Space, Time, and Theology in the Leibniz-Newton Controversy 9783110328301, 9783110327991

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Edward J. Khamara Space, Time, and Theology in the Leibniz-Newton Controversy

PROCESS THOUGHT Edited by Nicholas Rescher • Johanna Seibt • Michel Weber Advisory Board Mark Bickard • Jaime Nubiola • Roberto Poli Volume 6

Edward J. Khamara

Space, Time, and Theology in the Leibniz-Newton Controversy

ontos verlag Frankfurt I Paris I Ebikon I Lancaster I New Brunswick

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2006 ontos verlag P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN 10: 3-938793-26-0 ISBN 13: 978-938793-26-8 2006 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work Printed on acid-free paper ISO-Norm 970-6 FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by buch bücher dd ag

To Marie-Louise

Contents Editorial Foreword

i

Preface and acknowledgments

v

Abbreviations

ix

I. Leibniz’s Last Controversy with the Newtonians 1.1. Introduction: Leibniz's strategy

1

1.2. Plan of this work

8

II. Newtonian Absolutism 2.1. Introduction

11

2 2. Ten Newtonian theses

12

2.3. Getting at the core of Newtonianism

26

2.4. Theological repairs

31

III. Leibnizian Relativism 3.1. Introduction

35

3. 2. Summary statement of Leibniz’s theory of space

36

3. 3. The relativity of spatial position

39

3.4. Getting at the core of the Leibnizian theory

44

3.5. Some recent objections

47

3.6. Armstrong on absolute and relative motion

49

3.7. On an alleged impurity

52

3.8. Concluding remarks

54

3.9. Addendum: Leibniz on Time

55

IV. On Properties 4.1. Space, time, and individuals

57

4.2. Two classifications of properties

58

4.3. Defining the intrinsic

61

V. The Identity of Indiscernibles 5.1. Three grades of indiscernibility

65

5.2. The vagaries of PII.2

68

5.3. Inter-world indiscernibility

75

5.4. Appendix: Leibniz’s derivation argument

77

VI. The Nutcracker at Work 6.1. Introduction

81

6.2. Leibniz’s objections to absolute space

83

6.3. Objections to absolute time

86

6.4. The relevant texts in Leibniz

91

6.5. Appendix: John Earman on Leibniz

95

VII. Leibniz’s Verificationist Argument 7.1. The argument

99

7.2. A proposed interpretation

101

7.3. Clarke’s reply

103

7.4. An answer to Clarke

105

7.5. An unhelpful God

110

VIII. A Digression on Boethius: Eternity and Omniscience 8.1. Preliminaries

113

8.2. Boethius on eternity

114

8.3. Boethius’ account of omniscience

120

8.4. The refutation of Boethius

123

IX. Omniscience: Leibniz versus Clarke 9.1. How is omniscience possible?

127

9.2. The Newtonian account of omniscience

127

9.3. The Leibniz’s objections

130

X. Omniscience and Omnipotence: Clarke and Arnauld against Leibniz 10.1. Leibniz’s account of omniscience

135

10.2. ‘The paradox of omnipotence’

139

10.3. The Leibniz-Arnauld version

148

Bibliography

151

Editorial Foreword The series Process Thought features philosophical, interdisciplinary, and cross-disciplinary research on the analysis and application of dynamic categories; it is not tied to any particular school of research but aims to document what appears to be a theoretical paradigm shift in the making. The benefits of process-geared research are perhaps more easily acknowledged once the historical and systematic depth of the ‘process paradigm’ is thrown into greater relief. Thus we welcome with great pleasure into our series the monograph Space, Time, and Theology in the Leibniz-Newton Controversy by Dr. Edward J. Khamara—as an outstanding research contribution on Leibniz that also illuminates historical and conceptual foundations of process-based research. The focal point of this study, Leibniz’s refutation of Newton’s theory of ‘absolute’ space and time, is suited like no other to highlight that deep systematic concerns about the nature of individuality, becoming, and existence in space and time run counter to historical and methodological divisions in philosophy—the ontological implications of the processual reform initiated by Leibniz can be traced in the work of a long list of authors of all philosophical ‘denominations’, including Charles Renouvier, William James, Gabriel Tarde, Henri Bergson, James Ward, Alfred North Whitehead, Bertrand Russell, Ernst Mach, Albert Einstein, C. D. Broad, Roy Wood Sellars, Hans Driesch, Hans Jonas, Wilfrid Sellars, Gilles Deleuze, and most recently Nicholas Rescher. In addition, the study’s systematic topic, the still unresolved issue of the nature of space and time as relative vs. ‘absolute’ or ‘substantival’ is well suited to stress the interdependence of ontology and natural science. The following study can be profitably read in tandem with Gary L. Herstein’s recently published monograph, Whitehead and the Measurement Problem of Cosmology, which uncovers a peculiar methodological problem for astrophysics created by the ontological assumptions of Einstein’s theory of General Relativity and possibly avoided by Whitehead’s alternative interpretations of relativity and

spatio-temporal extension. The quarrel between Newton and Leibniz took place essentially in the years 1712–1716; it originated in the controversy around the authorship of the infinitesimal calculus but soon extended to broader issues in the philosophy of nature, revealing profound divergences. Newton championed space and time as actually infinitely divisible receptacles for a finite number of atoms in the void that constitute material bodies and are either in absolute motion or at absolute rest, governed by the laws of mechanics. All entities in Newton’s ontology enjoy the type of independent existence familiar from the Scholastic notion of substance. In contrast, Leibniz operated with a new notion of dependent or correlative existence, which he combined with a revival of the Aristotelian intuition of ‘inner drives’ in nature. In his New System of Nature (1695) Leibniz labels this combined move the ‘philosophy of organism’. There are three elements in particular in Leibniz’s approach that account for its actuality and significance for process-based or process-geared ontologies and metaphysics. First, Leibniz’s teleology is designed not as a wholesale replacement but as an extension and supplementation of a mechanist description of nature. His monadology rationally accounts for the constitutive opacity of nature at different levels of description. The scholarly legacy of this so-called psychical atomism includes a double-aced scala naturae, i.e., a ‘great chain’ of beings and of ‘levels of awareness’. To put it in present-day terms, Leibniz was the first to present the idea that so-called reductive or mechanist descriptions of nature are dependent on the level of mereological analysis. Relative to one level of analysis something might appear as an aggregate or reducible whole, and some of the developments of this whole appear as curious cases of ‘emergence’ unaccounted for by the laws of mechanics; relative to another level of analysis the phenomena can be seen to express a different law of development for a unified organic whole. Second, Leibniz’s relative theory of space and time unravels ii

the labyrinth of the composition of the continuum with the help of a simple tool: monads are actual discrete existents, while space and time are defined in terms of relations between actual existents; thus the continuity of space and time is merely potential. Leibniz formulated his strategy as follows: ‘Time, extension, motion, and the continuum in general, as we understand them in mathematics, are only ideal things —that is, they express possibilities, just as do numbers. Even Hobbes has defined space as a phantasm of the existent. But to speak more accurately, extension is the order of possible coexistence, just as time is the order of possibilities that are inconsistent but nevertheless have a connection.’1 The basic idea of defining space and time as ordering relationships between monadic events found its direct application in special relativity theory. Current research on ‘fourdimensionalism’ explores the ontology of events presupposed in this construction, often without taking sufficient guidance from, or issue with, Leibniz’s monadology and the idea of monadic activity. The following study discusses the prospects of a Leibnizian account of space, time, and motion with an eye to the present-day objections from philosophers of science and ontologists. Third, in Leibniz’s ontology individuality, numerical identity, and qualitative determinateness are analytically connected in his famous two identity principles. Any enterprise that is to replace the traditional focus on substances with a new focus on dynamic individuals (processes, events, activities etc.) must take its point of departure from a close consideration of the motives and the implications of Leibniz’s rejection of bare substrata. It is precisely here, at the difficult exploration of the deep structure of traditional category theory as partly endorsed and partly modified by Leibniz, that the penetrating arguments of the following study will provide unique guidance. Whitehead famously chose the label ‘philosophy of organism’ to characterise his own ontology and he claimed that ‘[i]n Western literature there are four great thinkers, whose services to civilized thought rest largely upon their achievements in philosophical iii

assemblage; though each of them made important contributions to the structure of philosophic system. These men are Plato, Aristotle, Leibniz, and William James.’2 Indeed, while structural elements supplied by the four mentioned philosophers can be found in many, also non-Whiteheadian strands of process philosophy, Leibniz is perhaps most palpably the common heritage of process thought in all its varieties. J. Seibt & M. Weber

1

Leibniz, Philosophical Papers and Letters. Translated by L. E. Loemker, Chicago, University of Chicago Press, 1956, p. 583 2 Modes of Thought [1938], The Free Press, 1968, p. 2

iv

Preface and Acknowledgments

This book draws on a number of previously published papers and articles, to which the various chapters are related as follows; fuller details of these previous publications are given in the Bibliography. Chapters I, IV, V, and VI are based on my article, ‘Indiscernibles and the Absolute Theory of Space and Time’, published in Studia Leibnitiana in 1988. Chapter II is new for the most part, though it borrows a little from my article, ‘Hume versus Clarke on the Cosmological Argument’, published in the Philosophical Quarterly in 1992. Chapter III is based on my article, ‘Leibniz’ Theory of Space: a Reconstruction’, published in the Philosophical Quarterly in 1993. Chapter VII is based on a paper entitled ‘On a Verificationist Argument in Leibniz (against absolute motion)’, presented to the Fifth International Leibniz Congress, which was held in Hanover in November 1988; the paper was published in the same year in the relevant volume of proceedings. Chapters VIII and IX are based on my article, ‘Eternity and Omniscience’, published in the Philosophical Quarterly in 1974. Finally, Chapter X is based on three previous publications. It incorporates a paper entitled ‘Leibniz and the Notion of Omnipotence’, which was presented to the Third International Leibniz Congress held in Hanover in 1977, and was published later in 1980 in Studia Leibnitiana Supplementa. It also makes use of some material included in my article, ‘In Defence of Omnipotence’, published in the Philosophical Quarterly in 1978. As well, Chapter XI embodies a paper entitled ‘"Adam and his Posterity": Leibniz on God’s Omniscience’, which was presented at an international colloquium with the theme ‘Leibniz and Adam’, held in the Hebrew University at Jerusalem in January 1992; the paper was later published in 1993 in the relevant volume of proceedings entitled Leibniz and Adam. As well

Chapter X makes use of an article entitled ‘Mackie’s Paradox and the Free Will Defence’, published in Sophia in 1995. I am grateful to the editors of the Philosophical Quarterly, Sophia, Studia Leibnitiana, and Studia Leibnitiana Supplementa, as well as the coeditors of Leibniz and Adam, for permission to make use of all this published material in the writing of this book. Special thanks are also due to the late Dr. Gerda Ütermöhlen, of the G.-W.-Leibniz Gesellschaft in Hanover; it was she who suggested to me many years ago that the themes sketched in my 1988 Studia Leibnitiana article could be expanded into a book. Most of the ideas embodied in the above publications came to me through teaching a course on the metaphysics of space and time in the philosophy department at Monash University, which was run for many years under the title ‘Space, Time and Causation’. I have learned a lot from the many generations of students who did this course, as well as former and present colleagues at Monash who, at one time or another, have shared this course with me; they include John Bigelow, John Collins, Lloyd Humberstone, Tom Karmo, and the late John McGechie. I feel particularly grateful to Lloyd Humberstone who has generously helped me with practically all my research since his arrival at Monash in 1975. But for him I would not have realised that there is an inter-world version of the principle of the identity of indiscernibles which is both true and important, an assumption which forms the backbone of the central part of this study. But my greatest debt is to John Bigelow, whose constant encouragement and enthusiastic interest in the metaphysics of space and time have sustained my efforts ever since I started working on this project. I am very thankful to him for the many happy and rewarding occasions on which we discussed the ideas of this book. Finally I wish to thank John Fyfield, Monima Chadha and Melva Renshaw for much needed help in formatting my manuscript to comply with the requirements of the publishers.

vi

Leibniz persisted in affirming that Newton called space sensorium numinis, notwithstanding he was corrected and desired to observe that Newton’s words were QUASI sensorium numinis. No, Sir; Leibniz was as paltry a fellow as I know. Out of respect to Queen Caroline, who patronised him, Clarke treated him too well. (Samuel Johnson, 1773) Leibniz’s argument against substantialism [i.e. Newtonian absolutism] is actually Clarke’s argument turned on its head. ... Nor is there any indication that this particular argument was explicitly constructed prior to the correspondence with Clarke, and the context strongly suggests that it was the product of opportunism and one-upmanship. (John Earman, 1989)

Abbreviations A&G

Ariew, R., and Garber, D., eds. and trans. G.W. Leibniz: Philosophical Essays, Indianapolis, 1989.

