TIME, SPACE AND PHILOSOPHY

TIMESPACE AND PHILOSOPHY

In this book Christopher Ray addresses fundamental issues in thephilosophy of space and time while avoiding daunting technicalitiesand jargon

Always careful to elucidate the philosophical problems associated withspace and time Ray examines the work of Newton Einstein Hawkingand other scientific giants and discusses the reactions of philosophers tothis workmdashfrom metaphysical worries about the nature and reality ofspace and time to questions about the status of relativity and its rivaltheories

He investigates the puzzling nature of spacemdashfrom the infinitesimallysmall to the unimaginably large the disturbing paradoxes of time andtime travel and the curious ideas of modern cosmologymdashfrom the bigbang and the possibility of creation ex nihilo to the quantum world ofblack holes

Christopher Ray is Assistant Professor in History and Philosophyof Science at Portland State University Oregon His published worksinclude The Evolution of Relativity (1987)

PHILOSOPHICAL ISSUES IN SCIENCE

General EditoWHNewton-Smith

THE RATIONAL AND THE SOCIALJRBrown

THE NATURE OF DISEASELawrie Reznek

INFERENCE TO THE BEST EXPLANATIONPeter Lipton

THE PHILOSOPHICAL DEFENCE OF PSYCHIATRYLawrie Reznek

MATHEMATICS AND THE IMAGE OF REASONMary Tiles

THE LABORATORY OF THE MINDJRBrown

Forthcoming

METAPHYSICS OF CONSCIOUSNESSWilliam Seager

TIME SPACE ANDPHILOSOPHY

Christopher Ray

London and New York

First published 1991by Routledge

11 New Fetter Lane London EC4P 4EE

Simultaneously published in the USA and Canadaby Routledge

a division of Routledge Chapman and Hall Inc29 West 35th Street New York NY 10001

Routledge is an imprint of the Taylor amp Francis Group

This edition published in the Taylor amp Francis e-Library 2003

copy 1991 Christopher Ray

All rights reserved No part of this book may be reprinted or reproduced orutilized in any form or by any electronic mechanical or other means nowknown or hereafter invented including photocopying and recording or inany information storage or retrieval system without permission in writing

from the publishers

British Library Cataloguing in Publication DataRay Christopher

Time space and philosophymdash(Philosophical issues inscience)

1 Space amp time Philosophical perspectivesI Title II Series

115

Library of Congress Cataloging in Publication DataRay Christopher

Time space and philosophyChristopher Rayp cmmdash(Philosophical issues in science)

Includes bibliographical references and index1 Space and time I Title II Series

BD632R39 1991 90ndash24118115ndashdc20

ISBN 0-203-01873-7 Master e-book ISBN

ISBN 0-203-19146-3 (Adobe eReader Format)ISBN 0-415-03221-0mdashISBN 0-415-03222-9 (pbk)

For Carol

CONTENTS

Preface xIntroduction 1

1 ZENO AND THE LIMITS OF SPACEAND TIME 5Introduction 5Divisibility versus indivisibility 6Infinitesimals and limits 11Thomsonrsquos infinite super-task 14The parallel task paradox 15Abstractions and the physical world 20

2 CLOCKS GEOMETRY ANDRELATIVITY 24Introduction 24My time and your time 33The paradox of the twins for ever young 36From twins to triplets 41Phantoms of perspective 44

3 TRAVELLING LIGHT 46Introduction 46Measuring the speed of light 49Absolute simultaneity 53Slow clock transport 57Spacelike travel a tale of two tachyons 60Just the two of us across the universe 66

4 A CONVENTIONAL WORLD 69Introduction 69When parallel lines meet 71

vii

Will the real geometry please stand up 74Convention and topology 79Dimensions 82The future of the universe 84The Cosmological Principle convention or fact 86The underdetermination of theory by data 90

5 NEWTON AND THE REALITY OFSPACE AND TIME 99Introduction 99Absolute space and time 100Matter in the Newtonian world 103Leibniz and relationism 105Clarkersquos defence of Newton 108Absolute motion without absolute space 113

6 MACH AND THE MATERIAL WORLD 116Introduction 116Machrsquos relationism 118Simplicity and science 120Positivism in action 122Can we see space 125Experiment and intervention 127

