The Shape of Space

This is a revised and updated edition of Graham Nerlich's classic book The shape of space. It develops a metaphysical account of space which treats it as a real and concrete entity. In particular, it shows that the shape of space plays a key explanatory role in space and spacetime theories. Arguing that geometrical explanation is very like causal explanation, Professor Nerlich prepares the ground for philosophical argument, and, using a number of novel examples, investigates how different spaces would affect perception differently. This leads naturally to conventionalism as a nonrealist metaphysics of space, an account which Professor Nerlich criticises, rejecting its Kantian and positivistic roots along with Reichenbach's famous claim that even the topology of space is conventional. He concludes that there is, in fact, no problem of underdetermination for this aspect of spacetime theories, and offers an extensive discussion of the relativity of motion.

The shape of space

The shape of space Second Edition

GRAHAM NERLICH Professor ofPhilosophy, University ofAdelaide, South Australia

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521450140 © Cambridge University Press 1994 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1994 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication data Nerlich, Graham, 1929The shape of space. Bibliography Includes index 1. Space and time. 2. Relation (Philosophy) 3. Science-Philosophy. I. Title. BD632.N45 114 75^4583 ISBN 978-0-521-45014-0 hardback ISBN 978-0-521-45645-6 paperback Transferred to digital printing 2008

To Andrew, David and Stephen

Contents Preface Introduction 1 Space and spatial relations 1 Pure theories of reduction: Leibniz and Kant 2 Impure theories of reduction: outlines 3 Mediated spatial relations 4 Surrogates for mediation 5 Representational relationism 6 On understanding 7 Leibniz and the detachment argument 8 Seeing places and travelling paths 9 Non-Euclidean holes 10 The concrete and the causal 2 Hands, knees and absolute space 1 Counterparts and enantiomorphs 2 Kant's pre-critical argument 3 Hands and bodies: relations among objects 4 Hands and parts of space 5 Knees and space: enantiomorphism and topology 6 A deeper premise: objects are spatial 7 Different actions of the creative cause 8 Unmediated handedness 9 Other responses 3 Euclidean and other shapes 1 Space and shape

page xiii 1 11 11 14 18 21 23 28 33 36 38 40 44 44 46 47 49 51 54 58 61 62 69 69

x List of contents

4

5

6

7

2 Non-Euclidean geometry and the problem of parallels 3 Curves and surfaces 4 Intrinsic curvatures and intrinsic geometry 5 Bending, stretching and intrinsic shape 6 Some curved two-spaces 7 Perspective and projective geometry 8 Transformations and invariants 9 Subgeometries of perspective geometry Geometrical structures in space and spacetime 1 The manifold, coordinates, smoothness, curves 2 Vectors, 1-forms and tensors 3 Projective and affine structures 4 An analytical picture of affine structure 5 Metrical structure Shapes and the imagination 1 Kant's idea: things look Euclidean 2 Two Kantian arguments: the visual challenge 3 Non-Euclidean perspective: the geometry of vision 4 Reid's non-Euclidean geometry of visibles 5 Delicacy of vision: non-Euclidean myopia 6 Non-geometrical determinants of vision: learning to see 7 Sight, touch and topology: finite spaces 8 Some topological ideas: enclosures 9 A warm-up exercise: the space of S% 10 Non-Euclidean experience: the spherical space S3 11 More non-Euclidean life: the toral spaces T2 and Ts The aims of conventionalism 1 A general strategy 2 Privileged language and problem language 3 Privileged beliefs 4 Kant and conventionalism 5 Other early influences 6 Later conventionalism 7 Structure and ontology 8 Summary Against conventionalism 1 Some general criticisms of conventionalism 2 Simplicity: an alleged merit of conventions 3 Coordinative definitions 4 Worries about observables

71 74 76 80 81 83 86 89 94 94 100 105 107 109 112 112 114 116 118 121 122 125 126 129 132 134 139 139 141 144 147 150 152 156 158 160 160 162 165 167

List of contents xi 5 The special problem of topology 6 The problem of universal forces 7 Summing up 8 Reichenbach's treatment of topology 1 The geometry of mapping S% onto the plane 2 Reichenbach's convention: avoid causal anomalies 3 Breaking the rules: a change in the privileged language 4 Local and global: a vague distinction 5 A second try: the torus 6 A new problem: convention and dimension 9 Measuring space: fact or convention? 1 A new picture of conventionalism 2 The conventionalist theory of continuous and discrete spaces 3 An outline of criticisms 4 Dividing discrete and continuous spaces 5 Discrete intervals and sets of grains 6 Grunbaum and the simple objection 7 Measurement and physical law 8 Inscribing structures on spacetime 10 The relativity of motion 1 Relativity as a philosopher's idea: motion as pure kinematics 2 Absolute motion as a kinematical idea: Newton's mechanics 3 A dynamical concept of motion: classical mechanics after Newton 4 Newtonian spacetime: classical mechanics as geometrical explanation 5 Kinematics in Special Relativity: the idea of variant properties 6 Spacetime in SR: a geometric account of variant properties 7 The relativity of motion in S R 8 Simultaneity and convention in SR 9 The Clock Paradox and relative motion 10 Time dilation: the geometry of 'slowing' clocks 11 The failure of kinematic relativity in flat spacetime 12 What GR is all about 13 Geometry and motion: models of GR Bibliography Index

172 176 177 180 180 183 183 187 188 192 195 195 196 200 203 204 206 207 212 219 219 222 225 229 233 244 248 251 254 257 263 268 272 279 287

Preface

First edition It has been my aim in this book to revive and defend a theory generally regarded as moribund and defeated. Consequently, I owe most of my intellectual debt to those against whom I try to argue in these pages. Above all, Hans Reichenbach's brilliant Philosophy of Space and Time has been the central work I wished to challenge and my model of philosophical style. My debt to the books and papers of Adolf Grunbaum is hardly less important. I am glad indeed to take this formal opportunity to express both my admiration and my debt to these writers who are subjects of so much criticism in this book. Quite generally, indeed, I acknowledge a real debt to all those philosophers whose work it has seemed important to examine critically in these pages. This book is the fruit of courses I have given for a number of years in the philosophy departments of the University of Sydney and of Adelaide. Among many students whom it has been an education to teach, I hope it is not invidious to mention Michael Devitt, Larry Dwyer, Clifford Hooker and Ian Hunt as particularly helpful in early stages. The whole of the typescript was read by Dr Ian Hinckfuss and Professor J.J .C. Smart (my first teacher in philosophy). Their comments and criticism have been invaluable. Various parts of the book

xiv

Preface

have been read and discussed with Robert Farrell, Bas van Fraassen, Henry Krips, Robert Nola, Hugh Montgomery and Wai Suchting. I am grateful to my colleague Dr Peter Szekeres for his patience with many queries I have put to him about General Relativity. Of course the responsibility for the errors and, perhaps, the perversity of the book is all my own. Parts of the book were written in 1973 when I was Leverhulme Fellow at the University of Hong Kong and a visitor at the University of Auckland. I am grateful to my colleagues in both institutions for their very agreeable hospitality. Chapter 2 first appeared under its present title in the Journal of Philosophy 70 (1973), 33751. I am grateful to Ms Iris Woolcock for the index and to her and Ms Alvena Hall for some bibliographical work. Mrs A. Bartesaghi typed several drafts and read the proofs and has been an invaluable aid in many ways. Alice Turnbull helped with the figures.

Second edition I have revised the book extensively for this edition. Much in the field has changed since I first wrote; the area has come to be influenced increasingly by the work of John Ear man, Michael Friedman, Laurence Sklar and Roberto Torretti. Like everyone who thinks about space and time I am deeply indebted to their splendid work. No longer do relationism and conventionalism dominate the field. While much that I wrote earlier seems relevant still, the need to rethink even some of the central arguments is plain. The main changes are as follows. The old chapter 9 has become chapter 4, so that all the geometry is introduced at an early stage. The new version attempts less but I think it is both more intuitive and formally richer than the old one. It focusses mainly on manifold structure, since that is all the structure which conventionalism allows space to have, but projective structure is stressed too because it plays a key role in going beyond the bare manifold. Chapter 1 is almost wholly new and takes account of versions of relationism which have emerged in the literature since the first edition appeared. The leading idea of the old chapter was that relationism is doomed to the hopeless enterprise of treating physical space as an abstract measure

Preface xv

space, a view which few recent relationists would endorse. The two general chapters on conventionalism have been drastically revised, developing the theme of the earlier version that Kant is the true ancestor of Reichenbach and that conventionalism has no plausible view of the relation of formal to empirical elements in physical theory. Chapter 8 became 9 and shortened its critique of Grunbaum. This allows a new last section on the ideas of Weyl, Ehlers, Pirani and Schild on how the metric of space is both evidenced and evinced by physical structures. Chapter 10 contains a new section arguing that simultaneity is no convention in Special Relativity and has been redrafted in a number of other ways. Every chapter has had some revision, drastic or trivial. But it is still the same book, arguing for a strong form of realism about space and spacetime and calling on the key idea of the shape of space as an explanatory tool of grace and power. My thanks are due especially to Jeremy Butterfield who read the first edition and made numerous useful comments on it and to Hugh Mellor and John Lucas for their encouragement over many years. The revisions were read by Michael Bradley and Chris Mortensen from whose advice the book and I have benefited. It is a pleasure to acknowledge the patient and good-humoured help of Birgit Tauss, Karel Curran and Margaret Rawlinson in preparing the text, the bibliography and the index.

Introduction Space poses a central, intriguing and challenging question for metaphysics. It has puzzled us ever since Parmenides and Zeno argued paradoxically that spatial division and therefore motion were impossible and Plato wrote of space in Timaeus as an active entity in the working of the world. It is easy to see why space is so problematic. On the one hand, we are drawn to make very powerful statements about it. Everything that is real has some spatial position. Space is infinitely large, infinitely penetrable and infinitely divisible. On the other, despite our confidence in these strong claims, space seems elusive to the point of eeriness. It seems to be largely without properties, apart from the few strong ones just recited. It is imperceptible by any mode of perception. It has no material property, no causal one, it does nothing. It seems to have no feature which we can learn about by observation. Arguably, it has a prominent role in natural science but it is far from obvious just what it is. Though being spatial is a mark of the real, space itself seems, paradoxically perhaps, unreal a mere nothing. This outlines some of the nodal points of an ancient debate in a rather artless way. If any of the strong claims is correct we can find out which only after carefully arguing through a range of intricate issues. A great light has been shed on the whole issue by developments in geometry and the sciences, especially in this century and the last. It has by no means dispelled every shadow. The main points of view on the ontological issue whether space is a real thing

2

Introduction

or a subtle fiction inherit long traditions in each of which we can trace, over the centuries, an increasing complexity and insight. The arguments have a power, elegance and depth unsurpassed elsewhere in philosophy. The beauty of the subject has drawn me to try to understand it and has yielded a source of constant delight and challenge which I hope you will share. This book is about two main philosophical questions: Is space something or nothing? If it is something, what is its structure? My answer to these questions is that space is a real, concrete thing which, though intimately linked to material objects by containing them all, is not dependent on them for its existence. It involves itself in our lives in a concrete and practical way when we try to probe and measure the world and understand how it works. We can hope to say a great deal that is surprising and informative about the structure of space and show how that explains things which must otherwise be accepted as sheer brute fact. I call this answer realism. Though that bold, short version of it is simple enough, the reasons for giving it and the explanation of what it means are not at all simple. Newton also gave a realist answer in his great books of mechanics in the seventeenth century. Ever since he tied this ontological question to those of physics, the discoveries of physical science have dominated the debate. In this century a new actor has taken the stage. Spacetime is of the same ontic type as space, though differing in a variety of ways structurally from the familiar geometry of Euclid. All of the philosophical arguments of the book apply to spacetime as well as to space. In chapters 9 and 10 it is singled out for special attention. A main interest in the idea of space and its role in physical science lies in its unique place in the system of our concepts. We think of space as the arena of those events which physical science tells us about. So it seems to loom large in perhaps the most developed, certain, precise and concrete part of the knowledge our culture may legitimately lay claim to. But it is, itself, perceptually remote: we can neither see it nor touch it. A main interest in epistemology lies in accounts of the order and genesis of our scheme of ideas. The role of perception in generating them has been much studied and emphasised. If space is a real structured entity with an explanatory role in natural science, then it enters the system of our knowledge by engaging the senses in a most indirect

Introduction

3

way, if it engages them at all. I argue that our intellectual imagination - our capacity to invent explanatory ideas, to understand them and exploit them - soars higher above a perceptual grounding than most of the philosophers who have probed the area think it can. Of course space would hold little interest for us if it did not provide a unity and system for what we perceive. There is no doubt that the perceptual imagination does help to fortify and even expand the intellectual imagination, as I hope to show. But, I argue, it is the prior grasp of spatial concepts which enables us to perform the feats of perceptual fancy, and not the other way round. The role of observation in our grasp of space is a subtle one. I don't think that I have succeeded in articulating the role in general terms, though it is never far from my mind in the following pages. Nevertheless, a very general conclusion is implied that scientific thought is not closely limited to what is observable and we should not try so to limit it. Some of the ancients spoke of space as The Void. This seems to have been meant as a subtle way of saying that space itself is nothing. There are only familiar things spatially related to each other in familiar ways. So began the long and illustrious tradition of answering the first, ontological, question in a relationist style. The bulk of what has been written on the subject is solidly relationist until quite recently. I have set my face against the run of the tide in this book. I argue for a realist conclusion. Space is an entity in its own right a real live thing in our ontology. Relationism comes in degrees and kinds (Lucas 1984 Appendix, gives a helpful account of the kinds). The sternest stuff is the doctrine that the idea of space is nonsense - no idea at all. Only talk about material things and their relations can be understood. Space is not a real independent, concrete thing. It is a fiction or a mere picture of some kind. The idea that it is real is born of confusion, a needless complication in our world picture; space is an impossible object and realism an unintelligible opinion. A milder view is that relationism is more plausible, neater, clearer than absolutism. Mildest of all is the claim that relationism is at least possible. We could say all we need to say, confining our statements to those about material things and their relations. But this implies no judgement about what it is best to do. It does not even claim that existing relationist theories actually work. I quarrel somewhat even with

4 Introduction

this mildest opinion. But I mainly argue that relationism must appear laborious, contrived and counter-intuitive beside the realist picture of space. A main problem in accepting the realist answers to the questions lies in understanding them. Much of the book is devoted just to that task. I assume, on the reader's part, no small maturity in the ability to follow a metaphysical argument. The philosophical stuff of the book is not at all introductory. It aims to challenge major relationist arguments in the field at present. But I take only a little for granted by way of knowledge of geometry and physics. I write mainly for someone like myself when I began to study the metaphysics of space - no novice in philosophy but naive in the ways of the geometer and the physicist. I should like to woo the general philosopher to pursue the study not only because it has so much to offer, but also because it risks becoming the exclusive preserve of those with a strong formal background who exploit it to the full. Though the best work is of great distinction, I believe that the field would benefit from the reflections of a wider audience. Chapter 1 attempts to make out the main lines of relationist positions and what motivates those who hold them. It aims to identify a crucial thesis about which the realist and relationist disagree. It is the claim that spatial relations are mediated ones. That means that two objects are spatially related only if they are joined by a path in space, either occupied or not. One key form of relationism takes for granted that spatial relations may hold unmediated: simply x is at a distance from y, or z is between x and y without need of a sustaining spatial link to carry the relation. This is a difficult and subtle issue. I doubt that conclusive arguments are available to carry either party in the debate to success, and I try to show why that is so. Nevertheless, I appeal to examples to motivate my point of view on the merits of the case. Although I feel that these are inviting appeals I am under no illusion that they are compelling. The arguments of later chapters are more decisive. They show the explanatory role of space as a whole, going beyond that of occupied points and spatial relations among them. I also begin here the theme that it is not so much its ontological type that makes space seem so mysterious but rather its geometric type. It is because we think of space as Euclidean that its elusiveness to perception is so striking.

Introduction

5

The first two chapters aim to show, in one way or another, the kind of yield for metaphysical arguments found in the study of geometry. A very few geometrical ideas are introduced quickly, sketchily but intuitively and put to work at once. This should serve to motivate the two more expositional chapters which follow them. In chapter 2 I take up an argument, or rather a challenge, to relationism which we owe to Immanuel Kant. How do left and right hands differ from one another? I reconstruct (one of) Kant's arguments for the view that it cannot be a property intrinsic to the hands that makes them different, nor can it be some relation which they bear to other objects nor to parts of space. Kant concluded that it must lie in a relation between a hand and space as a whole, in virtue of its topology - an aspect of its shape, as it turns out. But the argument needs to be reshaped since Kant was unaware of the consequences for the argument if we consider spaces with different topologies from Euclid's. These are introduced pictorially and used to show that handedness does indeed depend on the shape of its containing space considered as a whole, where shape is given a quite intuitive sense. Some relationist replies to this style of argument are considered and rejected. I conclude that although relationism can state that hands are handed it remains powerless to explain it in the luminous, insightful way open to realists. Relating the asymmetry of handed things to the shape of space as a whole makes handedness beautifully clear. These first chapters thus yield two examples of spatial phenomena (non-Euclidean holes and handedness) which can be explained by a style of argument which, while not causal, may yet call on the shape of space to explain events as well as states of things. It does so by concrete relations which space has to the things it contains. So there is a case for taking the shape of space to have a peculiarly geometrical role in explanation. The task of carefully making out in detail exactly what shape means when we apply it to space then lies before us. You may wish to skip these chapters either because the geometry is familiar to you or because it may be better to return to them later when it is clearer to what use I hope to put them. Chapter 3 begins with an account of metrical geometries which are alternatives to Euclid, introduces the important idea of curva-

6

Introduction

ture and articulates a main sense in which space has a shape. It then shows how these competing geometries fall within a wider and more general kind of geometry which gives, not a competing metrical account of space, but a view of deeper structures which the various non-Euclidean spaces of constant curvature share. The deeper geometry is projective. It explores what is common to all the geometries which share the idea of straight line (or geodesic), some axioms of incidence (about how points and lines are related to each other) and a quasimetrical idea of crossratio among points and lines. The story in chapter 3 is mainly a qualitative one with almost no reference to the mathematics in which the concepts of geometry are made precise and quantitative. Chapter 4 aims to introduce the mathematician's more articulate and general perspective on geometry and also to analyse the various levels of structure that make a space up. Apart from the more formal tools in use here, a main difference from the earlier treatment lies in the way in which the structures are defined locally on space and spacetime. The levels of shape are defined as infinitesimal concepts. The modest aim here is not to give the reader a capacity to do the mathematics but to read some of the formal sentences with a little understanding. Parts of the chapter assume a very elementary acquaintance with partial differential calculus. No proofs or calculations are offered, but some ideas are stated in the very compact notation which modern differential geometry offers. Various structures are developed as intrinsic to space; that's to say, they are taken to depend on nothing but spatial entities and properties for their understanding and not, for instance, on an understanding of whatever material objects may be in space. Various levels of structure are introduced as primitive; that's to say they are not defined in terms of other structures but are basic. Some are basic determinations of broader structures, somewhat in the way that 'red' is a basic determination of 'coloured', where 'red' does not mean 'coloured and F' for some term 'F' other than 'red'. I single out the most basic structure, the manifold (space with only its continuity and smoothness defined), for special treatment. This is partly because some philosophers take this to be the only structure which space or spacetime can actually have. Then I introduce a projective structure as intrinsic and primitive. Affine and metric structures follow so that a frame-

Introduction

work usable to study either space proper or spacetime is given. Chapter 5 returns to the problem of the relation between space and perception, beginning with Kant's opinion that the space of vision must have Euclid's geometry. Some modern discussions of this are unfortunately fixated on the challenge of a particular example. But it is a challenge which there is really no need to meet. This makes it clear that, to solve this problem, we must look into just what the geometry of visualisation in non-Euclidean spaces actually is. On some fairly natural assumptions, it is shown that in the more obvious non-Euclidean spaces, things would look very much as they do in Euclid's space. These natural assumptions are by no means compelling, but when we turn to others, there is no unambiguous answer to the question what our experience might be were we native to such spaces. Clearly, we need some principle which can integrate our experiences, but there is no single, simple answer to what the principle should be. Nevertheless, if we consider spaces which are topologically unlike Euclid's in being finite (but without boundaries) and we consider a series of experiences integrated over time, we find a strange but clear picture of perception in that kind of non-Euclidean space. But here we find a new methodological difficulty. It is called conventionalism. Conventionalists argue that, in any such case, we can redescribe our experiences so that, after all, they might just as well (for anything that observation tells us) take place in a Euclidean space. They conclude that there is no factual difference between spaces as regards their structure. The differences are all created by ourselves as conventions which render our theories of the world as simple as we can make them and as convenient as may be. Only the question of convenience is at issue in choosing a space in which to embed a theory, and not any question of truth. Chapters 6 and 7 dig deeper into conventionalism, the first in order to make out in more detail what the conventionalists are saying, the second to refute it. There are two main versions of conventionalism. Hans Reichenbach argues for the older version in neo-Kantian style. It is not entirely clear just what the argument is. Kant thought that sensation had to be subject to various formal principles before our experience could be intelligible and objective - before we could think of it as yielding a world which could be thought out articu-

7

8

Introduction

lately and understood as independent, somehow, of us. He thought that the formal principles were both a priori and inescapable. We could never make the world thinkable save through them. One formal principle was that space is what represents our perceptions to us as outer: we impose the geometry of Euclid on this form because we can do no other. Reichenbach appears to endorse the first aspect of Kant's thought but not the second. The mark of formal principles is not that they are a priori and inevitable, but that they are a priori and freely chosen by convention. Reichenbach argues through a number of intriguing examples to explain his view and convince us of it. But part of what makes the examples so elegant and searching also makes them puzzling. The argument that the conventions may be changed is apt to undermine the case that they are needed to make our experience objective and intelligible. A quite intelligible core emerges as retained amid the changing conventions. A crucial case concerns the conventions which Reichenbach thinks govern our free choice of a (smooth) topology for space. It is plausible that the way we understand what we perceive as objective and outer does depend on the basic continuity and smoothness of space and would not be intelligible without it. Further, if not even topological structure is real then conventionalism can legitimately join hands with the sort of relationism which claims that spatial relations make sense without mediating or sustaining paths. Not much attention has been paid to Reichenbach's arguments on topology, though his claims about it are surely both crucial and basic. A rather different version of how conventions determine all spatial structures richer than a continuous, smooth manifold has been argued, notably by Adolf Grunbaum. It depends on the claim that the continuous structure of spatial intervals does not permit us to find a metric which is intrinsic to space. Having arrived at an articulate picture of conventionalist views, the next task is to criticise them. Three main criticisms emerge in the general critique of chapter 7: conventionalism undermines its own motivations in leaving the alleged conventions looking like much more of a nuisance than they are worth; conventionalism depends on a thoroughly implausible view of observation and hence of the relation between geometrical theories and their basis

Introduction

9

in observation; Reichenbach's conventions for topology confuse several different ideas about what the conventionalist's tasks and achievements ought to be. Chapter 8 investigates the claim that the topology of space is open to purely conventional choice. I argue that Reichenbach's doctrine here is obscure. Made clearer, his arguments fall foul of a methodological principle made out in chapter 6: the alleged conventional differences result not in a redescription of the same objective world, but in a quite new one. On Reichenbach's own criteria, as far as one can make them out, changes in topological 'conventions' are factual changes and not empty ones. Chapter 9 investigates the question whether the metric of space rests on a conventional determination of the manifold structure of smoothness and continuity. Grunbaum's arguments that no space ordered by dense or continuous ordering can have an intrinsic metric are shown to beg central questions and to depend on an unnecessarily and confusingly abstract approach to whether the metric can be intrinsic. An account of how extension escapes Grunbaum's techniques, yet must be both intrinsic and primitive to space paves the way to showing how the metric can be both. But this does not yet solve the epistemological problem of how to tell what the metric of space actually is. I go on to show how the metric may be inscribed on spacetime by matter once we turn to General Relativity. This part of the argument depends on the analytic description of geometry given in chapter 4. It can be understood in the light just of a few qualitative ideas about spacetime. Thus the critique of conventionalism is complete: the ontological arguments that space cannot in fact be as we conceive of it are confusedly motivated and the epistemological doubts about its structure can be solved. The last chapter deals with the relativity of motion. I take the somewhat unusual course here of delineating the consequences of Special and General Relativity for rest and motion in largely classical ideas. That is, I assume that to take a system as at rest in a way that is satisfactory in metaphysics one needs to be able to give a global account of the whole of one's space and have this be consistent over an arbitrarily long period of time. While this is not incongruous with a spacetime perspective on the content of these theories - in fact it takes essential account of it - it is not framed quite

10 Introduction

in the local concepts which an analytic understanding of spacetime makes most natural. However, this approach allows me to introduce you to the Special Theory and, much more superficially, to the General Theory and to spacetime. The introduction aims to provide an algebra free and rather concrete account of as much of the theory as is needed to gain a useful perspective on the question of the relativity of motion as it relates the reality of space and to the explanatory role of its shape in modern physics. Though the account is, one might fairly say, rather artless I believe that it provides a useful perspective on the main issues. It uses the paradox of the relativistic twins to pursue the question whether and how far motion is relative in relativity theory. Most of the crucial issues about how the problem of motion bears on the reality of space or spacetime can be seen within this perspective. This is not to deny that analysis yields a deeper vision. But it does not provide a more decisive one. The argument of the book is, therefore, that space is a real live thing in our ontology. It is a concrete thing with shape and structure which plays, elegantly and powerfully, an indispensable and fruitful role in our understanding of the world.

