle certainly has the concept of point: points ('stigmai' or 'semeia') are talked about at many places in the Physics; in particular, the

:h~ti are the thin~~: :~~c:~i~h~;~~=~~:~i~;:v~~~e~~!~::~;~~h e~

whole passage to which text C belongs is very much concerned with points, and the word 'semeion' is used not only in the earlier part of it (notably at 262a22-24, b4 and b7) but also at several places in the later part of text C itself, e.g. 263a23-24: 'if anyone divides a continuous line into halves he uses one point as two ("toi heni semeioi hos dusi chretai")'. Such considerations seem sufficient to justify Ross's rendering': 'to traverse an infinite number of points, or be in contact with them', in the present instance, and in general the use of the word 'point' in translation of the phrases 'hapsasthai ton apeiron' (to touch an infinite number of points) in text A and 'eis to hemisu dein aphikesthai' (must arrive at the half-way point) in text B, and where similar phrases occur elsewhere in our texts. We turn now to question (1 ). Points can certainly be traversed (gone through, over or across) as well as touched, but so can extensions, e.g. distances. What is the appropriate object of traversing here? Our text A is part of a larger context in the later part of which (immediately after text A) the word 'diienai' (traverse) in one form or another occurs repeatedly; its object is always 'megethos' (magnitude, length, distance). Again, in text C (263a9) there is talk of the moving object traversing the whole something ('ten holen'); the something must be something extensive: traversing the whole point as opposed to traversing half of it would be incoherent. Perhaps 'gramme' (line) is the noun to be understood here. I suggest in any case that the noun required as object for 'traverse' in text A and the other relevant places must be one meaning some sort of extensive magnitude. 'Distance' seems a suitable choice. (It is worth noticing, incidentally, that in our texts and some adjoining passages the feminine adjective 'hemiseia' seems regularly to mean half-distance. Probably 'gramme', which occurs at 263b8 and also, with more obvious significance, at 263a27, is to be understood. The neuter 'hemisu', on the other hand, though it stands sometimes for half-way point ('semeion' perhaps being understood), does not always do so. An example ofthe exception is 'lou ... heterou teleute Hemiseos' ( ... end ofthe one half), 263b1; 'megethos' may be the noun, as severaHimes in 233bl-8, or 'mekos' (length).) · A careful reading of our four texts and their immediate contexts suggests, then, that in Aristotle's view there were two key expressions either of which could be used in stating the dichotomy argument; (1) traversing halfdistances, and (2) touching half-way points. Obviously these two expressions are closely related, and Aristotle probably, and perhaps correctly, regarded it as a matter of indifference which was used. Nevertheless they do represent different concepts; for although anything that traverses a halfdistance necessarily touches the point at which that half-distance terminates,

8

9

motion is motion over a continuous path, but m a contmuous pa

is an infinite number of halves, but only potentially, not actually. If one makes them actual one will get not a continuous but an mtermittent vement· which is what evidently happens in the case of the person :~a count~ the halves; the one point must be counted as two. For the end of one half-distance will be the beginning of the other If he do~s ~~t count the continuous distance as one but as two half-d~st~nces. o b: answer to the question whether an infinite nur;'ber o ~ mgs can .t traversed in time or in distance is that in a sense It c~ an m ~ sense 1 cannot. For if they exist actually it is not possible but bote.ntm~~n~~i~ then it is possible; the person who mov~s contmuo~~fkd :::~ For it is .. traversed an infinity of things, but not m an unqua I incidental to the line to be an infinite number of halves, but m essence and reality it is different.

{f

f

1

e is from the treatise De lineis insecabilibus (traditionally

~s~~i~~; t;~~~s~otle) in which the author deal~ wit~{~eli~:~u~:tss~~~ forward by those who hold that there are m IVISI argument is referred to in:

.

Text D (De lineis insecabilibus, 968a19-24) . se that the necessary result of Zeno's argument IS indivisible For make an infinite number of contacts, one y one a. ha tomenon'), in a finite time: but a moving body must hrst reach .the hai£-wa point ('anagke d'epi to hemisu proteron aphikneisthai to kinoum~non') of any distance, and there is always a half of any distance not absolutely indivisible.

:::~t~~::~::s~~~~ome

.

.

magnitu~e.

it(~~ i~~~:~~~~;~

ortance arises in connexion with the first few lines of

t.,::~~~': ~::e~':t:'!reira dielthein e hapsasth~i ton apeiron~~~: ~~~e:

adjective ('apeira, apeiron:) wi!~:: :~l~~r::::;ra:~ ~:~;,;~:~, c and D. In true (though the word varies) use the noun 'things' (to traverse an

tr~:'s\:ti~~;,: !~~~::g~~,I~~~ft~:~ leaves questions of interp:etation~ (1~

(2)":hat arebth~ L t g start with (2). The obvious answer seems to be points infm1te num er. e us

the converse is not true: the point can be touched without the distance

having been traversed. We shall look at several possible versions of the Dichotomy. In doing so it will be convenient in each case to present the argument in terms of an object or person A and a path, line or distance SF between a starting-postS and a finishing-post F, the argument having as its conclusion the proposition that A does not move from S to F. Since A may be any object and SF any distance anywhere, the argument if valid entails the paradox that motion is altogether impossible.

(a)

IffA moves from S to F, when it reaches Fit has touched one by . one m a mite time all the points in inf' 't . .

m illi e sequence of pomts.

Therefore,

A does not move from S to F. This is the argument as we shall be conside . . . ques.tion of whether it is really necessary::;,~:~~~ nex~ few ~ages .. The tune m (a) and (b) may be left till! t e e P rase m a flmte Th a er. e proposition (a) is not a starting-point but is itself th I . f argument the premisses of wh. h b ' e cone us1on o an IC may e stated as follows:

~ere is tanh inlffinite sequence Qp of points between Sand F (namely:

porn a -way from S to F, the point half-way from that point to F the point half-way from that point top and so on).

Figure 2.1: Distance for traverse

(a) (~)

If A moves from S to F it touches one by one in a finite time all the points in an infinite sequence of points, It is impossible to touch one by one in a finite time all the points in an infinite sequence of points.

2 ( )

(1)

~ff~~:~~::~~et;:ln~h;,n~reaches Fit has touched one by one in

~epr:::,tanh inlffinite frsequence QR of points between Sand F (namely

a -way omStoF the point half-way from S to that point the point half-way from S to that point' and so on). '

'

~ th ere. are two possible versions of the ar um~t ' progressive version goes as follows: g ~·· as a whole. The 1 ( )

~e;~~t~a~::~~ :r:~~n: ~P of points between Sand F (namely, the point half-way from that ~oint to F. the pomt half-way from that point to F and so on). '

A does not move from S to F.

10

'

h h . The infinite sequence Qp referred to in (1) i . shw at as been descnbed as a progressive sequence in which th ethsuccessive alf-way points get nearer and nearer to F There I·s h' owev · • er, ano er possib 'lity 1 should be one that regresses tow d S S I ' na':'e y that the sequence ar s · omeone statmg the ar ·t · h th' IS regressive sequence in mind ld . gumen Wit cou express the first premiss as follows:

to the conclusion:

In the argument so worded there is, however, a possible ambiguity, in connexion with proposition (~), which is avoided if the argument is expressed as follows:

m1 e sequence of points.

~to:~~=ib~e tot have touched one by one in a finite time all the

(b)

(1)

When all four texts are considered it seems reasonable to regard the Dichotomy as being, at least in its final and essential part, an argument from two propositions, the first of which states that if A moves from S to Fa certain process involving an infinite sequence is completed and the second of which states that such a process cannot be completed in a finite time. Aristotle, as we have seen, talks sometimes of the process as being that of touching the points in an infinite sequence of mid-points; at other times he makes it that of traversing an infinite number of halves. Later writers have sometimes taken the process to be an infinite sequence of events or the performance of an infinite sequence of tasks, each event or task being, e.g., the touching of a mid-point or the traversing of a half-distance. Following Aristotle rather than later writers, and setting aside initially the half-distance terminology.

There always remains a gap between A and F.

Therefore (3)

A never reaches F.

The two arguments are essentially the same and are similarly fallacious in a way which we can explain with reference to the Achilles. We first define the sequence of points: T.v T0, Tv ... ; and of times: t_ 1, t 0, tv ... , as follows: T_ 1 is the starting-point of Achilles, T0 that of the tortoise, and for every k greater than -1 Tk is the point at which the tortoise is when Achilles is at Tk-l; t_ 1 is the

31

time at which Achilles starts from T_1, and for every k greater than -1 tk is the time at which Achilles reaches Tk" Now the persuasiveness of the argument results from a latent ambiguity in the word 'always' in the intermediate conclusion (2). If (2) is to follow from premisses (1.1). (1,2), ... , 'always' must be taken to mean 'at every.time at which Achilles is at one of the points T_, T , Tv ... (i.e. at each of the times t_ 1, t 0, tv ... ) and of course at every relevant 0 earlier time'. On the other hand, if the conclusion (3) is to follow from (2), 'always' must mean 'at every time whatever (after and including the starting time t_

)'. 1

Unless these two interpretations are coextensive the argument

1s

fallacious. We shall now show, however, that they are not coextensive but that there are times later than all the times tv t 0, 11, ... 2. Attention will be confined to the case in which the speed of Achilles is twice that of the tortoise, but a generalization to other ratios can be made without difficulty. Suppose then that the speed of Achilles is twice th~t of the tortoise, and further that Achilles takes one unit of time (say one mmute) to go from h1s starting-point (T_1) to the tortoise's starting-point (T 0). The tortoise in the same period will traverse half the distance traversed by Ach1lles. The distance from T0 to T1is therefore half the distance from T_1to T0 and Ach1lles accordingly will take half a minute to traverse it. It can be seen that over each stage Tn_1 to Tn of the race (other than the first stage) the time taken by Achilles will be half the time taken by him over the previous stage Tn-2 to Tn-1· Let M be the time taken by Achilles over the first stage (from T_1 to To) 1 and in general let M; be the time taken by Achilles over the ith stage (T,_2 toT,_ 2 ). We have stipulated that, inminutes,M1~ 1. So M2 ~ 1/2, M3 ~ 1/2 , ··:, M; 1 ~ 1/2i-1, .... If the race starts at zero minutes by a clock, then the clock tunes in minutes at which the successive stages are completed by Achilles are: 1st stage: M1 2nd stage: M1 + M2 3rd stage: M1 + M2 + M3

·M +M +

n thst age .