Alexander

Alexander, H.G., ed. The Leibniz-Clarke Correspondence, Manchester, 1956.

C

Clarke’s replies in the Leibniz-Clarke Correspondence. References are given by reply and section; thus C.iii.2 = Clarke's third reply, section 2. The full text of the correspondence, with Clarke’s translation of Leibniz’s papers, is given in Alexander and Loemker (pp. 675-717), who omits Clarke’s fifth reply (presumably because Leibniz did not live to see it). The full texts, in their French and English originals, are given in GP.vii.352-440, and are reproduced in Robinet’s annotated edition. Spelling and punctuation have been modernised.

Couturat Couturat, Louis, ed. Opuscules et fragments inédits de Leibniz, Paris, 1903. GM

Gerhardt, C.I. ed. Leibnizens mathematische Schriften, Berlin and Halle, 1849-1863.

GP

Gerhardt, C.I., ed. Die philosophischen Schriften von Gottfried Wilhelm Leibniz, 7 vols., Berlin, 1875-90.

Hall & Hall

Hall, A.R., and Hall, Marie Boas, eds. and trans. Unpublished Scientific Papers of Isaac Newton, Cambridge, 1962.

L

Leibniz’s papers in the Leibniz-Clarke Correspondence. References are given by paper and section; thus L.iv.13 = Leibniz’s fourth paper, section 13. The French text is given in GP.vii and in Robinet; Clarke’s translations are given in Alexander and in Loemker; a modern translation of selections from Leibniz’s papers is included in Parkinson. In the quoted passages Clarke’s spelling and punctuation have been modernised, and I have not always adhered to Clarke’s translation of Leibniz's French text. Here and there I have departed from Clarke’s English version when his rendering seemed archaic or not close enough.

Loemker Loemker, L.E., ed. and trans. Leibniz: Philosophical Papers and Letters, 2nd edition, Dordrecht, 1969.

Parkinson

Parkinson, G.H.R., ed. Leibniz: Philosophical Writings, trans. by M. Morris and G.H.R. Parkinson, London, 1973.

Robinet

Robinet, A., ed. Correspondance Leibniz-Clarke, Paris, 1957.

Wiener

Wiener, Philip P., ed., Leibniz: Selections, New York, 1951.

x

Chapter I Leibniz’s Last Controversy with the Newtonians 1.1. Introduction: Leibniz’s strategy In the famous Correspondence with Clarke, which took place during the last year of Leibniz’s life, Leibniz advanced several arguments purporting to refute the absolute theory of space and time that was held by Newton and his followers. The main aim of this study is to reassess Leibniz’s attack on the Newtonian theory in so far as he relied on the principle of the identity of indiscernible. But before giving an outline of what I propose to do I must make a few remarks about Leibniz's own procedure in that controversy and the general strategy underlying the various objections that he raised. The first point to be made is that Leibniz met his opponents half-way, adopting what I would call an ad hominem stance. By this I mean that he granted for the sake of argument certain assumptions upheld by his opponents, which he himself regarded as false. In particular he granted the reality of spatially related material bodies, though his considered opinion was that material bodies are not irreducibly real, but merely ‘well-founded phenomena’ whose basis is the mind-like non-spatial monads. And he granted the irreducible reality of relational properties (particularly those involving spatial and temporal relations), though his official view was that relational properties cannot stand on their own but are always parasitic upon (or reducible to) non-relational (intrinsic) properties. In a real sense, then, Leibniz was here impersonating a philosophical position in which he did not ultimately believe. Yet it is this impersonated position that I want to restate and discuss, both with regard to his own relative theory1 and against the ab1

I owe this insight to C. D. Broad who pointed up this ad hominem stance on the part of Leibniz in his correspondence with Clarke (back in 1946), in an important article, ‘Leibniz’s Last Controversy with the Newtonians’; first published in Theoria, Vol. 12

solute theory of the Newtonians. How this impersonated position is to be reconciled with the non-spatial monadism to which he officially subscribed is a question which I shall set aside. Next we need to get clear about Leibniz’s starting-points. As is well known, Leibniz based his arguments against the absolute theory on the principle of sufficient reason (PSR) as well as the principle of the identity of indiscernibles (PII). And the question arises: how exactly did he take these two principles to be related? Two different positions may be distinguished in his letters to Clarke. Initially, and especially in his third letter, Leibniz’s arguments are cast in a way which suggests that he regarded these two principles as independent but equally important starting-points in his attack on the Newtonian theory. This position is also suggested by the following pronouncement, which occurs at the beginning of Leibniz’s fourth letter: These great principles of sufficient reason and of the identity of indiscernibles change the state of metaphysics, which by their means becomes real and demonstrative; whereas formerly it consisted of almost nothing but empty words.2

But in his fifth (and last) letter Leibniz stresses the paramount importance of PSR as his sole ultimate principle, and claims to have inferred PII from PSR alone.3 On his final view, then, PII is not an independent principle coordinate with PSR, but a subordinate principle logically derivable from the latter. Now, I think it would be wrong to regard this shift of position as a sign of confusion on Leibniz’s part; rather, it represents a conscious change of strategy in response to Clarke. It seems that Leibniz believed all along (1946); reprinted in his Ethics and the History of Philosophy, London, 1952; see especially p.187. This article is in my opinion still the most helpful introductory guide to that correspondence, and I have always regarded it as a model of how the issues involved should be separated and discussed. My only regret about it is that, as with his other historical writings, Broad does not supply any precise references to the texts that he covers. This has been remedied, to too small an extent, by R. S. Woolhouse, who reproduces a shortened version of Broad’s article in his collection of readings, Leibniz: Metaphysics and Philosophy of Science, Oxford, 1981, pp. 157-74. 2

L.iv.5; emphasis added.

3

L.v.21; cf. L.v.25 and 48. The argument which he used to effect the derivation of PII from PSR occurs at L.iv.3.

2

that PSR entailed PII, but did not find it necessary to stress this point at the start of the controversy. Initially, he adopted a provisional stance whereby he raised objections that relied on both PSR and PII, treating the two principles as if they were logically independent; for he did not expect Clarke to oppose either of them. But when Clarke made it clear in his third reply that while accepting PSR he rejected PII,4 Leibniz was led to change his strategy; he then abandoned his initial stance and insisted on his considered view that PSR entails PII. That this was a conscious change of strategy is reflected in a letter to Rémond, written a few weeks after Leibniz had despatched his fifth paper. Reporting on the progress of his controversy with Clarke, Leibniz tells his friend that he has ‘reduced the state of our dispute’ to simple acceptance or denial of PSR. Plainly, this view, if true, would drive Clarke into a corner: if PII is indeed logically derivable from PSR, then Clarke cannot consistently uphold PSR while denying PII. Nor could Clarke deny PSR without loss of face, or (one might even say) loss of intellectual integrity. For by that time Clarke had established his intellectual reputation largely on the claim that he could logically demonstrate the existence of God on the basis of PSR, which claim was stated in his famous Demonstration.5 Leibniz must have been well acquainted with this work by the time he wrote his fifth paper, where he gently reminds Clarke of the awkward position he would be in if he were to deny PSR: ‘I dare say that without this great principle one cannot prove the existence of God’.6 With his friends to whom he reported the progress of his controversy with Clarke, Leibniz was more explicit. The passage from his letter to Rémond to which I have just referred is worth quoting in full, as it throws light on how Leibniz himself saw the state of play by the time he sent his fifth

4

See especially C.iii.2. Clarke reaffirmed his position in his fourth reply, at C.iv.3-6. ‘Two things, by being exactly alike, do not cease to be two’ (C.iv.5-6). 5

The full title of Clarke’s work, which was widely admired, is Demonstration of the Being and Attributes of God, which was first published in 1705. The work was based on a series of Boyle Lectures which Clarke delivered as sermons at St. Paul’s Cathedral in 1704. I have examined Clarke’s argument, together with Hume’s objections to it, in my article ‘Hume versus Clarke on the Cosmological Argument’.

6

L.v.126.

3

paper, and as far as I know it is not available in English translation. Leibniz wrote to his friend: Monsieur Clarke and I are honoured that our dispute is passing through the hands of Princess Caroline. I have sent my fourth reply [i.e. fifth paper], and am awaiting his, on which I shall settle my account, for in my last reply I have stated my position more fully, in order to bring the matter soon to an end. He seems to claim that he is not acquainted with my Theodicy, and has sometimes forced me to repeat myself. I have reduced the state of our dispute to this great principle, that nothing exists or happens without a sufficient reason that it should happen rather than otherwise. If he continues to deny it to me, where would his sincerity be? And if he grants it to me, then farewell to the void, the atoms, and the entire philosophy of Monsieur Newton.7

In the covering letter to the Princess Caroline accompanying his fifth letter to Clarke, Leibniz was equally explicit: I shall see how Monsieur Clarke will reply to me. If he does not entirely accept the received great principle that nothing happens without there being a sufficient reason why it happens thus rather than otherwise, I could not help doubting his sincerity, and if he grants it then farewell to the philosophy of Monsieur Newton.8

A month later, having acknowledged receipt of two books by Clarke sent to him by the Princess with the suggestion that Clarke might be a suitable candidate to translate his Theodicy into English, Leibniz wrote to the Princess: I cannot judge of elegance in English, but I think I can at least judge of clarity of expression. Monsieur Clarke is certainly not lacking in it, but we shall soon find out whether it is accompanied by sincerity, and whether he is a man to hold out his hands to the truth; that would undoubtedly do him more honour than the detours which he might take to avoid it. If he continues to dispute my great principle that nothing happens without there being a sufficient reason why it happens, and why it happens the way it does rather than otherwise, and if he still claims that something can happen by a ‘mere will of God’ without any motive, ... he will have to be left to his opinion or rather to his own obstinacy. For it will be

7

GP.iii.678; quoted in Robinet, p.185; emphasis added. The letter is dated 19 October 1716; Leibniz’s fifth paper was despatched by the end of August. 8

The letter was sent along with Leibniz’s fifth paper, which was despatched by the end of August; quoted in Robinet, p.183.

4

difficult for him not to be touched by it in the depths of his soul, but I believe the 9 public will not let him off.