1 Space and spatial relations

1 Pure theories of reduction: Leibniz and Kant Two great men stand out among those philosophers who have wished to exclude space from their list of things really in the world. They are Leibniz and Kant. Each man held views at some time which differed sharply from views he held at others. To take our bearings in the subject, let us begin by giving an outline of one theory which we owe to Leibniz's Monadobgy and New System (1973) another which we owe to Kant's Critique of Pure Reason and a third which we owe to Leibniz's letters to Samuel Clarke, a defender of Newton in the great seventeenthcentury debate on mechanics. Leibniz's argument in the Monadology is not epistemological but purely metaphysical. It is about communication of influences among substances, and so about their causal and spatial relations. He was convinced that anything real must be a substance or its accident; that is, reality is comprised in things and intrinsic properties of things. Any apparent relation is either really an intrinsic aspect of a thing or it is nothing. Clearly enough, space and spatial relations are going to pose a problem. Leibniz's profound and elegant theory shows how to rid his ontology of space. Everything revolves round one nuclear idea: nothing mental can be spatial. Mind and mental attributes are a model for what is real but unextended. So Leibniz can regard the world as non-

12

1 Space and spatial relations

spatial if he can account for it all by regarding it as mental. This is an astonishingly bold and penetrating idea. It is worked out along these rough lines. Everything there is must be a monad, an unextended mindlike substance. Every monad has certain qualities which are to be understood on the model of perception. Monadic perceptions are relevant to spatial ideas for two reasons. First, they define a notional (ideal) relative 'position'. They are like photographs in that what appears on the film also defines a position from which the photograph was taken relative to the things that appear in it. The position defined does not depend in any way on where the photograph now is. Nor do the perceptions of monads need a place to define their notional position. But, second, perceptions are unlike photographs, since perceptions are mental whereas photographs are spatial. The monads together with their perceptions define, at any time, an ordering of copresence - a system of perceptions. This is space. But then space is not a real thing but only an ideal one, constructed out of monads. The construction is wellfounded on the qualities of monads, but is a mere phenomenon - an illusion. There could not be a real space across which these things are interrelated. To suppose there could would be to attribute to mental things properties which they cannot possibly have (see Leibniz 1898 especially §§51-64. See Latta's footnotes to these sections). This is a sketch, not a portrait, of Leibniz's theory. What I mean to capture in it is a blueprint for what a fully reductive picture of space must be. For Leibniz, no objects are spatial, and ultimately no real spatial relations hold among them. His theory is, therefore, a pure case of reduction and it is just its purity I want to emphasise. It contrasts with impure reductions in which space is reduced to extended objects with spatial relations among them. I call these impure reductions because spatial ideas remain, in some way, among the primitive ideas to which reduction leads. Unless we can extricate spatial relations from the taint of space itself, it must remain seriously unclear just where this gets us. So purity is a gain. But Leibniz clearly paid a very high price in plausibility to make his system pure. The world does not seem to be composed of minds and nothing else. He evidently thought the preservation of substance and accident metaphysics was temptation enough for us to pay the price willingly. But I think the metaphysics in question has a strong appeal only if you believe that classical Aristotelian logic is all logic. Insofar as he did

1 Pure theories of reduction: Leibniz and Kant

13

base his system on convictions about the nature of logic, not knowledge, his theory is pure metaphysics, not epistemology. However, Leibniz was not quite right about logic. Quantification theory now gives us a logic of relations which can be placed on foundations much more secure than Leibniz ever had for the syllogism. Further, it is a more powerful system. So it will be the cost of Leibniz's theory in plausibility that impresses us most. That remark is not intended to throw any stones at the consistency of Leibniz's system, nor at the purity of his reduction of space to objects. We can find another case of pure reduction by glancing into Kant's philosophy. Though he admitted a debt to Leibniz, the deepest roots of Kant's thought are epistemological. Kant believed that our experience was only of appearances, never of things themselves. There can be no reason to suppose that, in themselves, things really are as they appear to us. We can never get 'behind' appearances to check any resemblance between them and things as such. But, Kant argued, we do have reason to believe that, so far as appearances are spatial, they do not resemble the things that give rise to them. We impose spatiality on the flux of appearances as a necessary condition of making sensory input both digestible by intellect and so experienced as objective. The mark of this is our necessary knowledge of geometry prior to any experience; this would be inexplicable unless the source of spatiality lay wholly inside us. Kant was wrong, notoriously, in supposing that we actually do have any such knowledge. But what matters now is to understand the bold outlines of his theory, not to criticise it. The mechanics of the theory go something like this. Space is a pure form of perception which we bring to the matter of appearances and impose on it. Only by making appearances spatial can we experience them as objective, outer, interrelated and intelligible. But we make them spatial without the aid of empirical instruction from the appearances. Appearances come from things themselves and what they instruct us in is always a posteriori. So, if space is a priori it doesn't come from things. Space is purely a gift of the mind, not of things. It is ideal, not real. Things themselves are neither spatial nor spatially related (Kant 1953; 1961, pp. 65-101). We have come to very much the same place as before, but along a different path. The point of discussing Kant was to see that there are indeed different paths. His spatial reduction is just as pure as that of Leibniz. It is also pretty equally costly in plausibility. For Kant, things

14

1 Space and spatial relations

apart from our experiencing them are forever beyond our ken. We grasp appearances predigested by forms of a priori thought. In both these examples, we rapidly scaled, by different tracks, a peak from which we glimpsed vast systems, comprehensive, elegant, pure and improbable. These heights are too dizzying and the air too rarefied for us to pitch our tents here unless we give up hope of finding another place to live. The metaphor is not a refutation, of course. But I am at pains to persuade the reader that the plausible reductions are impure and pure reductions are implausible. The cloud of circularity hangs over impure reductions until we see how to detach spatial relations among things from space itself. Both Leibniz and Kant argue about what is intelligible and necessary in our thought about space. This is important for us later, as is the purity of their reductive aims. Let us first look deeper into Leibniz's thinking.

2 Impure theories of reduction: outlines In his famous correspondence with Samuel Clarke, Leibniz advances a view in the Fifth Paper which, on the face of it, is weaker than the one we just saw. He writes as follows (Alexander 1956, pp. 69-70): I will here show, how Men come to form to themselves the Notion of Space. They consider that many things exist at once, and they observe in them a certain Order of Coexistence, according to which the relation of one thing to another is more or less simple. This Order is their Situation or Distance . . . Those which have such a Relation to those fixed Existents, as Others had to them before, have now the same Place which those others had. And That which comprehends all their Places, is called Space. Here Leibniz seems to present the idea of space as a construct out of extended bodies with spatial relations among them. Space itself is nothing: that is the core claim of relationism. He goes on to explain these remarks about space by drawing an analogy with genealogical trees. In like manner, as the Mind can fancy to itself an Order made up of Genealogical Lines, whose Bigness would consist only in the

2 Impure theories of reduction: outlines 15

Number of Generations, wherein every Person would have his Place: and if to this one should add the Fiction of a Metempsychosis, and bring in the same Human Souls again; the Persons in those Lines might change Place; he who was a Father, or a Grandfather, might become a Son, or a Grandson etc. And yet those Genealogical Places, Lines, and Spaces, though they should express real Truths, would only be Ideal Things. It is obvious how the genealogical tree is both well-founded yet a mere phenomenon. Little else is obvious. Leibniz has been widely understood as arguing that we must acknowledge things and spatial relations among them, while repudiating space as an entity distinct from the things and relations which somehow transcends them. This yields a plausible idea, thus understood. However, Leibniz did not mean us to take him this way. Before I pursue that topic let me take a leaf out of the Third Paper, §5 of Leibniz's debate with Clarke. I have many demonstrations, to confute the fancy of those who take space to be a substance, or at least an absolute being . . . I say, then, that if space was an absolute being, there would be something happen for which it would be impossible there should be a sufficient reason. Which is against my axiom. And I prove it thus. Space is something absolutely uniform; and, without the things placed in it, one point of space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows (supposing space to be something in itself, besides the order of bodies among themselves) that 'tis impossible there should be a reason, why God, preserving the same situation of bodies among themselves, should have placed them in space after one particular manner, and not otherwise; why every thing was not placed the quite contrary way, for instance, by changing East into West. But if space is nothing else, but that order or relation; and is nothing at all without bodies, but the possibility of placing them; then these two states, the one such as it now is, the other supposed to be the quite contrary way, would not at all differ from one another. Their difference therefore is only to be found in our chimerical supposition of the reality of space in itself. Here, I suggest, Leibniz is offering us a quite different kind of

16

1 Space and spatial relations

argument. The genealogy argument is meant to convince us that not only space but also spatial relations make a mere picture just as the tree does. The latter argument, which HI call the detachment argument, aims to persuade us that spatial relations among things may, as a whole, be detached, extricated, from spatial relations which things have to space. It argues that spatial relations do not depend on space. The cloud of circularity hanging over impure reduction can be dispelled. The thrust of each passage is different. To see this, look at the passage in the Fifth Paper where Leibniz likens space to a genealogical tree. What can it mean? A family or other genealogical tree is a type whose tokens are drawings - that is, concrete objects of a certain spatial structure used to represent what is not a spatial structure at all but only a structure of blood ties. The token-tree (the concrete copy of the drawing) is unreal not because it is not a real drawing - of course it's that - but because what it represents is only isomorphic to it in structure and otherwise utterly unlike it. Now Leibniz was talking about the drawing, about 'those genealogical places, lines and spaces, which, though they should express real truth, would only be ideal things' (Fifth Paper, §47). What is ideal is just that the positions of the names, the lines connecting them, and the spaces the lines traverse correspond to no connecting tissue, so to speak, between parent and child, for only the people exist, not some stuff that lies between them. The token-tree is real as a drawing, but there's no real tree linking a family. But if that is the right reading, how can we apply the metaphor to space? Is space a representation, a type or token picture of some kind, of how things are which, though perfectly proper in itself, misleads us as to the real nature of what it represents? Is it a misrepresentation? Well, no! The family tree might mislead because its spatial relations represent quite another kind of relation. But how can we think of space as a token picture representing spatial relations? Only by the use of spatial relations themselves and by the use of lines, places and spaces to represent lines, places and spaces. The analogy can only bring it out as somehow s^representative and so veridical; in any case, neither ideal nor msrepresentative, since not a representation at all but the very thing itself. So what does all this mean? Leibniz surely has the metaphysics of monads in mind here, though he never expounds that philosophy directly in his letters to Clarke. The mental is our model for what is real but neither spatial or

2 Impure theories of reduction: outlines

17

temporal. Each monad is related to every other only by containing a mentalistic representation of every other. These representations are understood as analogous to sensedata in respect of ontobgical status. Like sensedata, they are not themselves spatially extended nor temporally extended either. Each monad mirrors all the rest of the world by percepts each spaceless and timeless in itself and, most importantly, each wholly internal or intrinsic to that monad in the non-spatial sense of dwelling in its (possibly idiot) mind. A monad is windowless the percepts it has of other things do not come from them to it along a path and through a window. Without this sustaining structure of paths, Leibniz takes it that the spatial relations themselves are ideal. A preestablished harmony (which is not a real relation) among the intrinsic percepts of all the monads is guaranteed by the existence of the supreme and perfect monad, God. Percepts are all qualities of monads, not relations', monads are related to other monads only by internal relations such as similarity and difference. So the places of monads, on this exuberant metaphysics, are indeed ideal. Our thinking of things as really related by spatial relations, not qualities of things, is mistaken though well-founded: illusory but not subjective. We find here an early version of the view that space is unreal because it is only a representation (indeed a misrepresentation) of what is real. So in this passage, Leibniz is telling us not merely that space is a fiction but that spatial relations are fictions, too. If Mars is at a distance from Jupiter then there really is no path between them, because there is no real spatial relation between them either, save in a confused sense to be glossed by Monadology metaphysics and dismissed. Otherwise the analogy with the genealogical tree falters. So the purity of his account of spatial relations was, for Leibniz a necessary condition of its intelligibility. Leibniz did not think that spatial relations can be extricated from space and retain their status as real relations. But the Third Paper takes spatial relations uncritically. It claims to identify a fallacious argument from them to a mediating entity. We will return to it in §1.10. What lies behind this first version of the representation theory is a complex (and implausible) theory of relations. It is this which entitles Leibniz to claim that his reduction begs no question about spatiality: it's the price paid for purity. As far as I know, modern representationalism simply raises no question and enshrines no view about relations generally nor about spatial relations in particular. Thus it begs what

18

1 Space and spatial relations

realists see as a central question: what is required for a spatial relation to hold?

3 Mediated spatial relations I begin with a sketchy classification of relations. Let me stress from the beginning that the realist is deeply committed to spatial relations. Everything about space depends on them, as we shall see. It is crucial that the nature of spatial relations be correctly understood. There is no dispute between realist and relationist whether the role played by relations is indispensable and central. There is dispute only about what role it is. We can't talk generally about relations without talking about what is necessary or possible - about modality. I try to clarify what my modals mean by calling on the device of possible worlds. We cannot depend on logical modalities, on truths of formal logic and definition. Some properties supervene on others. Property (stratum of properties) P supervenes on another Q iff P properties possibly differ only if Q properties do. That is, for any worlds w^ and w* w^ differs from w i n P properties only if it differs in () properties. Let us begin with internal relations. A is internally related to B if and only if in every world in which A exists B also exists and vice versa (I simply assume here the not altogether straightforward distinction between qualities and relations.) A monad is what it is because of the sum total of its properties and cannot exist except with every one of them. This allowed Leibniz to claim that all relations are internal. Consider next grounded relations. Grounding is a modal but not an analytic matter: grounded relations supervene on the natures of their relata, often only as part of some more complex state of affairs. A relation R is grounded in the nature of x and y iff xRy supervenes on some state of affairs of which x's being Q and / s being (7 are constituents. I regard causal relations as grounded: I enlarge on this shortly. Relations which are neither internal nor grounded are external relations. They are independent of the nature of their terms. Spatial relations are external.

3 Mediated spatial relations 19

There are between-mediated relations (more briefly mediated relations), which are our main target. Realism demands a quite different idea of spatial relations from relationism. One reason for taking space as a real thing is the strongly intuitive belief that there can be no basic, simple, binary spatial relations. Just such relations are the foundation of relationism, so long as its basic spatial facts lie in spatial relations among objects or occupied points of space or spacetime. Consider the familiar (though not quite basic) relation x is at a distance from y. There is a strong and familiar intuition that this can be satisfied by a pair of objects only if they are connected by a path. Equivalently, if one thing is at a distance from another then there is somewhere half way between them. Distances are infinitely divisible, whether the intervening distance is physically occupied or whether the space is empty. All such relations are mediated in two ways. We cannot define them and we cannot complete their semantic description for the purposes of model theory unless we show them to be mediated both by the relation of betweenness and by further terms among which the original binary relation also holds. To put this in a general form, we need something like the following, where S dummies for any binary spatial relation: x is S-related to y iff in all worlds in which xSy there is a z such that z is between x and y and xSz and zSy. Thus any spatial relation is mediated by a sustaining complex of relations involving both betweenness and itself. Betweenness is mediated by itself alone and thus counts as the foundation of all other spatial relations. Plainly this yields a dense ordering of spatial points. No spatial point can be next to another. But, of course, two material things can be next to one another when they touch. For a relationist this is a primitive relation. For realists it means that no unoccupied point falls between the two. More generally, I take it that any relation neither internal nor grounded is external. All relations are either internal, grounded or mediated. Realism is not wedded to these last claims and the classification in terms of which I frame them is rather coarse-grained. But the claims strike me as plausible. What justifies this realist idea is that we understand it but we don't understand how spatial relations can hold unmediated. We have a primitive intuition that space is infinitely divisible and fail to under-

20

1 Space and spatial relations

stand any claim to the contrary. We find it obvious. Whether that is a good reason to accept it is pondered in §1.6. How is spatial mediation built into theory? Mainly by the use of differential equations in which, since Newton, fundamental theories have always been, and still are, expressed. It underlies our best theories of order type for space and motion. At its most familiar, we intuit that for any distance between things there is a shorter distance: this is another form of the intuition that space is infinitely divisible. That ancient intuition guides the development of continuum theory and does not spring from it. It is an a priori intuition. Here I lean on Kant ('Transcendental Aesthetic', 1961) who argued that our conviction that space is infinitely large, infinitely divisible, unbounded (and so on) cannot be based on experience, which is too poor to carry the load and already presupposes space. Yet space is a concept which we cannot do without. Perhaps, in some dim sense, we bring it to experience. Has it been programmed into our genes in evolution because it has survival value? If so, then its safety may well lie in its accuracy. Kant thought its involuntariness and a priority showed that we could never revise it. But in this he was wrong. We go on developing it and learning about it all the time, both by inventing new forms of geometrical understanding (such as Riemann's) and by experiment and empirical theory. In short, the provenance of the concept is unique and problematic. How bare an intuition is it? Rather bare, I think, but we may gain a new perspective on it from the standpoint of our understanding of physical things. That view of the landscape might look different and more congenial, though I think we will see no new things or properties of space thereby. We have no understanding of physical things independent of our understanding of spatiality. (I use 'spatiality' as uncommitted to either realism or relationism). Every physical thing is extended: that includes causally and theoretically basic things. If we had a non-spatial grasp of matter we might try to elude the charge that spatial relations are tied to space by brute stipulation that spatial relations among things are neither grounded nor mediated. While this might be hard to understand it need not fall under suspicion of being circular. But as it is, the idea of physical things is already permeated with spatiality, so that it is not clear how much light we are shedding on our mystery, nor indeed that we are shedding any at all, by taking for granted physical things

4 Surrogates for mediation 21

and bare, binary spatial relations among them. The relationist usually concedes an important part of the case for mediated spatial relations: occupied spatial points are not next to one another. This intuition can't be reduced to material object relations, since objects can be next to one another: they can touch. But two points within an extended material volume are always mediated by a material path. Every such pair of points has other points between them and it is thereby that they have spatial relations. Without the path

there would be no spatial relations between them since there is no spatial entity (occupied or not) between them. No two points can be spatially related unless other points are likewise related. Finally, how shall we represent mediated relations in models of spatial theories? If we portray them as sets of ordered pairs we fail to model the structural feature of mediation. If we include separate axioms to state mediation then they leave it improperly external to the modelling of each relation itself. No realist can happy with that. The statement of mediation is not something distinct from the understanding of the relation itself. The semantics of any spatial relation involve an infinite ordered set. This will turn out to be of some importance later. Standard models of spacetime theories distort their metaphysics.