+M

··· n (n+l)ststage:M 1 +M2 + ... +M11 +Mn+l 1

2

=1 = 1 + 1 /z

=1+ 1/Z+ 1/z2

=1+ 1/2+ ... + 1/zn-l =1+ 1 /2+

...

1

11 1

=tn-1 1

11

+ /2 - + /2

=tn

These totals (1, 1 + 1/2, 1 + '/2 + ljz2, ... , 1 + 1/2 + ... + 1/zn-l, 1 + ... + 1;zn-1 + 1/2") which, as indicated above, are equal respectively to the clock times to, tv ... tn-11 tn, are respectively the partial sums Sv 52, ... 'sn-11 Sn of the geometric progression: 1, ljz, lj22, ... , lj2n-1, 1 /211 . It can be seen ~hat ea~h partial sum Sk+l (after. the first, i.e. after 1) is obtained by addmg to liS predecessor half the value of the last term of that predecessor; l.e. sk+1 (where 1 k>O) is obtained by adding to Sk (of which the last term 1s 1jzk-1) (' /zk- )+2 ~ ljzk. I

32

. The key to the proof that there are times later than all the times t_ 1, t0, t1, ... 1s the following proposition: (a)

Adding to sn half the value of its final term (i.e. adding half of I /zn-1) lS eqmvalent to addmg to sn half the difference between sn and 2.

A proof of (a) if one is required, may be spelt out as follows: The sum to n terms of a geometric progression is given by the formula: a(1-r")/(1-r), where a is the first term and r is the common ratio. Thus in the present case:

sn ~ 1(1-(1/2)")/(1-'/2)

~

(1-'/z")/('/z)

~

2(1-'/z")

~

2-2/z" ~ 2-'/zn-1

... (~)Now 1/zn-l ~ 2-2+1/zn-l ~ 2-(2-1/zn-1), from which and(~) it follows that 1;zn-l ~ 2-Sn. Therefore (a) adding to Sn half the value of its final term (i.e. adding half of 1/2"'') is equivalent to adding to S half the difference between sn and 2. n Since Sn+l is obtained by adding to Sn half the value of the latter's final term and by (a) this is equivalent to adding to sn half the difference between s~ and 2, it can be seen that, if sn is less than 2, sn+l is also less than 2, since to obtam Sn from 2 an addition of the whole difference would be needed. That is to say, if any sum in this sequence 5 11 5 2, ... is less than 2 its successor in the sequence is less than 2. But the first sum S1 (~1) is less than 2; therefore every sum Sk, however great k may be, is less than 2. Accordingly, every time in the sequence of clock times tv t0, t 11 ... is less than zero+2 minutes. Thus the second interpretation of 'always' ('any time whatever'), including as it does 2 minutes and all later clock times, includes times not covered in the first interpretation. So the argument is fallacious. We may close this discussion of the Achilles argument with a comment which perhaps may have some bearing on the question why this paradox has been so seductive. (Though attention is being confined to the Achilles similar considerations are relevant to the Dichotomy.) Aristotle does not tell us that the speed of Achilles is supposed to remain constant. Nevertheless that has nearly always naturally been taken for grant~d. It seems Worth pointing out that if that condition is dropped the paradox can disappear3. Suppose for instance that Achilles slows down in such a way that he takes one minute to traverse each stage, i.e. one minute to go from T_ 1 to T0, one minute to go from

T0 to T1, and so on; and that the tortoise too takes one minute over each of its stages, i.e. one minute to go from T0 to T1, one minute to go from T1 to T2 and so on, thus maintaining half the speed of Achilles in the sense that in each minute it covers half the distance that Achilles does in the same period, then indeed it will be the case that Achilles, though never halting and always moving at twice the speed of the tortoise, will never catch the tortoise. For, again, the tortoise will be ahead at each of the times t_ 1, t,, t, ... , but in this 33

case the two sets of times specified above in explanation of the ambiguity of 'always' will be coextensive. Achilles now spends one minute over ea~h stage of the race, and so the time t,_ 1 at which he completes the nth stage ':"Ill be n minutes after zero. So however great a number u may be, where u IS a number of minutes after zero, there is a time, e.g. tW' in the sequence t1, to, t1,

4 The Flying Arrow

... that is greater than u. Hence in this case there is no time that is later than

all the times in the set of times at which the tortoise is ahead.

Notes 1 2

3

Originally perhaps by C.S. Peirce; see Ushenko p. 157. , . . 'There are times later than all the times t_ 1, t 0, t1, .... That this IS something that needs to be shown was pointed out to me by Professor David Evans. Grattan-Guinness (p. 1) and Zinkernagel make the same point, the former referring to a suggestion by Sir Karl Popper.

34

In Book Vl of the Physics, which is concerned with continuity, time, space, motion and rest, Aristotle asserts and defends a number of propositions each involving one or more of these concepts. One such proposition (the meaning of which will be discussed later) is to the effect that a moving object cannot in the primary time of its motion be over against anything primarily. Aristotle states this proposition at page 239a lines 23 to 26 and immediately supports it in the passage ending at line 4 on the second part of that page, i.e. 239b, by an argument in which he claims that its denial would entail the consequence that a moving object is at rest. That of course would normally be accepted as a reductio ad absurdum, but Aristotle, evidently recalling that Zeno had argued paradoxically that an arrow in flight actually is motionless 1, refers now to Zeno, stating his argument and dismissing it as being based on a certain wrong assumption. This reference to Zeno, embracing the account of the Flying Arrow argument and Aristotle's comment occupies lines 5 to 9 of page 239b. Immediately thereafter Aristotle launches into a page-and-a-halflong digression in which he provides the conspectus mentioned earlier of Zeno's_ four paradoxes about motion. When he come;-to the third, which is the Flying Arrow, he deals with it very briefly in lines 30 to 33, simply referring back to his earlier statement and repeating his comment. These three texts, namely the statement and defence of the proposition about the moving object and the two texts referring directly to the flying arrow, may be regarded from our present point of view as a single unit; I shall call this unit the Flying Arrow passage, and its components may be referred to respectively as: the Preamble, the Central Text and the Resume2. The whole passage being quite short can be quoted in full but first it is necessary to say something about one of the textual problems alluded to in our Introduction (page 3).The first part of the Central Text, as it occurs in nearly all the manuscripts, may be translated in a fairly literal way as follows: 35

(A) But Zeno argues fallaciously; for if everything, he says, is always either at rest or in motion whenever it is over against the equal, and the

flying object is always in the now, then the flying arrow is motionless. No one has succeeded in making sense of this passage as it stands, and there is general agreement that some emendation is necessary. Two main alternative lines of emendation, each with subsidiary variations, have been followed by different editors. The first consists in omitting the words corresponding to 'either ... or in motion', the suggestion being that these have been inserted by a careless scribe under the influence of a number of occurrences somewhere in the vicinity of this passage of sentences identical in wording with, or closely similar to, the clause 'everything is always at rest or in motion'. The second strategy consists in leaving alone the words 'either ... or in motion' which are in all the manuscripts but inserting between 'motion' and 'whenever' words corresponding in meaning to 'but nothing is in motion'. There is fairly general though not complete agreement that either the phrase corresponding to' over against the equal' which occurs in lines 2-3 above should be repeated in the following line before or after 'in the now' or that the Greek as it stands can properly be taken to have the sense that it would have if those words were repeated. Translations of the text as emended in the two alternative ways referred to, account being taken of the point of agreement just mentioned, might go as follows:

(B) But Zeno argues fallaciously; for if, he says, everything is always at rest when it is over against the equal, and the flying object is so [i.e. over against the equal] always in the now, then the flying arrow is motionless. (C) But Zeno argues fallaciously; for if, he says, everything is always either at rest or in motion, and nothing is in motion when it is over against the equal, and the flying object is so [i.e. over against the equal] always in the now, then the flying arrow is motionless.

The arguments adduced in support of B and C respectively seem to be evenly balanced. I shall not attempt to discuss them but will adopt B. I suspend judgement as to the correctness or otherwise of the claim which has been made, certainly with some plausibility, that the essenllal character of the reasoning attributed to Zeno comes out the same whichever of these two suggested emendations is followed. Here now is a translation of the whole Flying Arrow passage (incorporating B):

36

PREAMBLE (239a23-b4) Since every moving object moves in a time and changes from something

to something, in the time in which it moves per se and not by moving in a part of the time it is impossible for the moving object to be primarily over against something. For if a thing, both itself and each of its parts, is in the same place for some time, then it is at rest. For this is how we use the term rest, namely when it is true to say that a thing, both itself and its parts, is in the same place at one after another of the nows. But if this is resting it is not possible for a changing thing to be as a whole over

against something in the primary time of its changing. For every time is divisible, so that it will be true to say that in a number of different parts of the time the thing, itself and its parts, is in the same place. For if we use the term rest not in this way but to mean that a thing is in the same place at a single now only, the object will be over against something for no time but only at the extremity of a time. It is true that in the now it is always over against something stationary; it is not however, at rest. For it is not possible for it either to move or to be at rest in the now, though it is true that it can be not in motion and can be over against something in the now. But in a time it is not possible for it to be over against something that is at rest; for if that were so the moving object would be at rest.

CENTRAL TEXT (239b5-9)

But Zeno argues fallaciously; for if, he says, everything is always at rest when it is over against the equal, and the flying object is so always in the now, then the flying arrow is motionless. But this is false; for time is not made up out of the indivisible nows, just as no other magnitude is made up out of indivisibles.