Nor should we be surprised that Leibniz should have changed his strategy in the course of this unfriendly controversy, in which, under royal pressure, he was forced to participate, and which in the end hardened into what has come down to us as a veritable dialogue de sourds. For he was faced with opponents whom he had learned by bitter experience not to trust, and who in the end lived up (or down) to his expectations. He was opposed to them on many issues besides their absolutist view of space and time, and they were determined not to understand him. Initially, he was very much in the position of a soldier in enemy territory, tentatively feeling his way around for a possible vantage point from which to fire his first shot. Or, to drop the metaphor, initially Leibniz was not sure what premisses he could safely take for granted, in the belief that they were acceptable to his opponents, as starting-points on which to base his objections. So we find him deploying his objections piecemeal, as the occasion arose, without meaning to exhaust his entire possible repertoire of arguments on a particular issue. Thus he says at the start of his attack on Newtonian absolutism, just before he propounds what I shall in due course call Argument II:10. I have several proofs to refute the fantasy of those who take space to be ... an absolute being of some kind; but for the moment I will only make use of the one required by the present occasion.11

I do not think that Leibniz’s attempt to derive PII from PSR is successful,12 but I will not elaborate this point here since it is not relevant to my present purpose. What I find interesting is that he sometimes stated the same objection in two versions: one relying on PSR alone, the other relying on PII alone.13 One is naturally led to ask: could Leibniz have made all the objec9

This letter is dated 11 September 1716; quoted in Robinet, p.184; cf. Alexander, p.197. 10 11 12

See section 6.2 below. L.iii.5. See section 5.4 below.

13

This is done most explicitly with what I call the temporal translation argument, summarised as Argument IV in chapter VI below, of which Leibniz gives two quite distinct

5

tions that he wished to make against the absolute theory of space and time on the basis of PII alone, thus dispensing with PSR altogether? I think he certainly could; for it is easy to see that to every PSR-based argument that he did advance there is a parallel PII-based argument which he could have used, even if he did not care to produce it. And what is more, I believe that had he done so, his case against the absolute theory would have been much more convincing. For, on the one hand, most14 of the relevant PII-based arguments invoke that principle by comparing whole worlds rather than parts of the same world; the former type of comparison involves an inter-world appeal to PII, the latter involves an intra-world appeal. And I accept the restricted inter-world version of PII on which most of these arguments rely; for although I want to reject the unrestricted PII, under several disambiguations, I shall nonetheless argue that the indiscernibility of whole worlds is a sufficient condition of their identity. And on the other hand, all of Leibniz’s PSR-based arguments rely on a theological version of that principle, to the effect that God never acts without a rational motive; yet strictly speaking this theological principle has no bearing whatever on the theory under attack, which is theologically neutral. The central core of the absolute theory are certain theses (which will be spelled out in the next chapter) about the ontological status of space and time and about their structure; and these theses can hold quite independently of whether or not God exists, and of whether or not he ever acts without a rational motive. It is true, of course, that the Newtonians combined their adherence to the absolute theory with certain beliefs about how God is related to absolute space and absolute time. But these beliefs, which Leibniz vehemently opposed, were not logically dictated by the absolute theory as such, but constitute a separable theological superstructure. Questions regarding the existence, size, or age of the material universe are evidently separable from the question whether the absolute theory is true. The Newtonians held that the material universe was finite in extent and had a beginning in time. Leibniz attempted to refute them on both counts, since he himself held the opposite views, namely that the material universe versions. In Leibniz’s statements of two other objections, summarised below as Arguments II and VII, the two versions are interwoven but are quite easy to separate. See references against the summaries of these arguments given below in section 6.4. 14

6

Five out of seven, to be precise; see chapter VI.

is infinite in extent and had no beginning in time. Yet it is clear that the absolute theory can be consistently combined with what Leibniz believed to be the right views about the size and age of the material universe; and that, moreover, there are some alternatives which never got considered. For example, what if the material universe had no beginning in time but were finite in extent? And what if no material universe existed at all? A proper attempt to refute the absolute theory must be comprehensive enough to cover all the possible views that one could have regarding the size, age, and existence of the material universe -- as is done in chapter VI. However, I must hark back to my earlier remark that, throughout his last controversy with the Newtonians, Leibniz, I believe, was literally impersonating a position in which he did not ultimately believe, a remark which, I suspect, must have raised the reader’s eyebrows. To which I hasten to add my firm conviction that in doing so he was not indulging in any form of deceit or dialectical impropriety. On the contrary, this is exactly how one ought in all honesty behave (i.e. conduct one’s side of the controversy) in the sort of situation with which Leibniz was confronted. To appreciate this point it must be emphasised that when, in the course of controversy, you propound an argument against an opponent, then, for your argument to have swaying power, you need not believe that its premisses are true; what is needed is that your opponent should grant your premisses or believe them to be true. And this is very much as it should be, especially when you propound an ad hominem argument against your opponent, or when you propound a reductio ad absurdum. (And for my purpose I think we can safely regard a reductio argument as a special case of an ad hominem argument.) Now it will be noticed, when in due course we reach chapter VI, that almost all of Leibniz’s arguments against absolutism are of the reductio form. So it is not surprising that they are all based on premisses which Leibniz himself did not actually accept, but was reasonably sure that they were acceptable to his opponents. The points I have just been making were in fact very forcefully made in recent years by the late Paul Feyerabend in a memorable piece entitled ‘Marxist Fairytales from Australia’.15 He there reminds his two misguided

15

This was Feyerabend’s reply to an article by J. Curthoys and W. Suchting, entitled ‘Feyerabend’s Discourse Against Method: a Marxist Critique’; both the article and Fey-

7

critics from Australia of a ‘basic rule’ and some of its ‘corollaries’, to which honest controversy must conform. In his own words, the Basic Rule says: ‘If an argument uses a premise, it does not follow that the author [i.e. the propounder] accepts the premise, claims to have reasons for it, regards it as plausible. He may deny the premise but still use it because his opponent accepts it and, accepting it, can be led in a desired direction’.16 Also relevant to my purpose is Feyerabend’s Second Corollary, which says: ‘In an argument against an opponent an author [i.e. the propounder] can use assumptions and procedures he has shown to be unacceptable elsewhere provided they are accepted by the opponent’.17 Notice that I prefer to speak of the propounder rather than the author of an argument, because one can author an argument (e.g. by merely entertaining it) without propounding it, i.e. without actually putting it forward against a real opponent. Here I follow a usage I owe to Frank Jackson, whereby propounding an argument is the counterpart of asserting a proposition in isolation.18 So I would rather express the vital point which Feyerabend rightly emphasises by saying that, in propounding an argument, the propounder is not himself committed to asserting any of its premisses; it is enough, for his argument to have swaying power, that his opponent should grant them or believe them to be true. Which is exactly what I take Leibniz to have done in his last controversy with the Newtonians. 1.2. Plan of this work The central task of this study, then, is to reassess Leibniz’s attack on Newtonian absolutism in so far as he relied on the principle of the identity of indiscernibles. As one distinguished writer has so nicely put it some years ago, Leibniz regarded that principle (PII) as a nutcracker with which he

erabend’s reply appeared in the same issue of Inquiry, Summer, 1977. References to Feyerabend’s reply are to the reprint in his collection of essays, Science in a Free Society, London, 1978; see especially the section entitled ‘A Guide for the Perplexed’, pp. 156-63. I am grateful to Richard Holton for drawing Feyerabend’s very pertinent remarks to my attention. 16

Ibid., p.156.

17

Ibid., p.158.

18

See Jackson, 1987

8

could crush Newtonian absolutism; and our question is, How far was he successful?19 Hacking’s own verdict is that Leibniz was indeed successful, but for reasons which I myself doubt whether the real Leibniz would have allowed. The success is achieved by endowing Leibniz with certain doctrines which the real Leibniz could not possibly have espoused without loss of philosophical identity. In particular, he is made to adopt conventionalism with regard to physical geometry and the laws of nature; i.e. the view that whatever geometry we ascribe to physical space and whatever physical laws we ascribe to nature are matters of convention; and, as the dictum goes, ‘there is no fact of the matter’. I propose, then, to reconstruct Leibniz’s attack on the absolute theory along these lines: to isolate, restate, and systematise all the PII-based arguments that he did advance or could have advanced against that theory, and assess their force. However, this reassessment cannot be delivered at once, but will have to wait until chapter VI; for there is a good deal of spadework that needs to be done before being in a position to deliver the goods, such as they are. I shall start (in chapter II) by giving an outline of the absolutist theory under attack, and then go on (in chapter III) to reconstruct Leibniz’s own theory and defend it against some objections. On both sides, I shall attempt to distinguish those theses which constitute the core of the theory from those that are peripheral and not logically dictated by that core. Next (in chapter IV) I introduce two classifications of properties; these classifications are then used (in chapter V, section 1) to distinguish three versions of PII: weak, middle, and strong. In the same chapter (section 2) there is a longish discussion of the middle version (called PII.2) as an intra-world principle, which is finally rejected. The indiscernibility of whole worlds is discussed in the same chapter, section 3, where it is argued that the strong version of PII (called PII.3) is acceptable as an inter-world principle, though it is to be rejected in its unrestricted form. These conclusions are finally brought to bear on seven PII-based arguments against the absolute theory, which are systematically discussed in chapter VI. With chapter VI my main task of reconstructing and assessing Leibniz’s PII-based objections to Newtonian absolutism is complete. The remaining four chapters are devoted to residual issues that figure in the Leib19

Ian Hacking, ‘The Identity of Indiscernibles’, p. 251.

9

niz-Clarke agenda. Chapter VII discusses yet another argument propounded by Leibniz, a variant on the last PII-based argument to which I refer as Argument VII, which is intriguing in that it is verificationist in style, a feature which seems to consort badly with his own rationalist position. I there suggest that this additional argument is best regarded as yet another ad hominem objection, and that, if we regard it as such, then it is more successful than its PII-based counterpart. The remaining three chapters (VIII-X) are devoted to the theological superstructure, which, though independent of the controversy regarding space and time, is not devoid of present-day interest, but ranges over some live issues much discussed in the current literature in philosophical theology. Chapter VIII is in the nature of a ‘digression’ on Boethius who deserves to be treated first because his position is similar to Leibniz’s but more fully developed; for like Leibniz, Boethius was led to place God outside time in a realm of non-temporal eternity. The Newtonians, on the other hand, held the opposite view, placing God in time, and interpreting his eternity as everlastingness, or better, sempiternity; that is to say, they believed that God’s eternity consists in his existing at all moments of time, and this holds whether we believe that time is finite in one or both directions, or infinite in both. Finally, chapters IX and X isolate and examine the rival accounts of omniscience given by Leibniz and the Newtonians, and the objections raised by Clarke and Arnauld against Leibniz’s views on God’s omniscience and omnipotence. I would add, as a postscript, that an important feature of the controversy, which is common to both Leibniz and the Newtonians and which (for historical reasons) I shall hold constant, is the assumption that space necessarily has a three-dimensional Euclidean geometry, and time a linear chronometry.

