Zeno's paradoxes [2 ed.] 0872205614, 9780872205611

Citation preview

Salmon, W. C. (Ed.). (1970). Zeno's paradoxes. Indianapolis: Bobbs-Merrill. TO MY DAUGHTER TORI Copyright© 1970 by The Bobbs-Merrill Company, Inc. Reprinted 2001 by Hackett Publishing Company, Inc. New material copyright© 2001 by Hackett Publishing Company, Inc.

All rights reserved Printed in the United States of America

06

05

04

03

02

2

01

3

4

5

6

7

For further information, please address:

Hackett Publishing Company, Inc. P.O. Box 44937 Indianapolis, IN 46244-0937 www.hackettpublishing.com

Cover design by Abigail Coyle

Library of Congress Cataloging-in-Publication Data

Zeno's paradoxes I edited by Wesley C. Salmon. p. em.

.

.

.

Originally published: Indianapolis: Bobbs-Merrdl, 11970]. Includes bibliographical references and mdex.

ISBN 0-87220-561-4 (cloth) -ISBN 0-87220-560-4 (pbk.) 1. Zeno, of Elea. 2. Paradoxes. 3. Space and time. I. Salmon, Wesley C. QC131 .Z46 2001 531--- Weiss used the idea of minimal sizes that could not be subdivided. He held this on empirical grounds, namely that there appears to be a minimal size in nature, i.e. the size of the electron. And in my earlier paper I put it forward that we cannot meaningfully speak of points below a certain minimal size. This type of approach, however, seems to be unjustifiable. Moreover it is not needed; the paradox can be resolved without assuming a minimal size. Put otherwise, we do not have to assume the finite divisibility of matter to resolve the paradox. Even if matter is infinitely divisible, the foregoing solution holds; for the points, in terms of which the paradox is stated, must be assigned a size, and therefore (despite our power to assign a smaller size indefinitely if we choose) we cannot pack more than a finite number of them into a finite distance.

87

they still, so long as they are physical and thus greater than zero, cannot be packed into a finite distance. And, if they are reduced to zero, they are no longer physical, but mathematical and no longer relevant. Nor can any device of 'infinitesimals' enable us to pack in an infinite number of them: 'infinitesi';'als,' 'vanishing quantities,' 'ghosts of departing quant1t1es of whatever minuteness greater than zero can always be amassed in too great numbers to be packed into a finite distance. This, I think, is the easiest way of seeing that Zeno's premiss cannot characterise a physical race: the 'and so on' is inapplicable because somewhere two neighbouring physical pornts. will touch each other and it will be impossible to subd1v1de the distance between them without altering the ass1gned s1ze of the points. These considerations obviously apply equally to the "Dichotomy." All this may or may not be what Black had in mind in (4) and (5); be that as it may, it seems to me to be needed for the complete resolution of the paradox. Black's argument, then, focussed attention on an important self-contradiction but he did not go on to use it. If I may put it so, he attended to Zeno's argument rather than to Zeno's paradox. §4. Summary and Conclusion. The idea that the limit of ~ninfinite series is attainable is a mistake. If a physical action IS rnterpreted by means of an infinite series, then the completion of the action is self-contradictory. Nonetheless, thoug,~ Zeno's premiss is self-contradictory, it retains plausiThe self-contrad1ct1on must convince us that an bility. infinite mathematical series cannot describe a physical distance, but we do not eliminate the paradox until we have seen in what way it fails as a description. To see this we draw attention to a contradiction between physical distance and mathematical distance given by an infinite mathematical series. This contradiction is brought out when we show that This is perhaps one of the reasons why dialectical philosophers could be~ heve that the physical world could be correctly and adequately described by self-contradictory categories. 11

•

J. 0. WISDOM

88

a physical distance can consist of only a finite number of physical points, i.e. points that have some size however small greater than zero. Hence the premiss fails to describe a physical distance at some place in the 'and so on': the physical description comes to an end when we have mentioned all of a finite number of points, but the mathematical description by an infinite series goes on without stopping. The peculiarity of the paradox lies in this, that the premiss which is false and even self-contradictory, gives a very ' strong impression of being true. Looking back over the order of the steps taken, we see that first the premiss was shown to be inapplicable (because self-contradictory) and second the explanation of this was found to lie in facts about physical points. But the most straightforward resolution would begin with these empirical considerations, after which we should go on to show that the premiss was inapplicable (because empirically false). In this way we could state the resolution in the simplest possible terms divested of allusion to subsidiary considerations, using on the idea of a physical point. It could be put as follows: A physical point, unlike a mathematical point, has some size, though this may be as small as we please. But, however small a physical point, since it has some size greater than zero, -an infinity of them cannot be packed into a finite distance. In particular an infinity of physical points cannot be packed to correspond to the .mathematical points described by an infinite geometric series. Hence an infinite geometnc series is inapplicable to a physical distance. /.e. a physical race cannot be described by repeated bisection, or Zeno's premiss is false.