7 EINSTEIN AND ABSOLUTE SPACETIME 131Introduction 131Machrsquos Principle 133Absolutely Professor Einstein 134Empty almost empty and rotating worlds 139Relationism and relativity an empirical view 143The hole argument and spacetime points 146

8 TIME TRAVEL 151Introduction 151Spacetime structure 154Back to the past 156Forward to the past 166Correlations and backwards causation 171

9 EINSTEINrsquoS GREATEST MISTAKE 176Introduction 176Space and infinity 177Einsteinrsquos universe 180

CONTENTS

viii

The cosmological constant did Einstein blunder 184Laws and theoretical change 187The Anthropic Principle 189

10 COSMOLOGICAL CONUNDRUMS 193Introduction 193The big bang a singular idea 196The beginning of time 199Inflationary cosmology something for nothing 204Black holes 209Cosmic censorship 211Determinism versus indeterminism 215

CONCLUSION RELATIVITYmdashJUSTANOTHER BRICK IN THE WALL 217

Introduction 217What is a theory 218The structure and scope of spacetime theories 221The last word 226

NOTES 229

SELECT BIBLIOGRAPHY 260

INDEX 263

CONTENTS

ix

PREFACE

This book presents my reflections upon a series of problems about timeand space Much discussed here has a long and distinguished heritage Ihave every reason for gratitude to both earlier and present generationsof scientists and philosophers for their exploration and clarification ofour ideas about space and time from Samuel Clarkersquos defence of Newtonto Hans Reichenbachrsquos empiricism from Aristotlersquos discussion of Zenoon motion to Hugh Mellorrsquos thoughts about time and time travel fromAlbert Einsteinrsquos revolutionary thoughts about matter and the source ofinertia to Stephen Hawkingrsquos equally startling discussion of the propertiesof black holes I also have every reason to thank those who helped invarious ways with this book reading part or all of the various drafts ordiscussing the ideas involved and giving so many valuable suggestionsand steering me away from error too often for me to have anything elsebut a marked sense of my fallibility I am especially grateful to HarveyBrown and Bill Newton-Smith of Oxford University Marthe Chandlerat DePauw University Carl Hoefer of Stanford University AlexanderRueger at the University of Oregon and Robert Weingard of RutgersUniversity I am grateful too for the assistance given to me by RichardStoneman and the editorial staff at Routledge and for their patienceAnd Carol Ray reading the manuscript as a non-specialist did morethan anyone to help me to clarify those ideas which were expressed tooclumsily or too technically So the merits of this book derive in partfrom the endeavours of others but the defects you must blame on me

Some of the material in this book is based on articles published injournals with revisions where appropriate and I am grateful to the editorsof the journals involved for allowing me to use thismaterial here Thecentral part of Chapter 1 appears as lsquoParadoxical tasksrsquo in Analysis 50 2(1990) the last section of Chapter 3 is based on lsquoCan we travel fasterthan lightrsquo in Analysis 42 1 (1982) the final section of Chapter 7 isbased on part of a review written together with Carl Hoefer of JohnEarmanrsquos World Enough and Space-Time in the British Journal for the Philosophyof Science 42 3 (1991) and much of Chapter 9 appeared in lsquoThe

x

cosmological constant Einsteinrsquos greatest mistakersquo Studies in the Historyand Philosophy of Science 21 4 (1990)

My thanks must go as well to Mr P never far from any centre ofactivity for his slumbering and I hope appreciative feline reflections onmy endeavours

Christopher RayPortland Oregon USA

PREFACE

xi

INTRODUCTION

Under the startling headlines lsquoRevolution in science New theory of theUniverse Newtonian ideas overthrownrsquo the New York Times reported in1919 the effects of Sir Arthur Eddingtonrsquos dramatic confirmation ofEinsteinrsquos General Theory of Relativity and its prediction that a lightray from a distant star would lsquobendrsquo in the curved space close to theSun

Yesterday afternoon in the rooms of the Royal Society at ajoint session of the Royal and Astronomical Societies the resultsobtained by British observers of the total solar eclipse of May29 were discussed The greatest possible interest had beenaroused in scientific circles by the hope that rival theories of afundamental physical problem would be put to the test andthere was a very large attendance of astronomers and physicistsIt was generally accepted that the observations were decisive inverifying the prediction of the famous physicist Einstein statedby the President of the Royal Society as the most remarkablescientific event since the discovery of the planet Neptune Butthere was a difference of opinion as to whether science had toface merely a new and unexplained fact or to reckon with atheory that would completely revolutionize the acceptedfundamentals of physics