10

Chapter II Newtonian Absolutism 2.1. Introduction In this chapter I want to give a critical account of Newtonian absolutism regarding space and time. I shall start by distinguishing and expounding ten different theses held by the Newtonians about the nature of space and time, and about the relation of space and time to God and the created material universe. Having done that, I shall attempt (in the next two sections) to get at the core of the Newtonian theory, to disengage those theses that are logically central from the rest, and argue that at least five out of the ten theses here distinguished are in fact peripheral and not logically dictated by that core. Finally (in section 4), I shall argue that the theological views held by the Newtonians introduce an incoherence into the nature of their theory of space and time, and go on to suggest a way out which would, I think, effectively remove that incoherence, though I believe it would not have been acceptable to Clarke (and Newton). My account will be based on Clarke’s replies to Leibniz, which are believed by modern scholars to have been vetted, if not dictated, by Newton himself.20 Indeed one suspects that the Newtonian theory became much clearer in Clarke’s replies, made under pressure from a critic of Leibniz’s stature, than it was before the date of the Leibniz-Clarke Correspondence.21 As well, my account will be based on Newton’s own earlier pronouncements in the philosophical parts of his Principia, and on his early and unfinished essay entitled ‘On the Gravity and Equilibrium of Fluids’ (hereafter 20

The part played by Newton behind the scenes in drafting Clarke’s replies to Leibniz is researched in an important article, published in 1962: see A. Koyré and I. B. Cohen, ‘Newton and the Leibniz-Clarke Correspondence’, Archives Internationales d’Histoire des Sciences, Vol. 15 (1962), pp. 63-126. 21

Cf. C. D. Broad, p. 172.

referred to as De Gravitatione).22 Some of the things which the young Newton says in that remarkable essay seem to me to throw a good deal of light on his mature view as to what is involved in absolutism with regard to space and time. 2.2. Ten Newtonian theses The Newtonians held a number of views on the nature of space, time, and the material universe, and on God’s relation to all these things, which may be summarised in ten theses as follows: (A1) The most fundamental thesis of Newtonian absolutism regarding space and time concerns their ontological status, which is expressed by saying that space and time are absolute rather than relative beings.23 In this claim the relevant contrast between the absolute and the relative is that between something that enjoys an autonomous existence as an individual, as against something whose existence is parasitic upon the existence of an individual. An apple as a whole, for example, is an independent existent and is in this sense an absolute being, whereas its shape, size and internal structure are relative beings, in that none of these features has being apart from the apple as a whole. On the Newtonian theory, space and time are absolute beings, in that their existence is not parasitic upon the existence of what we ordinarily regard as their occupants, such as material bodies, events and processes. The existence of material bodies, for instance, logically requires the existence of 22

This essay was first published in 1962 in Unpublished Scientific Papers of Isaac Newton, chosen, edited and translated by A. Rupert Hall and Marie Boas Hall (Cambridge, 1962). The essay is believed by its editors to have been written between 1664 and 1668, when Newton was 22-24 years of age (see Hall & Hall, op. cit., p. 90). In some of the quoted passages I have diverged from the Halls’ translation at the places indicated. 23

Note that I stick to Leibniz’s terminology in calling his own type of theory of space and time relative, rather than follow most contemporary writers on the subject who call it relational; cf. L.iii.4, and L.v.62. In the context of Leibniz’s relative theory of space, the term ‘relational’ strictly applies to places or locations relatively to the chosen frame of reference: any location is determined by its distance relations to the fixed items that constitute the frame of reference. But the Leibnizian theory itself, or indeed any antiabsolutist theory, is relative in that the existence of space is taken to be parasitic upon the existence of material bodies, as will be explained in the next chapter.

12

space for them to occupy and move in, but the existence of space does not logically require the existence of material bodies, or any other spaceoccupants. Similarly, the existence of events and processes logically requires the existence of time in which they have their dates and durations, but the existence of time does not logically require the existence of events, processes, or any other time-occupants. A region of absolute space is equally real, whether or not it is occupied; and similarly with any lapse of absolute time. (In my own preferred vocabulary, throughout this book, I call any part of space a region, and its magnitude a volume; and I call any part of time a lapse, and its magnitude a duration.) To elaborate this thesis, (a) we must distinguish between a material body and the region of absolute space which is occupied by it. For the same body could occupy a different region of space at a different time, and the same region of space can be occupied by different bodies at different times. The spatial region occupied by a material body at a certain time has the same shape and volume as that body. But, though they coincide, the outer shape of the body and the outer shape of the region are properties belonging to numerically distinct individuals, the body and the region that it occupies. And the same applies to their volumes: though equal in magnitude, the volume of the body and the volume of a region that it occupies at a certain time are different properties belonging to numerically distinct individuals, the body and that region. And similarly, (b) we must distinguish an event or a process from the lapse of time which it occupies. For different events and processes can occur at or during the same lapse of time, and the same event or process which occurs at or during a certain lapse of time could have occurred at or during a different lapse of time. An event or process which occurs at or during a certain lapse of time has exactly the same duration as the lapse of time in which it occurs; but, though equal in magnitude, the duration of the event or process and the duration of the lapse of time are properties that belong to numerically different items. Clarke is quite explicit about the spatial component of this thesis in his fifth and final reply:

13

Finite spaces [i.e. regions] are not at all the affections [i.e. properties] of finite substances [such as material bodies], but they are only those parts of infinite space in which finite substances exist.24

The Newtonians believed that the material universe is actually finite in extent; but, Clarke maintains, even if the whole of infinite space were fully, or continuously, occupied by matter, space would not then be a property of that infinite body; rather, the infinite mass of matter would still be merely in space, as finite bodies are actually in it. Says Clarke in the same reply: If matter were infinite, yet infinite space would no more be an affection [i.e. property] of that infinite body than finite spaces are the affections of finite bodies; but in that case the infinite matter would be, as finite bodies now are, in the infinite space.25

(A2) Space and time are continuous, i.e. infinitely divisible: there is no smallest region of space, and no smallest lapse of time. (a) Clarke holds that space is necessarily continuous, i.e. infinitely compact and divisible, in the sense that it is meaningless to suggest that there are minimal finite regions or granules of space which are not further divisible. And (b) a similar view is held about time. Time is necessarily infinitely divisible: it makes no sense to suggest that time is composed of minimal lapses or granules of time of finite duration but not further divisible. From the spatial components of the above two theses, i.e. A1(a) and A2(a), taken together, a further feature of space emerges, namely that regions of space actually and literally have all shapes and sizes, and that unoccupied regions of space actually have all the shapes and sizes which they might reveal to our senses if they were actually occupied. As the young Newton put it in De Gravitatione, there are everywhere all kinds of figures, everywhere spheres, cubes, triangles, straight lines, everywhere circular, elliptical, parabolical and all other kinds of figures, and those of all shapes and sizes, even though they are not disclosed to sight. For the material delineation of any figure is not a new production of that figure with respect to space, but only a corporeal representation of it, so that what was formerly insensible in space now appears to the senses to exist. For thus we believe all those spaces to be spherical through which any sphere ever passes, be24

C.v.40; Alexander, p.103.

25

C.v.41; Alexander, p.103; emphasis added.

14

ing progressively moved from moment to moment, even though a sensible trace of the sphere no longer remains there. We firmly believe that the space was spherical before the sphere occupied it, so that it could contain the sphere; and hence as there are everywhere spaces that can adequately contain any material sphere, it is clear that space is everywhere spherical. And so of all other figures.26

Commenting on this passage, the translators-cum-editors claim that Newton should here be understood to be saying that the parts of space have the various shapes and sizes only potentially. They write: Extension [by which they mean a region of space] is, in fact (though these are not Newton’s words), the potentiality of figure in space... [T]he figure of the material sphere does not, so to speak, imprint itself in shapeless space: rather the material sphere can be everywhere because the potentiality of extension to receive it exists everywhere27

But this is clearly a gross misrepresentation: Newton’s whole point is that before receiving the material sphere the region of space was, and always is, actually spherical; and similarly for any other shape. It seems to me that the very idea of a potential shape in space is completely alien to the Newtonian view. (A3) (a) A region of absolute space is immovable: it is necessarily where it is in relation to all other regions. And similarly, (b) a lapse of absolute time is immutable: it is necessarily when it is in relation to all other lapses of time. A region of space is necessarily immovable because its individual identity is determined by its position in relation to all other regions: every true attribution to it of a spatial relation to any other region forms part of its essence. Hence to suppose a given region of absolute space to have moved involves a contradiction. For it is to suppose it to have altered its spatial relations to other regions, and that would deprive it of its essence, and therefore of its individuality: it would be to suppose the given region numerically different from what it is, which is a contradiction. As Clarke puts it with regard to the whole of absolute space, ‘to imagine its parts moved from each

26

De Gravitatione, in Hall & Hall, p.133.

27

Hall & Hall, op. cit., p. 78

15

other is to imagine them moved out of themselves’. Says Clarke in his second reply: Space, finite or infinite, is absolutely indivisible, even so much as in thought (to imagine its parts moved from each other is to imagine them moved out of themselves).28

And again in his third reply: For infinite space is one, absolutely and essentially indivisible; and to suppose it parted is a contradiction in terms, because there must be space in the partition itself, which is to suppose it parted and yet not parted at the same time.29

And the case is similar with absolute time. Indeed the young Newton, in De Gravitatione, found the parallel point about time so obvious that he explicated the immobility of regions of space by stressing the complete analogy with the immutability of lapses of time: The parts of space are immovable30... [T]he immobility of space will be best clarified31 by [analogy with] duration [i.e. time]. For just as the parts of duration derive their individuality from their order, so that (for example) if yesterday could change for today, it would lose its individuality and would no longer be yesterday, but today; so the parts of space derive their individuality32 from their positions, so that if any two could change their positions, they would change their individuality33 at the same time and each would be converted numerically into the other. The parts of duration [i.e. time] and space are only understood to be the same as they really are because of their mutual order and position; nor do they have any element34of individuality apart from that order and position, which consequently cannot be altered.35

28

C.ii.4.

29

C.iii.3.

30

Substituting ‘immovable’ (immobiles) for ‘motionless’ in Hall & Hall.

31

Substituting ‘clarified’ (illustrabitur) for ‘exemplified’ in Hall & Hall.

32

Substituting ‘individuality’ for 'character' in Hall & Hall.

33

Ditto.

34

Substituting ‘element’ for ‘hint’ in Hall & Hall.

35

De Gravitatione, in Hall & Hall, p.136. I am grateful to Gavin Betts for correlating

16

The passage from Clarke’s fifth reply just quoted echoes what Newton himself had said in Principia, in the famous Scholium to Definition VIII: As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times [i.e. lapses of time] and spaces [i.e. regions of space] are the places as well of themselves as of all other things. All things are placed in time as to order of succession, and in space as to order of situation. It is from their essence or nature that they are places [i.e. positions in time and space], and that the primary places of things should be movable is absurd.36

The immutability of both the temporal and the spatial order may be likened to the immutability of the members in the numerical series of integers. It is of the essence of the number 3, for example, that it stands between the numbers 2 and 4, that it is five places ahead of the number 8, and two places behind the number 1, and so on. To suppose the number 3 to have relinquished its place in the number series and come to occupy a different place, between (let us say) 5 and 7, is a contradiction in terms; it is to deprive it of its individuality; it is to suppose it both distinct from and identical with the number 6. Similarly, it is of the essence of the year 1801 that it lies between 1800 and 1802, that it is exactly one century ahead of 1901, and so on; every one of these relational properties is essential to it. And for similar reasons, any region of absolute space is spatially immovable. Whence it follows that spatial relations, according to the Newtonian theory, hold primarily between regions of space, and only derivatively (or secondarily) between bodies as their occupants. When we say of two billiard balls, A and B, that they are now two feet apart, this should be read as saying that there are two regions, r1 and r2, which are immutably and permanently two feet apart, and which are now occupied by A and B respectively; A and B are now two feet apart in virtue of their now occupying r1 and r2 which are permanently two feet apart. And similarly with temporal relations. When we say of two events, C and D, that C is earlier than D, this should be regarded as a short-hand way of saying that event C occurs at a

the English translation in Hall & Hall with the original Latin text, and suggesting the noted amendments 36

Newton, Principia, new 1999 translation, p. 410.

17

moment of absolute time which is earlier than the moment of time occupied by the event D. In other words, the temporal relation of earlier and later obtains primarily between the fixed moments of absolute time, and only derivatively (or secondarily) between events that occur at those moments. And again, when we say of two events, E and F, that they are simultaneous, what we mean, according to Newtonian theory, is that the two events occur at one and the same moment in absolute time. (A4) (a) Space is actually, and not just potentially, infinite in extent: there is no boundary to absolute space in any direction. Similarly (b), time is infinite both in the direction of the past and in the direction of the future. Clarke emphasised this point with regard to space in his fifth (and final) reply: There is no such thing in reality as bounded space, but only we in our imagination fix our attention upon what part or quantity we please of that which itself is always and necessarily unbounded.37

A form of the view that space is only potentially infinite was held by Descartes, who expressed it by saying that space should be regarded as indefinite rather than infinite in extent.38 To this view the young Newton objects that it involves a misuse of the term ‘indefinite’. For the word ‘indefinite’ is never applied to that which actually is, but always relates to a future possibility signifying only something which is not yet determined and definite... Thus an indefinite line is one whose future length is still undetermined. And so an indefinite space is one whose future magnitude is not yet determined; for indeed that which actually is, is not [something] to be defined, but does either have limits or not and so is either finite or infinite.39

The idea that space is actually infinite is explained by Newton as follows: Space extends infinitely in all directions. For we cannot imagine any limit anywhere without at the same time imagining that there is space beyond it.40

37

C.v.38; Alexander, p.103

38

See Descartes, Principles of Philosophy, Part I, Sec. 26.