RESUME (239b30-33! ... the third is that stated above, that the flyingoarrow is stationary. It results from the assumption that time is made up opt of the nows; for that not being granted the reasoning will fail. Understanding the Flying Arrow argument involves interpreting two key expressions that occur in it: 'in the now' 3 and 'over against the equal'. We shall discuss these in turn. 'In the now'

The Greek is 'en toi nun'. Throughout the later part of Book IV and in Books VI and VIII of the Physics, 'nun', the ordinary Greek word for 'now', is used constantly with the definite article, sometimes in the singular and sometimes 37

in the plural, as an adverbial noun. It may be remarked at the outset that probably no special significance, at any rate no uniform special significance,

should be attached to the use of the definite article. Its use here is probably just in line with the common, though not unexceptional, Greek practice of using the definite article with proper names and abstract nouns (for example, 'ho chronos', meaning 'time'). I have kept it in the earlier translation of the Flying Arrow passage because the whole expression, in its various forms, is a rather unusual one and it seems desirable initially to give a fairly literal

be transferred mutatis mutandis to the other. The analogy applies also to the rela:1onsh1p between a continuum of motion and whatever in the case of

motwn corresponds to spatial points and temporal nows. The continuum analogy is implied throughout Book VI, chapter 1. It is apparent, for example, at 232al8-22: And if length and motion are thus indivisible it is neither more nor less necessary that tin:e. also be similarly indivisible, that is to say, be

rendering, but undoubtedly in many contexts a correct and idiomatic

composed of mdivlsible moments: for if the whole distance is divisible

translation would require the omission of the definite article or its replacement by something else, e.g. the indefinite article. What does the phrase 'the now' mean in our Central Text? It occurs there twice: once in a premiss of the argument, 'in the now the flying object always is so', and once (in the plural) in Aristotle's comment, 'for time is not made up out of the indivisible nows'. There are two possibilities. One is that by 'the now' Aristotle means a boundary or dividing point between two stretches or periods of time. The now in this sense has no duration. Though an element in a period of time it is not a part of any period of time in the way that an hour and a millionth of a second are both parts of a day. They both have duration, whereas the now has no duration. It is a dividing point, limit or boundary of time. We may use the expression 'temporal point' to represent this interpretation of 'the now'. The second possibility is related to a theory which some have held, though many would reject it. This theory (to be dealt with in Chapter 5) denies that time is infinitely divisible; it holds that there are little bits of time which though they have duration are so short in duration that they cannot be divided into smaller parts: they are partless. They are atomic durations. The word 'period' (shorn of any connotation relating to recurrence or to interval between phases, but just meaning 'stretch of time') may conveniently be employed. According to the first view then the premiss means that at every temporal point the flying object is over against the equal, and Aristotle's

and an equal velocity will cause a thing to pass through less of it in less lime the tune also must be divisible; and conversely if the time in which a thmg IS earned over the section A is divisible, this section A must also be divisible. Also 23lbl8-20: Either mag':'itude, time and motion are all composed of and divisible mto md!VIS!bles or none of them is. To come back to the meaning of 'now': a supporter of the 'temporal point' alternah~e m1ght pomt m the fust place to the many passages in which the now IS smd to be a limit or boundary of a period of time, or a dividing point between two limes. Book IV, chapter 13, for example, contains definitions of a number of temporal terms. The definition of 'the now' reads: 222a10-12: The now is the link of time, ... , for it connects past and future, and it is a limit of time, for it is the beginning of the one and the end of the other. There are numerous other passages in which 'the now' is clearly spoken of as bemg a boundary, limit or dividing point. Here are some:

comment means that Zeno is mistaken in assuming that time is composed of

temporal points. According to the second view, on the other hand, the premiss means that in every atomic period the flying object is over against the equal, and Aristotle's comment is that Zeno is mistaken in assuming that time is composed of atomic periods•. We shall consider which of these alternatives is correct, but first it is desirable to mention a principle strongly asserted by Aristotle which may be called his Continuum Analogy. This is the principle that the relationship between a spatial continuum (a line or a distance) and indivisible points ('stigmai') is exactly analogous to the relationship between a temporal continuum ('chronos', a time, a period of time) and the.indivisible nows; and that considerations and arguments which are shown to apply to either may 38

'

219a29: What is bounded by the now is thought to be,time. 222a17-19: So too the now is on the one hand potentially a dividing pomt of lime, and on the other the common boundary and unity of both parts. 233b33~234a3: The now in the primary sense also is necessarily md!VIS!ble ... For the now is something that is an extremity of the past (no part of the future being on this side of it) and also of the future (no part of the past being on this side of it); it is as we have said a limit of both.

39

Secondly, there are several passages ill which an emphatic contrast is drawn between the now and a time or a penod of hme. Smce 1£ the now lS not

a period of time it is not an atomic period of time, such passages seem to t~ll in favour of the 'temporal pomt' interpretation. Examples of passages m which the contrast is emphasized are:

234a22-b9: In this passage it is first argued that nothillg is moved ill the now. The conclusion is then drawn: Therefore it is in time that what moves moves and what rests rests.

237a4: [The last now] forms a boundary and that which is between the nows is a time. 218a8: But time does not seem to be made up out of the nows. It may be suggested, however, that the various passages cited or referred to

do not really give such strong support to the temporal pomt mterpretation as at first sight they may appear to. Even though Aristotle certamly refers to the now as a limit or boundary of a time, may he not shll be thmkmg of Jt as somethmg that though mdivisible has duration, just as the lilies which we thmk of as the boundaries of cricket or football fields, though very narrow marks, still have some width? Similarly, might it not be the case that the contrast between a time and the now amounts to no more than the contrast between a divisible and an mdivisible period of time? Such a suggestion, ill so far anyway as it might be illtended to show that by 'the now' Aristotle always means an atomic penod of hme, may ~e refuted by the following argument: (a) although some of Anstotle s statements about nows, or at any rate some of those containing the expression 'the now', do not presuppose the existence of nows, othe~s including some of those earlier qu.oted undoubtedly do have this presupposition; (b) Aristotle ~ertamly d1sbeheved m the ex1sten~e of atom1c times: he argues directly agamst atom1c hmes m 263b26 et s~qq, further h1s disbelief m atomic times seems a fair mference from the ContmuumAnalogy together with 206al7 where he says that the belief in atomic lilies ('atomaJ grammai') is easily refuted. Now from (a) and (b) it seems to follow indisputably that at least ill some cases where Anstotle us;s the. exp~ssJOn 'the now' he is using it with a meanmg other than that of atom1c penod of time'. Smce the only alternative seems to be 'temporal pomt', this argument establishes conclusively that ill those statements contammg the expressiOn 'the now' which presuppose the existence of nows Aristotl~ is usmg the expression to mean 'temporal pomt'. What we are mterested m_ however Js the occurrences of 'now' in our Central Text. As has been mentioned, there are two such occurrences one in the premiss of Zeno' s argument and the other m part of Aristotle's comment, and it is clear that, however it may have 1

40

been with Zeno, Aristotle is not committed in either context to the existence

of nows. Accordmgly, the argument propounded a moment ago cannot be employed to rule out the possibility that ill the Central Text 'the now' means 'atomic period' Can the question then be settled by some other means? Let us look at the Preamble. Here the expression 'the now' occurs, in one grammatical form or another, five times. In the contexts of all except perhaps the second of these occurrences the existence of 'the now' seems to be contextually implied; and this is true ill particular of the two occurrences ill the last sentence of the Preamble. In that sentence then, and elsewhere ill the Preamble, 'the now' must surely mean 'temporal point'. That bemg so would it not be very strange if five lines further down in the Central Text (which from Aristotle's pomt of view is a sort of parenthetical tailpiece or footnote to the Preamble) Aristotle should without any warning whatever employ the expression 'the now' ill an entirely different sense? That is the case for the 'temporal point' interpretation.s Although it may seem a strong case there are nevertheless some considerations which seem to tell in favour of the other, the atomic period interpretation. It may be true, and indeed I think it is, that there are no passages in which 'the now' obviously means 'atomic period'. There are, however, some passages which, taken at any rate by themselves (i.e. without consideration of their immediate contexts), may seem more readily mtelligible under the 'atomic period' mterpretation. This may be thought t~ be true indeed of our Central Text itself; certamly some commentators6, possibly the majority, have mterpreted 'the now' in its occurrences in the Central Text ill this way; and undoubtedly doing so has the advantage of givmg us a clear and straightforward mterpretation of Zeno's argument and, no less significantly, making Aristotle's comment and his evident assumption that it would be immediately seen to be relevant and conclusive readily understandable. There are two other passages that should be mentioned ill this connexion. These passages, each of which seems to have something in common with the Central Text, in a way foreshadowmg it ill the one case, reflectmg it ill the other, are: 23lb18- 232a17 and 240b8- 241a5. The former passage, which might be called the 'motiomby jerks' passage, seems to employ essentially the same core of reasoning (though in the context of a different argument with a different conclusion) as is encountered later in the presentation of Zeno's argument in the Central Text. It would seem that these two passages and the Central Text all hang together. Neither passage is at all easy but in each, taken by itself, the 'atomic period' interpretation of 'the now', though not absolutely required, would seem at first look anyway rather more natural. The arguments ill favour of the 'temporal pomt' mterpretation seem the stronger. It should be remembered, however, that our question has been what Aristotle meant by 'now'; and what Aristotle meant may not have been 41

what Zeno meant, even if 'now' ('nun') was indeed the term that Zeno used. In the remainder of this chapter, after discussing the meaning of' over against the equal' (the second key expression mentioned earlier), we shall be examining the value of the Flying Arrow argument as presented by Aristotle in the Central Text and shall then be assuming that 'now' there means

'temporal point'. The possible alternative, 'atomic period', interpretation, however, will not be forgotten, and will receive attention towards the end of Chapter 5.