(New York Times 1919)1

Later that year Einstein was invited to explain his ideas to the Britishpublic In a short article he presented the essential features of his theoryhe told the readers of The Times that lsquoIn the generalised theory of relativitythe doctrine of space and timehellip is no longer one of the absolutefoundations of general physicsrsquo (Einstein 28 November 1919)

1

TIME SPACE AND PHILOSOPHY

2

Our concepts of space and time already challenged by EinsteinrsquosSpecial Theory of Relativity were now under further attack from hisGeneral Theory Few understood the implications of Einsteinrsquos workin those early years Many found it hard to break free from the well-established Newtonian ideas But more and more the scientificcommunity embraced Einsteinrsquos theories Some of the initialimplications of both theories were hard to swallow the idea that timeis not an absolute framework and the possibility of a non-Euclideanuniverse in which the three internal angles of a triangle do not add upto 180 degrees Even Einstein found some of the implications hard tostomach his equations were consistent with the possibility of anexpanding universemdasha possibility which he initially rejected in a movewhich he came to regard as his greatest mistake And more surpriseswere to come as the theories were developed further the big bangtime travel and black holes all seemed to be consistent with the ideasof relativity theory

In this book we shall explore some of the major ideas and problemsbehind our views of space and time Most of the central questions aboutspace and time arise from considering the ideas of scientists such asIsaac Newton Ernst Mach Albert Einstein and Stephen Hawking Sowe must consider the essential features of the work on space and time bysuch scientists as these from speculations about how many dimensionsspace might have to the problem of infinitesimals from questions aboutwhether space and time are infinite to worries about the scientific statusof entities which cannot be seen from the ideas of black holes and thebig bang to conjectures about time travel We shall then be in a betterposition to understand the philosophical issues connected with all theseproblems

In Chapter 1 we shall look at the five paradoxes presented by theearly Greek philosopher Zeno His worries about the way we regardspace time and motion have a clear message for the way we think ofgeometry and its applicability to the physical world The problems ofgeometry are pursued further in Chapters 2 to 4 First we shall discussthe celebrated paradox of the twins and introduce the less well-knownparadox of the triplets we shall then investigate the importance of thespeed of light in relativity theory asking amongst other questionswhat happens when we relax the generally held convention that nothingtravels faster than lightand then we shall focus on the general

INTRODUCTIOIN

3

implications of relativityrsquos commitment to non-Euclidean geometriesIn Chapters 5 6 and 7 we shall look closely at the question of absoluteand relational space and time first through the arguments of Newtonand Leibniz and then through the ideas of Mach and Einstein Weshall see that the problems identified by Newton may be raised in bothNewtonian and relativistic contexts Chapter 8 focuses on the problemsand possibilities of time travel We shall discuss several ways in whichtime travel might be possible but we shall find that some of them mayinvolve logical contradictions or may require rather peculiar views ofthe physical world The problems of classical and modern ideas ofcosmology are addressed in Chapters 9 and 10 Particular attention isgiven to the cosmological constantmdashthe idea dismissed by Einstein as ablunder But we shall also review problems connected with black holesand the big bang The final chapter presents an overall impression ofthe status of claims about space time and motion how much shouldwe believe of the stories told to us by physicists when they seem tochange their minds so often

Throughout I have aimed to draw a balance between explainingthe physics and examining the philosophical assumptions argumentsand perspectives involved in the various physical accounts ahead Ihave tried to keep technical details to a minimum but sometimes theproblems which we meet cannot be grasped without at least someappreciation of the mathematical and geometrical ideas involved Wherepossible I have used diagrams to help the reader visualise the situationsbeing discussed In writing this book I have tried to provide acomprehensive up-to-date and accessible introduction to thephilosophy of space and time to help those without specialistbackgrounds in the physics of space and time begin to understand(and not just be dazzled by) some of the fundamental issues arisingfrom classical and modern ideas of space and timemdashissues which willalso introduce the reader to philosophical problems in metaphysicsthe theory of knowledge the philosophy of religion and the philosophyof science However I hope that many readers will regard this book asa starting-point for further studies in the philosophy of space and timeSo a select bibliography reviews the most important and helpfulliterature in the field And detailed notes to each chapter amplify thetext suggest further reading and point those wishing to engage infurther research in the right direction