39

De Gravitatione, Hall & Hall, p. 135.

40

De Gravitatione, in Hall & Hall, p. 133; cf. Clarke at C.v.38 cited above.

18

And he goes on to give an example: You may have in truth an instance of infinity; imagine any triangle [ABC] whose base [AB] and one side [AC] are at rest and the other side [BC] so turns about the contiguous end [B] of its base in the plane of the triangle that the triangle is by degrees opened at the vertex [C]; and meanwhile take a mental note of the point where the two sides meet, if they are produced that far: it is obvious that all these points are found on the straight line along which the fixed side [AC] lies, and that they become perpetually more distant [from A] as the moving side turns further until the two sides become parallel and can no longer meet anywhere. Now, I ask, what was the distance [from A] of the last point where the sides met? It was certainly greater than any assignable distance, or rather none of the points was the last, and so the straight line in which all those meeting-points lie is in fact greater than finite. Nor can anyone say that this is infinite only in imagination, and not in fact; for if a triangle is actually drawn, its sides are always, in fact, directed towards some common point, where both would meet if produced, and therefore there is always such an actual point where the produced sides would meet... And so the line traced by all these points will be real, though it extends beyond all distance.41

The example is intended to give us a firmer grip on the notion of space extending infinitely in a certain direction, in this case the direction AC. The straight line traced by the receding vertex until the two sides become parallel is infinitely long (‘greater than finite’): it has a first point, A, but there is no last point in the direction AC.42 Any point along this line is finitely distant from A; but the line ‘extends beyond all distance’, so that there is no point which is maximally distant from A; and it is therefore a mistake to assume that there is a last point which is infinitely distant from A. Indeed Newton’s discussion of the infinity of space strikes me as very modern: to say that space is actually infinite in extent is to say that there is no maximal distance between any two of its points, though any point is always finitely distant from any other. The idea of two points that are infi41

De Gravitatione, in Hall & Hall, p.134; the letters in square brackets are added for convenience.

42

This may be confirmed by performing Newton’s procedure in reverse: suppose we start with the same base but with the two sides initially parallel, and then imagine the moving side to turn by degrees towards the fixed side, and coming to meet it. It is clear that there would then be no first point where the moving side meets the fixed side to form a triangle.

19

nitely distant from each other is not only unnecessary but incoherent; but I can see no trace of this suggestion in Newton.43 (A5) We must distinguish between absolute motion and rest, on the one hand, and relative motion and rest, on the other; the former are defined in terms of absolute space, the latter in terms of relative space. We have a relative space, according to Newton, when we take a region of space around some perceivable body and regard it as immovable relatively to that body. A part of such a relative space is a relative place, and the ‘translation’ of a body from one relative place to another is relative motion. In ordinary life what is ‘popularly used for immovable space’, is in reality the relative space generated by the earth as our fixed frame of reference. It is immovable relatively to the earth, but, since the earth moves round the sun, that same space not only is movable but actually moves round the sun: which is another way of saying that our ordinary space moves within the relative space generated by the sun as a fixed frame of reference.44 But not all motion is relative. Newton also believes that, for example, the earth (and its relative space) moves absolutely, i.e. from one part of absolute space to another; and that, unlike a physical body or a relative space, absolute space and its parts are absolutely (logically) immovable. As Newton puts it: Absolute motion is the change of position of a body from one absolute place into another; relative motion is change of position from one relative place into another. Thus. in a ship under sail, the relative place of a body is that region of the ship in which the body happens to or that part of the whole interior of the ship which the body fills and which accordingly moves along with the ship, and relative rest is the continuance of the body in that same region of the ship or same part of its interior. But true rest is the continuance of a body in the same part of that unmoving space in which the ship itself, along with its interior and all its contents, is moving. Therefore, if the earth is truly at rest, a body that is relatively at rest on a ship will move truly and absolutely with the velocity with which the ship is moving on the earth. But if the earth is also moving, the true and absolute motion of the body will arise partly from the true motion of the 43

Contrast Jonathan Bennett, p. 132, where there is a suggestion that Newton may have thought of the last point where the two sides met as being a fixed point infinitely distant from A, in which case, we are told, ‘he was wrong’. 44

Principia, Scholium to Definition VIII, Section II; new translation, 1999, pp. 408-9.

20

earth in unmoving space and partly from the relative motion of the ship on the earth.45

The rival theory, that all motion is relative, has, as Clarke was to point out in his replies to Leibniz, certain ‘absurd’ consequences, which seemed to him strong enough to refute that theory. One consequence is that, if a body happened to be the only one that existed, it would, on the relative theory, be meaningless to suggest that it could be either in motion or at rest: the motion and rest of the whole material universe, taken as one body, would in that case be the same. Another ‘absurd’ consequence is that, if all matter outside a rotating body, such as the sun or the earth, were to be annihilated, it would at once become meaningless to say that it was rotating; and, presumably, in that case all the observable effects caused by the rotation of the earth would cease to take place. Says Clarke It is affirmed [by Leibniz] that motion necessarily implies a relative change of situation in one body with regard to other bodies; and yet no way is shown to avoid this absurd consequence, that then the mobility of one body depends on the existence of other bodies, and that [then] any single body existing alone would be incapable of motion, or that [then] the parts of a circulating body (suppose the sun) would lose the vis centrifuga [i.e. centrifugal force] arising from their circular motion, if all the extrinsic matter around them were annihilated.46

(A6) Absolute space and absolute time are, as such, causally neutral; and that is why, by themselves, regions of absolute space and lapses of absolute time are unperceivable; for, being causally neutral, they cannot affect a perceiver directly. Hence in common life we go by relative or perceivable measures of them, but philosophically we must admit absolute space and time as the things themselves as distinct from what are merely perceivable measures of them. Says Newton: But since these parts of space cannot be seen and cannot be distinguished from one another by our senses, we use sensible measures in their stead. For we define all places on the basis of the positions and distances of things from some body that we regard as immovable, and then we reckon all motions with respect to these places, insofar as we conceive of bodies as being changed in position with respect to them. Thus, instead of absolute places and motions we use rela-

45

Newton, ibid., Section IV; new translation (1999), p. 409.

46

C.v.31; Alexander, p. 101; cf. C.iii.4, C.iv.13, and C.v.52-3.

21

tive ones, which is not inappropriate in ordinary human affairs, although in philosophy abstraction from the senses is required. For it is possible that there is no body truly at rest to which places and motions may be referred.47

For this same reason (that space and time by themselves have no causal powers) the young Newton hesitated to call them substances; but neither could they be accidents. As he puts it, as regards space [‘extension’], in De Gravitatione: It is not substance ... because it is not among the proper dispositions that denote substance, namely actions, such as thoughts in the mind and motions in body. For although philosophers do not define substance as an entity that can act upon things, yet they all tacitly understand this of substances. ... They would hardly allow that body is substance if it could not move nor excite in the mind any sensation or perception whatever. Moreover, since we can clearly conceive extension existing without any subject, as when we may imagine spaces outside the world or places empty of body, and we believe [extension] to exist wherever we imagine there are no bodies, and we cannot believe that it would perish with the body if God should annihilate a body, it follows that [extension] does not exist as an accident inherent in some subject. And hence it is not an accident. And much less may it be said to be nothing, since it is rather something than an accident, and approaches more nearly to the nature of substance.48

I think the young Newton could have resolved his difficulties here by invoking Frege’s distinction between actuality (Wirklichkeit) and mere existence, the actual being something which besides existence has causal properties. Peter Geach, who in recent years has stressed the importance of this distinction, has this to say about it: A provisional explanation of actuality may be given thus: x is actual if and only if x either acts, or undergoes change, or both; and here I count as ‘acting’ both the inner activities of mind, like thinking and planning, and the initiation of changes in things. ... [But] I do not think this explanation ... can be developed into a definition. For it is not yet clear what counts as a thing’s undergoing a change. ... The notion of actuality is [in fact] needed to explain the difference between genuine changes, like the butter’s melting, and bogus changes, like the butter’s rising in price, and therefore cannot itself be defined in terms of change.

47

Newton, Principia, Scholium to Definition VIII, Section IV; new translation (1999), pp. 410-11.

48

Hall & Hall, p. 132.

22

All the same my explanation is likely to be of some use; for many of us have some intuitive grasp of this difference between real and bogus changes.49

It should be noted that what Geach and Frege call actual (wirklich) is contrasted, not with the merely possible or potential, but with what has existence or objectivity (das Objective) yet lacks causal powers. Frege insists on this distinction in several places, and illustrates it with some very interesting examples. Thus in The Foundations of Arithmetic he has this to say: I distinguish what I call objective from what is handleable ... or actual. The axis of the earth is objective, so is the centre of mass of the solar system, but I should not call them actual in the way the earth is so. We often speak of the equator as an imaginary line; but it would be wrong to call it an imaginary line in the dyslogistic sense; it is not a creature of thought, but is only recognised or apprehended by thought.50

This could provide a way out of the difficulties mentioned by the young Newton. Using the Frege distinction, a Newtonian absolutist could say that space and time are individuals that exist but are not actual: they are objective in the way in which the axis of the earth and its equator are objective, though neither has causal powers of its own. But whether this mode of existence disentitles Newtonian space and time from being called substances is a matter of linguistic legislation: for the term ‘substance’ has sometimes been used in a sense that requires Fregean actuality, sometimes not. (A7) Nevertheless the reality of absolute space and absolute time is established by the reality of absolute motion, which is defined in terms of absolute space and absolute time. The reality of absolute motion (as distinct from relative motion) can be established by its effects, particularly the existence of centrifugal force. One instance of this is the flattening of the earth at the poles. Another very simple case, described by Newton, is the famous bucket experiment, which can be performed in anybody’s backyard. Take a bucket half filled with water, and hang it up (from say, the branch of a tree), making sure that you can observe the subsequent changes in the surface of

49

Peter Geach, ‘What Actually Exists’, in his God and the Soul, pp. 65-66.

50

Frege, op. cit., 2nd ed., 1959, p. 35e. I am indebted to Richard Holton for this reference.

23

the water from above; and then twist the string a number of times, and leave it to untwist itself without further interference. The bucket will rapidly spin round its axis. At first the water will not spin; but gradually, as the water ‘picks up’, as we say, the spinning of the bucket , it will take up the spinning movement of the bucket; until -- and this is the final stage -- the water and the bucket will rotate as one rigid body. Now what is noticed in this experiment is that, at the beginning the surface of the water is quite flat even though the bucket is rotating, when the water has not caught up with the rotation of the bucket. Gradually the surface of the water forms a depression in the middle, and finally assumes a concave shape (which may be too high for the sides of the bucket, in which case the water will climb over the sides and cause a splash). Newton continues: The rise of the water reveals its endeavor to recede from the axis of motion, and from such an endeavor one can find out and measure the true and absolute circular motion of the water, which here is in the direct opposite of [i.e. inversely proportional to] its relative motion.51

(A8) The material universe as a whole is finite in extent and has a beginning (and an end) in absolute time. The temporal finitude of the created, material universe is implied by Clarke, when he says, in his fourth reply: It was no impossibility for God to make the world sooner or later than he did; nor is it at all impossible for him to destroy it sooner or later than it shall actually be destroyed.52

(A9) Absolute space and absolute time are necessary beings which exist in all possible worlds. They are held to be preconditions of all being, whether actual or possible. At the time he wrote his De Gravitatione, the young Newton gave a very clear articulation of this thesis: Space, in so far as it is a being, is a precondition of being.53 Nothing exists or can exist which is not related in some way to space. God is everywhere, created

51

Principia, new translation 1999, pp. 412-13..