4 Surrogates for mediation Leibniz repudiates both space and spatial relations. But it is important that he does not repudiate a surrogate form of mediation. Leibniz was convinced that there could be no vacuum (Leibniz 1973, pp. 82, 91, 158). The communion among monads, the 'spreading' of influence requires that, in the real ordering of copresence, between any two things there is another. The ordering of the monads themselves is logically mediated. Plainly Leibniz sees this as necessary both for harmony and community among monads as well as for the intelligibility of the well-founded phenomenon of spatiality. It is misleading to claim Leibniz as the ancestor of a crucial thesis of modern relationism: that there can be unmediated spatial relations. That is alien to his thought. A better candidate might be Locke (1924), Bk II, chapters, 13 and 14. Though Locke has no developed view of space, a famous passage (evidently drawn from Galileo) casts spatial relations as simply observable distances and directions among

22

1 Space and spatial relations

things. Locke never rejects mediation of the relations, but does not endorse it either. The main point to be made here is that unmediated spatial relations don't spring from a long-established tradition which needs no new justification. One form of relationism tries to mediate spatial relations not by spatial things but by possibilities (see Sklar 1974a, pp.281-285; Sklar 1985, §8). Suppose that x and y are at a distance but, as we would ordinarily say, nothing is between them. We are strongly drawn to concede that there is a path between them. Possibilist relationism offers us a surrogate for this intuitively obligatory mediation by path. It's possible that a physical thing z could lie between x and y. So spatial relations are mediated in the weak sense that possible physical points and possible relations among them (and among actual physical things) mediate. No actual space is needed to do the work. This project fails unless we make clear what sense of 'possible' this is. Consider a more explicit analysis of the idea that there is empty space between x and y: E: It is possible that there is an object z and z is between x and y and touches neither. If 'possible' means what it ordinarily means in philosophical contexts, vague though that is, E is certainly true whether or not there is empty space between x and y. This ordinary meaning can be expressed by 'It is logically possible' or 'It is conceivable'. But even if x and y are touching, it is logically possible and eminently conceivable that they could have been apart. Evidently this form of relationism cannot rest on that idea of modality. But what idea of possibility can it appeal to? Quite clearly the relationist cannot mean to ground the possibility of z's falling between x and y on the emptiness of the space between them. That would be circular, not reductive. But it remains unclear how to get the right kind of possibility which gives us the truth (without the circularity) of the analysing sentences in just those cases where we want them true. The force of this difficulty gets clearer as the cases which call for analysis get into more complex geometrical spaces. I will return to the point again when we can take more sophisticated examples (in §2.9). I am not here objecting to the use of modals. Troublesome they certainly are, but indispensable (see Nerlich 1973). Later in the chapter I traffic freely with just the sort of counterfactual conditional on which

5 Representational relationism

23

this kind of relationist calls in the reductive analysis of statements about space. I say how things would look and feel if space were locally curved. But this is no lapse into relationism-speak. Any counterfactual conditional needs a ground or truthmaker. What grounds my counterfactuals is the nature and especially the shape of space: they are true in every world which has the relevant kind of space (or true in the 'nearest' such world). When the relationist traffics with them he has to make it clear, somehow, that they are not grounded in the shape of space. Otherwise he moves in a curve not readily distinguished from a circle. His trajectory needs to be carefully watched. But my counterfactual claims are meant to illustrate and amplify my spatial ones. I appropriate part of the traditional arsenal of relationism, using its own weapons against it. Notice that the relationist also standardly endorses mediated spatial relations among material spatial points. An extended solid permits us to distinguish among the various occupied points, not merely among the distinct, independent material beings themselves. These are always at least densely ordered (Newton-Smith 1980, ch. VI). Paths traced out by moving things are taken to be mediated too, even if only by densely ordered distance and direction relations among things. This appears to be motivated in very much the same way as the realist's belief - by what makes sense to us. It lacks any motivation from within the relationist's programme itself. It is not clear why we should accept that the material relations need to be mediated if we may freely drop this requirement once it ceases to serve the relationist's turn.

5 Representational relationism The most formidable instances of modern relationist theories may be found in the work of Michael Friedman (1983) and Brent Mundy (1983). Both are distant kin to Leibniz's view of space as really a picture which, though illustrative is mainly illusory. For the moderns, the picture represents in as far as it portrays real physical things and the spatial relations among them; in as far as it portrays unoccupied spatial points having relations with physical things and with other unoccupied points, it is a misrepresentation. Let us see how Mundy takes space as a picture which portrays the real physical world in only a proper part. Take all physical things and

24

1 Space and spatial relations

assume an inner product structure for relations among them. The idea of an inner product derives from vector algebra and vector space. A vector, as we are concerned with it for the moment, is a directed distance from one point to another. In n-dimensional space, we can capture the directed distance structure by an ordered set of renumbers related to the coordinates or to the vector basis of the space. Think of two vectors A and B each giving a directed distance from the same point p. Then the inner product of these vectors is a single number which tells us the magnitude of A as it is projected onto B or vice versa. (Imagine two Vector arrows' which make an angle at p; then imagine an arrow from the tip of A which makes a rightangle on (perhaps an extension of) B. The length from p to this point of intersection on B is the projection of A on B. The inner product is thus the product of the length of each vector and the cosine of the angle between them.) The inner product of a vector with itself is the length of the vector, naively the distance along it from tip to tail. If the inner product of two vectors is zero then they are orthogonal (at right angles) to each other. Mundy's idea is that if we begin from the inner product structure among all material points we can find the vector structure among them. Given material points p and q, then the inner product structure of (pq) with itself will give us the length of the vector from p to q, while the inner product of the points (pq) and (pr) together with the lengths of these vectors will give us the cosine of the angle between them. Thus from the simple scalar numbers of the inner product we will get the more complex and more geometrical structure of a vector space. This will not be a whole vector space since it yields vectors only among the occupied points. Now we proceed to 'fill in the gaps' and construct the fullblown vector space. This is a process which embeds the concrete partial vector space within the full, abstract vector space and justifies us in using it to describe the occupied points and their relations. One has to show that the physical inner product structure and derived vector relations do have the formal properties which will allow us to embed it in a standard vector space of appropriate structure. Then one has to show that this embedding is unique up to an isomorphism: that is, roughly, to show that other embeddings differ from this only in ways which amount to differences in the way coordinates are conventionally assigned. Mundy (1983) shows how to carry this programme out in full,

5 Representational relationism 25

given that the physical system is simple enough to be embedded in a Euclidean space. One then regards the embedding space as an ontically dispensable, though useful, representation and only the concrete, physical, embedded relations as real. So, in this simplest, Euclidean case, we have a relationist reductive programme which we know how to carry out in full and rigorous detail. He shows that the construction extends to Minkowski spacetime. No earlier reductive programme can claim as much. In what way is the representation useful? It enables us to call on the theoretical resources of the abstract space in which the occupied points have been embedded. I believe that this must mean the purely mathematical resources of the theory and I take that to be Mundy's intention. So the embedding space has a use only as an instrument of calculation. If it has another use, it would need to be specified with great care, else it will remain unclear that the theory used is not, after all, the realist's theory in its concrete sense of real spatial objects (occupied or not) and the structure in which they exist. Once those resources are called upon we are firmly committed to their realist use and cannot properly discard them as having played no ontologically significant role. I argue that the theory is used in its concrete, not just in its purely instrumental sense, in later chapters. I risk misrepresenting Mundy's theory by picturing vectors as arrows. Arrows are directed paths, exactly what he seeks to repudiate. The inner product structure of a vector with itself is not to be understood as the length along the imaginary arrow from one physical point to another. The information at any point is simply a set of names and numbers: there is a number and a pair of point names for each pair of physical points, including pairs with the same point as each member. What is constructed out of this is, once again, not directed paths linking the occupied points one with another, but vectors as ordered triples of numbers. I voice again a difficulty in understanding this if the numbers are not measures of paths. As far as I do understand it, the intention is this: the numbers characterise the spatial relations but do not replace them. The relations simply hold in a way that doesn't depend on mediating paths. The aim of the construction is not to help us to understand unmediated spatiality but rather to show how to embed the concrete relational structure uniquely in the abstract one. It seems impossible that the information at the material points should

26

1 Space and spatial relations

ever be gathered independently of (spatiotemporal) paths linking the points. Of course these paths would be (or would have been) occupied ones, even if only by the light rays along which other things are seen. But the occupied 'gathering' paths need not coincide with the vectors. If my friend tells me where something is relative to me, then obviously the paths along which that information passed need not coincide with the shortest path from it to me. But the epistemology of this structure depends on information being passed by mediated causation which itself seems to rest on spatial relations being mediated, too (see §1.6). This relationism risks begging the realist's question that betweenness and mediation are the foundations of all spatial relations. Now this allows the relationist to state all the claims he ever wants to make. A crucial question, however, is what it allows him to explain and whether his pruned down structure is robust enough, for instance, to nourish speculation and invention within given space and spacetime theories. I will be arguing in later chapters that the whole structure of space, whether occupied or not, plays such roles, which go well beyond providing for a theory of calculation. Mundy notices other problems which are best mentioned here. In non-Euclidean spaces, we need to distinguish the constructed vector space from the physical spaces in which it is defined at each point (see §4.1). The vector space is only a tangent space, not part of the real space. If the space is constantly curved this looks straightforward in principle. If the curvature is variable (as in General Relativity) then, as we shall see, we cannot relate the vector spaces at different points save relative to a path between them along which the vectors are parallel-transported (§4.3). This does raise, not just technical problem with carrying out the programme, but a problem of principle. It looks like appealing to mediated spatial relations. Friedman offers us a similar theory. He does not endorse it, but he does describe it at some length (1983, pp. 217-23) There are differences in detail. These have consequences which do not matter here (see Mundy 1986; Catton and Solomon 1988; Earman 1989 for detailed comparison). Where Mundy talks of embedding, Friedman begins with a model of the realist's spacetime theory in which the relations described by laws of the theory are given for all spacetime points, occupied or unoccupied. Friedman then considers the restriction of the model and its relations to a submodel: just the set of occu-

5 Representational relationism 27

pied points and relations among them. If the model is well defined, then the restriction must also be well defined. The spacetime realist regards the restriction as a submodel of the model and incomplete. However, the relationist sees the restriction as the primary model which is embedded in the extended pictorial model. The extended model is not included in the relationist's ontology. It is merely a picture of things. This misrepresents the debate. The relationist account of things is not part of the realist account, nor is his model a restriction of an acceptable realist one. Standard model construction simply fails to render the core issue in debate here (see p. 9). The main point cannot lie simply in theorems of model theory or of measurement embeddings. It is a metaphysical problem. Relationism just helps itself to a favourable solution. Neither Friedman nor Mundy sees a problem with the spatial relations themselves. They simply take them for granted. The realist is seen as wishing, for no obvious reason, to add unoccupied points to models. The plausible realist claim to mediation, the fundamental role of betweenness and of mediating paths which motivates the claim, drops from sight. The relationist pays scant heed to the prior question of his entitlement to his practice. We are then invited to consider the grounds on which to prefer the one story to the other. Mundy thinks that there are no grounds for realism. Friedman thinks that there are good, but subtle, grounds for it: it unifies spacetime physics thus making its explanations of things easier to understand and, more importantly, allowing the realist's theory to be better confirmed by evidence than the relationist theory is. Debate about the nature of spatial relations is aborted. Friedman reads Leibniz as taking spatial and temporal relations for granted (1983, pp. 62-3). True, Leibniz's theory of relations hardly surfaces in the correspondence with Clarke. It's certain, however, that Leibniz rejected spatial relations as illusory, well-founded though the illusion is. (Earman also considers this point in his (1979), but gives a sense to it with which I disagree in Nerlich 1994, §9.) He accepts mediation as grounded in the structure of monads. Ungrounded, unmediated relations have no place. He would have deemed them unintelligible.

28

1 Space and spatial relations

6 On understanding So much the worse for Leibniz perhaps. Indeed, we can fairly expect a reply to all this which is characteristic of recent Anglophone metaphysics. We can expect the relationist to repudiate the need to understand. Theories of meaning, meaningfulness and logical necessity have fallen on lean times. The charge that some doctrine is or is not meaningful is too likely to be merely intuitive and ungrounded. Debates about understanding risk collapse into mere arm-waving. Many of them fail. That is the story of a great deal of philosophy in this century. Arguments from what makes sense have not been regarded as robust for some time now. This is a two-edged sword. Neither side in the controversy may appeal to the strangeness, the unintelligibility of space. Empty space is meaningful, conceptually possible, the world could well have lots of it. A relationist has to show not merely that we can cut down the realist's ontology to relationist size, but that we should: that the rich and possibly accurate picture of a world in space not merely may but does delude us. He must convince us that in dismissing unoccupied spatial points there is no appeal to them to explain the nature of the world. Unless the role of the space 'picture' is very explicitly confined, he risks making serious ontic use of what he claims to repudiate. What Mundy and Friedman exploit is a kind of algorithm of reduction. One role of algorithms is to remove the need for us to think, invent and understand. In logic, algorithms as procedures of decision, remove the need for genius and inspiration. They are a means of getting thinking out of logic and replacing it with mechanism. Bacon thought that scientific method ought eventually to 'level wit' in the sciences and reduce the whole enterprise to mechanical rule. Of course, this was never accomplished; clearly, it never will be accomplished. I suggest that, in the two forms of relationism just described, a main thrust is to get the thinker out of metaphysics by placing a formal procedure before us and resisting the claim that there is any problem of understanding the use of it. Explanation itself requires that we understand. We can't deploy arguments from explanation while repudiating complaints that one can't understand. The issues here are difficult, somewhat intangible, and the going methodology of them unsatisfactory. I will appeal to

6 On understanding

29

examples to argue a case that unmediated relations need to be explained. My case strikes me as, in one sense, rather winning. But I do not claim that it compels assent. Consider another mediated relation, causality as it was conceived of classically. Cause is grounded in the nature of the causal state of affairs and in the nature of the effect which that produces. The philosophical literature on causality is dominated by the question just how to conceive of the grounding of causal relations (see French et al. 1984; Armstrong 1983; Mackie 1974). For much the most part the question of mediation - the question of action at a distance - is not discussed in depth. It is apt to be conceded that actual causes are mediated but that mediation is no necessary condition of causality. Typically, causal efficacy is seen as propagated from the cause to the effect along intervening states of affairs, each of which is causally linked to later, more distant states. It is mediated by a continuous causal chain or path. Causal mediation is not necessary, but in as much as it is even possible, it surely depends on mediated spatio-temporal relations. Causal mediation is plausible because the spatial (and temporal) mediation is guaranteed. Let's now consider the paradox of Einstein, Podolski and Rosen, the inequality theorem of Bell and the confirmation of Bell's theorem by experiment and observation. Two particles in the singlet state of spin zero, each with spin one half, are separated arbitrarily far. A measure of the spin of one of them along a given direction determines the spin of the other relative to that same direction. Bell's theorem shows that no initially local 'coding' of the particles is consistent with the predictions of quantum mechanics for an ensemble of spin measurements on such particles. The determination of the spin on one particle by the measurement of the other is not carried by any local, that is, any mediated, connection. This is by no means a straightforward instance of unmediated causality since not a plain instance of causality at all. My point is that philosophical discussions of locality, of causal relations without benefit of mediating causal states, leaves one totally unprepared for the deep consternation which confirmation of the Bell results has caused among physicists. Both the structure of the difficulty and the reaction of physicists to it are shown with great force and clarity in Mermin (1985). Surely this suggests very strongly indeed that the mediation of causal (quasicausal) relations lies deep among the tenets of modern science. Not

30

1 Space and spatial relations

so deep as to be ineradicable when confronted with a recalcitrant reality, but deep nevertheless. The dismay, as Mermin documents it, seems rather clearly to be of this kind: if the world contains these non-local connections then it resists understanding. And, if it does, that is because mediated spatial relations are so intuitively compelling, or so I suggest. In any event, doesn't the social phenomenon within physics, the fuss about non-locality (to call it that), suggest not a naivety among physicists about the non-necessity of causal relations which their more astute philosophical colleagues foresaw centuries ago, but rather the opposite? I suggest that our own classifications are inadequate when they suggest that what we cannot show as a logical necessity is something which we may freely jettison on the artless algorithmic ground that it yields a leaner ontology. Getting the thinker, the understander out of ontology in these ways may prove to be less sophisticated than it is apt to be represented. Quine (1960) spoke of the deep entrenchment of some items in the web of our beliefs. But I do not think that this metaphor is either adequate or well worked through (see Nerlich 1973). We need something like the idea of a satisfactory world, one which though it is logically possible, resists understanding, baffles inspiration and insight and gives rise to endless and irresoluble metaphysical conflict. Quantum mechanics tells us that we live in some such world. Newton's world of absolute space is perfectly possible but not at all satisfactory, for reasons discussed in §10.4. So where space, too, is concerned the philosopher's insouciance toward unmediated spatial relations may prove to be naive and premature. If the crunch were ever to come from the recalcitrant world of experiment, we might find that, all along, the relation was understood to be mediated: dismay in physics might parallel that over Bell's inequalities. Of course, this is nothing like a proof that spatial relations are mediated. However disconcerting we might find the failure of this intuition, nothing necessitates its truth. The negative message is also awkwardly clear from the dismay within physics. Non-local connection looks set to stay. The discussion ought to lead us to note well, however, that there is no empirical evidence against mediated spatial relations, and rather a lot for them. Getting the need for thinkers and understanders out of some enterprise which has been the preserve of genius and inspiration has

6 On understanding

31

to be a substantial gain. Obviously, getting the thinker out of some enterprise is not itself algorithmic - it needs imagination of a high order. One sees that quite clearly in Friedman and Mundy. But much that we may be tempted to think of as advancing the cause of philosophy merely deletes the need to understand (see Sklar 1985, p. 10 for another example). Nor am I denying that appeals to intuition and understanding have impeded progress. Early resistance to the nonEuclidean geometries, to the relativity of time, and the like, show how fruitful it can be to break through the barrier set by the demand that new ideas be already intelligible. Nevertheless the risk here is that we will mistake for solutions to problems devices that have merely the form of solutions. In our case, we risk ditching mediated spatial relations on the dubious basis of grasping the mere form of unmediated ones. Another example suggests how little mere form may give us. Take a mediated grounded relation like seeing or smelling. I know what seeing is, and I can even tell you a good deal about what I know, as long as I can tell you how it is mediated by propagated light waves (photons or whatever). If someone asks why seeing couldn't be an unmediated grounded relation, I have no answer. Maybe it can be shown that no unmediated relation could possibly count as seeing, but I don't fancy the job of showing it. It is not a plain truth of metaphysics that seeing could not be another sort of relation than it is. Leibniz thought of seeing as grounded but unmediated. Roughly, a pre-established harmony between things seen and seers of them ensures that (almost always) perceivers have percepts of a thing when and only when (i) the state of the thing seen matches the seer's percept and (ii) the thing can be 'reached' (here one plugs in Leibniz's grounded, unmediated story about motion from monad a to monad b) along the 'path' of perception (which, equally, is not the mediating path of a light ray, monads being 'windowless'). There are, of course, the 'mediating monads' in the logical order of copresence. One has a special story of some sort about delusion and illusion. (I do not count God's pre-established harmony as a mediating entity.) This isn't a downright silly story, since it is part of an interesting global metaphysics. It is bizarre, perhaps, but at least one knows what one is up against. Suppose someone now asks why seeing couldn't simply be a grounded unmediated relation, something that simply relates seer to thing seen with no supporting story telling how the one

32

1 Space and spatial relations

sees the other. (One would have to disallow that the perceiver's state alone constitutes the perception, for an inner representation will be delusive unless the relation of perception holds too.) Is it naive or obstructive to say that one doesn't know how to reply to this since we don't really know what is being put to us? For instance, does it count that one doesn't quite know whether the orientation of eyes must fit into the story? I don't think it naive. I don't know what this 'theory' of seeing is really supposed to be for it makes no new move in the game of articulating the concept of seeing. It might suggest how to make a new move, but it does not count as making one. In any event it sheds no light at all on actual seeing. To clinch that, I owe you a comprehensive theory of understanding but can't deliver it. Understanding is an ability; in a slogan it is a kind of know-how about knowing-that. I offer three remarks about understanding. First, it is not just a deductive or proof-theoretic capacity within a theory: one can identify the postulates and theorems of an uninterpreted formal system and grind out its logic, but one can't understand any such system. Second, knowing the semantics of a theory ought to include relating its assertions, somehow, to perception and action. But saying just how is subtle as well as important. Shortly, I will be illustrating our grasp of spatial statements by reciting some counterfactual conditionals about what might be seen or felt under this or that spatial conditions. This does not constitute our understanding nor anatomise it, since the conditionals make sense only given the prior understanding of spatiality. The perceptual content does something like inscribe our understanding on a possible world (as in §9.8). It fortifies it, articulates it. Third, perhaps it also extends our understanding since the pictorial grasp is generative of other pictures, other capacities of various kinds. At any rate, a mark of understanding is the power to connect, to anticipate, to imagine and invent. The role of perceptual ideas in the course of this remains elusive and intricate. The modern relationist accepts so small a burden of explanation. Leibniz at least felt he had to tell some story or other to make us understand what the grounded, unmediated relation of seeing could possibly be. But modern relationists recognise no such need. What is the upshot of all this so far? Recent work has shown that there is a reductive algorithm of an ingenious kind which can be applied to spatial realist ontology. The formal problem of showing that

7 Leibniz and the detachment argument

33

there is such a thing has been largely solved. Given that we have the algorithm, it would seem that Ockhamist principles will always justify us in using it. The recent history of physics as it concerns non-locality suggests that this may be incautious and even naive. It seems to be rather clear that physicists would never have accepted non-locality, the failure of causal mediation, on merely Ockhamist grounds. So, again, it seems that something more than the mere form, the mere reductive algorithm, is needed to give us metaphysical grounds for regarding unmediated spatial relations as in good conceptual order. But this does not establish that they are not in good order. Whether they are or not waits on further studies. The representationalist offers no argumentative reply to the intuitive claim that spatial relations are mediated. If the claim is true, representationalism begs the question against realism. Whether that charge is a just one is a question which we have no well-established principles for answering. Mediated causality is deeply intuitive but false. Yet it is hard to see how we can fail to rely on understanding in constructing theories. What can't be understood is not likely to be useful since, though it may state things, it can't explain them. However, banging our heads hard against the shut door of unintelligibility has sometimes let us into new rooms for the imagination which eventually become furnished with new understandings.