TIME SPACE AND PHILOSOPHY

4

The ideas of space and time provide us with a rich and rewardingfield of study The challenge which faced Newton and Einstein maybe shared by everyone We may not have their genius but we canshare their insights And these insights can give us a better appreciationof the role of philosophy as it meets the problems of science

1

ZENO AND THE LIMITSOF SPACE AND TIME

INTRODUCTION

We typically think of space and time as three dimensions plus oneMathematicians tell us that each dimension may be continuously sub-divided But they also tell us that we may construct model universeswith rather different properties We may have other structures whichmay not be continuously sub-divided And to complicate matters wemay construct worlds with whatever dimensionality we please So canwe really chop lsquorealrsquo space and time up as small as we like

The pre-Socratic philosopher Zeno of Eleamdasha Greek settlement inSouthern Italymdashis said to be responsible for five lsquoparadoxesrsquo which wrestlewith the properties of space time and motion The main focus of Zenorsquosparadoxes is the lsquosmall-scalersquo character of space and time Is this small-scale structure really continuous or is it lsquoindivisibly atomisticrsquo or lsquodiscretersquoin some sense If threedimensional space is a continuum then we maycontinuously and indefinitely sub-divide its parts But if space or timeare discrete in some way then any process of sub-division will have adefinite limit Aristotle gives a brief and perhaps incomplete account ofthe first four paradoxes in his Physics and Simplicius discusses the fifthin his commentary on Aristotle1 Zeno is thought to have produced hisideas around 460 BC We shall review Zenorsquos discussion and we shallfind that these paradoxes do identify some real difficulties for ourlsquocontinuumrsquo view of space and time

Many mathematicians and philosophers believe that a thoroughacquaintance with the mathematics of the continuum should besufficient to dispel any worries that might arise from Zenorsquos paradoxes

5

TIME SPACE AND PHILOSOPHY

6

But the problems raised by Zeno live on and somewriters includingthe philosopher Wesley Salmon and the theoretical physicist RogerPenrose advise against any uncritical and complete acceptance ofthe role of the continuum in our physical theories2 A related problemsuggested by James Thomson in 1954 concerns the paradoxicalnature of any super-task consisting of an infinite number of tasks Ishall argue that this problem is genuinely paradoxical on themathematiciansrsquo own terms But I shall not join Zeno in rejecting thereality of a complex diverse world I shall merely question the extentto which mathematics and geometry may serve as an adequate modelfor the physical world

Imagine that we have two theories about the way objects move inthe world One theory assumes that space and time may becontinuously sub-divided The other denies this But also imaginethat both theories are perfectly consistent with every measurementand observation we can possibly make If we can actually constructsuch an empirically impeccable rival to the lsquocontinuumrsquo theory thenwe might begin to wonder about the status of the continuum Wemay be willing to admit that it gives us an extremely useful way oforganising our experience But should we believe that the world isreally like that The advantage of mathematics is that it helps us tothink clearly about those structures which we believe to be the actualstructures of the world but the problem with mathematics is that itallows us to generate all sorts of weird and wonderful possiblestructures for the world The job of sorting out which if any weshould accept as the lsquorealrsquo picture is left to the physicist Andsometimes the choice is far from straightforward

DIVISIBILITY VERSUS INDIVISIBILITY

Zenorsquos paradoxes of space time and motion attack the very idea of thedivisibility of space and time We begin by imagining a distance or atemporal duration which is divided by two and we imagine that theprocess of division is continued Why may we not imagine that theprocess could continue indefinitely Zeno tells us that any assumptionthat the process could go on indefinitely will lead us into logicalcontradictions But he also argues that any assumption that the processhas some definite limit also leads us into just as much trouble The firstfour paradoxes reveal the dilemma