52

C.iv.15.

53

This is my deliberate amendment of Hall & Hall, whose translation of this sentence reads: ‘Space is a disposition of being qua being’. For the noted amendments to the translation of Hall & Hall in the passages from De Gravitatione here cited, I am indebted to discussions with Gavin Betts and Tom Robinson.

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minds are somewhere, and body is in the space which it occupies; and whatever is neither everywhere nor somewhere does not exist. And hence it follows that space is a consequence of54 the first existence of being, because when any being is postulated, space is postulated. And the same may be asserted of duration [i.e. time]. For certainly both [space and time] are preconditions of being or attributes according to which we denominate quantitatively the presence and duration of any existing individual thing. So the quantity of existence of God was eternal in respect of55 duration, and infinite in respect of the space in which he is present; and the quantity of existence of a created thing was as great, in respect of duration, as the duration since the beginning of its existence, and in respect of size of its presence as great as the space belonging to it.56 (A10) In relation to God, however, space and time are not substances but properties: and constitute respectively the divine attributes of immensity (or omnipresence) and eternity (i.e. sempiternity).

God, therefore, does not exist in space and time in the same sense as created things and events. The latter God chooses to create in a certain place and at a certain time. Says Clarke: God does not exist in space and in time, but his existence causes space and time. And when, according to the analogy of vulgar speech, we say that he exists in all space and in all time, the words mean only that he is omnipresent and eternal, that is, that boundless space and time are necessary consequences of his existence; and not that space and time are beings distinct from him, and in which he exists.57

Until recently, it used to be thought that A10 was a theological doctrine added to the Newtonian theory by Clarke himself, and could not be assumed to have been acceptable to Newton;58 but the research done by Koyré and

54

Rather than ‘an effect arising from’, as in Hall & Hall.

55

Rather than ‘in relation to’, as in Hall & Hall.

56

Hall & Hall, pp. 136-7.

57

C.v.45; cf. C.iv.10.

58

Cf. Alexander, p.xxviii: ‘Clarke... puts forward a theory of his own--that space and time are neither substances nor relations, but attributes. This theory also appears in Clarke’s Boyle Lectures [i.e. his Demonstration], but there is no evidence that it was held by anyone else on the Newtonian side.’

25

Cohen, which was published in 1962, established that Newton himself did subscribe to those theological doctrines.59 2.3. Getting at the core of Newtonianism But though they are offered as a Newtonian package deal, not all the above theses are essential to the absolute theory. To get at the core of that theory let us grant the central thesis (A1), and try to determine how much of the rest is logically bound up with it. I want to argue that the real core of Newtonian absolutism is constituted by the conjunction of the first five theses that I have listed above, namely A1-A5, and that the remaining five theses (A6-A10) are all peripheral. To do so, let us go through the remaining nine theses, A2 - A10 inclusive, and ask with regard to each: Is it logically dictated by the central thesis (A1)? I will take each of these theses in turn, going backwards from A10 to A2. Our first question, then, is: How is the theological thesis A10 logically related to the central thesis A1? The answer is that not only is A10 not entailed by A1, but it is in fact inconsistent with A1. For A1 proclaims that space and time are absolute beings, that they are individuals enjoying autonomous existence, whereas A10 proclaims that they are really properties of a certain individual, namely God. But if they are properties of God, then in the relevant sense, they are relative rather than absolute beings. So A10 is inconsistent with A1. But A10 is also inconsistent with A9, which says that space and time are preconditions of existence; so if God exists, then he must be in space and in time in exactly the same way as created beings exist in space and in time. Leibniz was quick to latch on to this inconsistency in the overall Newtonian story, and in his fifth paper he does not mince his words: God’s immensity [according to Clarke] makes him [actually] present in all spaces [places]. But if God is in space, how can it be said that space is in God, or that it is his property? We have heard of the property being in the subject, but never of the subject being in its property. In the same way, God exists in every time [according to Clarke]; how then is time in God, and how can it be a property

59

See A. Koyré and I.B. Cohen (1962), pp. 63-126.

26

of God? Such expressions never make sense (ce sont des alloglossies perpétuelles).60

In his fifth reply, Clarke remained unperturbed, and merely reasserted A10; but four years later, when Des Maizeaux was preparing his own edition of the Correspondence, Clarke seems to have realised that Leibniz had put his finger on a genuine difficulty, and made Des Maizeaux insert the following note in his preface: Monsieur Clarke has asked me to warn his readers that ‘when he speaks of infinite space or immensity and infinite duration or eternity, and gives them, through an inevitable imperfection of language, the name of qualities or properties of a substance which is immense or eternal, he does not claim to take the term quality or property in the same sense as they are taken by those who discuss logic and metaphysics when they apply it to matter; but that by this name he means only that space and duration are modes of existence of the Substance which is really necessary, and substantially omnipresent and eternal. ... When we speak of things which do not fall under the senses it is difficult to speak without using figurative expressions’.61

But this is sheer obfuscation. In section 4 of this chapter I shall suggest a way out in which Newton-Clarke could have avoided this ontological inconsistency which A10 seems to introduce into their view of the relation of space and time to God. Next let us turn to A9, the thesis that space and time are necessary beings that exist in all possible worlds. Is this entailed by A1, the thesis that space and time are absolute beings? I think not. I cannot see that A1 is incompatible with the doctrine that all existence is contingent, a doctrine upheld by such philosophical giants as Hume and Kant. Leibniz himself held the existence of space and time to be contingent; but, I believe that his own relative theory as such leaves it open whether space and time exist contingently or necessarily: if all their occupants exist contingently, then (on the relative theory) space and time exist contingently, and if at least one of their occupants exists necessarily, then space and time exist necessarily. Why, then, should the absolute theory of space and time be incompatible with their contingent existence? A Newtonian absolutist could hold that in every 60

L.v.45.

61

Quoted in Alexander, p. xxix.

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possible world in which there are material objects there are also absolute space and absolute time in which material objects exist; the existence of material things entails the existence of space and time, but that does not render the existence of space and time logically necessary. Nor does their theological superstructure compel the Newtonians to assume that space and time exist necessarily. They could regard God as a contingent but eternal being, and say that in every possible world in which God exists, he is responsible for everything contingent other than him, which exists in that world. As for A8, the thesis that the material universe is finite in extent, and has both a beginning and an end in time, this is certainly not logically dictated by the central thesis A1. I need not argue the point, but the absolute theory is compatible with the opposite view, namely that the material universe is infinite in extent, and has no beginning (or end) in time. What about A7, the view that certain experiments, such as the bucket experiment, establish the existence of absolute motion, and so not all motion is relative? This is in fact a highly controversial thesis, and I beg leave not to discuss it in any detail. All I need is to assert is that the state of the discussion of such experiments in the literature leaves the question open, so that A1 need not entail A762. And what about A6? If in saying that space and time are causally neutral we restrict causality to physical explanation, then I think the causal neutrality of space and time is generally granted on both sides of the controversy as to whether space and time are absolute or relative beings. However, if we extend causality to cover topological as well as physical explanation, then it has been claimed (e.g. by Kant, and recently by Nerlich) that only the absolute theory can accommodate the distinction between incongruent counterparts (e.g. my right hand and my left hand). The strength of this argument is that it is purely topological, and does not depend on any contingent physical theory, such as Newton’s. But is it compelling? For my purpose I do not want to regard these issues as settled, and so I will say that A6 is not part of the core.

62

Two recent discussions of these experiments are to be found in Dainton, pp. 173-180, and in Vailati, pp. 126-132.

28

Next, let us turn to A5, the view that we must distinguish between absolute and relative motion, and that not all motion is relative. Is that logically dictated by A1? I should say, ‘Yes certainly’. Two things are worth noting here. The first is that Leibniz, in his critique of absolutism, may be said to have turned the tables against Clarke and Newton. The latter seem to pride themselves on the claim that their theory makes sense of the supposition that a finite material universe could move in space, whereas the Leibnizian relative theory could not make sense of this supposition. To which Leibniz replied (and this will be amplified in chapter VII) that, on the contrary, the supposition that the entire material universe should be moving in a certain direction should not make sense, and on his theory it does not. The second point regarding A5 that I wish to make is this. Intuitively, we want to say that it is part and parcel of the notion of material thinghood that a material object is movable, or mobile: it may in fact be always at rest, but it should be logically possible for it to occupy a different region of space from the one that it does in fact occupy at a certain time. This intuition is encapsulated in the traditional view that mobility is one of the so-called primary qualities of a material object. It is perhaps a consideration in favour of the absolutist theory that it leaves this intuition intact; whereas the rival Leibnizian theory introduces a reservation. To say of the sum-total of all space occupants (the material universe as a whole) that it is moving makes no sense, for there is nothing left over relatively to which the material universe as a whole could be said to move. However, in a way, the traditional view about mobility is retained by the relative theory, but in this form: a finite space-occupant is always movable, provided the relativist conditions of mobility are met. This would then exclude the supposition that the entire material universe should be moving, on the grounds that the supposition does not meet the conditions of mobility, as laid down by the relative theory, for nothing is left over to act as a fixed frame of reference relatively to which there is to be movement. In the end, the difference between absolutism and relativism on this account does not seem as big as it initially does. And is A4 to be regarded as part of the core? Intuitions may differ, but I should say, ‘Yes, provided it is borne in mind that (along with both parties to the controversy) we are assuming space to have a Euclidean geometry and time to have a linear chronometry’. Given this assumption, can space be absolute and be finite in extent? Can time be absolute and have a

29

beginning, or come to an end? G. E. Moore, for example, felt quite certain that the answer to all these questions was No.63 Next let us focus on A3, that the parts of space and time are necessarily where and when they are. Is this part of the core? I should say it certainly is. Now Clarke claimed that the immobility of the spatial order renders it improper to speak of space as divisible; strictly speaking, absolute space has no parts and must be regarded as ‘essentially indivisible’.64 Thus he says in his fourth reply: Parts, in the corporeal sense of the word, are separable, compounded, ununited, independent on, and movable from, each other; but infinite space, though it may by us be partially apprehended, that is, may in our imagination be conceived as composed of parts, yet those parts (improperly so called) being essentially indis65 cerpible and immovable from each other, and not partable without an express contradiction in terms, ... space consequently is in itself essentially one and abso66 lutely indivisible.

But this is unnecessary and highly misleading. The central point is that regions of absolute space are essentially immovable. To insist, as Clarke here does, that absolute space is indivisible because its parts are immovable is a confused way of pointing up the contrast between the essential immobility of a region of absolute space and (what Clarke took for granted) the essential mobility of a material object. Absolute space, or any region of it, is certainly divisible, in the sense that it contains smaller parts that make it up; and this is a sense that applies to material bodies as well. But divisibility into smaller parts does not entail mobility of such parts. (It should be added that Clarke’s implicit assumption here, that a material object is essentially mobile, intuitively plausible though it is, cannot fully and unqualifiedly be

63

See his Some Main Problems of Philosophy, chapters 9-10.

64

C.iii.3; cf. C.ii.4.

65

There is an unfortunate misprint in Alexander’s edition, page 48, line 10, where ‘indiscernible’ should read ‘indiscerpible’, the latter meaning ‘incapable of being torn to pieces or dismembered’.