I/

Tasks and JAMES THOMSON

Super-

Tasks "To complete any journey you must complete an infinite number of journeys. For to arrive from A to B you must first go from A to A', the mid-point of A and B, and thence to A", the mid-point of A' and B, and so on. But it is logically absurd that someone should have completed all of an infinite number of journeys, just as it is logically absurd that someone should have completed all of an infinite number of tasks. Therefore it is absurd to suppose that anyone has ever completed any journey." The argument says that to complete a journey you must do something that is impossible and hence that you can't complete a journey. It may seem that this argument is valid; and then, since the conclusion is absurd, we must deny one of the premisses. But which? Each has a certain plausibility. To some, it is more plausible that you can't complete an infinite number of journeys than that you must. These people infer the falsity of the first premiss from the truth of the second premiss and the falsity of the conclusion. To others it is more plausible that you must complete an infinite number of journeys than that you can't. These people infer the falsity of the second premiss from the truth of the first premiss and the falsity of the conclusion. The first party says "You couldn't, but you don't need to"; the second party says "You must, but you can." James Thomson, "Tasks and Super-Tasks," Analysis, XV (1954-55), 1-13. Reprinted by permission of the author, and the editor and publisher of Analysis. Footnotes have been renumbered to number consecutively throughout the article.

I I.

~-·

,,

89

JAMES THOMSON

90

This division was neatly illustrated in some recent num-

bers of Analysis. Professor Max Black 1 argued that the expression 'an infinite number of acts' was self-contradictory, and thus affirmed the second premiss. Unfortunately, he was not entirely convincing in his rejection of the first premiss. Messrs. Richard Taylor2 and j. Watling• rejected Professor Black's arguments for the second premiss, and at least part of their reason for doing so was that they were impressed by the plausibility of the first premiss. Unfortunately, they were not entirely convincing in their rejection of the second. Luckily we need not take sides in this dispute. For the argument stated above is not valid. It commits the fallacy of equivocation. There is an element of truth in each of the premisses; what the elements of truth are, it is the purpose of this paper to explain. Let us begin by considering the second premiss. Is it conceivable that someone should have completed an infinite number of ,tasks? Do we know what this would be like? Let us say, for brevity, that a man who has completed all of an infinite number of tasks (of some given kind) has completed a super-task (of some associated kind). Then, do we know what a super-task is? Do we have this concept? It is necessary here to avoid a common confusion. It is not in question whether we understand the sentence: The operation so-and-so can be performed infinitely often. On the contrary, it is quite certain that we do. But to say that some operation can be performed infinitely often is not to say that a super-operation can be performed. Suppose (A) that every lump of chocolate can be cut two, and (B) that the result of cutting a lump of chocolate in two is always that you get two lumps of chocolate. It follows that every lump of chocolate is infinitely divisible. Now I 1

2 3

"Achilles and the Tortoise" [reprinted above]. "Mr. Black on Temporal Paradoxes'' [108]. "The Sum of an Infinite Series" [117].

91

su,ppcJse that one of the assumptions A and B is false. For a molecule of chocolate is a lump of chocolate, and A is false, or it is not, in which case the result of cutting lump of chocolate in two is not a lump of chocolate, then B is false. But the conjunction of A and B is cerconsistent, and so it is certainly conceivable that a be infinitely divisible. But to say that a lump is in•mnlfehys;cal relation of later than, as well as an explanation of the relations be-

the two kinds of treatment, see Adolf Grlinbaum, Philosophical ProbSpace and Time [139], chaps. 7-8, and "The Anisotropy of Time," in and D. L. Schumacher, eds., The Nature of Time {Ithaca, N.Y.: Cornell uopmslly Press, 1967), pp. 149--186.