7 Leibniz and the detachment argument In §1.2 I quoted the Third Paper of Leibniz's correspondence with Clarke in which he objects, not to space and spatial relations as mere phenomena but rather urges that we can extricate spatial relations among things from spatial relations of things to space. If the argument works it proves that the practice of taking spatial relations among things as unmediated is innocent of circularity and question-begging after all. So this detachment argument is crucially important Leibniz wants to grant space to realists and then show that the gift is empty. It should make sense to change all the spatial relations between things and space while retaining the spatial relations of things to one another, but it doesn't. Thus the realist is doomed to concede that the new state of affairs is different from the old, but fails to find a way to state this difference in terms of the privileged way of stating matters;

34

1 Space and spatial relations

that is, in terms of spatial relations among things. If it can't be stated in this privileged way, Leibniz thinks, then it's unintelligible. The major conclusion of the argument clears the relationist of the otherwise plausible charge that spatial relations presuppose space itself. Leibniz suggests a number of ways in which we could change the relation everything has to space while leaving all the spatial relations of things to each other just the same. The simplest of these, though Leibniz does not mention it, is to double the size of everything. The quantity of space occupied by everything is changed but the relations of one thing to another are not. Hence, the argument would have us concede, we may detach the whole system of thing-thing spatial relations from the wider system of thing-space spatial relations. So where we might have thought that the former system of relations was logically dependent on the latter, we see that this is illusory. The thing-space system has no role to play. So the entity which founds that system, space, has no role and should drop out of our theory of the world. This argument and the ensuing critique of it is developed more fully in Nerlich 1994, chapter 6. What can we say about the detachment argument of the Third Paper? It is invalid. You can double the size of everything in Euclidean space without changing the spatial relations among things. You can't do it if space has another geometry - or another shape, as we will call it later. It does not hold for simply any space, as Leibniz requires. It is easy to see that if space has the structure of constant positive curvature, then things would change their shape when they change their size. Such a space has a close 2-dimensional analogue in the surface of a sphere. Clearly, as the size of a triangle inscribed in the surface changes, so will its shape. Now this does not mean just that Leibniz's premise is false; that he simply didn't know that there are geometries other than Euclid's. It means that his argument is invalid, whether the space in question is Euclidean or some other space. For he takes it that thing-space spatial relations play no distinctive role in determining thing-thing spatial relations. But they do. The symmetries inherent in Euclidean space play just as distinctive a role in conserving the shape of things under increase in size as the structure of non-Euclidean spaces plays in changing shapes. It is true that in the Euclidean case, when we change the thing-space relations by changing the thing-thing relations the

7 Leibniz and the detachment argument 35

result is a similarity transformation - all geometrical properties of things are similar. But when we change thing-space relations by changing the other term of the relation, that is, changing the Euclidicity of the space, it is clear at once that the relation, though conservative, was far from a trivial one. The way in which Euclidean space eludes perception because of its symmetries may lead us to wish for an argument which would allow us to drop it from our ontology. It may seem to sharpen the edge of Ockham's Razor. But we must not confuse having a motive to banish imperceptibles with having an argument for banishing them. The detachment argument is powerless to show us how to rid our ontology of space. Leibniz gave us a reason of principle for thinking that his detachment argument must work. It was not that space is Euclidean. It was this: Space is something absolutely uniform . . . one point of space does not absolutely differ in any respect whatsoever from another point of space. (Third Paper) I do not dispute this claim, but it does not yield Leibniz the principle of his argument. Doubling in Euclidean space leaves thing-thing relations invariant not because all points of the space are alike, but because of the way the points are spatially related to each other. In non-Euclidean spaces, where doubling changes shape, it is still true that the points are absolutely similar. But they are differently related. Leibniz thought that the assemblage of the indiscernibly different points made space what it is. It does not. The point that must be grasped here is that no realist can dispute the dominant explanatory role of spatial relations. They, not the nature of the points they connect, do the work. At times relationists seem almost to argue as if the realist must reduce spatial relations to some inscrutable property of spatial points absolutely or to some property of space as a whole, as if its structure did not consist in the relations among its parts. Some realists may also lapse sometimes into some such view. It is profoundly wrong. Does this make the relationist's task easier? Hardly! He still has to show how spatial relations can hold without space to mediate them. He has to show that he can do all the explanatory work with only occupied points which the realist can do with the whole of space. The case for mediated spatial relations is strong. The continuum is

36

1 Space and spatial relations

well entrenched and well understood. No one has shown how to extricate spatial relations from their attachment to space. Unmediated spatial relations are certainly strange. So far, nothing makes it plausible to reject mediated spatiality other than the elusiveness of paths and other spatial

things to perception. That and the advantage of economy, which is easily overrated. We will turn to the question of perception next.

8 Seeing places and travelling paths This is not enough to upset the relationist's applecart. I need to tackle a core motive of the theory and show that it is naive. The motive is this: space is inaccessible to perception. It motivates a core claim: the reason why space is imperceptible is that it isn't there to be perceived. It is invisible because unreal. The real is equated with what's observable in principle. So I now turn to make some observations on how space and perception are linked. I will aim at three conclusions: (1) that space is perceptible in principle; (2) that not its ontic but its geometric type may make it elusive to perception; (3) that the shape of space explains various possible phenomena which are not explained merely by reporting them in terms of unmediated relations. It does so in a peculiarly geometric, non-causal, style. We are wont to say that you can't see space or places and you can't touch them either. But paths, regions, places and space (empty or not) involve themselves in our perceptions. If I see Mars at a distance, then there is a path across which I look; an explanation of what goes on in my seeing Mars involves the path and the places it joins just as surely and concretely as it involves the planet, the traversing photons, the retina and the visual cortex. I can move my hand along part of that path. I can look along your sight-path (whether or not you see something along it) from another angle, so that the path lies in a plane of my vision (one I see along). I may see thereby that the path is empty, or see the heat source that troubles the air through which the path passes, or see where to raise a dust that might obscure your sight along the empty path and so on. These familiar facts give substance to the idea that, when two things are spatially related but not in contact, then there is something between them, a path or region, empty or not, which we can trace, obstruct, or clear. Consider places-at-a-time. I can see that there are two things exactly

8 Seeing places and travelling paths

37

alike in the telescope's field of vision without moving my eyes from the place of one to that of another. We do not say that I see the places-at-the-time when I do that, but only that I see that there is something or nothing in these places. But I can distinguish things solely by distinguishing places-at-a-time, even though the places aren't visible objects and even though I need no eye movement to distinguish them. Thus spatiality permeates vision. It also permeates other modes of perception. Proprioceptive and kinaesthetic perception is primitively spatial. Even hearing is primitively, if not very accurately, spatial. Causally, hearing is directed by the sound's reaching one ear just before it reaches the other. We do not perceive the minute time-gap but only the direction of the sound. There is an immediate intuition of direction, one might then say. Here again paths, places, directions in them engage with perception in a way we can only describe as concrete. Let me compare and contrast the role of photons in perception with the role of places, directions and paths. Photons are not perceptible entities, not even in principle perceptible; they cause seeing but, due to the way they cause it, can't themselves be seen. This does not mean we can never see what causes our seeing - everything we see causes our seeing it - but it's how things cause it that matters. We can see that photons are about the place but we can't see them. They can never be at rest. They have no mass (though that has been questioned). These seem to be oddities of their ontic type. But they are certainly concrete, for all their perceptual elusiveness. That is because they have positions (even if indeterminate ones), they move, they fill space (a little) and they (more strictly, the events of their absorption or emission) cause things to happen. Now, as I have been saying, paths are involved in the explanation of what and how we see things. But they don't cause our seeing things. Because of the way light rays (made of photons) lie along paths, paths couldn't be seen if anything else was to be seen. (I mean that they couldn't be opaque visibles if anything else was to be seen. But I go on to conjecture a sense in which they are seen.) Places have positions - indeed they are positions. They do not move but have relations of a concrete sort - being briefly occupied, for instance - tied to the motion of things. They do not fill space, though they may be filled by things and they are certainly spatially extended. They have no

38

1 Space and spatial relations

mass. In these ways the concreteness of paths and spaces is somewhat like that of photons. Yet they seem metaphysically utterly unlike. Now while indeed there are clear metaphysical differences between them I will argue that these seem more significant than they really are. It isn't the real ontological differences between photons and places that strike us. It's the difference between photons and particular geometric kinds of places that impresses us. The kinds of places I mean are Euclidean.

9 Non-Euclidean holes As a preliminary to examining the differences between places and photons, let me spell out further how the idea of concrete engagement of paths with perception works so that we have a clearer idea just what the difference is - if the one I focus on does have the metaphysical relevance I think it has. (You will find a more extended treatment of the themes of this section and the next in Nerlich 1994, chapter 7.) I'll deal with just two examples, though their number and kind could easily be multiplied. Talking counterfactually about unoccupied spaces of strange geometries is not relationism, but illustrated realism. Stating, about actually empty places, how things would be were they occupied is illustrative talk about their properties, occupied or not. The stories are grounded in the geometry of the local spaces. The imaginary percepts merely trace the actual geometry out. Let's suppose that, nearby in an otherwise flat space, there is a football-sized volume within which the curvature sharply varies. (All you need grasp in order to follow the examples is that light lines will have no parallel paths to follow through regions which aren't Euclidean.) It will be a non-Euclidean hole. Since the curvature is zero everywhere round this hole but not zero inside it, it has to contain both positively and negatively curved regions. Linear paths which are parallel outside the hole converge and diverge again inside it (depending on just what kind of hole it is). So, as we look at distant things that lie beyond the hole, the photons by means of which we see them sometimes pass through the hole and sometimes not; things will change their appearances as the visual angles subtended by the various paths change, just as things change their shapes and shimmer in a heat haze or when seen through some inhomogeneous physical medium

9 Non-Euclidean holes 39

like uneven glass. Now as we circle the hole and look round it and through it, we would see that the distorting region lies now in this direction from us, now in that, and we would soon see how distant it was and how big. In fact would it not be just like detecting a (smaller) warp in very clear glass, seen only by the distortions of vision that it causes? If all this came about, would we say, not just that we see that there is such a hole, but that we see the hole itself yast as we say of warps in clear glass? I think we well might. So when we look at Mars and see that it is at a distance along a path, we might come to say not merely that we see across the paths which the photons take but that we see part of the paths themselves - the hole through which the paths lie. And were there to be many holes and many flat intervening spaces, might we not come to say that we see places quite generally? Even if we did not go so far, it seems that we might at least say that since Mars is at a distance from Jupiter, then not only is there an entity involved besides the planets but that it is an entity with essentially visual properties - visually distorting ones - and hence quite certainly concrete. Helmholtz showed that only spaces of constant curvature permit free mobility; that is, if the space is variably curved then a thing would have to change its shape in order to move from one region to another of different curvature. It would not be freely mobile, therefore. Squash a flat piece of paper onto the surface of a sphere and it wrinkles up and overlaps itself; if it were elastic, some parts of it would have to stretch or shrink. Since we ourselves are reasonably elastic we could move about in a space of variable curvature, but only by means of distorting our body shapes into non-Euclidean forms. We would have to push to get our bodies into these regions, for only forces will distort our shapes. If the curvature were slight, the rheumatism might be easy and bearable; if acute, fatally destructive, just as if you fell into a black hole. Let us suppose that the changes are noticeable and the effort to move into the hole perceptible too. Then we could feel nonEuclidean holes. They would be more or less obstructive, some of them downright barriers to progress. We could palpate their contours and ache with the pressures of keeping our hands in the parts of deepest curvature. If that does not show the hole concrete, what could possibly count as concrete?

40

1 Space and spatial relations

10 The concrete and the causal Non-Euclidean empty holes are not just perceptual. They are concrete spatial things that are parts of the empty spaces between material things. They intervene; they mediate. The core claim is false. Now when, in our world, we look at empty places and try to see them and palpate them, they're invisible and intangible. That's a contingent fact. So our space is imperceptible in the same way as clear air in a jar is. We know where to look and feel to see that it's invisible. That we see and feel nothing there shows that the perceptual qualities of our space are different from the perceptual qualities of other spaces. It's in a particular geometric kind of space that empty places and paths are imperceptible. Space is not of an ontic kind which makes it non-visual in the way that the ontic kind of numbers makes them non-visual. So we do see, quite empirically, that invisible places and paths mediate spatial relations among material objects. We feel the bland symmetries of our space. The relationist picture of spatial relations is not consistent with our empirical understanding of them. It is tempting to say here that places, empty or not, have causal powers. These explain what happens perceptually even when nothing much happens. That is what the relationist story omits from the world. But I hesitate to say so. When I push to get my hand into a non-Euclidean hole, I don't push against the hole. The push into the hole has to push my body parts against one another (or apart) so as to change the spatial relations among them and give my hand a non-Euclidean shape that can be in the hole. My body's molecules move along linear paths when unforced, but once the geometry makes the paths converge or diverge I have to exert external forces on myself, else the internal elastic forces that keep me in shape will not permit the convergence or divergence. My hand's shape won't change if these internal forces are the only ones at work, and the hand won't move into the hole either. So I don't pressure space and it doesn't pressure me. I can't push, pull or twist it; nor can it do that to me. Yet I feel the hole distinctly - or so I am supposing we could pointfully say. So I think that the part a hole plays in explaining my pains and pushings is not causal. Many who accept my examples think that it is causal (Mellor 1980; Le Poidevin 1992). If they are right, that is an obvious boost to the idea that space is a real, concrete, mediating

10 The concrete and the causal

41

thing - it's causal. But I persist in thinking that space is not causal for a somewhat subtle reason. There is still no consensus yet on what we mean by cause. The debate whether some explanations are non-causal can still take on a definite sense in the context of definite theories, though the basic idea of cause is not tied to any one of them. In classical mechanics (and, perhaps less clearly in relativity mechanics too), causes are forces. An explanation of a process is causal if it tells about forces. Thus the explanation of linear motion at constant speed that it is force-free is a non-causal explanation. Now you might jib at saying that there is any explanation at all here, so I make my point negatively. Galileo's revolutionary insight, fundamental to classical mechanics, is just that uniform motion in a straight line is not caused, not forced. There is no causal explanation of it. So if that is true of the puzzle he resolved, then the geometrical structure of the hole is not a causal agent in my struggle to get into it. The explanation of the struggle is that there isn't the space (isn't the shape) there for me to fit my hand or body into without the struggle. The structure of the hole independently of my filling it explains things. But not causally it says nothing about how the hole pushes me or how I push it. It is geometrical explanation. It is analogous to the situation with a rotating flywheel. The angular momentum of the flywheel is conserved, yet it rotates under tension. If it rotates too fast it will burst in pieces. Here one may resort to the fiction of inertial or centrifugal forces. The real explanation why the wheel rotates under tension is that its molecular parts are being pulled in from their force-free linear trajectories (pointed out by the tangent vectors of their motion at any instant) into curvilinear and hence accelerated paths by the elastic forces of the other molecules which surround them. The linear structure of space determines where the force-free paths lie. Comparison of linear and non-linear trajectories plays a role in explaining the tension. But space exerts no force on the flywheel and plays no causal role in the explanation. Whether we say here that the explanation is causal or not matters less than the recognition that the two styles of explanation differ. I take geometrical explanation, a kind that calls on local or global shapes for space, to be a kind of explanation all its own. It is like causal explanation in that it explains events, and like it in that the explanans needs to be understood as a real concrete thing for the

42

1 Space and spatial relations

explanation to make sense. Either account will support realist ontic conclusions. Here we do find a notable difference between photons and paths or places; photons are causal. Another is that though places have location, like photons, they have no ontologically marked bounds as particles do. Particles end where particle stuff ends and other stuff or no stuff begins (roughly). Where a place ends is a largely arbitrary matter, since places are surrounded by other places and have no ontic bounds, even when they are sharply curved and whether or not they are filled by matter. But I see no reason in this to query the concreteness of holes and other places nor to question their entityhood nor their mediating role. They look just like well paid-up card-carrying things. So, again, whenever one thing is spatially related to another, the ontic situation is that something intervenes - something concrete which engages in concrete ways with particular perceptions and experiences. We can see or feel what this mediating entity is like. If that is correct then a fundamental tenet of relationism is false. Spatial relations are mediated and empty paths and places are empirical things. Conversely, suppose that x is at a distance from y and no material z is between x and y. Consider this counterfactual conditional: if you were between x and y, you would be able to see or feel the structure (probably feel just its blandness) there. To accept this is to concede that spatial relations are mediated. Not to accept it would be inconsistent with accepting the many counterfactual consequences of much of our theory of the world which we concede without hesitation. The conditionals of a realist theory of space are no less sound than those based on gravitational theory which tell us how things would fall if unsupported. It seems to me a highly plausible concession. But, again, we must not mistake our concession for a reductive account of what the place is. The space founds the conditional, not vice versa. To sum up: we began by looking for a theory which offers us a pure reduction of space to what has nothing to do with space. The theories we found first were ingenious but alarming: both were disconcertingly mentalistic in different ways. We then pondered a way to postulate things and spatial relations, reducing space to no more than these. But I argued that any such impure reduction begs a clear and critical question: how can spatial relations be both ungrounded and unmediated? The claim that no such relation is intelligible runs into

10 The concrete and the causal

43

the example of the causal relation in quantum mechanics, which seems rather certainly not a mediated one. I ruminated on the significance of the fact that this caused something of a crisis of understanding in physics, concluding that, in the absence of any experimental evidence against mediated spatial relations, their intelligibility is a good reason for accepting them. I looked somewhat askance at the risk of trivialising problems by defying understanding them: we may accept the mere form of an answer for the answer itself. Then I considered a leading motive in relationist thought: space is imperceptible. This, we found, is naive. Change the shape of space and one immediately changes perceptions. The explanatory role of geometry leaps at once to hand and eye. The best motive for relationism seems to be that we can use the Razor on it. I hinted that wielding it for its own sake is a dangerous pastime. The way ahead, then, is realism. But we must not think that realism can dispense with spatial relations. They are indispensable to any understanding of what space is. And mediation by space is indispensable for them.

2 Hands, knees and absolute space

1 Counterparts and enantiomorphs My left hand is profoundly like but also profoundly unlike my right hand. There are some trifling differences between them, of course, but let us forget these. Suppose my left hand is an exact mirror-image replica of my right. The idea of reflection deftly captures how very much alike they might be, while retaining their profound difference. We can make this difference graphic by reminding ourselves that we cannot fit left gloves on right hands. This makes the point that one hand can never occupy the same spatial region as the other fills exactly, though its reflection can. Two objects, so much alike yet so different, are called 'incongruent counterparts'. In my usage that phrase expresses a relation among objects, just as the word 'twin' does. Thus, no one is a twin unless there is (or was) someone to whom he is related in a certain way. Call this relation 'being born in the same birth'. Then a man is (and has) a twin if and only if he is born in the same birth as another. Let us call the relation between a thing and its incongruent counterpart 'being a reflected replica'. Then, again, a thing is an incongruent counterpart if and only if it has one. My right hand has my left hand, and the left hands of others, as its incongruent counterparts. If people were all one-

1 Counterparts and enantiomorphs 45

armed and everyone's hand a congruent counterpart of every other, then my hand would not be (and would not have) an incongruent counterpart. But incongruent counterparthood gets at something further and deeper than twinhood does. No actual property belongs to me because I could have been born in the same birth as another. But all right hands do share a property: that's clear from the fact that incongruent counterparts of them are possible. Though there are incongruent counterparts of my left hand, it is not their existence that makes it left. It appears to be enough that they could exist for my hand to have this property. Here again I am using the possibilities which one style of relationism is wont to call on for reductive purposes to illustrate realist claims about space. The possibility can hardly be the property unless it is some kind of causal property or disposition more broadly. It certainly does not seem to be that. Let us express this new further and deeper idea by calling hands 'enantiomorphs' and by saying that they have handedness. Then that is not an idea that depends on relations of hands to other things in space, as incongruent counterparthood does.We can show this in various ways. A well-worn method is to invent a possible world, say, one where all hands are right, as I did a moment ago. A more vivid possible world is one that contains a single hand and nothing else whatever. This solitary hand must be determinate as to being either left or right. It could not be indeterminate, else it would not be determinate on which wrist of a human body it would fit correctly - that is, so that the thumb points upward when the palm touches against the chest. Enantiomorphism, then, is not a relation between objects that are in space. Let us not make too much mystery out of this. We can say how hands, or anything else, come to pose a problem: they have no centre, axis or plane of symmetry. The sphere has all three; many common shapes have at least a plane of symmetry, the human body being an example. But the failure of symmetry yields no difference between left and right hands, since each is asymmetric. Let us grant that asymmetry is an intrinsic property of things that have it. It gives just a necessary condition of enantiomorphism. What is sufficient is a relation between an asymmetrical thing and a space.

46

2 Hands, knees and absolute space

2 KanVs pre-critical argument These ideas originate in Kant, of course. Though enantiomorphs crop up in the Prolegomena, they feature in and around the Critique as illustrating that space is the form of outer sense. But in an early paper (Kant 1968, reprinted in Van Cleve and Frederick 1991) of 1768, 'Concerning the Ultimate Foundations of the Differentiation of Regions in Space', he used them to argue for the ontological conclusion that there is absolute space. It is this earlier argument that this chapter is all about. Kant rightly regarded his paper as a pioneering essay in Analysis situs, or topology, though an essay equally motivated by metaphysical interests in the nature of space. He recognised only Leibniz and Euler as his predecessors in this geometrical field. The argument has found little favour among geometers and philosophers since Kant first produced it. I guess no one thought he was even halfright as to what enantiomorphism reveals about space. Kant himself has second and even third thoughts about his argument. My aim is to show that his first ideas were almost entirely correct about the whole of the issue. This is not to deny that Kant's presentation is open to criticism. He was ignorant of several important and relevant facts which emerged later. A clear and lively account of many of the facts can be found in Gardner (1964). But knowledge of these leads me, at least, only to revise the detail of his argument, not to abandon its main structure and content. In the remainder of this section I will set out a version of Kant's argument I try to follow what I think were his intentions closely and clearly, but in such a way as to indicate where further comment and exposition are needed. Hence this first statement of the argument is rather bald ('Internal relation' here means just a relation among the parts that make a thing up). Aj Any hand must fit on one wrist of a handless human body, but cannot fit on both. A^ Even if a hand were the only thing in existence it would be either left or right (from Aj). A3: Any hand must be either left or right (enantiomorphic) (from A2). By Left hand and right hand are reflections of each other. B2: All intrinsic properties are preserved under reflection. B3: Leftness and lightness are not intrinsic properties of hands (from

3 Hands and bodies: relations among objects 47

B4: All internal relational properties (of distance and angle among parts of the hand) are preserved in reflection. B5: Leftness and Tightness are not internal properties of hands (from Cj: A hand retains its handedness however it is moved. C2: Leftness and Tightness are not external relations of a hand to parts of space (from Cj). D: If a thing has a non-intrinsic character, then it has it because of a relation it stands in to an entity in respect of some property of the entity. E: The hand is left or right because of its relation to space in respect of some property of space (from, A3, B3, B5, C2, D). Kant places several glosses on K He claims that it is a hand's connection 'purely with absolute and original space' (1968, p. 43) that is at issue. That is, space cannot be The Void, a nonentity, since only something existing with a nature of its own can bestow a property on the hand. Again, to have made a solitary right hand rather than a solitary left would have required 'a different action of the creative cause' (p. 42), relating the hand differently 'to space in general as a unity, of which each extension must be regarded as a part' (p. 37). The premise D is not explicit in Kant. It is, quite obviously, a highly suspect and rather obscure claim. But I think Kant needed something of this very general nature, for he did not really know what it was about space as a unity that worked the trick of enantiomorphism. It will be possible to avoid this metaphysical quicksand if we can arrive at some clearer account ofjust what explains this intriguing feature.