'To be over against the equal' The Greek is 'einai kata to ison'. This expression is used in the premiss of Zeno's argument in the Central Text. The phrase 'to be over against' ('ein~i kala') comes to our notice first in the Preamble, and in order to mterpret 1t, 1! seems necessary to begin by analysing the argument of that section. It will be convenient to set out part of the Preamble again with certain clauses labelled. Since (a) every moving object moves in a time and changes from something to something, (b) in the time in which it moves per se "':d not by moving in a part of the time it is impossible for the movmg obJect to be primarily over against something. For (c) if a thing, both 1tself and each of its parts is in the same place for some time then 1t 1s at rest. (d) That is how we use the term rest, namely when it is true to say that a thing, both itself and its parts, is in the same place at one after another of the nows. But if this is resting (e) it is not possible for a changmg thmg to be as a whole over against something in the primary time of its changing. For (f) every time is divisible so that it will be t~ue to say that (g) in a number of different parts of the same llme the thmg, 1tself and its parts, is in the same place. (h) .... (i) But in a time it is not possible for it to be over against something that is at rest. For G) If that were so the moving object would be at rest. The passage is not fundamentally difficult but it is expressed in what is perhaps a rather confusing way. In interpreting it we have to make certam assumptions, and among those that seem reasonable are the followmg: (i) The concept that is under consideration throughout is being over against something primarily: in (e) 'over agamst something m the pnmary time of its changing' means (i.e. is short for)' over agamst somethmg primarily in the primary time of its changiftg'. Similarly in (i) below. (ii) 'itself and each of its parts' in (c), and similar expressions in (d) and (g) mean the same as 'as a whole' in (e).

(iv) The translation 'in the same place' in (c), (d) and (g) gives the right sense, though the Greek phrases 'en toi autoi', 'en t'autoi' mean literally

'in the same' and contain no word for place. (v) The passage (h) of three sentences, represented by dots, between (g) and (i) is a digression from the argument. (vi) 'in the primary time of its changing' (e) means the same as 'in the time in which it moves per se and not by moving in a part of the time' (b).

The meaning of 'the primary time of moving of an object 0', if not sufficiently explained in (vi) above, may be illustrated as follows: if a train on Wednesday afternoon makes only a single journey and that a non-stop journey beginning at 2 p.m. and ending at 3 p.m., then Wednesday afternoon is a period of time in which the train moved but not the primary time of its movement; the latter is 2 p.m. to 3 p.m. The primary time (i.e. period of time) of a certain movement of an object 0 is a period of time all of which is occupied by that movement and within which all of that movement takes place. The argument in the Preamble aims to show that a moving object cannot as a whole be over against something primarily in the primary time of its moving (b,e). It can be expressed as follows: (1) Every moving object moves in a, period of time [as distinct from in a now: see 234a24]. (a)

(2) Every period of time is divisible into parts [which are bounded by nows].(f) Therefore, from 1 and 2, (3) If a moving object 0 is as a whole over against something primarily in the primary time of its moving it is as a wholefver against that thing primarily at different parts of the period of time into, which the primary time of its moving is divisible, and so .is over against it primarily at different nows. (4) To be over against something primarily at different nows is to be at the same place at different nows. (g) [(g) is being interpreted as logically a conflation of (3) and (4).] (5) To be in the same place at different nows is to be at rest. (c,d)

Therefore, from (3), (4) and (5),

(iii) 'changing' in (e) effectively means 'moving'.

42

43

(6) If 0 is as a whole over against something primarily in the primary time of its moving it is at rest. (i,j) But (7) 0 (being a moving object) is not at rest. Therefore, from 6 and 7,

For the purpose of analysis if we make certain assumptions we may express

(a), (b) and (c) a little more formally as follows: a = Whatever is over against the equal is at rest. (It is being assumed that the always in (a) is pleonastic) b = At every now the flying arrow is over against the equal. ('arrow' instead of 'object' is tidying up. The assumption is that 'always in the now'= 'at every now') c=

(8) 0 is not as a whole over against something primarily in the primary time of its movement. (b) Given this analysis of the argument we can see what the key expression of the Preamble must mean. 'einai kala ti proton', which we have been rendering as 'to be over against something primarily' means to be directly opposite or directly above a certain spatial expanse (Wicksteed and Cornford's translation (at 239a31) is 'exactly cover') or to fill a certain space exactly. We may think, for example, of a boat moored at the bank of a canal, the bank being marked off in short sections by a number of trees or posts. The boat is over against a section Sl- S2 of the bank primarily provided that the bow is directly opposite one end of the section (e.g. Sl) and the stern IS directly opposite the other. It is beyond our immediate purpose to consider certain questions that a reading of the Preamble naturally brings to mind: for example, what IS It to be over against something but not primarily so? what exactly IS the distinction that seems to be implied, between bemg over agamst somethmg as a whole and being over against something in part? These questions are discussed in Appendix C. What is essential to our purpose is to explain the key expression in the Central Text, namely 'over against the equal' ('kala to ison'). This is done in terms of 'over against primarily'. To say that an obJect 0 is over against the equal is to say that it is primarily over against a space which is equal in size to 0; for example the boat on the canalis over agamst the equal if it is primarily over against a section of the bank which !S equal m length to the boat. We are now ready to examine the Flying Arrow argument itself, as it is presented by Aristotle. . . For convenience of reference we begm by settmg out the argument as given in the Central Text, but with each clause labelled: If (a)

everything is always at rest when it is over against the equal, and (b) the flying object is so [ave~ against the equal] always in the now, then (c) the flying arrow is motionless.

44

The flying arrow is at rest. (The assumption is that 'is motionless' in

(c) means the same as 'is at rest' in (a).) So we have the following argument: (1) (2)

Whatever is over against the equal is at rest. At every now the flying arrow is over against the equal.

Therefore (3)

The flying arrow is at rest.

Let us refer now to our explanation of the expression 'over against the equal'. When we do so we see that (2) is ambiguous. It may mean either: There is a space equal in length to the flying arrow which at every now the flying arrow is over against. or (2b) At every now there is a space equal in length to the arrow which the flying arrow is over against. (2a)

For (2a) the space in question must be the same at every now. For (2b) it may be different at different nows. Now (2b) is a plausible proposition, but if it gives the sense in which (2) is to be taken then the arrow may be at different places at different times, and this means that (3), if it is properly deducible from (2) must be understood in what would commonly be regarded as an unnatural sense according to which the arrow can be at different places during the time that it is at rest. We have been looking at the argument as reported by Aristotle without regard to his criticism, but we must now inquire in the light of that criticism what Aristotle's interpretation may have been. Aristotle considers that he has refuted the argument when he claims that it is false that time is made up of nows. Now the argument nowhere (either in Aristotle's statement or in the above reformulation) asserts and uses as a premiss the proposition that time is made up of nows. Aristotle must therefore be of the opinion that this proposition is a necessary but tacit assumption underlying the argument. Let us try to state what he may have taken the complete argument with this assumption made explicit to have been. Before doing so, however, we may note that Aristotle's comment is not what we might have expected in the light of the immediately preceding passage, i.e. the Preamble. One thing that

45

he has affirmed in that passage is that although in the now a moving·object is over against something it is not thereby at rest, for in the now an object is

neither at rest nor in motion; only in a period of time ('en chronoi'), he implies, is an object either in motion or at rest. In view of this affirmation we might have expected Aristotle to have directed his criticism of Zeno's argument against the major premiss, i.e. (1), and pomted out th~t that proposition is not universally true but needs to be qualified. It 1s n~t universally true that whatever is over against the equal1s at rest; for what IS in the now over against the equal is not at rest. The most we are entitled to

say is that what is in, i.e. throughout, a time over against the equal is at rest. But now perhaps we can see how Aristotle thought that the assumptwn that time is made up of nows is necessary for Zeno's argument. For he IS assuming that if time is made up ofnows, more specifically if a period of tin:'e Pis made up only of nows then if any proposition is true at every now w1thm the period P it is true throughout P. Thus if the flying arrow IS over agamst the equal at every now in P it is over agamst the equal throughout P. But whatever is over against the equal throughout P 1s at rest. Therefore throughout P the flying arrow is at rest. Zeno's argument as construed by Aristotle may therefore be formally stated in full with all assumptwns made explicit, as follows: Let P be a period of time that the arrow is in flight. (1) Every period of time is made up solely of nows.

Therefore (2) Whatever is true at every now in a period of time is true throughout that period. Therefore (3) If at every now in P the flying arrow is over against the equal, then throughout P the flying arrow 1s over agamst the equal. (4) At every now in P the flying arrow is over against the equal. Therefore (5) Throughout P the flying arrow is over against the equal. (6) Whatever throughout Pis over against the equal is at rest throughout P. Therefore (7) The flying arrow is at rest throughout P. Aristotle thinks that the argument fails because (1) is not true. But it may be contended that there is a different error and that, even if it should be true that time is entirely made up of nows, Zeno's argument still does not establish his conclusion that the flying arrow is at rest and so motionless. Th1s error IS m essence the same as that which we detected in the argument as we previously construed it, in disregard of Aristotle's criticism. To show this let us begin by simplifying the expression of the argument, supposing that suitable rules of 46

inference have been adopted, governing the moves from (1) and (4) to (5) and from (5) to (7). What we are now left with is the following: (1) (4)

Every period of time is made up solely of nows. At every now in P the flying arrow is over against the equal.