ZENO AND THE LIMITS OF SPACE AND TIME

7

1 Achilles and the tortoiseZeno asks us to imagine a race between Achilles and a tortoise inwhich the tortoise is allowed to start first After an agreed time Achillessets off in pursuit Although it seems entirely obvious that the race isa mis-match and that Achilles will all too soon overtake the tortoiseZeno raises a doubt in our minds For in order to overtake the tortoiseAchilles must first reach the point where the tortoise was when Achilleswas given the signal to start in pursuit Let us call this first point PBut when he reaches point P the tortoise will now be a little furtheron at point Q Achilles now must reach Q if he is to catch the tortoiseYet when he arrives at Q the tortoise is still ahead at R When Achillesgets to R the tortoise has reached S The race continues just like thisevery time Achilles reaches the tortoisersquos last lsquostaging-postrsquo the tortoisehas moved further on to a new post Of course the distance betweenthe two gets shorter and shorter all the time But Achilles is alwaysbehind So despite first appearances Achilles cannot even catch letalone overtake the tortoise

2 The racecourse (or dichotomy paradox)Here Zeno not only argues that an athlete would never finish saya 100-metre race it also seems that the athlete could not even getstarted To reach the end of the track the athlete would first haveto reach the 50-metre point Having run 50 metres the athletewould now have to reach the half-way point between the 50-metrepoint and the finish line That would take the athlete to the 75-metre mark But now the athlete would have to reach the half-waypoint between this mark and the finish No matter how far theathlete gets down the track there would always be yet anotherlsquohalf-wayrsquo point to reach between the point where the athlete isand the finishing line So the athlete would get closer and closerto the end of the track but would never actually reach the finishFor there would be an infinite number of half-way points ahead ofthe athlete This might seem bad but an associated argumentimplies that the race would not even begin For to reach thefinishing line demands that the athlete would first need to reachthe 50-metre mark and to reach the 50-metre mark demands thatthe athlete would already have reached the 25-metre point and toreach that point would require that athlete to have got to the 125-metre mark and so on As we keep dividing the distance by two

TIME SPACE AND PHILOSOPHY

8

we get closer to the startingline but we never actually reach it Andwe may divide these distances an infinite number of times So toreach the end of the track there would be an infinite number ofdistances to run through Indeed no matter how short the trackthere would always be an infinite number of distances ahead Theathlete would be stuck at the start To go any distance at all theathlete would have to run through an infinite number of distancesmdashand how could that be possible

3 The arrowTake a high-speed photograph of an arrow in flight and you may findit hard to disagree with Zenorsquos assertion that such an arrow occupiesexactly that space which is equal to its own shape and size We seemto have captured the arrow at an instant of time At such an instantthe arrow is motionless If it were not motionless the instant of timecould be sub-divided now the arrow is here now there Yet the entireflight of the arrow could be captured in a series of instantaneousphotographs At every instant the arrow is motionless There is notime between the instants for the arrow to move on to the next instantFor such a time would be composed of instants itself So how can analways motionless object move

4 The moving rows (or the stadium)Imagine a stadium in which a column of soldiers passes a columnof soldiers at attention so that each step brings every soldier in themoving column into line with the next comrade in the stationarycolumn a third column of soldiers is also moving but in the oppositedirection so that with each step the soldiers here also are broughtinto line with the next comrade along in the stationary column seeFigure 1 (p 9) With each step each soldier in each moving columnencounters one comrade in the stationary column but two comradesin the oppositely moving column Now imagine that each soldierrepresents an indivisible minimum unit of length and that eachstep represents an indivisible minimum unit of time Surely wecan ask the question at what instant and in what position did thetwo moving columns align so that each soldier was alongside thenext (rather than the next-but-one) soldier in the adjacent movingcolumn If we can sub-divide the time for the step and the spacebetween steps there is no problem at all For they will meet afterhalf a step But we have supposed that there is no such thingas half of one of our units of length or timemdashsince they are

Figure 1 Zenorsquos moving rows or stadium paradox

indivisible minima So either the question is unreasonable (and whyshould this be) or we are wrong to suppose that space and timeconsist in indivisible minima

In the first two paradoxes Zeno tries to illustrate the absurdity of believingthat a line may be divided up into progressively smaller chunks ad infinitumAnd there is something seductive in his argument For how can I movefrom A to B when I first must move to some point in between Andwhatever point I choose and no matter how many times I do this thereis always going to be yet another point in between Zeno warns us againstsaying that sooner or later I must reach the smallest possiblelsquoindivisiblersquodistance For this discrete view of space too will generate