ADOLF

174

tics of the thermodynamic behavior of classes of systems each of whose members is quasi-isolated for limited time provides a criterion with precisely these qui red properties. For the numerous relevant details, I must refer the reader to other publications." Our analysis has shown that we are absolved from necessity of answering "how" a succession of events can cur by exhibiting a discrete sequence of occurrence. freeing ourselves from the limitations of the I criterion of time-order by means of the constructive elabo tion of an alternative, autonomous physical criterion, becomes clear that the dense temporal ordering of the stituent point-events of a motion is no obstacle whatever either its inception or its completion in a finite time. thus to ask "how" the motion can occur despite the temporal order, or what the runner does immediately after, leaving his point of departure or immediately prior to ', ing at his destination, is seen to be entirely unwarranted. Once the physical intelligibility of temporal denseness established, the following can be shown: There is no reason to infer that a temporally dense set of physical must be of infinite duration by virtue of thus being aPnS£,,', than for concluding untutoredly that a spatially dense set , physical points must be of infinite spatial extent on the: strength of its denseness.

Modern Science and Refutation of the Paradoxes of Zeno

17S

time are not quantized or granular in standard quantum

: every point of continuous physical space is a paten. y defined position of, say, an electron, and separately, every instant of a continuous time is potentially the ,, 1 of a physical event, speculations about hodons and ccllro•nc>n< notwithstanding. And since the space and time of quantum mechanics are each dense, the ordinal time-metrical features of the refutation of Zeno's "C)jd1oton1y" and "Achilles" which I have outlined are not superfluous or invalid by that theory, even though paradoxes involve quantum-mechanically proscribed, ,W•ell··dE,fiiim'd classical particle trajectories. In regard to speculations envisioning a genuine quantizaof space and time which repudiates their mathematical "'"'""'uity altogether, it is well to be mindful of the followperceptive comment by H. Weyl: So far, the atomistic theory of space has always remained mere speculation and has never achieved sufficient contact with reality. How should one understand the ~etric relations in space on the basis of this idea? If a square is

built up of miniature tiles, then there are as many tiles along the diagonal as there are along the side; thus the diagonal should be equal in length to the side.1o

QUANTUM THEORY

Standard quantum theory has discretized or quan some physical properties whose counterparts in c~l~~~~~~~~;, physics were mathematically continuous. When c with reports of speculations about incorporating minimal distances ("hodons") and times ("chronons") in some elaborated future form of the theory, this kind of quantization insinuated the belief to some that standard quantum chanics affirms the atomicity of space and time. But l6

Ibid.

a H. Weyl, Philosophy of Mathematics and Natural Science [130], p. 43. For a i of the Democritean conception of mathematical atomism and its · in antiquity, see S. Luria, "Die lnfinitesimaltheorie der antiken i " in Que/len u. Studien zur Gesch. d. Math. Astr. u. Phys., Abt. B,

JJ {Berlin: Springer, 1933), pp. 10&-185.

Zeno' Metric ADOLF GRUNBAUM

lena's Metrical Paradox of Extension

177

thinkers as Aristotle, 3 Cavalieri/ Tacquet/ Pascal, 6 Bolzano/ Leibniz, 8 Paul du Bois-Reymond," and Georg Cantor/ 0 to mention but a few. Thus, William James wrote: If, however, we take time and space as concepts, not as perceptual data, we don't well see how they can have this atomistic constitution. For if the drops or atoms are them~elves without duration or extension it is inconceivable that by adding any number of them together times or spaces should accrue. 11

Parad

... that being should be identified with the consummation of an endless chain of units (such as "points"), no one of which contains any amount whatever of the being (such as "space") expected to result, this is something which our intellect not only fails to understand, but which it finds absurd. 12

§1. THE PROBLEM

It is a commonplace in the analytic geometry of physical space and time that an extended straight-line segment, hav~ ing positive length, is treated as "consisting of" unextended points, each of which has zero length. Analogously, time tervals of positive duration are postulated to be agJgre·galtes of instants, each of which has zero duration. Ever since some of the Greeks defined a point as . which has no part," 1 philosophers and mathematicians . ·.· questioned the consistency of conceiving of an extended continuum as an aggregate of unextended elements. On long list of investigators who have examined this question i the context of the specific mathematical and philosophical theories of their time, we find not only Zeno' but also From Modern Science and Zeno's Paradoxes (Middletown, Conn.: W•esl