Therefore (5) Throughout P the flying arrow is over against the equal. Therefore (7) Throughout P the flying arrow is at rest. Let us examine (4) and (5) more closely in the light of our definitions. As before we see that (4), essentially identical with our earlier (2), is ambiguous. It may mean either (4a)

There is a space equal in length to the arrow which at every now the arrow is over against;

or (4b) At every now there is a space equal in length to the arrow which the arrow is over against. For (4a) to be true the space has to be the same one for each different now. For (4b) to be true the space maybe different atdifferentnows. Now consider (S) Throughout P the arrow is over against the equal. This may mean either (Sa)

There is a space equal in length to the arrow which throughout p the arrow is over against;

or (Sb) Throughout P the arrow is over against a space equal in length to the arrow. Let us look more carefully at the conclusion (7). We naturally take it to imply that the arrow is in the same place throughoufi?. So taken it follows from (Sa) but not from (Sb). (Sa) in its tum comes from (4a) but not from (4b). (4a) however seems to be just untrue. So if the conclusion (7) is taken in the natural sense the argument fails even if, contrary to the view of Aristotle, time is made up of nows. Is it possible to take the conclusion (7) in any other sense? Well, if (4) is taken in the sense of (4b) then the space which the arrow is over against may be different at different nows. (S) will be taken in the sense of (Sb) which does not differ in meaning from (4b ), since it means 'At every now in P the flying arrow is over against a space equal in length to the arrow', and again the space which it is over against may be different at different nows. So (7) will

47

be taken as meaning 'At every now in P the flying arrow is at rest', but since it comes from (5b) the place in which it is at rest may be different at different nows: at one now it may be at rest in one place and at another now at rest in a different place. However, according to Aristotle's view, which seems right7, it

5 The atomic theory of space and time

makes no sense to say that an object is at rest in a now. Aristotle is here certainly taking now in the sense of temporal point. So if 'now' is interpreted

as 'temporal point' (7) cannot be taken in other than the natural sense, and the argument accordingly fails. What if 'now' is interpreted as 'atomic period'? This will be considered towards the end of the next chapter which will be concerned with some of the possible implications of a rejection of the infinite divisibility of space and time. Notes 1

In Aristotle's statements of the argument (239b5-9 and 239b30-33) he appears to be using the expressions 'is at rest' ('eremei'), 'is motionless' ('akineton einai') and 'is stationary' ('hesteken') interchangeably.

2

Vlastos's view that in reconstructing Zeno's argument it is necessary

3

4 5

6

7

to look also at a certain non-Aristotelian source is discussed in Appendix D. Vlastos has argued that the expression 'en toi nun' was probably not in the Zenonian original (Vlastos (1966, [1]), pp. 6-7). That is not universally accepted, but in any case, as indicated earlier we are concerned primarily with the paradoxes as interpreted by Aristotle. Idonotfind plausible Lear's view (Lear, p. 91) that 'to nun' means 'the present moment'. For an argument based similarly on the continuity between the two passages see Pickering, pp. 254-255. For example Vlastos; see his argument innate 20a (Vlastos (1966, 1), p. 9) [;note 21 in Allen and Furley version]. In opposition to Aristotle's view that an object cannot be either moving or at rest in a now it has been pointed out that there are good senses in which an object can be said to be moving (or doing something that entails moving, e.g. accelerating) in a now, i.e. at an instant. However, these senses are secondary, depending on primary senses of moving and resting in respect of which Aristotle's view is right. (For more detailed and further discussion see Vlastos (1966, 1), pp. 13-16, Barnes (1982), pp. 280-281, Owen, pp. 158-162.)

48

!

Aristotle held that space and time are infinitely divisible (Physics 231a21-b17 and elsewhere), and according to Sir David Rossi is the first person known to have stated this proposition clearly. Historically the opposing view that space and time are not infinitely divisible but have ultimate indivisible elements has taken one of two forms according as the ultimate elements are thought to be: (a) dimensionless spatial points and dimensionless temporal mstants, or (b) hnes and distances so short that though they have some length they are indivisible and are not exceeded in shortness by any line or distance and periods of time which despite having duration are indivisible and are not exceeded in shortness by any period of time. (a) however will not be under consideration in this chapter, since it is not the view of those commentators referred to in the previous chapter who interpret 'now' in the Flymg Arrow argument as meaning atomic period; nor again is it what, as we shall see~ some believe to be the assumption underlying Zeno's argument in the Stadmm. Two remarks may, however, be made in passing: (i) it was the Vtew taken by some of the so-called 'indivisibilists' writing in Latin in the 14th century 2; (ii) in recent centuries at least mathem~ticians have tended to assert two propositions both of which were rejected by Aristotle: that a line can be composed entirely of points, and that that is no bar to infinite divisibility. Non-scientists, and perhaps this is true of some scientists also, tend to have no very elaborate or developed ideas of space and time, and for the most part would probably assent to the proposition that space and time are infinitely divisible. Infinite divisibility is part of what we may call the ordinary man's conception of space and time, or simply the ordinary view or theory. The opposing view, which in the variant (b) mentioned earlier, is our present subject, is criticized, insofar as lines are concerned, in an Aristotelian treatise (previously referred to in Chapter 2) which is entitled in Latin De 49

different ~ategory) are the forms of atomism that specially concern us now: th~ ~tom1c theones of space and time. The fragments that remain of the wntmg~ of the earliest atomists, Leucippus (floruit circa 440 BC) and Democntus (c. 460 - c. 370 BC) are uninformative about their atomistic doctrmes, and we are dependent on what later writers have told us. It seems doubtful whether they went b~y?nd material atomism, though they may have espoused an atomism of mmrmal parts. There is no indication that they held that there were atoms of space or time. With regard to Epicurus (342 to 270 BC) we are better off: though only a fraction of his allegedly voluminous works remam we have in his Epistula ad Herodotum (Letter to Herodotus)) a summary account of his atomistic theory. The Letter to Herodotus is concerned mainly with material atomism, but how far it goes beyond this is not entirely clear. At least one passag~ 4 has been taken to imply minimal parts atomism, and there are apparent md1c~hons of atomism with respect to time and motwn. Further mdeed, Srmphcius does attribute an atomism of magnitude, trme ~d m~twn to th~ Ep1cureans 5 . Withal, however, there is nothing either

lineis insecabilibus and in Greek Peri atomon grammon~ both these titles being

naturally rendered in English as 'Concerning indivisible lines'. From the Greek title as well as from many references elsewhere in the Aristotelian corpus to the belief in 'atomoi grammai' we may take the word 'atomic' and refer to the theory we are going to examine as ~the atomic theory of space and time'. The joint phrase is a convenient one, since although conceptually there are distinct atomic theories of space and time the two go naturally together. For terminological guidance it may be added that the terms 'extension' and ~magnitude' are sometimes used instead of 'space' in this and similar contexts; and the meaning of 'discontinuous' may be sufficiently indicated by saying that 'the theory that space and time are discontinuous' is to be regarded as equivalent to 'the atomic theory of space and time'. We shall be examining additionally a related theory about motion. This could be referred to as the atomic theory of motion, and might naturally be taken, on the analogy of our definition of the atomic theory of space and time, to be the theory that motion is not infinitely divisible but that every movement consists ultimately of indivisible movements which are little movements so short that they are not exceeded in shortness by any movement. However, though that may have been the nature of the theory that Aristotle had in mind and rejected when asserting the infinite divisibility of motion as well as of space and time, it will be seen later that there is a special account of motion which seems to be entailed by, or at least to fit in best with spatial and temporal atomism, and which is to some extent misrepresented if it is thought of as exactly parallel to the atomic accounts of space and time. So we should regard the expression 'the atomic theory of motion' as meaning 'the theory of motion that accords best with spatial and temporal atomism'. The expression 'the atomic theory of space and time' may suggest that we are to be considering a well-developed theory of which a full exposition exists. This would be misleading. No classical text expounding fully and succinctly an atomic theory of space and time has been preserved, and whether any such exposition ever existed must be doubtful. It may, however, be worth while, in illustration of the limitations of our knowledge, to set out briefly some facts about the views of ancient and mediaeval atomists. Broadly, there are three varieties, or rather levels, of atomism. The basic atomic theory is material or corporeal atomism; according to this view everything is composed of particles that cannot be cut up or in any way divided into physically separate parts. A refinement of material atomism, which can be called minimal parts atomism, maintained that the corporeal atoms though not physically breakable, consisted ultimately of parts that were theoretically distinguishable from one another, but were minimal in the sense that each did not contain within itself any even theoretically distinguishable parts.3 Different again (and perhaps to be regarded as in a

50

In Ep1cur_us s own wnhn?s, as we have them, or in what later writers say about Ep1curus or the Ep1cur~ans, that could be interpreted as a complete theory. The same IS true of Ep1curus' s follower, the Roman writer Lucretius (94 to 55 BC), .whose magnificent poem, the De rerum natura (On the nature of thmgs) IS mspued by the doctrines of his master. One writer who can certainly be named as having advocated, or at least contemplated, atomism of space, time or motion is Diodorus Cronus (fl. c. 300 BC) We know about him from fragments and from what we are told by Sextus Empiricus (2nd century AD) and others. He was evidently aware of som~ of the .Important features of the atomic accounts of space and time, parhcularly ~respect of the implications for motion', but again nothing like a full expos!lion of a theory has survived. That there was in the fourth or third century BC something in the nature of a school of spatial atomists is evidenced by the existence of the treatise De hnets ~~secabilibus. Unfortunately the author was hostile to the doctrine of mdiV!S!ble lines; he is content to point out difficttlties involved in the doctrin.e, and is . not concerned to consider how, by. modification or otherwise, they might be surmounted7. We have been speaking mainly about Greeks in the classical and Hell~nistic periods. In a different age and culture there was some espousal of spahal and temporal atomism by Muslim philosophers within the period 800-1200 AD, and the best approach to their ideas (apparently even for those wh? know Arabic 8 ) seems to be through the Jewish philosopher Marmomdes 9 (1135-1200 AD). Again we are handicapped by the fact he was opposed to the atom~sts and ,though well aware of certain objections that had been made to the1r theory, and of some of their replies, was not specially

t-· !~

II;' i'; :

51

concerned to consider whether any fully coherent theory could be

divisibility, and therefore that the respective atomic theories with their

constructedlO.