Moving rows paradox Two rows (X and Z) move by a stationary row (Y) asshown In the top diagram X1 and Zl are in adjacent columns X1 to the leftand Z1 to the right An instant later X1 and Z1 have shifted their positions sothat they are still in adjacent columns but with X1 now to the right of Z1 asshown in the lower diagram Zenorsquos problem is this when and where wereX1 and Z1 in alignment vertically Given that the change of position tookplace in the shortest possible time we cannot say that they were in line in halfthis time And because the change of position involves the shortest possibledistance we cannot say that they were in line when they had moved throughhalf this distance

9

TIME SPACE AND PHILOSOPHY

10

problems as demonstrated by the fourth paradox Some writers approachZenorsquos paradoxes with confidence saying that just a little modern calculuswill be sufficient to dispel any worries which the paradoxes may produce3

Ian Stewart identifies the central issue in Zeno as the way we think ofinfinitesimal quantities and says that only in the last hundred and fiftyyears or so have we begun to see the problem in a way that helps us toresolve the paradoxes without too many qualms Stewart asks

Can a line be thought of as a sequence of points Can a plane besliced up into parallel lines The modern view is lsquoyesrsquo the verdictof history an overwhelming lsquonorsquo the main reason being that theinterpretation of the question has changed

(Stewart 198766)4

Mathematicians now seem to have few worries about continuous sub-divisions What has changed is their attitude towards infinitesimalquantities Such quantities are not regarded as extensionless points inspace or in time If we regard points as having no extension then we fallvictim to Zenorsquos fifth paradox that of pluralitymdashsaid by GEL Owenand others to be Zenorsquos primary concern and to underlie the other fourparadoxes5 Indeed Owen argues that we should regard the paradoxesas providing a coordinated attack on the reality of space time and motionThe first two paradoxes challenge the idea that space and time can becontinuously sub-divided and the second two attack the notion that thereare indivisible minima of space and time so that Zenorsquos overall judgementmay be summarised thus lsquono method of dividing anything into spatialor temporal parts can be described without absurdityrsquo6 The fifth paradoxdiscourages us from regarding the end result of some continuous sub-division as either an extensionless quantity like a point or a quantitywith some definite if minute extension 5 The paradox of plurality

Zeno according to Simplicius asks how even an infinite numberof extensionless distances could add up to a finite distance andhow an extended body can consist of an infinite number of parts(geometrical points) which themselves have no extension sucha distance or such a body must be infinitely smallmdashie it mustbe just like its constituent parts extensionless7 Yet if we allow

ZENO AND THE LIMITS OF SPACE AND TIME

11

these constituent parts to have some finite sizemdashhowever smallmdashthenthe body must be infinite in size8

Owen points out that this paradox taken together with the first fourmay be seen as providing reasons for Zenorsquos view of the world as asingle global entity rather than as made up of parts whether theseare indivisibly small or continuously divisible As soon as we start tosub-divide we run into difficulties So the sensible thing to do is toresist the temptation to divide the world up at all Zenorsquos world is asingle body which may not be sub-divided in any way withoutabsurdity

INFINITESIMALS AND LIMITS

Must we accept Zenorsquos conclusions The answer seems to lie in ourattitude towards the lsquoendrsquo result of an unending process of sub-divisionto the idea of infinitesimals It is a mistake to regard them as havingsome lsquoconstantrsquo value whether this be the lsquozerorsquo of extensionless objectsor points or whether it is the non-zero value of the shortest possibledistance or time In both cases we would fall straight into one or other ofZenorsquos traps We need a different approach if we are to avoid the trapsaltogether The way out was first suggested by the French mathematicianCauchy in 1821 he introduced the idea of a limit and the notion of theinfinitesimal was absorbed into this more coherent concept9 And somethirty years later Weierstrass showed that we could move the debatefrom the realm of geometry to that of arithmetic from ideas of spatialand temporal distances to those of functions Instead of talking aboutever-decreasing distances along a straight line we could talk with a littlemore rigour about infinite series converging on limiting values in termsof functions and real numbers

The problem may be highlighted by considering how we shouldanswer this question what speed does the athlete have at any giveninstant If we think in terms of infinitesimals with a lsquozerorsquo value thenthe equation for the speed of an object (distancedividetime) collapses intononsensemdashthe speed of any moving object considered in this way willalways be zero divided by zero So instead of saying that we may describethe motion of the athlete by reference to infinitesimal distances and timeswe should calculate the speed of the athlete at any instant in terms ofhow the object is moving in the immediate neighbourhood as shown by the