3 Hands and bodies: relations among objects A number of relationists have replied to Kant's argument by claiming flatly that a solitary hand must be indeterminate as to its handedness. But it is very far from clear how a hand could possibly be neither left nor right. The flat claim just begs the question against Kant's lemma A, which argues that a hand could not be indeterminate as to which wrist of a human body it would fit correctly. Nevertheless, there is an influential argument to the effect that this lemma is itself a blunder. It is instructive to see what is wrong with this relationist contention. Kant's reason for claiming that it cannot be indeterminate whether

48

2 Hands, knees and absolute space

the lone hand is right is perhaps maladroit, because it suggests that the determinacy is constituted by a counter factual relation to a human body. The suggestion is hardly consistent with Kant's view that it is constituted by a relation between the hand and space. However, it provides Remnant (1963) with a pretext for charging Kant with an essentially relationist view of the matter. (The paper is cited as definitive by Gardner 1964 and Bennett 1970. It is reprinted in Van Cleve and Frederick 1991, pp. 51-60.) He takes Kant to be offering the human body as a recipe for telling whether the lone hand was left or right. He shows quite convincingly that even the introduction of an actual body into space will not decide whether the hand is left or right. For suppose a handless body is introduced. The hand will fit on only one of the two wrists correctly, i.e., so that when the arm is thrown across the chest with the palm touching it, the thumb points upward. But this does not settle whether the hand is left or right unless we can also tell whether it is on the left or right wrist that the hand fits. But we can be in no better position to settle this than to settle the original question about the single hand. Possibly we are in a worse position because the human body (handless or not) is not enantiomorphic (except internally, with respect to heart position, etc.). Even if every normal human body had a green right arm and a red left (so that it became enantiomorphic after a fashion), that would not help. For incongruent counterparts of it, with red right and green left arms, are possible. News that the hand fits the green wrist enables us to settle nothing about its Tightness unless we know whether the body is normal. Settling this, however, is just settling handedness for a different sort of object. Thus, it is concluded, a solitary hand is quite indeterminate in respect of handedness. What Remnant shows, in these arguments, is that no description in terms of relations among material things or their material parts ever distinguishes the handedness of an enantiomorph. This is an objection to Kant only if his maladroit reason is construed as a lapse into the view that what makes the hand left is its relationship to a body. Since this casts his reason as a contrary of the conclusion he draws from it, the construal is improbable. Remnant takes Kant to have been guilty of the blunder of supposing that, though it is impossible to tell, of the single hand, whether it is left or right, it is quite possible to tell, of a lone body, which wrist is left. It would have been a crass blunder indeed. But Kant never says that we can tell any of these

4 Hands and parts of space 49

things. In fact, he denies it. Insofar as Remnant does show that relations among the parts of the hand and the body leave its handedness unsettled, he confirms Kant's view. His attack on Kant succeeds only against an implausible perversion of the actual argument. Kant's introduction of the body is aimed, not at showing the hand to be a right or a left hand, but at showing that it is an enantiomorph. It shows this perfectly clearly. For the hand would certainly fit on one of the wrists correctly. It seems equally certain that it could not fit correctly on both wrists. Whether it fits a left wrist or a right is beside any point the illustration aims to make. Though this expository device may suggest a relationist view of enantiomorphism, it does not entail it. It is simply a graphic, but avoidable way of making the hand's enantiomorphism clear to us. The idea that a hand cannot be moved into the space that its reflection would occupy is certainly effective, too. But it is less striking and less easily understood (I shall say more about it later). Kant was quite well aware that this method was also available and, indeed, used a form of it in his 1768 paper. Thus the objection to the determinacy of handedness in solitary objects fails.

4 Hands and parts of space John Earman (1971), reprinted in Van Cleve and Frederick (1991, pp. 131-50) charges Kant with incoherence. No relation of a hand to space can settle handedness. Kant appeals to a court that is incompetent to decide his case. It is useful to look into Earman's objection. As I understand Earman, he argues as follows: we can plausibly exhaust all relevant spatial relations for hands under the headings of internal and external relations. Kant takes the internal relations to be solely those of distance and angle which hold among material parts of the hand. But these do not fix handedness, since they are invariant under reflection, but handedness is not (B5). What makes Kant's argument incoherent is that the external relations (to the containing space) cannot fix it either. For the external relations can only be those of position and orientation to points, lines, etc., of space outside the hand. (Incidentally, whether these parts of the container space are materially filled or not appears to make no relevant difference.) But for every external relation a right hand has to a part of

50

2 Hands, knees and absolute space

space, a left hand has just that relation to a similar part, as reflection makes quite clear. In any case, changes in these relations occur only if the hand moves through the space. However, although movement through the space can alter these relations, it cannot alter handedness (C^). Evidently none of these relations settles the problem. This presents us with an unwelcome parody of Kant's argument: since handedness is not settled by either internal or external relations in space, it must consist in the hand's relation to some further welldefined entity beyond space (Earman 1971, p. 7; Van Cleve and Frederick 1991, p. 137). That is an unlucky conclusion. We will have to think again. Earman offers us a way out of the wood. The spatial relations we have been looking at do not exhaust all that are available. He suggests that we say, quite simply, that being in a right configuration is a primitive internal relation among parts of the hand. If we could say that, it would certainly solve the problem in a very direct fashion. It would be a disappointing solution since we cannot explain the difference between left and right by appeal to a primitive relation. Nor, I suggest, do we understand what this relation could be. But, in fact, there is no such relation, as I will try to show in §§2.6 and 7. However, my interest in the present section lies in other things that Earman has to say. He frankly concedes that Kant was well aware of the kind of objection (as against the kind of solution) that he offers, so that, in this respect, his criticism is 'grossly unfair'. For Kant did insist that it is 'to space in general as a unity' that his argument appeals. Earman says (p. 8; Van Cleve and Frederick 1991, p. 138) that he does not see how this helps. But that merely invites us to take a closer look. We can get some grip on the idea of 'space in general as unity' by taking a quick trip into geometry. We need not make heavy weather of the rigour of our journey. A rigid motion of a hand is a mapping of the space it fills, which is some combination of translations and rotations. Naively, a rigid motion is a movement of a thing which neither bends nor stretches it. Such mappings make up an important part of metrical geometry. A reflection is also a mapping of a space, and, like a rigid motion, it preserves metrical features. This last fact, so important for Kant, is easily seen by supposing any system of Cartesian coordinate axes, xy y and z. A reflection maps by changing just the sign of the x (or the y or the z) coordinate of each material point of the

5 Knees and space: enantiornorphism and topology

51

hand. In short, it uses the y-z (or the x-z or the x-y) plane as a mirror. Thus it preserves all relations of distance and angle of points in the hand to each other, since only change of sign is involved. Thus the lemma Bv H4, Bb is sound. Now we can express the idea of enantiomorphism in a new way which has nothing to do with possible worlds or with the relation of one hand to another (actual or possible) hand or body. It does, however, quantify over all mappings of certain sorts. We can assert the following: each reflective mapping of a hand differs in its outcome from every rigid motion of it. That is a matter of space in general and as a unity. (Thus lemma C can be gained virtually by definition.) This quantification over the mappings seems to have nothing to do with any object in the space; not even, really, with the hand that defines the space to be mapped. Though this terminology is much more recent than Kant's, the ideas are old enough. I see no reason to doubt that they are just what he intended. Space in general as a unity is exactly what is at issue.

5 Knees and space: enantiomorphism and topology Kant was right. The enantiomorphism of a hand consists in a relation between it and its containing space considered as a unity. This is more easily understood if we drop down a dimension to look at the problem for surfaces and figures contained in them. We need to see how hands could fail to be enantiomorphs. Imagine counterpart angled shapes cut out of paper. They are like but not the same as L's, since L's are directed. Let me call them knees, for short (but also, of course, for the legitimacy of my chapter title). They lie on a large table. As I look down on two of them, the

Fig. 2.1.

52

2 Hands, knees and absolute space

thick bar of each knee points away from me, but the thin bar of one points to my left, the thin bar of the other to my right. Though the knees are counterparts, it is obvious that no rigid motion of the first knee, which confines it to the table's surface, can map it into its counterpart, the second knee. Clearly, this is independent of the size of the table. The counterparts are incongruent. The first knee I dub left. The second is then a right knee. Their being enantiomorphs clearly depends on confining the rigid motions to the space of the table top, or the Euclidean plane. Lift a knee up and turn it over, through a rigid motion in three-space, and it returns to the table as a congruent counterpart of its mate. That the knees were incongruent depended on how they were put on the table or how they were in the space to which we confined their rigid motions. A different picture is more revealing. Suppose there is a thin vertical glass sheet in which the knees are luminous angular colour patches that move rigidly about. They are in the sheet not on it. Seen from one side of the sheet, a knee will be, say, a left knee. But move to the other side of the sheet and it will be a right knee. That is, although the knees are indeed enantiomorphs in being confined to a plane of rigid motion, any knee is nevertheless quite indeterminate as to being a left rather than a right knee, even granted the restrictions on its motion. It could hardly be clearer, then, that nothing intrinsic to an object makes it left or right, even if it is an enantiomorph. Our orientation in a higher dimensional space toward some side of the manifold to which we have confined the knee prompts our inclination to call it left, in this case. It is an entirely fortuitous piece of dubbing. Hence, if the knees were in a surface of just one side (and thus in a non-orientable manifold) they would cease to be enantiomorphic even though confined to rigid motions in that surface. (Thus Bs is correct independently of Bx and B% and despite Earman's objection.) There is a familiar two-dimensional surface with only one side: the Mobius strip (see fig. 2.2). If we now consider knees embedded in the strip, they are never enantiomorphic. A rigid motion round the circuit of the strip (which is twisted with respect to the three-dimensional containing space) turns the knee over, even though it never leaves the surface. However, the Mobius strip might be considered anomalous as a space. It is bounded by an edge. We need the space of Klein's bottle, a closed finite continuous two-space (see fig. 2.3). This

5 Knees and space: enantiomorphism and topology 53

Fig. 2.2 Mobius strip.

Fig. 2.3 Klein bottle.

surface always intersects itself when modelled in Euclidean threespace. But this does not rule it out as a properly self-subsistent twospace. It can be properly modelled in four-space, for example. Rigid motion of a knee round the whole space of Klein's bottle maps it onto its reflection in the space. Knees are not enantiomorphic in this onesided, non-orientable manifold. They are indifferent as to left or right. Let us say that here they are homomorphic. These general results for knees as two-dimensional things have parallels for hands as three-dimensional things. It seems to be pretty clearly the case that, as a matter of fact, there is no fourth spatial dimension that could be used to turn hands 'over' so that they become homomorphs. No evidence known to me suggests that actual space is a non-orientable manifold. But it cannot be claimed beyond all conceivable question that hands are enantiomorphic, and it is not too hard a lesson to learn how they could be homomorphic. So Kant's conclusion A3, a principal lemma, is false. But this has nothing to do with whether there is one hand or many. It has nothing to do with an obscure indeterminacy that overtakes a hand if there are no other material things about. It is false because spaces are more various than Kant thought. Thus, whether a hand or a knee is enantiomorphic or homomorphic depends on the nature of the space it is in. In particular, it depends on the dimensionality or the orientability, but in any case on some aspect of the overall connectedness or topology of the space.

54

2 Hands, knees and absolute space

Whether the thing is left depends also on how it is entered in the space and on how the convention for what is to be left has been fixed. Kant certainly did not see all this. Nevertheless, it should be obvious now how penetrating his insight was.

6 A deeper premise: objects are spatial Clearly enough, Kant's claim that a hand must be either left or right springs from his assumption that space must have Euclidean topology, being infinite and three-dimensional. The assumption is false, and so is the claim. This suggests an advantageous retreat to a more general disjunctive premise for the argument, to replace Ag. Rather than insist that the hand be determinately either left or right, we insist rather that it be determinately either enantiomorphic or homomorphic. Thus, if there were a handless human body in the space, then either there would be a rigid motion mapping the hand correctly onto one wrist but no rigid motion mapping it correctly onto the other; or, there would be rigid motions, some of which map it correctly onto one wrist and others which correctly map it onto the other wrist. Which of these new determinate characters the hand bears depends, still, on the nature of the space it inhabits, not on other objects. The nature of this space, whether it is orientable, how many dimensions it has, is absolute and primitive. What underlies this revision of Kant's lemma A is the following train of ideas. We can dream up a world in which there is a body of water, without needing to dream up a vessel to contain it. But we can never dream up a hand without the space in which it is extended and in which its parts are related. To describe a thing as a hand is to describe it as a spatial object. We saw the range of spaces a knee might inhabit to be wide; the same goes for spaces in which a hand might find itself. So dreaming up a hand does not determine which space accompanies it, though Kant thought it did. But it does not follow that there could be a hand in a space that is indeterminate (with respect to its global connectivity, for example). We can describe a hand, leaving it indeterminate (unspecified) whether it is white or black. But there could not be a hand indeterminate in respect of visual properties. Like air in a jar, even an invisible hand can be seen to be invisible, so long as we know where to look. (I am here shuffling under a prod, itself invisible

6 A deeper premise: objects are spatial

55

here, from David Armstrong.) No considerations mentioned yet admit of a hand that could be neither enantio- nor homomorphic. There seems no glimmer of sense to that expression. But I spoke just now of there being a wide range of spaces that hands or knees might occupy. What does this mean, and does it offer a route for the relationist between the alternatives I am pushing? What it means is that we can describe a knee, for example, as a mass of paper molecules (or continuous paper stuff) which is extended in a certain metrical two-space. We can regard this two-space as bounded by extremal elements that make up edges (or surfaces for the three-space of a hand). These elements define the shape the mass of matter is extended in by limiting it. A wide variety of global spaces can embed subspaces isomorphic (perhaps dilated) with our knee- and handspaces. In short, a kneespace is a bit of our ordinary space. Nothing mooted here is meant to suggest that a hand might somehow be taken from one space to another while being spaceless in the interim. Let us, for a moment, toy with the idea that a kneespace need not be a subspace at all, but that it could just end at its material extremities without benefit of a further containing space. Does the hand or knee become indeterminate as to enantiomorphism if we consider it just in its own handspace or kneespace? Kant evidently feared that it would, and it might seem that he was right. Can't we argue that, for the hand or knee to be either enantiomorphic or homomorphic, there must be enough unified space to permit both the reflection and some class of rigid motions to be defined in it? Otherwise the question whether any rigid motion maps the hand onto its reflection does not have the right kind of answer to yield either result. But this is wrong. Wrong, first, because the relevant property of the object (or of the space it fills) is its asymmetry. This property is itself defined by the motions of the thing in some containing space, but let us not quibble with the relationist about this. Secondly, what counts is not whether the particular object has a reflection or whether it can be rigidly moved in the kneespace, but whether suitable objects in general do. It depends on the kneespace, not on the object which fills it. Both handspaces and kneespaces are orientable spaces. This is easy to see by imagining a much smaller hand or knee in the space and considering its reflections and the class of its rigid motions in that limited space. Clearly a hand in a handspace is enantiomorphic. This does not mean that hands are intrinsically enan-

56

2 Hands, knees and absolute space

tiomorphic. It means that handspaces are certain kinds of spaces. Even in the trivial case where the space of the object is the same orientable space which makes it handed, it depends on the relation of the filled space to the container space. Any hand must always lie in a handspace at least as a subspace, but there is no necessity about which space it is a part of. Nor does there seem to be anything to say about the hand as material which could determine even so small a thing as whether the space extends beyond the matter of the hand or is confined to it. Neither handedness nor homomorphy depends on the nature of the object's non-spatial character. Dreaming up a hand means dreaming up a space to contain it, of which the being and nature are independent and primitive. It is introduced in its own right as a well-defined topological entity. An oddity here is that, though a hand filling its handspace can be a dilated incongruent counterpart of a small hand in its space, we could never compare or contrast one hand in its own handspace with another in its own handspace. That requires mutual embedding in a common space. I conclude that leftness is not a primitive relation with respect to which hand parts are configured. I touch on this topic again in §2.7. But handspaces or kneespaces as anything other than subspaces have, so far, been mere toys of our imagination. The thought that space could just come to an end is one at which the mind rebels. Let us express our distaste for such spaces by calling them pathological. More technically, I believe that we find a space pathological when it can be deformed over itself to a point, or to a space of lower dimension. Topologists call such spaces contractible (see Patterson 1956, p. 74). The sphere as a three-space is deformable through its own volume to a point, but the surface of the sphere, as a two-space, cannot be deformed over its own area to a point; so it is not pathological or contractible. The Mobius strip can be deformed across its own width to a closed curve (of lower dimension). It was for this reason that I moved, earlier, to Klein's bottle as a non-contractible proper space. I want to do more now than toy with the idea of pathological spaces. It was an important conviction in Kant's mind, I think, that pathological spaces cannot be the complete spaces of possible worlds. Though I can think of no strong defence for this deep lying conviction, which most of us share, I can think of an interesting one. At least, it interests me. Why might one think that space cannot have boundaries?

6 A deeper premise: objects are spatial

57

The thought that space cannot simply come to an end is ancient. The argument was that if, though impossible, you did come to the end you could cast your spear yet further. This challenge is quite ineffective against the hypothesis that there is just nowhere for the spear to go. But what the challenge does capture, rather adroitly, is the fact that we cannot envisage any kind of mechanics for a world at the point at which moving objects just run out of places to go. What would it be like to push or throw such an object? We can't envisage. It would be unlucky if this boggling of the mind tempted us to regard pathological spaces as contradictory. To go Kant's way on this is to go the way of synthetic necessary truth. Nowadays, the prospects for following that road are dim. But there are prospects, more tangible and, perhaps, a bit brighter, since Kripke's semantics for modal logic. It is tempting to pursue this defence of Kant's inference from pathological spaces to global ones which properly contain them. It would be a protracted and rather tangential undertaking, however. The best defence for the disjunction enantiomorphic/homomorphic is to argue, as I did earlier, that hands and knees must be either homomorphic or enantiomorphic whether their containing spaces are pathological or not and even in the degenerate case where the containing space is identical with the space one of them fills. This rests the disjunction on the deeper premise that hands are spatial objects and there can be no hand without a space in which it is extended. Nevertheless, I am inclined to offer the suggestion that, when it is our task to conceive how things might be, as a whole, we should ask for what I will call an unbounded-mobility mechanics. (I intend the phrase to recall Helmholtz's Tree mobility', 1960, pp. 652-3, which he used to express the possibilities of motion in spaces of constant curvature.) That would rule out pathological spaces for possible worlds. But the issue by no means depends on this suggestion Before I move to consider some relationist responses to this argument, let us clear up a small puzzle. If the secret of handedness lies in the topology of space, why does so much importance attach to metrically rigid motions of the hand? The answer is that little importance actually does attach. Ironically the essence of handedness is not captured in the example of hands. The property we really want is one I'll call deep-handedness. It is a topological property and so not destroyed by any continuous deformation of the thing which has it. But the leftness of a hand is destroyed by general continuous deformations. We

58

2 Hands, knees and absolute space

can smoothly distort a left into a right hand. The property we want, intrinsic to the things that have it, is topological asymmetry. We see an example of this in the difference between the Mobius strip and the paper strip without the twist. No continuous distortion of the one will ever create or eliminate the twist. They differ in a deeply-handed way. But the usual way of conducting the debate is graphic and not likely to mislead us much. For an entertaining and instructive account of the relation between handedness and vision in a traversable non-orientable space, see Earman (pp. 243-5) and Harper (pp. 280-6) in Van Cleve and Frederick (1991); also Weeks (1985), §4.

7 Different actions of the creative cause So far I have not let the relationist get a word in edgeways. But his general strategy for undermining our argument is pretty obvious. He must claim that there cannot be a space that is a definite topological entity unless there are objects and unmediated relations that define and constitute it. Of course, he has to do more than simply to assert his claim; he has to make it stick. A number of strategies have been used for the purpose. Let me begin by looking at a problem which the relationist might seem to face but which he repudiates. The problem is for the relationist to show how bringing objects into the picture can settle topological features of a space in some way - for example, settle its orientability. How could topology supervene on the natures of objects? Why might this seem to be a problem? The relationist needs some surrogate for mediated relations. That suggests grounding them, somewhat in the style of Leibniz's internalized relations. The advantage of this would be that it would yield us an understanding of spatiality, even if only a strange one. A model will help to focus our ideas. Suppose there are two strips of paper, one white and the other red. We cut several knees out of the red paper and we intend to embed these in the white strip by cutting appropriate shapes out of the white paper and fitting the knees into them. Clearly, once we have made a cut in the white strip to admit the thick bar of a knee, there are two directions in which we can cut to admit the thin bar.