claims that there are indivisible lines and indivisible durations are inherently

Another movement in favour of an atomistic doctrine with regard to space at least occurred in the early fourteenth century, and aroused considerable controversy, largely focused on geometrical problems. For the most part the indivisibilists, as they were called, maintained that the atoms of which space is composed were unextended (lines were composed of points, surfaces of lines and three-dimensional shapes of surfaces) and so fall outside the scope of our consideration, as explained at the beginning of this chapterll. There was at least one exception: Nicolas Bonet (Nicolaus Bonettus or Bonetus)l2. Unfortunately Bonet's own writings are not easily accessible, but I have not found in the literature about them any indication that they are likely to contain any valuable theoretical presentations. It may be said then, that (with a mild reservation concerning Bonet) in the whole area of ancient and mediaeval atomism, although we can find scattered bits of theorizing, we do not have any thorough exposition by a writer disposed to explore sympathetically the possibility of a consistent and comprehensive development of the basic ideas. So much for earlier periods. Atomism of the relevant kind has found favour also in recent times. Some physicists have indeed suggested that space and time may be discontinuousl3, and the terms 'hodon' (from the Greek word for 'road') and 'chronon' have been invented to apply respectively to the smallest unit of space and the smallest unit of time, but this seems to be a suggestion thrown out rather tentatively under the influence of the quantum theory : there does not appear to be a well-developed theory supplying answers to possible objections. It may be mentioned also that among a number of French writers who towards the end of the nineteenth century participated in a vigorous discussion of Zeno' s paradoxes one at least, Fran.;ois Evellin, espoused the theory of minimum elements. Evellin was aware of and discussed some of the inherent problems; his conception of the atomic theory of movement will be discussed later. It should be evident after this broad survey that we are not dealing in the present chapter with any well-known historically actual theory. The aim is rather to point out certain problems implicit in the view that space and time are discontinuous and to make some suggestions about how a theory involving some sort of solution to these problems might be developed. The title of the chapter, particularly insofar as it concerns space, should be understood in the light of these remarks. The atomic theories of space and time will be discussed separately, but it is necessary first to deal with certain questions about coherence that apply in a similar way to both theories. Some would maintain that the concepts of a spatial distance or a line and of a temporal period or duration both include

52

absurd. However, the line of thought from which such a view results may be faulty. To take time first: durations are regularly associated with numbers (100 years, 3 days, 10 minutes, 1 second); and since any number, however small, can be divided by any whole number (for example, by 2), resulting in a new number, a proper or improper fraction within the system of rational numbers, it is e~sy to get into the way of thinking that to each resulting numencal fractwn, however small, there must correspond a fractional duration; and then of course it follows that since there is no smallest fraction there is no shortest duration. Similar thinking may lead to the rejection of the atomic theory of space. But in both cases the reasoning may be faulty because of the fals1ty of the premiss. Consider (a wider version of a question raised in Chapter 2) what can be meant by a fraction or division of a spatial distance or ~period of tirr;e. In the case of space can anything else be meant by dividing a !me than puttmg a mark, or a number of marks, of some kind between the two ends of the line? Time is not in all respects the same, since one cannot set about making marks within a period of time that is already complete. Nevertheless, ':'arks of one kind or another can be made within a period of hme that IS bemg passed, and so a division of the period is effected; for example, the jerks and the ticks of the second hand of a watch are visible and audible marks that divide a period of time, e.g. a minute, into parts. If this is the sort of meaning that has to be given to the division of a spatial distance or of a period of time, then an essential premiss of the argument may be false, for what guarantee can there be that to each numerical fraction there corresponds a possible physical division of a distance or a duration? Space We look first at the atomic theory of space, and enquire what must be the spa hal characteristics of a world in which there is a minimum and indivisible unit of length, which may be called a minimum space unit or SU. We shall assume, there being no reason to question this, that for the atomic theory space is three-dimensional. It will be sufficient, however, for us to consider a flat lwo~dimensional cross-section of space which we may think of as being m a honzontal plane. In this cross-section what we shall call objects or bodies are flat, two-dimensional shapes which represent the bottoms or tops or other cross-sections of objects or bodies in the three-dimensional space of which this is a cross-section. We looked a moment ago at a possible argument purporting to show that the atomic theories of space and time are internally inconsistent, and it was suggested that that argument need not confidently be regarded as cogent. We may now ask with respect to the atomic theory of space whether, even if 53

internally consistent, it is consistent with certain common notions about

space which it would be difficult to abandon. A few instances may be mentioned in which common sense intuition, combined in some cases with elementary geometrical reasoning, is in conflict with the atomic theory of space. ABCD (figure 5.1) represents a square with side 1 SU in length. The theorem of Pythagoras can be used to show that the diagonal AC is more than one SU and less than two SUs long. If then we mark off along AC a segment AE 1 SU long, the remaining segment EC must be a line less than 1 SU long. But according to the atomic theory this is impossible: no line is shorter than 1 SU. In the above reasoning we have been using the theorem of Pythagoras, and it might be objected that we have no right to do so since that theorem belongs to Euclidean geometry, essential to which is a proposition that directly contradicts the basic tenet of the atomic theory, the proposition namely that between any two points on a line there is a third point. If the theorem of Pythagoras does indeed depend on that proposition, that is a fair criticism, but what may be regarded as common sense geometrical reasoning, apparently independent of the proposition in question, seems to lead to a similar result. In figure 5.2 ABCD is again a square with side 1 SU in length and diagonal A C. AB is produced to E, so that BE = AB. Also BE = CB. A point F is marked off along AE so that AF =AC. It seems obvious that F will be beyond AB. Again, it seems obvious that AC is shorter than AB + BC. Hence F cannot be as far along AE as E, for if it were AC( =AF) would be greater than or equal to AB + BC. Therefore the point F lies between B and E, and so divides a line 1 SU in length, which again according to the atomist is impossible. A more simple example is seen in figure 5.3. AB is a distance of 1 SU measured along a curve, say an arc of a circle. It seems obvious to

common sense that a straight line chord can be drawn joining A and Band that the distance between A and B will be shorter along the chord than along the arc, i.e. shorter than one SU, again in contradiction of the atomic theory. These examples show that the atomic theory as we have been stating it is in conflict with common and Euclidean notions about space. If it is to be a theory that is at all worth considering it must draw in its horns to some extent and make something less than the absolute claim that there is a shortest length. One way in which it might seek to do so will now be described. It has been stipulated that we are confining attention to a flat cross-section of space. Let us now suppose that this cross-section is covered by an imaginary grid or network. The holes of the network must be of uniform size and shape. There may be various possibilities for the uniform shape but it will be convenient to suppose that the grid is a network in which all the holes are square. The concept of such a grid is all-important for the atomic theory

54

D

c

A

B

Figure 5.1: A shorter than shortest?

D

C

~--·-/l'',

/

..

/ /

j'

', '

\\

A

B

F

Figure 5.2: Indivisible divided?

A

B Figure 5.3: Another shorter than shortest?

55

E

as we are envisaging it. In the first place, the theory stipulates that all measurements are to be made along lines of the grid, up or down or across,

and the reduced claim of the theory is that there is a minimum and indivisible length (an SU) that can be so measured. Any two adjacent parallel lines of the grid are one SU apart, measured along a line of the grid that cuts them at right-angles, and any distance measured must begin and end at a junction of two lines of the grid. In the second place, the theory has a view about the shapes and positions of objects which is definable by reference to the grid. Each square is either full or empty, and every square occupied at all by part of an object is fully occupied. It is all rather like a knitting pattern or a design for a canvas-based rug, or like certain kinds of electronic illuminated pictures or writing. Every object has an outline every side of which coincides with part of a line of the grid and ends at a line of the grid. It can be seen that each side of an object is a straight line which is at right-angles to two adjacent sides and has a length which is an integral multiple of the minimum unit of length, i.e. is an integral number of SUs. The characteristics that have been mentioned of objects in respect of shape are those which an object has in reality. To the eye, however, many objects may appear to have quite different characteristics. Thus some sides of objects may appear to be curved, others to be sloping in relation to other sides rather than at right-angles to them. These are illusions. An object which appears circular has in reality a shape similar to that shown in figure 5.5, and one whose surface has the shape of a right-angled triangle (with the arms of the right angle lying along lines of the grid) is in reality similar to that in figure 5.4. In both figures it is to be understood that in the stepped sides the steps are so tiny that what is in reality regularly or irregularly serrated appears to the ordinary eye as a smooth line, curved or sloping. We pass on now to the topics of time and movement. Although our main discussion of movement will necessarily follow that of time, there are certain features of the atomic theory of movement which it will be convenient to draw attention to at this point. Accordingly our discussion of motion will be in two parts separated from one another by a section on time.

Figure 5.4: Atomic theory slope

Figure 5.5: Atomic theory circle

12

Movement We must think about two basic kinds of movement: translation and rotation; the two may occur independently or in combination. In simple translation (translation without rotation) an object moves from one place to another, and the movement is uniform throughout the parts of the object. Consider the minimum parts of an object each of which occupies initially one of the smallest squares of the grid. If in the movement of the object as a whole one particular part moves so many squares across and so many squares up or down, say x squares to the right and y squares down, every other part does 56

Figure 5.6: Atomic theory clock face

57

exactly the same. The shape of the object remains unchanged. In simple rotation, on the other hand, the object as a whole stays in the same place but

its parts change position and in different ways. A clear example is a wheel, e.g. a roulette wheel, which rotates on a fixed axle. Though the wheel as a whole remains in the same place, its parts move but not uniformly. For example, if the wheel has spokes the outer parts of a particular spoke move through a greater distance in a given time than the inner parts: they move at different rates.

two events, ~r h:o intervals, each of one TU in duration, can overlap one

another. For If this happened there would be the absurdity that one end of each e~ent would s~r~e to divide the other event into two parts, thus causing a dtvtswn of an mdivtstble TU. This means that for an understanding of the atomic theory of time we need something like the imaginary grid in the case of space, an imaginary standard clock ticking away TUs, i.e. indicating in some way the beginning of each TU, and the beginning and end of every event being exactly synchronous with the beginning or end of some tick.