66

C.iv.11-12

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accommodated into Leibniz’s spatial relativism. An obvious counterexample, already mentioned, which became a bone of contention in the controversy, is the material universe as a whole, considered as a single material object. On the Leibnizian view of space, this material object would not be mobile, because there would be nothing left over to play the role of a frame of reference relatively to which it could move.) Finally, A2, the thesis that space and time are continuous and infinitely divisible, seems to be part of the core too. Intuitively, it does not seem to make sense to suggest that absolute space and absolute time could have an atomic structure, that there could be minimal regions of space and minimal lapses of time, each with a finite thickness but not further divisible. That there are no minimal regions of space or lapses of time seems to be a conceptual truth. Nor should we take the absolutist theory to be committed to the view that space ultimately consists of extensionless points or that time ultimately consists of durationless instants, views which some regard as incoherent. For how can any number of extensionless points make up a finite region, or any number of durationless instants make up a finite lapse? If, despite these apparent difficulties, such views about the ultimate parts of space and time are held to be coherent, then I should say that they are compatible with both the Newtonian and the Leibnizian theories. The issues involved here are too large, and I shall say no more about them. 2.4. Theological repairs In the last section I argued that A10 introduces an inconsistency into the Newtonian view regarding the relation of space and time to God. However, I do not think that the inconsistency is beyond repair, and would like to suggest a possible way out. On a tidier view, Clarke (or Newton) might have held that space, time, and God are three distinct absolute beings, that God exists in space and in time just like material bodies, and that his necessary existence renders the existence of absolute space and absolute time also necessary. Thus there would be three distinct necessary and absolute beings, and the existence of space and time would not entail the existence of God.

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I think Clarke would not have accepted this reconstruction of his position, because he believed he could prove that there cannot exist more than one necessary being. His argument may be paraphrased as follows: A necessary being must be unique, since there cannot be more than one necessary being. For suppose A and B were distinct necessary beings. Then it would be possible for either of them to exist alone. But if it is possible for A to exist alone, then it is possible for B not to exist at all, contrary to the supposition that B is a necessary being (and vice versa). Hence if A is a necessary being, then no other 67 being can be necessary.

This argument, however, is fallacious: to say of A and B that they are numerically distinct does not imply that either of them could exist alone; what is needed is that the existence of at least one of them does not entail the existence of the other (in an intuitive, hard-to-define sense of 'entail'). This could be explained by reference to necessary propositions. The two propositions (P) that every husband has a wife, and (Q) that two and two make four, are both necessary, and yet neither of them entails the other; and they are in this sense logically independent. It would be fallacious to argue that for P to be independent of Q, it should be possible for P to be true and Q false, but that is impossible since Q is necessarily true. Similarly if Space, Time, and God were three distinct individuals, each of the three propositions S, T, and G, which severally assert their existence, would be a necessary truth, and yet the three individuals would still be numerically distinct, inasmuch as G entails both S and T,68 S entails T, but neither T nor S entails G.

67

Clarke’s own wording is as follows: ‘To suppose two (or more) distinct beings existing of themselves, necessarily and independent from each other, implies this plain contradiction: that each of them being independent from the other, they may either of them be supposed to exist alone, so that it will be no contradiction to imagine the other not to exist, and consequently neither of them will be necessarily existing.’ (Demonstration of the Being and Attributes of God, Proposition VII, 9th ed., p. 47.)

68

This is on the amended Newtonian account of the relation of God to space and time, suggested above, in which we assume that God occupies absolute space and time in exactly the sense in which material objects do, and that space entails time, but perhaps not vice versa.

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I will end this chapter by summarising three main conclusions that have emerged. The first is that the logical core of Newtonian absolutism is constituted by the five initial theses, A1-A5. Secondly, this logical core is compatible with a non-modal contingent variety of the theory whereby the existence of space and time is taken to be contingent, and also with a modal variety, in which case we should add thesis A9 to the five theses that constitute the core. And thirdly, the theological superstructure of the theory is logically flawed, but this flaw could be remedied by admitting three distinct necessary beings, namely space, time, and God, and regarding God as occupying space and time in exactly the same way as ordinary things are in space and in time.

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Chapter III Leibnizian Relativism 3.1. Introduction In this chapter I offer a careful restatement of Leibniz’s relative theory of space, which parallels the account I gave in the last chapter of the Newtonian theory. Here again I shall try to isolate what I call the core of the theory from its periphery; or, in other words, I shall try to distinguish those theses which are logically central to the theory from those which, though held by Leibniz himself, are not logically dictated by the central theses. Having done that, I go on to defend the theory against some recent criticisms. (The discussion of time does not figure in Leibniz’s five letters, but it will be worthwhile to see what his view of time was like, through a study of what he wrote on the subject before he corresponded with Clarke; this will be done in the final section of this chapter.) But before offering my restatement, I want to remind the reader of two relevant points which I have made before. The first concerns my own approach: an important feature of the controversy which I shall hold constant, since it was common to both sides, is the assumption that space necessarily has a three-dimensional Euclidean geometry. The second point concerns Leibniz’s own approach: it is that in his controversy with the Newtonians he met his opponents half-way, adopting what I have called an ad hominem stance. By this I mean that he granted for the sake of argument certain assumptions upheld by his opponents which he himself regarded as false. In particular he granted the reality of spatially related material bodies, though his considered opinion was that material bodies are not irreducibly real but merely ‘well-founded phenomena’ whose basis is the mind-like, non-spatial monads. And he granted the irreducible reality of relational properties (particularly those involving spatial and temporal relations), though his official view was that relational properties cannot stand on their own but are always parasitic upon, or reducible to, non-relational (intrinsic) properties.1 In a 1

Cf. C.D. Broad, ‘Leibniz’s Last Controversy with the Newtonians’, reprinted in his

real sense, then, Leibniz was here impersonating a philosophical position in which he did not ultimately believe. Yet it is this impersonated position that I want to restate and discuss under the label ‘Leibniz’s relative theory of space’. How this impersonated position is to be reconciled with the nonspatial monadism to which he officially subscribed is a question that I shall set aside. 3.2. Summary statement of Leibniz’s views With these points in mind, we may restate Leibniz’s views on space, as they emerge in the Clarke Correspondence, in eight different theses (R1-R8) as follows: (R1) The central thesis of Leibniz’s relative theory of space and time concerns their ontological status: it asserts that they are, as Leibniz put it, ‘relative beings’, in that their existence is parasitic upon the existence of things which we ordinarily regard as their occupants. Thus if there were no material bodies, there would be no space; and if there were no events or processes, there would be no time.2 -- The remarks which follow are for the most part concerned with Leibniz’s relative theory of space; his doctrine of time is not as fully articulated. (R2) The theory holds that spatial relations (e.g. being 3 feet distant from) obtain primarily between simultaneously existing material bodies or physical objects. Spatial relations are held to be primitive and irreducible; but what they primarily relate are coexisting physical objects and not regions of space, for in reality there are no such things. Thus Leibniz declares in his Third Paper: For my part, I have stated more than once that I hold space to be something purely relative, as time is: that I hold it to be an order of coexistences, as time is an order of successions. For space denotes in terms of possibility an order of

Ethics and the History of Philosophy, London, 1952, p.187. 2

In his Fifth Paper Leibniz declares that ‘space, taken apart from things, ... has nothing actual about it’ (L.v.67); cf. L.iii.4, L.iv.41, and L.v.62. For abbreviations used in these notes see pp. ix-x.

36

things which exist at the same time, in so far as they exist together, without con3 sidering their particular ways of existing.

Thus Leibniz’s theory is reductive with regard to the existence of space, as a separate container, but not with regard to spatial relations. (This contrasts with the Newtonian view that spatial relations obtain primarily between fixed regions of absolute space, and only derivatively between any physical objects that occupy them. However, on both the Newtonian and the Leibnizian views, spatial relations are taken to be primitive and irreducible.) (R3) The theory requires a frame of reference, consisting of a set of actual physical objects, relatively to which spatial positions can be assigned to other physical objects, whether these other objects are actual or merely possible. Leibniz elaborated the theory in two stages: by first defining ‘sameness of spatial position’ in terms of spatial relations to a frame of reference, and then defining ‘place’ (i.e. spatial position) in terms of being in the same place at different times. And he regarded space as a collection of places, so defined. This feature of Leibniz’s theory will be elaborated in the next section.4 (R4) Though all motion is relative (in virtue of the relativity of place), there is room for a distinction between what Leibniz calls ‘true motion’ and ‘mere change of relative situation’. For there to be true motion within Leibniz’s theory two necessary conditions have to be satisfied. X moves truly relatively to Y if and only if: (i) X changes its situation relatively to Y; and (ii) the cause of that change of situation lies in X rather than in Y. An example of true motion in this sense would be that of a man (W) walking towards another man who is seated (S). Here both S and W undergo a change of situation relatively to each other; but whereas S undergoes a mere change of situation relatively to W, W is in true motion relatively to S, since it is W who is the cause of this change of situation.5 In a well-known passage6 Leibniz characterises the distinction he has in mind as follows:

3

L.iii.4.

4

See L.v.47, and L.v.104.

5

This example is borrowed from Henry More; see his letter to Descartes dated 5 March 1649 in Descartes, Correspondance, ed. C. Adam and G. Milhaud (Paris: P.U.F., 1963), vol. 8, p. 177; see also More’s letter of 23 July 1649, in the same volume, pp. 246-47. I

37

... I grant that there is a difference between a true absolute motion of a body and a mere relative change of its situation in relation to another body. For when the immediate cause of the change is in the body, that body is truly in motion, and then the situation of other bodies relatively to it will be changed in consequence, though the cause of that change is not in them.

This passage, as we shall see, has generated a good deal of misunderstanding. Leibniz’s use of the term ‘absolute’ in connection with the kind of true motion which he takes his own theory to allow, is certainly unfortunate; and I shall avoid using it in my own exposition. For what he means by true motion is still relative, and must be distinguished from Newtonian absolute motion which presupposes the existence of absolute space and time as autonomous entities. (R5) Matter is continuous (i.e. infinitely divisible), and the material universe is infinite in extent. However, it is more reasonable to assume that the material universe had a beginning, so that time is not infinite with respect to the past. (R6) There is no vacuum, i.e. there are no empty pockets of space, within the material universe; and similarly there are no empty lapses of time.7 (R7) Space and time are contingent beings, since their existence is parasitic upon the existence of created things (physical objects and processes) which are always contingent beings.8 (R8) God is not in space and time. If there were no created beings there would be no space or time, but God would still exist as a necessary being. Thus God’s existence is non-spatial, and he is eternal in a non-temporal or timeless sense.9

owe the example and the references to G. H. R. Parkinson, ‘Science and Metaphysics in the Leibniz-Newton Controversy’, p. 106. 6

At L.v.53.

7

See L.ii.2, L.iv.PS (Alexander, pp. 43-5), L.v.24, and L.v.33.

8

See L.v.63.

9

See L.iv.41, L.v.79, and L.v.106.

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This completes our outline of Leibniz’s views on space. The question next arises: Are we to regard these eight theses as constituting a ‘package deal’, so that to accept Leibniz’s theory one must accept all of them? My answer is No: that we must isolate the core of the theory from the rest. The core of the theory is constituted by the conjunction of R1, R2, and R3, which are logically inseparable. R1 declares that the existence of space is derivative and depends on the existence of physical objects; R3 explains exactly how this dependence is to be understood; and R2 is presupposed by R3. But the remaining theses (R4 - R8) are all logically independent of that core; and in a moment (in section 4) I shall argue the point in each case. 3.3. The relativity of spatial position Before I do so, however, we need to clarify the central theses R1 and R3. How exactly does the existence of space depend on the existence of material bodies? About half-way through the Correspondence, Clarke got the impression that Leibniz was identifying space with matter. And against this alleged identity Clarke argued that, since (as he thought) space exists necessarily and is both eternal and infinite in extent, ‘those who suppose matter and space to be the same’ are committed to the view (which he found absurd) that the material universe too exists necessarily, and is both eternal and infinite in extent.10 To which Leibniz replied: I do not say that matter and space are the same thing; I only say that there is no space where there is no matter, and that space by itself is not an absolute reality. Space and matter differ as time and change (le mouvement). However, these things, though different, are inseparable. But it does not at all follow that matter is eternal and necessary, unless we suppose that space is eternal and necessary -11 an altogether unfounded supposition.