7 Different actions of the creative cause

59

Well then, a space is orientable if itcan be covered by an array of directed entities in such a way that all neighbouring entities are likedirected. (Notice that a space's no/Zrorientability is never constituted by how it is covered.) In our case, the grounding question is, perhaps, whether we can make the white strip orientable or non-orientable by entering the knees in some way which makes them enantiomorphs but does not depend on how the white strip is already made up. The answer is that these entries have no bearing on the matter at all. Suppose our white strip is joined at its ends to form a paper cylinder. Then we can cover it with knees in such a way as to illustrate its orientability. But clearly this illustrates orientability without constituting it. That is obvious once we see that we can cut across the strip, give it a twist and rejoin it without changing the way any knee is embedded. What we now have is a Mobius strip which is non-orientable. Some pairs of neighbouring knees will be oppositely directed. That is solely a matter of how the space (the white strip) is pathwise connected globally. It is quite irrelevant how many knees are embedded in the strip and how their shapes have been cut out for them. Of course, twisting the strip is tantamount to changing the spatial relations which the embedded knees bear to one another. How could it be otherwise? So changing their relations in that way equally entails that the white strip is twisted. No doubt, as Ear man (1989, p. 144) suggests might quite generally be true, an appropriate spatial relation just of one knee to itself will entail that the strip is twisted. Here again, the relationist might be able to state what the realist states. The deeper question is whether he can explain what he states so as to yield the same lucid insight into what constitutes handedness as the realist can. This makes it look, more than ever, as if the space as a definite topological entity can only be a primitive one; that its nature bestows a character of homomorphism, leftness or whatever it might be, on suitable objects. My conviction of the profundity of Kant's argument rests on how sharp and graphic a challenge hands pose. I see no way to accept the very robust intuition that a lone hand has some relevant character and deny that this rests on a relation between it and some space with a definite global shape. As always, of course, that might mean just that I still have lessons to learn. But so do relationists. The difficulty of our going to school with open minds on the matter is strikingly shown in Jonathan Bennett's paper (1970), reprinted in Van Cleve and Frederick (1991, pp.

60

2 Hands, knees and absolute space

97-130). His paper is devoted largely to the question whether and how an English speaker whose grasp of the language was perfect, save for interchange of the words 'left' and 'right', could discover his mistake. He has to learn it, not ostensively, but from various descriptions in general terms relating objects to objects. Bennett concludes that the speaker could not discover his error that way, though he could discover a similar error in the use of other spatial words, such as 'between'. It is a long, careful, ingenious discussion. But it is a pointless one. Bennett states on p. 178 and again on p. 180 that in certain possible spaces (some of which we know) there may not be a difference between left and right. So how could one possibly discover 'the difference' between left and right in terms of sentences that must leave it entirely open whether there is any difference to be discovered? That cannot be settled short of some statement about the overall connectivity of the space in which the things live. The same may be said, moreover, of Bennett's discussion of the case of 'between'. For, given geodesical paths on a sphere (and Bennett is discussing air trips on earth, more or less), the examples he cites do not yield the results he wants. The familiarity of relationist approaches appears to fixate him and prevent the imaginative leap to grasping the relevance of those known global spatial results which clarify the issue so completely. No doubt relationism has some familiar hard-headed advantages. But it can also blinker the imagination of ingenious men. It can make them persist, despite better knowledge, in digging over ground that can yield no treasure. Empiricist prejudice is prejudice. Let us get back to my paper strip and the knees embedded in it. It is a model, of course. It models space by appeal to an object, one which also defines what is essential to the space. That is, the subspaces, paths, and mappings that constitute the space are modelled by the freedoms and limitations provided by the constraint of keeping objects in contact with the surface of the modelling body (or fitted into it). This modelling body is embedded in a wider true space, which enables the visual imagination to grasp the space as a whole. Realists do treat space just like a physical object in the sense of such analogies. A space is just the union of pathwise connected regions. The model also shows us a distinction already mentioned, in a clearer light. Whether an object is enantiomorphic or not may depend, in part, on its symmetries; spheres are homomorphs, even in Euclidean space. It also depends on the connectivity of the space, stan-

8 Unmediated handedness 61

dardly, in global terms. But, what differentiates a thing which is an enantiomorph from one of its incongruent counterparts is a matter of how it is entered into the space - how we cut the hole for it in the white strip. Whether we call a knee in an orientable strip a left knee or a right is wholly conventional; it does not really differentiate the knees themselves at all, but simply marks a difference in how they are entered. So we have three elements in the account of handedness: (1) an asymmetrical object, (2) a space, either orientable or not and (3) a way in which the object is entered in the space. The idea of entry is a metaphor, clearly. It springs from our ability to manipulate the knees in three-space, and turn them there so that they fit now one way, now another into the model space. Once in the model space, they lose that freedom or mobility and are left only with that which determines enantiomorphy. It is not easy to find a way of speaking about this which is not metaphorical. But a very penetrating yet not too painfully explicit way of putting the matter is Kant's own, though I believe it to be still a metaphor. The difference between right and left lies in different actions of the creative cause.

8 Unmediated handedness If the relationist's problem is to ground enantiomorphy on the nature of objects, then it is plainly a hopeless one. However, it is not likely that he will accept the problem. The relationist may respond in the way discussed in §§1.5-6. He will simply take for granted as unmediated any relation among things which the realist mentions as mediated. If his task is as easy as this, then it is no task at all. For consider again my model of the embedded knees in the paper strip. When the strip is twisted to break the orientability of the space and make the knees all homomorphs, the mediated spatial relations among the knees are all changed too. That's all that the twist does! The realist can't claim to tell us what non-orientable spaces are like without calling on (mediated) relations among things and places. Nothing else can make a difference between one kind of space and another. All the relationist would need to do is populate his space with enough objects to yield the realist's mediated spatial relations among things, and the job is done. Nothing about our model suggests that the question is

62

2 Hands, knees and absolute space

begged by introducing several objects. Alternatively, if the challenge of the example is to tell a persuasive tale limiting himself to some set of objects - the lone hand, say - then he need not admit that the hand has any definite property other than that given it by the material space which the hand itself occupies - enantiomorphy, as we saw. Other properties' are illusions bred of picture thinking (see Earman 1989, p. 144, and in Van Cleve and Frederick 1991, p. 242). If it were that easy, it would be hard to see why the debate ever got past the first round. For all the relationist need do is to mimic what the realist says, but repudiate the mediation of the relations among the objects. Here again, without some argument to extricate spatial relations from their attachment to space itself, the tactic risks begging the question by making significant use of what it wants to repudiate as useless.

9 Other responses Let us consider some other forms of objection which make the relationist's task less trivial. Lawrence Sklar claims that the lone hand affords no special challenge to relationism. Consider fig. 2.4(a) and (b). Call the first an opposite-sided and the second a same-sided figure. Sklar claims (1985, pp. 238-9; reprinted in Van Cleve and Frederick 1991, pp. 176-7) that these pose exactly the same problems as are posed by the lone hand. But the difference in the figures as wholes lies quite plainly in differing relations among the parts: we can describe this in terms of purely local relations. They are not alike up to reflec-

(a)

(b) Fig. 2.4.

9 Other responses

63

tion as left and right hands are; their differences resist any local description. If we move either of these figures as a whole (i.e. as deformable surfaces), no continuous motion will map either onto the other whatever the dimensions of the space. Only by moving the parts separately will we be able to get the outside bar inside, no matter what the topology of the space (I owe this observation to Chris Mortensen). Hands are neither left nor right, enantiomorphic or homomorphic intrinsically. Suggestions that there is some intrinsic orientation of the hand are obscure. What is internal to the hand is a matter of the external spatial relations of its parts to one another. It is no matter of relations between things and what is outside them. The relations internal to the hand (the external relations of distance and angle among its parts) are the same for left and right hands. All intrinsic properties are preserved in reflection. Other relations described by Sklar as intrinsic all involve some relation of a hand to the wider space outside it and are not intrinsic. The difference between a left hand and a right is a difference in the way the hands are embedded in some wider space containing both. The difference exists only if the space does. What is the peculiar virtue of hands? They are the most familiar and domestic of things. Recognising the difference between left and right in our daily lives is trivial, but describing and analysing it is not. It is surprising and revealing that that this domestic difference takes us at a bound to a deep, global property of space without intervening relations of a hand to any other object. We must relate the hand to space as a unity, shaped in some definite way. It shows how very simply handedness can direct us to the global shape of space and how a feature which looks as if it lies in mere relations of one object to others, does not lie there. Kant throws down a gauntlet to relationists. The argument is a challenge rather than marking out a specific style of realist reasoning or posing a novel difficulty for relationist methods. Its merit is that the lone hand seizes attention: the challenge is immediately, intuitively obvious. Realism explains all this through the shape of container space. The relationist can state it, but can't meet the challenge to explain until he finds something to say which equals the elegance, the explanatory and generative power, and the illumination of the realist's story - in short, when he provides something we can understand and which aids invention and imagination just as well. It needs something more than a mimicry of realism which, once realism gives the answer,

64

2 Hands, knees and absolute space

throws away all that makes sense of the whole story: mediated spatial relations. The remnant with which the relationist leaves us sheds light only because we first saw it as part of a realist's picture, a picture which the relationist seeks to erase once it has served its turn. We might seek to avoid reference to space as a unity by resorting to an idea of local enantiomorphy (see Sklar, p. 181, Earman, pp. 239-40 and Harper, pp. 272-3, all in Van Cleve and Frederick 1991). A hand is locally enantiomorphic iff there is some neighbourhood which is large enough to admit reflections of the hand but does not permit the hand to be mapped onto the space of its reflection. This tactic fails. First, you still have to quantify over all the rigid-motion paths in some space or other. Obviously - trivially - you can take just a neighbourhood as the space in question and ignore its wider containing space. That does nothing to cancel the realist character of the definiens. A neighbourhood is just the same sort of metaphysical entity as the total space it is part of. And you still have to talk about all of this neighbourhood. Nor does this tactic succeed in trivialising the reference to space globally since the 'property' is not a real property: local enantiomorphy (or local any other cognate 'property') is recessive relative to the global property it corresponds to. For consider. My actual concrete left hand is 'locally enantiomorphic' relative to a neighbourhood, a part of space, which contains at the moment nothing else but it (let's suppose). But relative to that neighbourhood, it is neither left nor right, since that can be determined only relative to the way it happens to be embedded in our wider space (in which there are other hands and a convention for the use of 'left'). True, but trivial. It does not mean that there is some acceptable sense in which it is not a left hand. It is left in any sense which we can take seriously. Its character relative to the wider total space always dominates over any local definition. If our assumption that space globally is orientable turns out false, then it is not, in fact, a left hand at all. Again, the global character dominates and cancels the local one. There is no real property of local handedness. Kant's challenge is to explain the dominant character, the actual property, which, intuitively, even a lone hand has: that's why what counts is space in general as a unity. Of course, in the generous region of spacetime open to our observation, we find that we cannot move a left hand into congruence with a right.

9 Other responses

65

The relationist should begin with asymmetry, as I suggested at the end of §2.1. At least that looks like an intrinsic property. Even if it, too, must be defined by the motions of the object in some space then at least we can hope to confine the space to the occupied region filled by the object itself. Asymmetry conveys no blatant realism about space. Nor is it a recessive property as local handedness is. It does not imply enantiomorphy by itself: thus it is not liable to suspicion that our question may be begged. Yet given an orientable space, asymmetry implies dominant enantiomorphy. So I give the relationist asymmetry for free. The challenge he must meet is to give us a luminous account of the whole story about hands which is plausible independently of his general position, one as transparent as realism provides so elegantly. Let us reconsider the kind of relationism which wants to replace spatially mediated relations with a surrogate mediation by means of possibilities. Lawrence Sklar has suggested the course such an account of things might take, though he does not claim that the story is free of difficulties. I see the problems as much more damaging than he does, and I wish to look at them mainly to reinforce the charge, laid in §1.4, that it threatens circularity. Sklar (1985, p. 239 and Van Cleve and Frederick 1991, p.177) sees the problem that the possibilities invoked may depend on the nature of space itself, so that the whole enterprise risks circularity. He does not probe the risk in detail. But it is profitable and generally illuminating to do just that. Let us consider how it works for orientability. Sklar's way of using modal expressions somewhat obscures the issue, as I will try to show. He speaks throughout of possibilia: possible objects and possible continuous rigid motions. Now, unlike Sklar, I do not think that a relationist can afford to quantify over motions (i.e. refer to them), possible or not. Any absolutist whose blood is even faintly red, will insist that continuous rigid motions are a kind of spatial entity. They are paths, continua of points, and space itself is nothing but their sum. All such relations are mediated. But, quite apart from running this risk of a warm welcome to the absolutist camp, we ought not to use the adjective 'possible' when we might use the sentence operators 'possibly' or 'it is possible that'. The operators make issues very much clearer. So a relationist should analyse (or reduce) the expression 'orientable' in some such style as this:

66

2 Hands, knees and absolute space

O: It is possible that there are two hands a and b such that a is beside b, but it is not possible that a and b move rigidly with respect to c and d etc. so that b will stand to c, d etc. as a now does. Here, we do not quantify over motions and we avoid 'possible' as an adjective in favour of operators on sentences, open or closed. It is obvious that the relationist still has the job ahead of him. If these modal expressions mean what they ordinarily mean in philosophical contexts, vague though that is, 0 is just flatly false. I take the ordinary meaning, among philosophers, to be also expressed by 'it is logically possible' or 'it is conceivable'. In that case, the negative clause of 0 must be false just because non-orientable spaces are logically possible or conceivable. 0 is false, even if we construe the modals in the common sense of 'for all I know'. For all I know, our space is not orientable and I guess that this is so for all Sklar knows, too. So any relationist who wants to stay in the field owes us some account of what his modal expressions mean. There is no straightforward way of taking them. It is not enough for the orientability of space that 0 be true. It might be true for any of a hundred reasons that are consistent with a non-orientable space. It might be true because it is impossible to move objects relative to others, or because it is impossible to move them rigidly. We can look for less perverse reasons than these. It might really be the case that 0 is true because, although our spacetime yields only a finite non-orientable space for every projection out into a space and a time, still, for all of these projections, the universe has a finite timespan, too short for circumnavigating the space in the limits imposed by the speed of light. So even if we grant 0 true in some sense, the relationist must make sure that this is the sense he means. Sklar allows reference to and quantification over possible continuous rigid motions as 'perfectly acceptable relationist talk'. I think this obscures at least this last objection. For, if we speak of possible motions, this fixes our gaze on the motions (the paths) as what we must look to. So we forget about the speed of light, in my earlier case, because the relevant paths are there in spacetime. This takes some of the heat off 'possible', over which the charge of circularity looms black in the preceding paragraph. This might even be proper, if we take motions as entities. But it's as plain as the nose on my face, that this commits us to realism. It is not hard to make the risk of circularity clear if the relationist talks as I say he should. One way of doing it is to see modal ideas as

9 Other responses

67

Kripke does. We assume a range of worlds. Some are accessible to others in virtue of resemblances among them, expressed in the true necessary sentences of the modal language. For example, granted the standard philosophical sense of modals mentioned before, all the worlds accessible to ours resemble each other in having non-pathological spaces. Different senses of modal expressions correspond to different relations of access, that is, different respects of resemblance among worlds. Which worlds are in the range of a modal operator taken in a given sense depends on respects of resemblance among worlds. What we want the relationist to tell us is which range of worlds his operators cover; that is, how his worlds resemble each other. It is not clear what he can reply without saying that the worlds at issue are just those alike in having orientable spaces. That is circular analysis and no reduction. By contrast, the orientability of space does determine the handedness of hands, for it determines which paths there are in a space which a hand might take. It is a genuinely explanatory idea. Spatial relations, I suggest, explain enantiomorphy only by way of entailing the orientability of the containing space, and it is through that understanding that we come to grasp handedness. Orientability looks rather abstruse, especially when we see it as intended to apply to a weird thing like space as a whole unitary entity. But recourse to the concrete models I used make it clearly a simple, concrete property when ascribed to material things. Among twospaces, I spoke of table tops, cylindrical and Mobius strips of paper, Klein's (glass?) bottle and so on. It is easy to see how these things differ in the relevant ways. They differ in shape. The Mobius strip is like the paper cylindrical strip, except that it has a twist in it. Orientable spaces, like the twospaces of the plane and the sphere, clearly differ in shape in many ways. So, clearly, do non-orientable spaces, as Klein's bottle and elliptic two-space show. In each case we see, from the shape of the space, whether asymmetrical things in it will be handed or not. It is the shape which carries the primary load of explanation. Orientability is a disjunctive property, like being coloured (red or green or . . . ). It is a very general topological aspect of shape, like having a hole. Nevertheless, it is a concrete, explanatory idea. So I claim this advantage: I can explain handedness simply by relating the shape of hands to the shape of the space a hand is in. It strikes me as a very simple

68

2 Hands, knees and absolute space

and direct explanation, given that we can speak of the shapes of spaces. The kind of explanation which the shape of space offers us is not causal. Perhaps that is a main reason for dissatisfaction with it on the part of relationists. But, at this point, the offer of a geometric explanation rather than a causal, mechanical one looks too vulnerable to the charges of being strange and, essentially, empty. The idea of the shape of space is nebulous, remote, abstract. We must give it substance and familiarity. Explanations from the geometry of space are too different from our common paradigms of explaining to draw any strength from work they might do in a variety of other connections. What else can the shape of space explain? Unless we find it other roles besides its part in handedness it will simply perish from isolation and remoteness from the familiar intellectual civilisation. Let us begin the long task of showing just what a shape for space means, in detail, and of showing its power to explain the world to us.

3 Euclidean and other shapes

1 Space and shape Naively, we think that left and right hands differ in shape. But we just saw that there is actually no intrinsic difference of this kind between the things themselves; there can only be a difference in the way they are entered in a certain kind of space. Hands can differ if their containing space is orientable. If it is not they will all be alike however we enter them in it. We found this out by looking at the spaces defined by a paper strip from outside it. A paper cylinder is orientable, but if we cut it and twist it we can change it into the non-orientable Mobius strip. It alters what things do when they move in the space defined by the strip. This looks as if we can say that the difference in the spaces is a difference in their shapes. The shape of space is to play an explanatory role in our theory of the world. That is what we need to make some sense of. But making sense is just the problem. Can we properly speak of the shape of space? Now it seems all very well to say that space has a shape when we can regard it as defined by a strip of paper, the surface of a ball or of an arbitrarily far extended table top. We see it from outside in a space of higher dimension and it is visibly twisted, curved or flat from that vantage point. How are we to understand someone who talks of the shape of the three-dimensional space we live in? (I go on to speak of two-spaces, three-spaces etc. in giving the dimensions of a space. I use

70 3 Euclidean and other shapes

'£ 2 ', '£3' etc. to speak of Euclidean two- or three-spaces.) We cannot get outside the space. We have no reason to think it is embedded in a space of higher dimension. We directly see a shape only if it is a limited subspace which our vantage point lies beyond. Yet though the subspace has fewer dimensions than the space which contains it, it need not have any limits in its own dimensions. Any Euclidean plane is an infinite two-space though it will certainly be limited in threespace. The spherical surface, by contrast, is a finite two-space. But it still has no two-space limits. No point of the surface is an end-point. Unlike the plane, it 'comes back on' itself. The plane and the sphere seem to differ in shape, therefore, not only as to how they are limited in container three-space, within which we view them, but also in themselves - that is, from inside. That looks as if it might make eventual sense of the idea we want of shape from inside a space. Let's set some tasks. First, we want to identify a geometric sense in which space has a shape from inside. That needs an excursion into geometries other than Euclid's. I offer three guided tours, so to speak. I begin the survey of Euclidean and other geometries with a sketch of their histories, which suggest a crucial role for the concept of parallels. This section is short and easy. Next, §§3.3-6 develop the idea of the intrinsic shape of a space. This is rather geometrical and rather detailed, but pictorial and fairly easy. The idea of the shape of space is central to the philosophical argument of the book and you ought not to be content with a less complete grasp of it than these sections provide. Thirdly, there are four sections on projective geometry. Again, these are rather geometrical and detailed. The argument is a bit harder, but still pictorial. Perhaps the going is roughest in §3.9, which touches on analytical ideas. You might skip this, though it does tie up some loose ends. Again, there is philosophical point to all this. Projective geometry unifies our view of the geometries of constant curvature (including Euclid's). But it also introduces the important concepts of transformation and invariance, which underpin the concept of spatial structure, a key concept of this book. If we are to do any effective work with the concept later (from Chapter 7 onwards), we need to understand just what projective geometry is and what role transformations play in it.

2 Non-Euclidean geometry and the problem ofparallels

71

2 Non-Euclidean geometry and the problem of parallels Let us look at the history of non-Euclidean geometries. 1 Euclid began his Elements (1961) by distinguishing three groups of initial statements: there are definitions of the various geometrical figures; there are axioms supposed to be too obvious to admit of proof; there are postulates which admit of no proof from simpler statements, but which are not taken as obvious. Euclid saw these initial statements as providing the foundations for a deductive system of geometry. This insight was one of the great path-breaking ideas in Western thought. The path has since been broadened and smoothed into an arterial highway through our intellectual landscape. Some features that Euclid valued strike us, now, as ruts or hillocks; they no longer appear in deductive systems. It is not fruitful to distinguish between axioms and postulates: axioms are now seen as sentences containing more or less familiar general terms. They are chosen severally, because they seem clear and probable, corporately, because they seem mutually consistent and fruitful. Usually, some of the general terms in the axioms will be substantives (for example 'line' or 'set'). These will tell us which domain of study we are making systematic. Others will not be substantives but predicates and relations (for example '. . . intersects . . .'). These will tell us which features of things in the domain are under investigation. There will be some definitions in the system, but no attempt is made to define the primitive terms that occur in modern axioms. These carry the weight of linking the system to the things it is about. If these primitive terms are left undefined, we can reinterpret them as true of other domains and other features of the things in them, without dislocating the proof theory of the system. Our later models of non-Euclidean two-spaces will make use of this fact. Finally, we can classify the axioms of Euclid's geometry and those of other geometries into five main kinds: axioms of connection, of continuity, of order, of congruence and of parallels. Many different geometries will have axioms of the first four kinds in common. Euclid made an interesting remark about parallels in his fifth postulate. Putting the matter simply, what he said amounts to this: 1 This section also comes closest to a quick view of the non-Euclidean geometries as rival axiomatic systems. For a more comprehensive account and for detail on the subject, the classic texts are Bonola (1955) and Somerville (1958).