We shall see that the atomic theory of movement in g,eneral is at variance

with our ordinary ideas. This is very markedly the case in respect of rotation. The theory has to say that rotating objects change their shape as they rotate; for otherwise their edges will (at least for most of the time) cut across the lines of the grid obliquely, involving distances not measurable in exact

Motion

On the face of it time measurement is applied to two different kinds of thing: on the one hand to the duration of an event, on the other hand to the interval between the beginning or end of an event and the beginning or end of another event. However, the duration of an event is the interval between its beginning and its end; so we can say that really all time measurement is of the interval between termini of events: for example, between the beginning and end of the same event or between the end of one event and the beginning of another. What the atomic theory holds is that in time measurement of whatever variety there is a minimum time unit (a TU) such that that the duration of any interval is an integral number of such units. It follows that no

We return now to movement. According to what we may call the ordinary view motion is essentially continuous. What this means is that if an object moves and its motion lasts for a certain period P, then there is no interval i, however short, within the period Pin which the object is not moving. In a loose sense we may think of certain phenomena as exhibiting discontinuous motion. If a train goes from A to B in an hour and stops en route for two minutes at each of three stations, the process as a whole may be thought of as discontinuous movement. Again in some electric clocks the hand moves jerkily: there is an iteration of a period of rest followed by a very quick movement to a new position. But according to the ordinary view to refer to these phenomena as discontinuous movements is only a manner of speaking. The reality is periods of genuine movement, long or short, alternating with periods of rest; and within the periods of genuine movement the movement is continuous in the sense explained. Can the atomic theory of space and time accept the theory that motion is essentially continuous? According to the atomic theory of space every object either occupies one of the smallest squares of the grid or has parts each of which does so. We may think now either of one such minimal part among all the parts of an object made up of a plurality of minimal parts or of a separate complete object which occupies a minimum square. The latter will be more convenient. We suppose accordingly that there is an obje~ct 0 occupying a minimum square of the grid, and so being square in shape with a side one SU in length. Let us suppose that it moves from this square to the next along a line or column of the grid, and that after one TU the move is complete: 0 occupies the next square of the grid. What account may the atomic theory give of this phenomenon? On the face of it there may seem to be two possibilities. One is for the theory to say that 0 has been moving continuously throughout the TU. One objection to this account is that there is awkwardness, at the best, about the phenomenon of crossing. Two objects 0 and P cross one another when, for example, having started in adjacent columns of the grid, each arrives in the column in which the other was (figure 5.7). At some instant of time 0 and P must have been directly

58

59

numbers of minimum units. Consider (figure 5.6) a clock face which is set normal to the grid with the 12-6 diameter coinciding with a vertical line of the grid. Let us suppose that the minute hand of the clock is rectangular in shape and that it is pivoted at its bottom right-hand corner, with its righthand edge, which is to indicate the time, pointing to 12 o'clock. In this position, of course, each side of the hand and each end is a straight line. Now let the hand move round towards 3 o'clock. According to the orthodox view the shape of the hand is unaltered, and at 1.30, for example, the right-hand (now the lower edge) cuts diagonally across the squares of the grid. The atomic theory cannot have this. It has to say that as the hand moves round its shape changes so as to conform to the lines of the grid. Focusing on the lower edge alone we may say that as it moves from 12 o'clock to 1.30 it becomes increasingly, and from 1.30 to 3 decreasingly, serrated, until at 3 o'clock it is again a straight line. Figure 5.6 illustrates successive shapes of the timeindicating edge at different times. To avoid unnecessary clutter the other side and the ends of the hand and the rim of the clock face have all been drawn as they would be under orthodox assumptions.14 Time

opposite one another. If both have been moving continuously at the same speed this must have been when each was half in one column and half in

another; i.e. when 0, for example, occupied one half of the width of one column and half of the width of the other. But the width of each column is one SU and there is no such distance as half an SU. It follows that the atomic theory cannot allow simultaneous continuous movement by 0 and P in opposite directions. If continuity is retained, the movements of 0 and P must be made consecutive. 0 first moves to the right and comes opposite toP, and P then moves to the left into the column vacated by 0. (Or the other way round) Adopting this expedient has consequences that are difficult if not impossible to accept. It means for one thing that the movements not only of 0 and P but of all objects in the two columns in question, and indeed of all objects in every pair of columns and every pair of lines, and so of all objects in the world must be held to be co-ordinated by some sort of pre-established harmony, or however, in such a way that crossing does not occur. However, there is a more fundamental and more simple objection to the hypothesis of continuous movement (within the assumptions of the atomic theory). This is that the concept of continuous movement throughout a TU is incoherent. For it can only mean movement in every part of a TU, and a TU by definition has no parts. As we have just said, the concept of continuous movement under the assumptions of the atomic theory appears to be incoherent. And even if this is not so, the consequences of supposing that there can be continuous movement and that yet crossing can be altogether eliminated seem completely unacceptable. We turn to an alternative account in which, although the basic concept (unlike that of continuous motion per se) is unnatural there is no obvious incoherence or awkward consequence. If continuous motion is ruled out the atomic theorist is left with the alternative theory that motion is essentially discontinuous. On the face of it there are two forms which such a theory may take. One is that motion is essentially cinematographic. Everyone is familiar with phenomena that can fittingly be said to be instances of cinematographic motion. They occur when a supposedly moving object occupies different places at different times. If the

Figure 5.7: Crossing

60

places are suitably related and the times are suitably related, an illusion of motion is given. A cinematographic film of a motor car moving from A to E is

of course a sequence of still photographs in which an image of the car appears at a sequence of positions between A and E. The same is true of any moving object but in view of our interest in Zeno's third paradox it will be suitable to take the example of an arrow. A film of Zeno's flying arrow according to this theory would be a sequence of still photographs in which the image of the arrow appears successively at a sequence of positions between the bow and the target. However, the image does not move. The image of the car at B is a distinct individual from the image of the car at A, though similar to it. The image of the arrow a yard away from the bow is a distinct individual from the image of the arrow at the bow, though similar to it. Another example of cinematographic movement occurs when we have a row of electric light bulbs illuminated successively, and the impression is given of a single light moving along the row IS. From the orthodox point of view, of course, these and other so-called cinematographic movements are cases of apparent rather than real motion: real motion, which is continuous, is distinct from though simulated by such apparent motion. According to the atomic theory, however, in the version we are now to consider, it is not the case that cinematographic motion is found in only a few special phenomena such as those that have been mentioned. On the contrary, the theory affirms, all motion is essentially cinematographic. For example, it is not just that the motion of the image of the arrow in the film is cinematographic: the very motion of the arrow itself (of which the film is a representation) is cinematographic. Now this is an extremely bold theory. For it entails that just as the image of the arrow in any one frame of the film is an individual entity distinct from, though qualitatively similar to, the image of the arrow in the preceding frame, so the arrow itself at any one place in its flight is an individual entity distinct from (though qualitatively similar to) the arrow at preceding places in its flight. In other words the arrow, and similarly every object normally regarded as a continuant, is being (in the words of Berkeley's hypothetical critic in section 45 of the Principles of human knowledge) 'every moment annihilated and created anew'. ~ Let us spell out the theory in relation to a minimal object 0 (a tiny bit perhaps of the arrowhead) which occupies initially a minimal square A1 of the grid, and which moves without change of direction along a single line of the grid, coming to rest finally in another minimal square B, not adjacent to A1• The cinematographic account of this movement may be stated in two propositions: (1) the movement of 0 from A 1 to B consists in the occupancy by 0 one after another of a sequence of squares A 1, A2, .•. , B, starting with the occupancy of A 1 and ending with that of B, each occupancy lasting for an integral number, greater than or equal to one, of TUs; (2) proposition (1), though a convenient way of talking, is misleading to the extent that there is 61

in reality no single object 0. What there is (figure 5.8) is a sequence of individual objects: 0

1r

0

2, ... ,

and so on, distinct from, though qualitatively

similar to, one another, such that 0 1 occupies square A1 for a time and then ceases to exist, whereupon without any interval of time 0 2 comes into existence and occupies A 2, the next square in the sequence for a period before in its turn ceasing to exist and being followed by 0 3 in the next square A,, and so on. This reformulation (which may be called the basic cinematographic theory) is intended to spell out what is meant by saying that 0 is every moment annihilated and recreated. It conveys the mam 1dea, but it is arguable, as will be seen later (page 67), that the words 'without any interval of time' are not right and that a modification is required which consists in replacing proposition (2) by the following: (2) proposition (1), though a convenient way of talking, is misleading to the extent that there is in reality no single object 0. What there 1S (hgure 5.8) is a sequence of individual objects: 0 1, 0 2, ••• , and so on, distinct from, though qualitatively similar to, one another, such that 01 occup1es square A 1 and then ceases to exist, whereupon after a minimal interval of time 0 2 comes into existence and occup1es A 2, the next square m the sequence, for a period before in its turn ceasing to exist and being followed by 0 3 after another minimal interval in the next square A,, and soon. In the foregoing we have stipulated that the minimal object 0 should be thought of as occupying a sequence of positions: A1, A 2, ••• , B. Each position was to be a minimal square of the grid, but it was not stipulated that each of the squares should be adjacent to its predecessor or successor in the sequence, nor again that this should not be the case. Now there are here two possible variants of cinematographicity for the atom1st to choos~ between, which may be referred to respectively as single-space and multiple-space cinematographicity. Suppose we use the term 'jump' to apply to. the behaviour of 0 as described above: '0 jumps from square X to square Y' 1s to

A A A. A.