Figure 2 Distance-time graph comparing Achilles with tortoise idea of velocity

mathematical function describing the athletersquos motion By consideringsmaller and smaller neighbourhoods we typically reach a limiting valuefor the functionmdashthe lsquoinstantaneousrsquo speed We get our answer byconsidering what happens as we approach the instant not by asking whatis happening at the instant Similarly we consider whether or not Achillesovertakes the tortoise and whether or not the athlete may run from A toB by thinking in terms of what happens as Achilles approaches the tortoiseand as the athlete approaches the end of the racecourse see Figure 2(above)

So using these ideas we may give the following provisional responsesto Zenorsquos worries about a continuum which may be continuously sub-divided 1 The functions describing Achillesrsquo and the tortoisersquos motions show

that when Achilles is in the immediate neighbourhood of the tortoiseAchillesrsquo speed is greater than that of the tortoise and he thereforeovertakes it10

2 When Zenorsquos athlete attempts to run from A to B the athlete

Although Achilles starts the race after the tortoise because his speed isgreater than that of the tortoise he overtakes the tortoise at the pointshown The speed of Achilles (distancedividetime) rather than the decreasingdistance between the two is the key to the problem

12

ZENO AND THE LIMITS OF SPACE AND TIME

13

will indeed need to cover half the distance then a furtherquarter and so on but the function describing the series ofdistances run by the athlete converges upon a natural limit thetotal distance AB

3 The idea of an instantaneous lsquosnapshotrsquo of a moving object (eg anarrow) does not by itself carry with it any idea of motion only whenwe consider its immediate spatio-temporal neighbourhood may wegive any sense to the idea of a moving arrowmdashbut once we do this theidea of motion is quite coherent (many action photographs showmoving objects against a blurred background to convey the idea ofspeedmdashsuch pictures would capture the arrow not at an instant butover a short period of its flight)

But we still face some important questions

1 Are we entitled to say that a converging infinite series has a sumgiven the fact that an infinite sequence has a limiting value Foralthough we may agree that the limiting value of the sequence of partialsums

is 1 we do not thereby have sufficient reason to say that theseries

has 1 as its sum We might accuse mathematicians of fudging the issuewhen they assure us that the limiting value of such a sequence is alsothe sum of a related series So although we may say that the sequenceof distances from A run by the athlete has AB as its limiting valuethis need not commit us to the view that the series may be summedat all For such a summation seems to involve an infinite number ofadditions and we might regard such an addition as at best implausibleor at worst logically impossible

2 To what extent may we say that the mathematical concepts employedabove apply to the physical world For even if we allow that aninfinite series may have a mathematical sum this is no reason for usto agree that we may apply this procedure to the physical world withimpunity For example we may use the idea of a simple arithmeticalsum when adding one quantity of money to another but when adding

TIME SPACE AND PHILOSOPHY

14

velocities in the Special Theory of Relativity a different procedure isrequired So we might ask to what extent are mathematical andgeometrical concepts and structures strictly true of the physical worldWhy should abstractions apply literally to the physical world

These questions are now addressed by Thomson who challenges us tocontemplate a super-task consisting of an infinite number of tasks

THOMSONrsquoS INFINITE SUPER-TASK

James Thomson asks us to imagine a lamp which may be switched onand off an infinite number of times in a finite time If we set aside thequestion of whether or not it is physically possible for an infinite numberof such tasks to be performed in a finite time we may still ask withThomson whether or not it is logically possible We may easily imagineThomsonrsquos reading-lamp with its switch in the off position and himswitching it on then off then on and so on If it is switched on at timezero and off after one minute and on again after another 30 secondsand off again after a further 15 seconds and so on then we mightthink that after two minutes we would have completed an infinitenumber of switching operations But Thomson asks

at the end of the two minutes is the lamp on or offhellipItcannot be on because I did not ever turn it on without atonce turning it off It cannot be off because I did in the firstplace turn it on and thereafter I never turned it off withoutat once turning it on