72 3 Euclidean and other shapes

Through any point not on a given line there exists just one line which is both in the same plane as the first line and nowhere intersects it. Even the early commentators on Euclid singled out this statement as specially doubtful and bold.2 But they rightly felt that something like it was needed if geometry was to get far. In short, they kept the postulate because it was fruitful. What caused their uneasiness, presumably, was that the postulate generalises over the whole infinite plane. That clearly goes beyond any experience we have of lines and planes, since our observations are confined to limited parts of these spaces. But the statement grips the imagination powerfully because it is very simple. Rival hypotheses about parallels were invented later, but Euclid's simple insight was so dominant that the rival theories all originated in attempts to reduce every alternative idea to absurdity, so as to prove Euclid's postulate indirectly. Gerolamo Saccheri's influential work was devoted to this end. He exhausted the rival possibilities by taking a quadrilateral ABCD with two right-angles at A and B and with sides AD and BC equal. He could prove angles C and D equal without the help of Euclid's parallels hypothesis. But he could not prove that each was a right-angle. One way of identifying all the hypotheses in the field, then, is to assert the disjunction that these angles are both acute, both obtuse or both right-angles. So Saccheri and later writers speak of the hypothesis of the acute angle, the hypothesis of the obtuse angle and the hypothesis of the right-angle (Somerville 1958, chapter I, §8; Bonola 1955, §§11-17). D

Fig. 3.1. 2 This form of the postulate is called Playfair's axiom. See Somerville (1958), chapter I, §4.

2 Non-Euclidean geometry and the problem ofparallels

73

Fig. 3.2.

It was shown early that the hypothesis of the obtuse angle entails that straight lines have only a finite length. That lines are infinite was an axiom. This seemed encouragingly absurd, so efforts to demolish the hypothesis of the acute angle went forward with some optimism. But the rewarding breakdown never appeared. Gauss, Lobachevsky and Bolyai each got the idea, independently, that a consistent, intelligible geometry could be developed on the alternative hypothesis. Gauss never published his work, though it was found among his letters and papers, so Lobachevsky and Bolyai count as the founders of the subject. Taurinus took the first steps towards a trigonometrical account of the acute angle geometry. He found he could use the mathematics of logarithmic spherical geometry, the radius of the sphere being imaginary. That is, the radius was given by an imaginary (or complex) number; one that has i = ^ T as a factor. This introduces a pervasive constant k, related to this radius, into the mensuration. In this geometry, there are two parallels, pl and pv to a given line q through a point a, outside it. Here neither parallel intersects the other line, but approaches it asymptotically in one direction of the plane. (Asymptotes are curves which approach one another in a given direction without either meeting or diverging, however far produced.) The hypothesis of the obtuse angle entails that every line through a intersects q in both directions. Its trigonometry also gives rise to a pervasive constant k (Somerville 1958, chapter I, §11; Bonola 1955, §§36-^8). This glance at rival axiom sets for geometry highlights the role of axioms of parallels. The new axioms entail that both versions of nonEuclidean geometry permit no similar figures of different sizes. The acute angle theory makes the angle sum of a triangle always less than two right-angles, but this * defect' depends on the area of the triangle

74 3 Euclidean and other shapes

Fig. 3.3.

and the constant k. Larger triangles have greater defect and the size of other figures also affects their shape. A similar result holds in obtuse angle geometry, but with the angle sum always in excess of two right-angles. However, to characterise the space by its parallels structure is to talk about it only globally. We say something about line intersections in the whole plane, for example. Yet there is reason to think that spaces which are alike in this global way may be more significantly different overall than they are similar. Further, the parallels character does little to make sense of the idea of a shape for the space, even though it is global in scope. It turns out that a space may be Euclidean in some regions but not in others; it is then said to be variably curved. That sounds more like an idea of shape. It seems that we have not dug deep enough. Let us get a more general and a more theoretical look at spaces from a new angle: space curvature.

3 Curves and surfaces Think, to begin with, of spaces we can stand outside. Then three simple ideas are clearly basic to the theory of curvature: (1) in general, spaces vary from point to point in how they are curved; (2) the simplest curved spaces are circles, being one-dimensional and uniformly curved at each point; (3) small circles curve more sharply than big ones, so we can make the reciprocal of its radius a measure of the curvature of a circle. Now think of any arbitrary one-dimensional continuous curve in the plane. We can measure how much it is curved at each point if we can find a unique circle which fits the curve near the point better than any other circle does. It is called the osculating cir-

3 Curves and surfaces 75

Tangent plane

Fig. 3.4.

cle and it is picked out by a simple limiting process. Take a circle which cuts the curve at the point p in question and in two others. Move the other points, 1 and 2, as in fig. 3.3, arbitrarily closer and closer along the curve toward the point in question. This generates a series of circles which has a unique circle as its limit. The curve at the point has the same curvature as this circle. We simply take the reciprocal of its radius. One-dimensional curves in three-spaces are a bit more difficult,but we can ignore them. Let us turn directly to curved surfaces.3 The problem of defining the curvature at a point for a surface, or for a space of still higher dimension, is more complex. We want a strategy which will let us apply the ideas that work so simply in the case of plane curves. We can find it by thinking about various planes and lines which we can always associate with any continuously curved surface. All the curves in the surface which pass through the point have a tangent in space at the point. It is remarkable that all these lie in a plane. So, for every point in a curved surface, there is a tangent plane to the surface. Given the tangent plane, we can find a unique 3 A fuller geometrical treatment, but still introductory, may be found in Aleksandrov et al. (1964), chapter 7, and Hilbert and Cohn-Vossen (1952), chapter 4. It covers the material in §§3-4.

76 3 Euclidean and other shapes

line which is both perpendicular to the tangent plane and which passes through the point on the surface. This is the normal N to the surface at the point. Now, consider any plane which contains the normal (as in fig. 3.4). It will be perpendicular to the tangent plane and will cut the surface in a section which contains the point. The section will give a continuous curve in its own plane and we can settle the curvature of this curve at the point in question exactly as we did before. The curvature of all these sections will describe the curvature of the surface at the point completely. But things get simpler at a deeper level. First, choose whichever direction of the normal pleases you as its positive sense. Then call the curvature of the section positive if it is concave round the positive direction of the normal and negative if it is concave round the negative direction of the normal. If the plane section is a straight line the curvature of the section is zero, of course. Now these normal-section curves will usually have different degrees and signs of curvature. Gauss proved that unless all sections give the same curvature, there will be a unique maximum and a unique minimum section of curvature and that the sectioning planes containing them will be orthogonal (per-

pen-dicular to one another). This result greatly simplifies the whole picture. It gives us two principal curvatures, kx and k^. These two curvatures completely determine how the surface is curved at the point, since we can deduce the section in any other plane, given its angle with respect to the principal planes.

4 Intrinsic curvatures and intrinsic geometry But the principal curvatures are still just a means to yet deeper ideas about curvature. We can use them to define a mean or average curvature, though this is not of direct concern to us. It is a useful concept in the study of minimal surfaces, for example. If we throw a soap film across an arbitrary wire contour it will form a minimal surface - one of least area for that perimeter. But we do not want to ponder this average of the principal curvatures, but rather their product. It is called the Gaussian or intrinsic curvature, K, of the surface at the point. K - kjk2.

4 Intrinsic curvatures and intrinsic geometry 77

If kx and &2 have the same sign then their product J^will be positive and the surface near the point will curve like a ball or a bowl. If h^ and &2 differ in sign, the product l£will be negative and the surface near the point will curve like a saddle back. If either k^ or &2 is zero, K will be zero and the surface is not Gaussianly curved at the point, but can be developed from the plane. Perhaps you can see that a plane parallel to the tangent plane (but below it) would be cut by the space of fig. 3.5 in an ellipse-like curve; whereas a plane parallel to the tangent plane (but above it) in fig. 3.6, would be cut in curves like a hyperbola. Thus the geometry of constant positive curvature is called elliptic geometry; that of constant negative curvature hyperbolic geometry. The idea of developing the plane is intuitively very simple. It is just a matter of bending, without stretching, a plane surface into some other surface. We may also cut and join to get what we want. For example, you can develop a cylinder from the plane by cutting out a rectangle, rolling it into a tube and joining the appropriate edges. The surface is now curved in three-space, in an obvious sense. But what will its principal curvatures be? They will be the same everywhere. Given any point on the surface, one plane through it will intersect the cylinder in a line (i.e. of zero curvature) and other planes will give sections either all positive or all negative, depending on how we picked the direction of the normal. Whatever direction we choose, the linear section will be either the minimum or the maxi-

Fig. 3.5. +N upward, kx-, k2-; K + +N downward, k^, k2+; K +

Fig. 3.6. +Nupward, kt-, k2+; K+N downward, k^, k2-; K-

78 3 Euclidean and other shapes

mum of all the curvatures. The section which gives us the circle as its curvature must therefore be the other principal section. So, for every point, the Gaussian curvature will be zero, since everywhere one principal curvature is zero. Gaussian curvature is not the obvious sense in which the cylindrical surface is curved in three-space. Now, not every continuous development of the plane will roll it up into a cylinder. We may get something like an arbitrarily scuffed, undulating or even rolled-up carpet. (A real rolled-up carpet will also bend in the middle, whereas the plane would not. This is because real carpets also stretch a little.) But however variably curved up it is, in the ordinary sense, we can approximate it as closely as we please with a series of strips from parallel cylinders of varying radius. So the Gaussian curvature at each point on the arbitrarily scuffed carpet is the same as on the cylinder; that is, the same as on the plane. The curvature is everywhere zero. Let a patch on the cylinder be any contractible space on the surface; it will be bounded by a single closed curve. Now the geometry intrinsic to any patch on the cylinder or the carpet is exactly the geometry of a patch on the Euclidean plane. That is the significance of zero Gaussian curvature. If we do not confine ourselves to patches, we will get closed finite lines on the cylinder corresponding to straight lines on the plane, because we joined the edges of a rectangle in developing the cylinder. Euclidean geometry of the full plane does not allow that, of course. Nevertheless, the geometry intrinsic to the cylinder counts as the same as the plane's, because it is the same for each patch. But what does it mean to speak of the geometry intrinsic to a surface? Infinitely many, but not all, of what were straight lines both in the plane and in three-space have rolled up into circles, or other curves, in order to make the cylinder. They are now curves of threespace. The same is true for the general case of the scuffed carpet. But if we look at all the curves in these developed surfaces, it is obvious that the former straight lines can still be distinguished from the rest. Each is still the shortest curve contained wholly in the surface between any two points that lie on it. Having agreed that they are curves of three-space, we can hardly call the 'lines' straight without running a serious risk of ambiguity. So they are called 'geodesies' instead. We can ferret out geodesies in surfaces much more complex than those we can develop from the plane. In general, a curve is a geodesic of a

4 Intrinsic curvatures and intrinsic geometry 79

surface if, no matter how small the segments we take out of it and however we pick them out, the curve traces the shortest distance in the surface between neighbouring (sufficiently close) points. (A more careful account of geodesies in given in §4.3.) So it is a curve made up of overlapping shortest curves between nearby points all within the surface. On the sphere, geodesies are great circles or their arcs. (A great circle contains the point diametrically opposite any point on it. Circles of longitude are great circles, but only the equator is a great circle among circles of latitude.) For any two points of the surface, the geodesic lies on a great circle joining them. In general, pairs of points cut great circles into unequal arcs. But both count as geodesies since they are made up of overlapping curves of shortest distance between neighbouring points. There are geodesies across arbitrarily and variably Gaussianly curved surfaces, such as mountain ranges, too. Now, to return to the geometry intrinsic to the curved surface, it is just the geometry given by the mensuration of its geodesies. Perhaps it is getting clearer why the surfaces developed from the plane should keep its intrinsic geometry. Although we bent curves, we did not stretch any. Bending means leaving all arc lengths and angle measures unchanged. So the distances between points in the surface are still the same, measured along geodesical curves, though not if measured across lines of three-space outside the surface. In surfaces more complex than the plane, where neither principal curvature is zero at every point, it may still be possible to bend the surface. In these cases, surfaces which can be bent into one another are called applicable surfaces. Obviously the principal curvatures will change when we bend the surface about. But it is a remarkable fact that their product, the Gaussian curvature, does not change for any point on the surface. This means that surfaces of different Gaussian curvature cannot be transformed into one another by bending, but only by stretching. If we stretch the surface we change area, lengths and angles; this destroys the whole geodesical structure in one area or another. It clearly changes the intrinsic geometry of the surface. So different Gaussian curvatures mean different geometries for the surface as this is given by the geodesies set in it. The product of the principal curvatures reveals something more deep-seated in the space at the point than the principal curvatures alone would tell us. It tells us what the surface is like from inside.

80 3 Euclidean and other shapes

5 Bending, stretching and intrinsic shape Each principal curvature describes, primarily, how the surface is bent with respect to the higher dimensional space outside it. Clearly, that concerns the shape of the two-space, but from outside as it stands in three-space. It is a question of the limits of the surface in the containing space. But, as we saw, the surface need not be limited in its own dimensions, whether it is infinite like the plane or finite like the sphere. So there is clearly a more general, and clearly a deeper, idea of the shape of the surface which is captured by the Gaussian curvature. A carpet is rightly described as a flat surface, even if it happens to be scuffed or rolled up either right now, or rather often, or even always. A hemisphere can be quite freely bent with respect to its surrounding space, yet there is a general notion of its having a shape which does not change when we bend it. Consider the familiar and, I think, quite uncontrived example of the human shape. The body surface is a finite, continuous, unlimited two-space (if we overlook the holes, for the sake of simplicity and, perhaps, decency) which we regard as a quite particular shape. It has a very intricate, variable, Gaussian curvature. But think of the wide array of postures and attitudes in which we quite ordinarily identify this single shape! Thus we find the same human shape in a person bending, crouching, running, sitting, standing to attention or with arms akimbo. But men differ in shape from women, fat people aren't the same shape as thin ones, and so on. (The example is not quite perfect, of course - the skin stretches a bit, which allows for more mobility, but the inner bones are rigid, which allows for less. I cannot see that these blemishes affect the point, however.) We can equally well distinguish a general shape and a particular posture for it on behalf of two-spaces generally. Shape is intrinsic to the surface, posture is not. Gaussian curvature tells us everything about shape, but in most cases, nothing much about posture but the range open to it. Gaussian curvature is inside information. Given the main aim of this chapter - to make sense of the idea that space has a shape - that is an important conclusion. Clearly, enough, if we can tell the intrinsic geometry from the Gaussian curvature, then, conversely, we can get this curvature out of the geometry as measured from inside the space. We need to look at the geometry in patches round each point and discover how the

6 Some curved two-spaces

81

space is intrinsically curved everywhere. That will tell us its shape without our needing to discover its posture first by hunting out principal curvatures. Suppose we start with the geometry of some threespace and look at the measurement of distances between points along geodesies in various volumes (hyper-patches) containing different points. We may find that the geometry varies from point to point, and we can assign a Gaussian curvature to the space at each point as the measurements suggest. It is easy to see in this case that the curvature neither tells us whether there is a containing four-space in which the three-space has a posture, nor, of course, just how it is limited and bent (postured) if there is a four-space. The sense in which a threespace has an intrinsic shape is precisely the same as the sense in which a two-space like the surface of the human body has one, independently of how it lies in containing three-space. So a space is an individual thing with the character appropriate to individuals; it has a shape. We still need to give this concept some percepts, for threespaces, at least in imagination. But telling how a sharply curved threespace would impinge on our nerve endings if we were inside it must wait till the geometric stage is set more fully. We can quickly look at some suggestive and interesting two-spaces - from outside, however.

6 Some curved two-spaces There are surfaces in 2% which provide very simple examples of what we have been discussing. I have already mentioned geodesies of the sphere and it is fairly obvious that the spherical surface is a two-space whose Gaussian curvature is everywhere constant and positive. Though the saddle back of fig. 3.6 is a negatively curved surface, it is not constantly curved. If the curvature of a saddle back is to be everywhere negative, it must vary in degree. However, the pseudosphere is an example of a two-space in Es which has constant negative curvature (see fig. 3.7), though it has a family of finite geodesies on it and a singularity at the central ridge. But any patch on the pseudosphere outside the singularity has the geometry of constant negative curvature. Unluckily it is not so enlightening about negative curvatures as the sphere is about positive curvature. An interesting example of a simple space of variable curvature is given by the toroid, the surface of a doughnut or anchor ring (see fig. 3.8). The curvature is clearly

82

3 Euclidean and other shapes

Fig. 3.7.

Fig. 3.8.

positive at point a and negative at point b. Therefore it must be zero somewhere, namely at all the points of either of the two circles where the toroid would touch any flat surface on which it rested. One of these circles is shown in the figure. How do these ideas connect with the earlier investigation of parallels? You will not be surprised to hear that the Gaussian curvatures K can be related quite directly to the trigonometric constant k discovered by Taurinus and pervasive in both acute angle and obtuse angle geometry. On the sphere, every pair of great circles intersects (twice, in fact, in polar opposite points) so there are no parallels on the sphere, just as there are none in obtuse angle geometry. Every triangle on the sphere (composed of arcs of great circles) exceeds two right-angles in its angle sum, the defect being more the smaller the ratio of the triangle's area is to the total area of the sphere (that is, roughly, the larger the triangle). On the pseudosphere the 'horizontal' geodesies of the figure obviously approach each other asymptotically in the direction away from the ridge, but this singularity prevents a simple representation of the pair of parallels permitted in the geometry of Bolyai-Lobachevsky. On the toroid, there is a sort of Euclidean parallelism. We can find a geodesic on the toroid such that, given a point outside it, there is just one geodesic that nowhere intersects the firstin fact, we can find an infinity of such geodesies. But how different this surface is in its intrinsic geometry from what we find on the plane! Within this rather extensive view of spaces of continuous curvature, it is very useful to single out those which projective geometry deals with. Let us turn to that.

7 Perspective and protective geometry

83

7 Perspective and protective geometry4

Fig. 3.9.

Spaces of constant curvature lie at the centre of geometry's field of interest. Two reasons explain this. First, were intuitions about physical realism. Helmholtz (1960) and Lie (Somerville 1958) established that only spaces of constant curvature allow the free mobility of all objects. Clearly, a thing in the two-space defined by an eggshell cannot move freely about it without changing its shape. If the curve in fig. 3.9 were a paper patch lying on the eggshell, then it would tear if we tried to fit it onto flatter areas near the middle of the shell, or it would wrinkle if we tried to fit it onto the more acutely curved regions near the 'pointed' end. If it were to slide along the space then it would either have to deform like an elastic thing (stretch, not merely bend) or it would resist the motion. Our experience of how things move in space suggests that they are very freely mobile. Until General Relativity, constant curvature seemed necessary for any space if it were to be regarded as physically real. A second, perhaps more important reason, for interest in spaces of constant curvature is the unified view of them which projective geometry presents to us. The subject is a thing of beauty in itself and will give us some concepts we can put to work later. Let us take a quick look at it. If an artist draws a perspective picture of a scene, what appears on his paper are recognisable likenesses of things. Obviously enough, 4 For a detailed, introductory treatment, see Aleksandrov et al. (1964), Part 2, chapter 3, §§11-14; Adler (1958), Parts 2 and 3; Bonola (1955), appendix 4; and Courant and Robbins (1941), chapter 4, give synopses of the material. These cover the material in §§7-9.

84

3 Euclidean and other shapes

that is because the figures on the paper share some geometric characteristics of their subjects. But the illusion of perspective also depends on some departures from the shape of the objects portrayed. What must the artist alter and what must she preserve? Think just of the simple problem of putting on a roughly vertical canvas a perspective representation of shapes and patterns on a single horizontal plane. The problems of translating three-space objects into flat outlines hold no interest at present. The painter is to project one plane onto another, the lines of projection all passing through a common point, her eye. What happens is that lines on the horizontal plane are projected into lines on the canvas (the projection plane); non-linear curves are projected into non-linear curves (with exceptions we can afford to ignore). Any point of intersection of lines (or curves) is projected into a point of intersection of the projections of the lines (or curves); if two points are on a line (or curve) then the projections of the points lie on the projection of the lines (or curve). But distance between points is not preserved. This is clear from the striking and important fact that the painter draws a horizon in her picture. This is not a portrayal of the curvature of the earth and the finite distance from us of the places where the surface recedes from sight. Even for the horizontal Euclidean plane the artist who puts a perspective drawing of a horizontal plane onto a vertical one must draw a unique horizon across it. She represents an infinite distance by a finite one. The role of the horizon brings us back to the question of parallels. Lines which are parallel in the horizontal plane and do not intersect in it will be projected into lines which converge on points that lie on this line of the perspective horizon. Sets of parallel lines that differ in direction will be projected into lines that converge on different points on the line of the horizon. A real line on the artist's canvas represents an improper idea; there are infinitely distant 'points on the plane' at which parallels 'meet' and all these 'points' lie on a single straight 'line' (or 'curve') at infinity. Projective geometry therefore completes the planes it studies with points and curves at infinity. We could never come across such points in traversing the plane and nowhere on it do parallel lines begin to converge, let alone meet. What justifies the addition of improper elements is, first, that sense can be made of it in analysis and, second, that it suggests itself so naturally in the elementary ideas of perspective which we have just put through their paces.

7 Perspective and protective geometry

85

Fig. 3.10.

But our view is still not general enough. What an artist does is to use the space on her canvas above the line of the horizon to portray things that do not exist in the horizontal plane. She pays the price of representing only that part of the horizontal plane which is in front of her (fig. 3.11). Since she looks in only one direction, the protective apparatus is really just by rays (half-lines) from the eye 'through' the canvas to points on the plane. But if we both take the whole pencil of lines on the point (her eye) and give a geometer two infinite planes to play with, we will get an unrestricted view of what plane projective geometry is about. (The pencil of lines on a point is just the infinite class of lines that contain it. We can also speak, dually, of the pencil of points on a line as the class of all points contained by the line.) What will be mapped onto points above the line of horizon h will now be points in the horizontal plane behind the plane of projections, as fig. 3.11 illustrates. It also shows us that, not only will the improper curve at infinity map into a real line of the projection plane h but, conversely the real line L of the horizontal plane will map only into the improper line at infinity of the projection plane. For the plane through P and L runs parallel to the projection plane. So the line at infinity is generally changed in projective geometry, and so, therefore, are parallels. On the complete planes of projective geometry (or on the projective plane, as we will soon be entitled to call it) we can say that every two lines join on just one point, and that every two points join on just one line. For now even parallel lines join on points at infinity. This gives rise to a very elegant and important principle of duality between points and lines for the projective geometry of the plane. Each theo-

86

3 Euclidean and other shapes

Fig. 3.11. Points A' and B' on the projection plane are the images under projection through P of the points A and B on the horizontal plane.

rem about points has a dual theorem about lines, and for every definition of a property of points there is a dual definition of a property of lines. So it is useful to speak of points 'joining on' lines of both pencils of lines and pencils of points. In the projective geometry of threespaces, planes and points are dual (and lines are self-dual, trivially).