··rrrrr·1·1

Figure 5.8: Cinematographic movement

62

mean that after 0 1 has occupied X and ceased to exist 0 2 comes into existence in occupation of Y. If the atomist in his account of motion says that 0 may

jump only one square at a time that is single-space cinematographicity; if a jump of more than one square is allowed that is the multiple-space variant. Which variant to opt for seems to be a problem for the atomist. On the one hand, he will naturally wish to keep as close to the orthodox non-atomic theory as is possible without sacrifice of basic principles. According to the orthodox theory ,of course, an object cannot move along a straight line from X toY without passing through every point at which the line XY might be divided, and the nearest the atomist can come to this is to insist that 0 must occupy in turn every minimal square from X to Y. On the other hand, in this single-space cinematographic theory there is a difficulty for atomism the essence of which has been pointed out by Sorabjil'. If 0 can jump only one square at a time, its maximum speed, and this will be true of any object, will be one SUper TU. Now suppose 0 is not a minimal-sized but a large object in relation to which another object is moving; suppose for instance that 0 is a ship along the deck of which a smaller object P is moving in the same direction as 0. Then if 0 is moving at the maximum speed of one SUper TU (in relation to the grid, of course), then P must move in one TU over more than one SU, thus infringing the supposed requirement that jumps are limited to one minimal square, i.e. one SU, at a time. There may be more than one possible way by which the atomist might seek to avoid this difficulty, but the simplest might be to abandon the requirement that jumps be limited to one SU only, i.e. to adopt the second variant. Such essentially is the cinematographic theory. It was indicated earlier that it is one of two alternative versions of the atomist's theory of the discontinuity of motion. We now turn briefly to the second. This may be described as single-space cinematographicity (in its basic formulation) with the element of perpetual annihilation eliminated. It is the identical individual 0, and not a sequence Ov 0 2, ... , and so on, that occupies successively A1, A 2, .•. and so on, without any interval between its complete occupancy of one square and its complete occupancy ohhe next. This theory, which seems to have been the view of the French writer Fran

Then

Figure 6.3: Position 11 (Mansfeld)

which indeed is the one he uses (on page [336]18) for the starting position in what I shall be calling the supposed potential argument. Mansfeld maintains that Zeno based his argument leading to the paradox about time on the initial conditions stated at the beginning of the Stadium passage, i.e. in I, together with a proposition which is recoverable from V, in particular from the second sentence of V, ' ... Each of these two is alongside each individual other for an equal time', and Mansfeld argues reasonably that the two referred to are C1 and B1. Here now is Mansfeld's statement of Zeno's argument (page [335]17): This recovered premiss, taken together with the initial conditions stipulated by Zeno in(!), entails that B1 has to be alongside (or to pass) C1 in time t, or that B1 in I passes C1. Cv however, is also moving, at the same speed as B1 (and as the other equal bodies concerned), but in the opposite direction; therefore, C1 in I, gets as far as the last of the B's. 82

(R') Each of C1 and B1is alongside each individual other for the time 1. R' and!' entail that (i) B1 has to be alongside (or to pass) C1 in timet.

(ii) C1, however, is also moving, at the same speed ~s B but in the 11 opposite direction; therefore (iii) C 1 in t gets as far as the last of the B's [B2]. Consequently (iv) cl has passed all the B's in t, whereas

(v) B1 in I has passed only one C, i.e. half the C's. 83

Since

~ssump~,i~n is th~t each "bodyo is coextensive with the place or

(vi) tis the time which, according toR', a moving body needs to pass one and only one other body moving in the opposite direction,

square 1t occupies. A move can only be from one such place into another....

and (from iv) (vii) C 1 has passed two such bodies in t, and (viii) C1 only needs pass one body:

I

h

t to

(ix) its "time is half".

ix is Aristotle's wording in 240a12. His statement of Zeno's paradox is that half a given time is equal to its double, which is expressible in the form: I/ 2t ~ t. We may spell out here how by an obvious extension of the above reasoning this consequence is obtained.

viii is equivalent to (x) Time taken by C1 to pass one body~ 1/2 t. ButfromR

Body, motion, space, and time, therefore, in the world of Zeno's Stad1um, are t~eated. as being discrete, not continuous. The bodies represent spal!al umts. Motion comes in jumps, each of which corre~ponds to one such spatial unit: a body, according to the recovered prem1ss, moves one unit of space (one "square") equal to itself. Time hcks away m corresponding units: the unit of time is the time needed to get from one spatial unit into the next. This is what we have called the atomic theory of space and time, and discussed m Chapter 5. Mansfeld uses it in support of Zeno in the following passage (page [336]18):

Ze~o, of course, could also have argued that there is no time t during wh1ch C1 and Bl are alongsid~ one another. When both move past each other at equal speed, covermg the minimum distance (one body) durmg I, the following sequence occurs:

(xi) Time taken by C 1 to pass one body~ t. Therefore from x and xi (xii)l/zt.~

t.

The argument as a whole is fallacious. Consider the sequence ending in iv. iv follows from iii all right, but iii does not follow from the earlier propositions. To get as far as the last of the Bs C 1 would have to pass two Bs, and this would have taken it the time 21, not just I; for it can be seen that according toR' (and particularly clearly as spelt out in vi, and as Mansfeld' s italics emphasise) I is the time that a moving body takes to pass one and only one other body moving in the opposite direction. Accordingly, the deduction of iii, and so of iv, vii, viii, x, xi. and finally the paradox xii, is invalid. An argument could be invalid, of course, and nevertheless have been advanced by Zeno. However, in this case the proposition iii not merely doesn't follow from the premisses: it is incompatible with them. The argument seems to be more than just invalid: it is not even specious or plausible. Mansfeld is certainly aware that the argument is in some way defective. He says (page [335]17): This paradox is feasible only on the assumption that motion consists of a series of moves, as of pieces on a chess- or draught-board: .... A further 84

~ ~

->

->

Figure 6.4: Sequence of positions '

There is no time between 11 and 12, so there. is no time either at which th following situation occurs: e

l

C1 B2 B1

Figure 6.5: Non~occurring position

85

c2

But this is against common sense: surelyr B1 and C 1 must be alongside one another before CI can be alongside B2 (and BI alongside C 2 )! Common sense demands that [the situation shown in the latter diagram] occurs in t, and the recovered premiss not only allows this, but positively demands it. But CI is moving as well, and at equal speed. Common sense, thereforer demands that it advances one spatial unitr viz., one body, and so posits it alongside B2, which, following Bv has, of course, moved too. Consequently, C 1 is alongside B1 and B2 simultaneously, which is patently absurd: the situation presupposed by Zeno' s paradox cannot be represented in a diagram! Zeno has proved his point, viz., that motion cannot occur.

but completely implausible. They cannot reasonably be attributed to Zeno, and accordmgly Mansfeld's attempt to establish his thesis, that Zeno's fourth paradox argument involved two rows of bodies only, must be deemed unsuccessful. We tum now to another account of the manoeuvre. Aristotle's account of the starting position, which by the ordinary interpretation is taken to indicate the figure 6.1 position, could alternatively and quite naturally be read as indicating the position shown in figure 6.6 below.

/A /A /A /A I

Since the passage just quoted is said to show how 'Zeno ... could have argued', we may refer for convenience to the argument it presents as 'the supposed potential argument', distinguishing it from the previous argument (that reformulated in I', R, R' and i- xii) by calling the latter 'the supposed actual argument'. Concerning the supposed potential argument the two main comments that I have to make are the following: (1) Even if we assume for the sake of argument that the reasoning set forth is sound on the basis of the atomic hypothesis, it has not been shown that this was the reasoning us~d by Zeno to reach the paradoxical conclusion in his fourth argument. For Anstotle says firmly that Zeno's paradox was that half a given time is equal to ~ts double, whereas the paradox arrived at by the reasoning we are lookmg at IS a different one, namely that C 1 is alongside BI and B2 simultaneously. Possibly the former could be derived from the latter, but that is something that has not been done but would need to be done before Mansfeld' s contention could be accepted. (2) I think, however, that the reasoning is not sound. Let us consider it. There are two propositions concerned: (i) in the l!me t the movement of the two rows of bodies results in the situation shown in the right-hand part of the upper diagram, where C 1 is alongside B2; (ii) in the time t the movement results in the situation depicted in the lower dmgram, where C 1 is alongside B1. Now, it is certainly true that, on the hypothesis ~f the atomic theory, i is correct. But the use of ii must surely be quesl!oned. It IS being used as a premiss (based on common sense), but it is incompatible with i and so with the atomic theory from which i is deduced. So the paradoxical conclusion which we may call iii, namely that C1 is simultaneously alongs1de B1 and B2, is being deduced from two incompatible propositions. But from two incompatible propositions any proposition can be deduced (by use of the tautology p---7[-p---?q] (if p then if not p then q)): in other words no deduction so made has any value. In the light of my comments on the supposed actual and the supposed potential arguments I conclude that both arguments are not JUSt fallacwus

Ross gives ~areful consideration to this possibility but, after a complicated argum~nt mvolvmg a diversity of factors, concludes that despite .its attracl!veness 1t cannot be sus tamed. Much of his discussion is concerned with detailed linguistic and textual questions, but a crucial factor is the following. On the basis of Con 2 he takes the final position, as do most though not all commentators, to be that shown in figure 6.2. but if the Bs and the Cs start fr~m the figure 6.6 p?sition it is impossible for them to reach the figure 6.2 posihon; for by the hme the row of Bs and the row of Cs are directly opposite one another they will both be out to the right of the As. It follows that Zeno cannot have meant figure 6.6 to be the starting· position. One of those who are in favour of 6.6 is Jonathan Barnes. He differs from Ross in his judgements both about the linguistic and textual questions and about the logical consideration just mentioned. He sums up concerning the former as follows: ' .. : adoption of figure 5 [corresponding to 6.1 above] reqmres two changes m the text of [Physics 239b39-240a18], one of which is mostimplausible'6 . With regard to the latter he points out that, although it is certamly true that under the postulated conditions the 6.2 position could not be arrived at if the start is from position 6.6, nevertheless Zeno may have mistakenly thought that it could; and he sets forth an argument, basing it closely on the Stadium passage, which he thinks Zeno used to show that the figure 6.2 position was reached. We shall state and examine this argument,

86

87

IB IB IB I B I

->