(Thomson 19545)11

And this Thomson tells us is contradictory He concludes that wecould not in principle carry out such a super-task Sainsbury says inParadoxes that Thomsonrsquos conclusion is unwarranted12 He begins bydistinguishing between those moments when the switching tasks arebeing performed (the T-series) and that first moment after the super-task has been completed (T) Following Benacerrafrsquos argument inlsquoTasks super-tasks and the modern eleacticsrsquo13 Sainsbury then maintainsthat lsquofor any moment in the T-series if the lamp is on at that time thereis a later moment in the series at which the lamp is off and vice versarsquo

ZENO AND THE LIMITS OF SPACE AND TIME

15

However nothing follows from this about whether the lamp is on oroff at T for T does not belong to the T-series The T-series is closed atone end (time zero) and open at the other lsquoendrsquo This means that althoughwe can identify a first task at time zero we cannot give any sense to thenotion of a last taskmdashthe openness of the T-series guarantees the possibilityof the tasks continuing an infinite number of times The time T doesnot occupy any point at this open end and is therefore independent ofthe T-series But since T is independent of the T-series Sainsbury pointsout that our lsquospecification of the task speaks only to members of the T-series and this has no consequences let alone contradictoryconsequences for how things are at T which lies outside the seriesrsquo(Sainsbury 198815)

So Sainsbury concludes that Thomson fails to demonstrate that theidea of a super-task is logically absurd Two clear implications ofSainsburyrsquos argument are

1 that the lamp is either on or off at T and2 that we cannot say either at the beginning of the super-task or once it

is under way just how things will turn out at T

Hence there is no way to predict the state of the lamp at T whateverits state at the outset

THE PARALLEL TASK PARADOX

Suppose we ask an operator to carry out the super-task twice insuccession in exactly the same way each time Then there is no reasonto suppose on Sainsburyrsquos view that the lamp would be in the samestate at each of the moments T after the two tasks are completedOtherwise we would always be able to predict the final state of thelamp We may sharpen this problem as follows Imagine now twolamps and one operator for each lamp We ask both operators toattempt Thomsonrsquos super-task at the same time Both lamps are offand at time zero both operators switch their lamps on After oneminute the operators switch the lamps off after 30 seconds bothlamps are switched on together after 15 more seconds the lamps areoff again and so on If we grant the point that T lies outside theseries we may also grant that at T each lamp will be either on oroff But are we forced to conclude that both lamps will be in thesame state Given that the operators begin together and continue

TIME SPACE AND PHILOSOPHY

16

together with the lamps flashing on and off in unison we might expectthem to finish together with the lamps in precisely the same stateHowever what happens during the T-series is we are toldindependent of what is the case at T As Sainsbury claims nothingconcerning T follows from our specification of the task to be carriedout because this specification relates only to times within the T-seriesThe fact that the lamps are initially in the same state is irrelevantsince the moment at the start of the super-task lies within the T-series And the fact that the operators continue together does nothelp because our instructions to them relate solely to times withinthe T-series So why should we expect the lamps to be in the samestate at T We are left with the unsatisfactory conclusion that twopeople always in step during an infinite sequence of tasks may be outof step immediately after the sequence has lsquoendedrsquo

Such a conclusion seems to involve us in rather more than anempirical puzzle about the way things will turn out with such lsquoparallelrsquosuper-tasks There seems to be a rational if not a logical inconsistencyIf the super-tasks run in parallel and in step at all times during theseries then we have no reason at all to suppose that this patterncould be broken when the super-tasks are over By accepting thedistinction between the T-series and the (independent) time T thenthere is also now a reason to suppose that the pattern may be brokenat T when the tasks are over The two lamps may be in differentstates at T because there is no connection between the states of thelamps during the switching operations and their states at T Giventhis lack of connection the chances of the lamps being in the identicalstates at T seem to be the same as the chances of them being indifferent states at T And this appears to be an acceptable reason forbelieving that the pattern may be broken Hence acceptance of theindependence of T from the T-series leads us to a direct conflictwith our apparently reasonable expectation that two lsquoparallelrsquooperations should always remain in step not just up to the end of atask but also when the task is over

The difficulty involved in this problem seems to derive from thefact that there is no limiting behaviour of the situation to which wemight appeal in order to dissolve the puzzle When Zenorsquos athleteattempts to run from A to B at a uniform speedmdashfirst passing throughthe half-way point then the three-quarter-way point and so onmdashthemathematical function which describes the athletersquos progress can be