8 Transformations and invariants Before we embark on this rather more general perspective, I must emphasise that the axioms of incidence and the theorems of duality make up a classical projective geometry. It is only in this that we need the line at infinity, the projective plane and so on. The axioms of incidence are motivated by study of which geometries of constant curvature are like Euclid's in all axioms but those of parallels. In the much wider range of spaces we will go on to consider, these incidence axioms need play no part in projective structure in spaces or spacetimes. But linearity and cross-ratio are still important. We want to continue to steer wide of the holy ground of analytical methods of investigating space. Still, we can get a more general and much deeper insight into the projective geometry of n-spaces (that is, spaces of n dimensions) if we ponder the elements of the theory of transformations of a space. There is no doubt that this theory draws its strength from the ease and accuracy with which it can be pursued analytically. But we can still profit from a qualitative view of transformations. The projective geometry of spaces of three or more dimensions is the study of a certain group of transformations of the space, and it does not require an outside space across which projection is carried on. A transformation is simply a function or mapping defined

8 Transformations and invariants

87

over the points of any space which carries each point into some point of the space (usually a different point, of course). In perspective drawing we map one space (plane) onto another by visual linear projection. But if we now rotate the projection plane round the line in which the two planes intersect (I in fig. 3.11) till both coincide, we can see that the mapping might now be regarded as carrying the first plane into itself by mapping points on the plane into other points of that same plane. We can regard the projective geometry of n-spaces generally as the study of certain transformations of the space into itself. This is why one speaks of the projective plane. Clearly enough, these transformations are 1-1 and continuous, except for mappings of finite into infinite points and vice versa. But, as the idea of perspective projection suggests, the transformations are linear as well. So projective geometry is the study of all those transformations of a space into itself which are linear and (for the most part) 1-1 and continuous; the projective geometry of a three-space is the study of such transformations of three-space into itself. When an area of geometry is defined by means of the transformations it studies, these are not picked at random: they always form a group. This means that the product of two transformations (the result of following one with the other) must also be a member of the collection of transformations. So projective geometry will describe perspectives of perspectives. Further, for each transformation of the collection there is an inverse transformation which reverses it and is also in the group. It follows that there must be an identity transformation, which changes nothing. It is given by the product of a transformation with its inverse. It leaves everything the same. Some properties of objects in the space will be preserved whichever transformation of the group we use on the space. But other properties, length and angles in projective transformations for example, will be altered by some members of the group. The first sort of property is called an invariant of the group of transformations; the second a variant property. Projective geometry can be seen as the study of invariants of the projective group of transformations. In fact, every geometry can be regarded as the study of the invariants of transformations of some definite group or other, as we will see in more detail later. What are the invariants of projective geometry? They include the ones we saw before: lines are invariant, being always transformed into lines. Intersections are invariant: if two points are on one line then

88

3 Euclidean and other shapes

the transformed images of the points lie on the images of the line; if two lines are on (intersect at) a point then the images of those lines are on the image of the point. Projective transformations are called collineations for this reason. There are similar invariant incidence properties for points, lines, planes etc. for spaces of more than two dimensions. Distance proper is a variant of the projective group, as our earlier glance at perspective drawing showed. However, a related notion of the cross-ratio of collinear points is invariant. If we take four points ABCD on a line then the ratio (AC/CB)/(AD/DB)(written (ABfCD)) remains the same for all projective transformations of the points. The cross-ratio is a key to projective structure (the other keys are linearity and intersection). We can also set up a cross-ratio between four lines through a point (as the duality of points and lines would suggest), which is preserved under projective transformations. A number of elegant theorems in the subject refer to and exploit the cross-ratio (of which a very accessible account is to be found in Courant and Robbins 1941, chapter 4). Now consider again just classical projective space. As we saw in perspective, the line at infinity is not invariant under these transformations. Nor, therefore, are parallels. The improper line at infinity may easily get mapped into a real finite line on which parallels will be mapped as intersecting. Analytically it is an advantage to let there be improper points at infinity, which can vary under projective transformation. In fact, as Poncelet (Adler 1958, §§6.7, 6.8) was the first to see, the whole subject takes on a unity and profundity if we also include imaginary and complex points and study the whole complex ra-space. It then turns out that if we regard the line at infinity (or the plane at infinity for three-spaces) as some kind of conic, and treat it as a special element of complex space, the non-Euclidean geometries of constant curvature, and Euclidean geometry too, can be shown to be different branches of one overarching subject, classical projective geometry. We can then understand the point of a well-known remark of Arthur Cayley's: Trojective geometry is all geometry' We can catch a glimpse of these ideas in the next section. It will not be worth our while to go into the geometry of the complex plane to grasp just how this works, since we will soon go beyond the standard non-Euclidean geometry of constant curvature. The following discussion should be quite intuitive.

9 Subgeometries ofperspective geometry

89

9 Subgeometries of perspective geometry

Fig. 3.12. A correspondence between three pairs of elements from the pencils of lines pt and p2 determines a protective relationship between the pencils. This defines a conic (Adler 1958, §§9.2, 9.9).

A conic in Euclidean metrical geometry is some section of a cone. Different sections give rise to a variety of curves among which we should notice the ellipse, the parabola, the hyperbola and the straight line, considered as a degenerate conic produced when the sectioning plane is tangent to the cone. But this last sentence trades in metrical ideas; we want to characterise a conic by appeal only to projective properties. Consider two points, px and pv and the pencils of lines through them. (See fig. 3.12.) Pick any three lines (o^ bv (\\ a^, bv %) from each pencil and pair them off at random. Label their points of intersection (al> a^), (bv b2), {(\, ^). Then the five points, pv p2 and the three just mentioned, determine a projective correspondence between all the lines of the two pencils. That is, take any other line \ in the pl pencil: its projectively corresponding line ^ in the p% pencil is the one which has the same cross-ratio to a^ b2 and ^ as ^ has to av \ and (\. The points of intersection of corresponding lines in the pencil generate a curve. This is a conic since it is fixed byfivepoints and since any line intersects it in at most two points. It resembles the familiar conic of metrical geometry in both these ways. But projective geometry does not distinguish among the various conies as metric geometry does. It is also possible, dually, to generate the same curves by the pencils

90

3 Euclidean and other shapes

of points on two lines and a projective correspondence between three pairs of such points. The lines joining the points in the correspondence will define the conic by generating all its tangents. This fact hints that our principle of duality for points and lines in the projective plane may be extended to conies: the dual of a point on a conic is a tangent to the conic. Given any theorem about points, lines and conies, if each element is replaced by its dual, the result is also a theorem It is obvious that if we select some subgroup from a group of transformations, then this smaller set of transformations will change fewer properties of elements in the space. That is, the subgroup will have more invariants. Now, in fact, Euclidean geometry (and also the nonEuclidean geometries we looked at earlier) is defined by transformations which are a subgroup of the projective ones. The problem is: how shall we pick these out? It's at this point that we need the complex plane. But let us look at something simpler instead of it. Take a circle in the plane and consider all the projective mappings of the plane into itself which leave this conic unchanged. They will transform the points inside (or outside) the circle into other points inside (or outside) the circle. It is only the transformations of the inside points that concern us now. Now we want to define the idea of a distance between points in such a way as to yield the result that, under any of the projective transformations we have allowed, the points are transformed into others which are the same distance apart according to our definition. (This may strike you as a bizarre tactic, but wait a bit!) To do this, regard the points on the circle itself (which we'll call the absolute conic, since we do not transform it) as composed of points at infinity. Then we will use the fundamental projective idea of the cross-ratio to define our distance measure. In fact we will use its logarithm, which yields a satisfactorily additive distance function. Now lines of the plane (other than tangents) intersect the absolute conic in two such points 0 and U in fig. 3.13. Given two points on such a line, A and B, we can define the distance between them as the logarithm of the cross-ratio (OU, AB) multiplied by a constant. The absolute will be infinitely distant from any point inside the conic, according to this metric. The constant allows us to pick units at will. Thus d(AB) = dog(OU, AB). Note that we define the distance AB by the cross-ratio of these points with points on the absolute conic. When distance is defined in this way we get just the metric geometry of constant negative curvature.

9 Subgeometries ofperspective geometry

91

Fig 3.13.

What is bizarre in this is that the circle represents points at infinity, points outside it represent nothing at all and the 'metric' of the inside points is a distortion of the metric on the page where the drawing sits. We need to get more comfortable with all of this. We have been looking at the projective transformations as carrying us from one metrical plane to another (or back into the same one) so that, in general, the new figures and relations change metrically but don't change in properties which we have now learned to regard as projective. But it is crucial for a grasp of later philosophical arguments that we countenance something more sophisticated than this. We need to think of projective space as one in which the only properties are projective. That is to say, when we map the lines, points and conies back into the plane, no relations or figures or any other properties of our space's geometry change because the metric properties are simply not there to be changed. So, in particular, there is no such difference as that between, say, a circle and an ellipse. We must envisage our projective space as containing conies which are neither ellipses nor circles: their properties are exhausted in those shared by ellipses and circles. Now the philosophical points which depend on our under-standing this are not ones which I want the reader to accept: far from it! But conventionalism and the debate about it make no sense unless this makes sense. So I urge you to ponder the idea with care. Projective space has no metric which any definition can distort. So we can't sensibly worry that the log of the cross ratio perpetrates an outrage on what is there in projective space. Until we impose the definition, nothing is there.

92

3 Euclidean and other shapes

This still leaves us with the circle of points at infinity and the points outside them. Here we would find an advantage in moving to a coordinate discussion of our topic and seizing the opportunity to exploit the rich structure of the complex projective plane, of which the real protective plane is a subplane. For this allows us to specify an absolute conic by complex coordinates and by means of our log cross-ratio metric, specify a metric on the whole real subplane. Different choices of this sort yield us metrics on the real subplane for any of the nonEuclidean geometries. So there is no finite conic inscribed on the real plane with irrelevant outside points. Of course, this only hints at the structure somewhat broadly, but the discussion overall should give you some insight into the deep projective unity among the various metric geometries of constant curvature. I said earlier that the axioms about incidence in projective geometry are closely related to the Euclidean ones, except for an extension allowing parallels to intersect on the line at infinity. So the incidence axioms of projective geometry stipulate that two lines join on just one point and this axiom will carry over into elliptic geometry. This is a geometry of constant positive curvature. The earlier model of a twospace within Es having constant positive curvature was the spherical surface. But it is clear enough that the surface of a sphere is not a classical projective space. On the sphere, any two lines join on two points. Further, the dual axiom of incidence, that every two points join on one line, is false on the sphere: any pair of polar points join on infinitely many lines. The trouble is that the sphere is a kind of double space; there is twice as much of it as we need. We can get half of it in two ways: one is simply to identify each point on the sphere with its polar point. An equivalent way is easier to visualise - take a hemisphere (see fig. 3.14) and identify diametrically opposite points

Fig, 3.14.

9 Subgeometries ofperspective geometry

93

on its equatorial section as suggested by the figure. This provides a good model of the elliptic plane. Lines join on just one point; points join on just one line. Incidentally, this plane is non-orientable, just as the Mobius strip is. Let me rehearse what we have knitted our brows so tightly over up to this point. First, we looked at geometry graphically and historically by thinking about the question of parallels; then we went into the matter of curvature, discovering what it means to speak of the shape of a space as an inside matter given by its Gaussian curvature at all points, and distinct from any posture it may (or may not) have; then we looked into projective geometry to find a deeper unity underlying the sharp differences among the spaces of constant curvature. We will need all this to understand how to grasp the idea of a shape for space in the intuitive sensory imagination. Much of the survey will be useful to us in other ways as well. The models of non-Euclidean space fall well short of what we want from an intuitive glimpse into these spaces. The geodesies of the twospace of the sphere, elliptic hemisphere and toroid are visibly curved in Es and the logarithmic length measure used in the model of hyperbolic two-space makes no concessions to vision, despite remarks of Reichenbach (1958, p. 58). We have made progress on the way, but we are still far from home.

4 Geometrical structures in space and spacetime

1 The manifold, coordinates, smoothness, curves In this chapter I explain some ideas of geometry from a different standpoint and with different aims. The last chapter mainly showed ways to visualise geometry. It moved within a somewhat classical ambit of spaces with constant curvature and rather restricted axioms of incidence. It is time for a broader canvas. This will be more useful if it is tied to more mathematical techniques, to show the advantages of mapping spacetime points into the real-number space of n dimensions, R1. With luck and application, you will gain some reading skills which will carry you through some of the mathematical parts of the literature. This approach also relates spatial and spacetime geometries together in a fairly intuitive way. Exploiting the notation of partial differential calculus yields a view of local geometric structure, of neighbourhood or infinitesimal structure. Later chapters do not lean very heavily on this one, but you should find it useful to read it. You could skip it and still get rather a lot from the rest of the book. The most accessible books on these topics that I know are Lieber (1936) which starts at a quite elementary level and takes a clear path to sophisticated concepts, Schutz (1980; 1985), the latter being especially useful. They cover a great deal more than is aimed at here. Sklar

1 The manifold, coordinates, smoothness, curves

95

(1974b) and Angel (1980) are also very helpful. Friedman (1983) and Torretti (1983) are excellent, but advanced, philosophical and mathematical treatments. In chapter 3 various metrical geometries were found to be subgeometries of projective geometry. The situation was like this: projective geometry is the study of a certain set of properties Pi which are projective invariants; i.e. properties of items in a space subjected to a certain group of projective transformations PT. Euclidean geometry studies a group of metrical invariants Ei of a transformation group ET, hyperbolic geometry studies the invariants H{ of the hyperbolic group HT The groups £ T and // T are proper subgroups of P T . Every transformation of ET or of HTis a transformation of P T , but not vice versa. Conversely, invariants of the metrical geometries are a larger class than the projective invariants, which they obviously include. This suggests the idea of different geometries in the sense of different levels of structure studied, with metrical geometry being the richest structure, and projective structure poorer but more basic. That is what will be meant, in this chapter by different geometries. But how far can this idea be generalised? Felix Klein (1939) proposed (in his Erlanger programme) that it stretched across the board of geometries. Every geometrical level can be defined by specifying its group of transformations and, thereby, its invariants. Very roughly, we get a hierarchy of geometries like this: DIFFERENTIAL TOPOLOGY PROJECTIVE GEOMETRY AFFINE GEOMETRY METRICAL GEOMETRY (Euclidean) in order of decreasing generality and increasing structure. Of course, there will still be spaces which differ from one another within any of these levels - different topologies, for instance, or the same topology but dissimilar projective features. But given any space, we can find these different levels of structure in it. Affine geometry has parallelism as an invariant but preserves neither length nor angle. It is, therefore, only an intermediary stage between projective geometry and Euclidean metrical geometry. Conformal geometry (§9.8), which studies invariance of angularity, does not fit straightforwardly into this hierarchy. Let us now turn to the task of describing these levels of structure in

96

4 Geometrical structures in space and spacetime

spaces of rather general type, general enough to include the spacetimes of special and general relativity, which we will meet when we turn to the problem of space and motion in chapter 10. This will involve us in using the methods of partial differential calculus to pick out and characterise various structures in space which give us a more local idea of its shape at various levels and of the way in which that can be put to theoretical and explanatory uses. I begin with the most minimal structure, a differential manifold (differential topology), the barest of candidates for physical space. We then move on through richer structures to arrive, at last at the full metric of Riemannian spaces. These are not the only conceivable spaces, but cover all of serious interest to philosophers of physical space and spacetime. First we begin from the most basic level of invariance under the broadest group of transformations the diffeomorphisms, which are smooth topological mappings of space or spacetime into itself (Schutz 1980, §§2.1-2.4). Assume that space, or the manifold M, is a collection of points with a continuous topology: its points are ordered as the real numbers are. Or rather, vice versa, since the theory of the continuum owes so much to geometry and the idea of spatial relations as mediated ones. Since the global topologies of spaces may vary widely, we begin with an assumption that they are all alike in being made up of pieces each like Euclid's space in its topology: whole spaces are stitched together out of patches from such neighbourhoods. Most of our general description tells about these neighbourhoods: it is local. This leaves us free to say later how the patches might be specifically made up into wholes topologically like the sphere, the torus, or the plane etc.. But in every case, where the patches are stitched together, the region where they overlap has the same smooth topology as the parts it joins. Each neighbourhood of M can be coordinated with i?", the set of all n-tuples of real numbers. Here is a way to do that. I assume that you already understand a path as a one-dimensional part of space. Let us suppose that a neighbourhood can be covered by n families of paths: no path in a family intersects another member of it, but each member of a family intersects each path in any other family in just one point. Number the families from 1 to m use V as an index ranging over the index set, i = (1, . . . ,n), to identify the family. Thus each point lies on one path from each family. Number the paths in each family using real numbers so that the ordering is continuous. Nothing

1 The manifold, coordinates, smoothness, curves 97

else is required of the numbering. This gives us a coordinate system or chart for the neighbourhood. It maps from the points of /^dimensional space into an open set of If1. The map has an inverse. The coordinates of a point p are written thus: x{(p) =(x1(p), . . . , xn(p)) or, more simply and generally {#,}. Our total space is described by an atlas of coordinate charts of every neighbourhood. The coordinates {xf£ of a point p in a new coordinate system S' are a continuous differentiate function (a diffeomorphism) of the coordinates {x]t in the old system S. Assume an inverse for this function. The relations which take us from S to Sf or back can be written out in any of the following ways

Notice that each new coordinate is a function of each of the old coordinates. x\ is a function of xl and of x% and so on. If we consider the new coordinate differentials dx\ then each is a sum of the partial changes due to changes in each of the old coordinates. For example, dx' 3 - 1 is the change in x\ per unit change in xs which must then be multiplied by the total change in xs, dx^. This then gives us, for a 3 dimensional space j 1

d

dx\

d

dx\

d

dx\

d

There are three such equations, one for each index in the index set. These can be written in general form as equations for dx'i9 just like the equation above, except that the variable x\ replaces x^ on the left. A further simplification is to lump all the sums on the right hand side together whenever there is a repeated index in a term, as there is in each of these. The convention is that one then sums on all the indices in the index set whenever the variable index is repeated on the right. Thus above the equation condenses to dx\1 = 3—' dxax- J

where the repeated index j instructs one to make up the sum of terms for all values of the index. In more complex formulas the compact notation simplifies things enormously. But one does have to bear in mind how long the expansion may be. For a four-dimensional spacetime this equation schema represents four distinct equations

98

4 Geometrical structures in space and spacetime

each with a sum of four terms on the right. There would be no value in a ticket for each point, so that for any change of coordinate systems the transformation function will find it again, unless we foresaw something interesting to say ultimately about the points. As Schrodinger puts it, 'Our list of labels would amount to a list of (grammatical) subjects without predicates; or to writing out an elaborate list of addresses without any intention even to bother who or what is to be found at these addresses' (1963, p. 4). This is not to say that the points are different from one another in themselves. They are indistinguishable, each taken by itself. Yet each may have some different non-intrinsic property in terms of what is at it. There may be a field of force, a temperature, a density of charge and so on. A single number that tells us what is at the point is called a scalar. A smoothly varying assignment of scalars to each point is a scalar field. But a point may have a geometrical property which says something about the space around it. For instance, curvature varies continuously through the space, as we saw in the case of surfaces (§3.3), so we want to specify the property at each point This means attaching a number to the point which belongs to it in virtue of the intrinsic geometric properties of the limiting region round it. The new number, also a scalar, will be like a predicate which is true of some point or other, whereas the coordinate numbers are simply an arbitrary, non-descriptive name of the point. (Lieber 1936, chapters XIII-XX) The smooth topological structure of the space and the families of paths has allowed us to map the space M into R*. We can now use the mapping in the other direction so that features of the real number space can be used to describe features of the space itself which would otherwise elude a precise account. The real number space has more structure than the physical space because the numbers have intrinsic differences which characterise each uniquely, whereas the points don't differ intrinsically from one another. Since we assigned numbers arbitrarily save for the ordering, the magnitudes, differences and sums of real numbers don't correspond to the distances along or between paths, nor does the fact that the points on a path may have the same Ah coordinate mean that it is straight. That means only that it is one of the paths chosen from the ith family on which no condition of linearity was placed. But of course if a point p is between q and r on some path, we can expect the coordinate differences among the points to reflect the fact that pq and pr are shorter parts of the path than qr is.

1 The manifold, coordinates, smoothness, curves

99

If a neighbourhood is smooth, this means roughly that paths can be inscribed in it everywhere and in all directions so that each path has a tangent at all its points. Of course, we can also find continuous paths in it which don't have tangents everywhere: paths with cusps or corners. This makes it plausible that smoothness is a property intrinsic to space; that is, constituted by purely spatial features of purely spatial things. Smoothness is made precise through analytical techniques. We can use the mapping into R1 to describe smoothness in terms of the idea of differentiating a function. Suppose we consider a scalar field, a continuous assignment of single numbers to each point in a neighbourhood which tells us, let's say, the temperatures everywhere. Then the neighbourhood is smooth if the scalar field function can be differentiated one or more times. That is, there is a small change in the function for any small change in the coordinates of the point. An n-dimensional space M composed of neighbourhoods all of which are differentiable and smooth is called a differential manifold. This means that, when the charts of these neighbourhoods are linked together in an atlas, the areas of overlap connect the pieces smoothly. That is the basic structure from which we begin. Nothing less structured is of any conceivable interest as a physical space. Cusps or holes in the manifold are interesting, but we can omit that complication in this general sketch. They do not weaken the basic overall structure of space. Once we have coordinates we can describe a path by its equation in the coordinates; by the function into if1 which describes the coordinate numbers of all its points. But one can talk more articulately about curves rather than paths. A curve is a continuous function from an interval in R into the manifold; that is, onto some path in it which is a part of M (An interval is a set of numbers, such as the unit interval (0,1) of all real numbers n such that 0

Adopting Fx as a reference frame makes it natural to read this table down the columns taking the O clocks as at rest and synchronised, each (? being checked against the series of rest clocks as it goes past. The moving clocks are seen to run slow. Adopting F2 as a reference frame makes it natural to read the table across its rows, taking the (? clocks at rest and synchronised, each C1 being checked against a series of rest clocks. The moving clocks are seen to run slow. It is also clear that, for either frame, the moving clocks are out of synchrony with each other. It is interesting to glimpse how the symmetry of the slowing of moving clocks emerges in a new example, especially one where the very same table is used for each choice of frame. But the main point in this example was to be a spatial one. Various things about these nine comparisons make it clear that the compari-

0)

n