This crucial passage may be read in two different ways, yielding what I would call an extreme and a moderate version of Leibniz’s theory. (i) On the extreme version, Leibniz rules out the possibility of unoccupied places. The word ‘where’ is to be given a literal, locational reading as indicating a

10

C.iv.15.

11

L.v.62-63; emphasis added.

39

spatial position; and what Leibniz is saying here is that there is no spatial position unless it is occupied by some material object. (ii) On the moderate version, Leibniz does not rule out the possibility of unoccupied places; what he does rule out is the possibility of a spatial world containing no material objects at all. The word ‘where’ in the above passage is to be taken nonspatially as a mere logical ‘if’; and what Leibniz is saying here is that if there are no material objects at all then there is no space, but if there are material objects then it is possible to have unoccupied as well as occupied places. The extreme version of Leibniz’s theory is open to some pretty strong objections to be discussed later (in section 5).12 But I believe the moderate version to be truer to the Leibniz of the Clarke Correspondence when we take the totality of his views on space into account. A weighty point in my favour is that the moderate interpretation permits a fuller articulation of Leibniz’s package deal, distinguishing the issue of ontological dependence (R1 and R3) from that of the non-existence of a vacuum (R6); whereas the extreme interpretation conflates these two issues. Yet if I am right in claiming that Leibniz was in fact impersonating a position which, he thought, the Newtonians should find worth arguing against, it would be very wise of him to take the question of ontological dependence first, in the weak sense, before raising the question of the existence or non-existence of a vacuum. For these reasons it seems to me that the moderate version is at least a defensible reading of Leibniz’s position, and in what follows I shall assume that it is the correct interpretation of his theory.13 Our next task is to see how exactly the thesis R3 is to be elaborated under the moderate interpretation. To simplify Leibniz’s detailed account, we may restate it in terms of point-particles rather than material bodies. The

12

Among Leibniz’s precursors as spatial relativists, Aristotle undoubtedly upheld what I have called the extreme version of the theory. Thus he believed that there is a maximal finite distance at any time because the material universe is (and cannot but be) finite in extent, and the maximal distance is dictated by the maximally distant point-particles of the material universe at any time. On this and related issues, see the illuminating article by William Charlton, ‘Aristotle’s Potential Infinites’, in Linsay Judson (ed.), Aristotle’s Physics: A Collection of Essays (Oxford: Clarendon Press, 1991). 13

Here I am indebted to John Bigelow.

40

introduction of point-particles is intended to be a mere simplifying device: it does not commit the theory to the existence of point-particles, which Leibniz would certainly reject. He held that the infinite divisibility of matter does not entail the existence of material points as its ultimate constituents.

Let W, X, Y, Z be four distinct point-particles, not in the same plane, which preserve all their mutual spatial relations unchanged during a certain period of time, T. These would then constitute a frame of reference (F) relatively to which places may be assigned to other point-particles during the period T. Now let A and B be two other point-particles, and t1 and t2 be an earlier and a later moment falling within the period T. And suppose that at t1 A stands in certain spatial relations to each of the four items in the frame F (A will stand in a definite distance-relation to each of the four items in the frame F). Suppose next that, at t2, A no longer stands in those same relations to the frame F, but that B does. Then we can say that, relatively to the frame F, B at t2 is in the same place as A was at t1. Thus to say that B at t2 is in the same place as A was at t1 is to say that B at t2 stands in the same spatial relations to the frame F as A did at t1. Let R be the set of relations to the frame F which obtains in either case. Next suppose that at a still later moment t3 (also falling within the period T) no actual particle stands in the relations R to the frame F. (This stage is not explicitly stated in Leibniz, but I take it to be essential to what I have called the moderate version of his theory. Leibniz did not recognise this third stage because he did not believe in the existence of a vacuum; but, as I shall argue later, this view is not dictated by the core of his relative theory.) Even so, it would be true to say that if a particle C were to stand in the rela-

41

tions R to the frame F, then C at t3 would have been in the same place as A was at t1 and B was at t2. And in general, throughout the period T, there is a place, P, a point-position, which is entirely determined by the set of relations R to the frame F. Thus the place, P, is an abstraction, an ‘ideal thing’: it is simply the possibility of a point-particle standing in the relations R to the actual point-particles which constitute the frame F. Different sets of spatial relations to the frame F will determine different places, each of which is a different possibility. And according to Leibniz, space is the set of all places relatively to our frame of reference. The above reconstruction is largely based on what Leibniz says in his Fifth Paper, particularly the following passage: ....To give a kind of definition, place is that which is said to be the same for A and for B when the relation of coexistence between B and [W, X, Y, Z], etc., entirely agrees with the relation of the coexistence which A previously had with those same bodies, supposing there has been no cause of change in [W, X, Y, Z], etc. It may also be said, without entering into particulars, that place is that which is the same at different moments for different existents when their relations of coexistence with certain other existents, which are supposed to continue fixed from one of those moments to the other, agree entirely. And fixed existents are those in which there has been no cause for any change of the order of their coexistence with others, or (which is the same thing) in which there has been no mo14 tion. Lastly, space is that which results from places taken together.

We are now in a position to grasp the sense in which (according to R1) the existence of physical objects is necessary for the existence of space. This does not rule out a world with unoccupied places; what it does rule out is a spatial world in which no actual physical objects exist. The frame of reference must consist of actual physical objects; and this is enough to bestow reality on a whole space with every place in it. For a real space, according to this theory, is a set of places, a set of locational possibilities relatively to an actual frame of reference; and given an actual frame, all the possibilities of being situated relatively to that frame are also given. This at once guarantees, a priori, both the continuity and infinite extent of relative space.15

14

L.v.47; in the bracketed bits I have substituted my own symbols for Leibniz’s.

15

Cf. W. H. Newton-Smith, pp. 38-42.

42

In the above account and the accompanying diagram I have assumed that the minimum frame of reference needed by the Leibnizian theory to generate a three-dimensional space is one consisting of four distinct pointparticles that are not in the same plane. It is interesting to see what happens when this minimal frame of reference is diminished. (i) The most exiguous frame of reference would consist of a single point-particle, Q. In such a spatial world the space generated would consist of half an infinite straight line bounded by Q as its end-point; for any point-position which is at a certain distance x from Q will be identical with any ‘other’ which is at the same distance. Of course, if we had a three-dimensional space then, given the point Q and a precise value for x, there would be an infinity of points all of which would be at distance x from Q, namely any and every point lying on the surface of a sphere of radius x with Q as its centre. But with the point-particle Q alone as our frame of reference, there is no way of distinguishing one point-position on such a sphere from any ‘other’ point-position; so that the entire surface of any such sphere will collapse into a single point-position. (ii) What if we increase our frame of reference to two distinct pointparticles, Q and R ? If we had a three-dimensional space there would be an infinite number of planes each passing through the points Q and R; but given the point-particles Q and R as our only reference points, there is no way of distinguishing one such plane from any ‘other’, and so they all collapse into a single plane. But further, any point-position on that single plane which is at distances x and y from Q and R respectively, will be identical with its ‘mirror image’ on the ‘other side’ of the straight line determined by Q and R. Hence we should then have a space consisting of half an infinite plane which is bounded by the straight line determined by the point-particles Q and R. (iii) And if we have a richer frame consisting of three noncollinear point-particles, Q, R and S, they will generate half a threedimensional space bounded by the plane that is determined by these three point-particles; for any point-position determined by the distances x, y, and z from Q, R and S respectively, will be identical with its ‘mirror image’ on the ‘other side’ of that plane. (iv) To avoid this halving of three-dimensional space, we need a fourth point-particle which is not on the same plane as Q, R and S. Thus to get a full three-dimensional space, we need a frame of reference consisting of at least four point-particles that are not on the same

43

plane; the existence of four distinct non-coplanar point-particles is in fact both necessary and sufficient for the generation of a full three-dimensional space.16 These remarks about minimal frames of reference are not echoed in the Leibniz-Clarke Correspondence, but in a remarkable paper entitled ‘Metaphysical Foundations of Mathematics’ (and believed to have been written in the same year, 1715),17 Leibniz had this to say: Absolute space is the completely filled locus, or the locus of all loci. From a single point, nothing results. From two points a new structure results, viz., the class of all points whose places in relation to the two given points are uniquely determined, i.e., the straight line which goes through the given points. From three points the plane results, i.e., the locus of all points whose places in relation to the three given points, not lying in the same straight line, are uniquely determined. From four points not lying in the same plane, absolute space results. For every point in relation to the given four not lying in the same plane, has its place 18 uniquely determined.

3.4. Getting at the core of the Leibnizian theory Assuming, then, that the conjunction of R1, R2 and R3 (so interpreted) constitutes the central core of Leibniz’s theory, I now want to argue that each of Leibniz’s remaining theses (R4 - R8) is logically independent of that core. And to show this in detail, I shall take R4 - R8 in reverse order. To begin with R8, the thesis that God is not in space or time, it is clear that this is not dictated by the core of the theory, which is theologically neutral. That space and time are relative beings is independent of whether or not God exists, and is also independent of whether or not he is a temporal or spatial object. Only if we assume, with Leibniz, that everything spatial and temporal is created by God and exists contingently, will it follow that God, as the creator and the only necessary being, exists outside space and time. And the case is similar with R7, the view that space and time are contingent beings. R7 would follow only if we assumed, with Leibniz, that

16

Cf. Andrew Newman, pp. 200-02.

17

Included in Wiener, pp. 201-16 = GM.vii.17-28.

18

Wiener, p.207. Clearly Leibniz is not here using the term ‘absolute space’ in the Newtonian sense.

44

whatever is in space and time exists contingently. But the core of the theory is compatible with the existence of necessary spatial and temporal objects. Thus if God exists and God is a necessary but temporal being, then the existence of time, though relative, will be necessary in virtue of the necessary existence of God as a temporal being. This point is worth dwelling on, since at least one influential writer has considered the type of relative theory espoused by Leibniz as constituting a good reason for placing God outside time. Richard Sorabji writes: ... A very big difference is made by the doctrine, attributed ... to a good many thinkers from Philo to Augustine, that time had a beginning, along with the moving creatures that God created, since time cannot exist in the absence of motion. This immediately makes it impossible to say that God exists in time. For, first, this would now imply that he too, like time, had a beginning and a finite past. And, secondly, it would imply that he depended for existence on his own creatures. For he could not exist without time, nor time without motion and the moving creatures created by him. This is a very strong reason indeed for thinking 19 God timeless.

But all this is a non sequitur. A relative theory of time such as Leibniz’s can certainly be combined with the view that the material universe had a beginning in time (as Leibniz did believe), but without having to place God outside time. Assuming with Leibniz that God is a necessary being and everything else exists contingently, all one has to do is assume that before the creation of the material universe God existed in time from all eternity, so that, in virtue of God’s necessary existence as a time-occupant, the existence of time would also be necessary; but time would be relative none the less. And similarly with R6: the theory does not logically require the denial of a vacuum; for, on the moderate version, the possibility of unoccupied places is not ruled out. Leibniz’s real reasons for denying a vacuum were in fact theological: one reason being that ‘the more matter there is, the more God has occasion to exercise his wisdom and power’.20 R5 too is not dictated by the central core, which is compatible with matter being either continuous or discontinuous, and also with the material

19

Richard Sorabji, Time, Creation and the Continuum, p. 254.

20

L.ii.2.

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universe being either finite or infinite in extent. Leibniz’s reasons for holding that matter is continuous and infinite in extent were theological and speculative. ‘There is no possible reason’, Leibniz declares, ‘that can limit the quantity of matter; and therefore such a limitation cannot take place’.21 And the same argument is used for the infinite divisibility of matter; for ‘what reason can be assigned for limiting nature in the process of subdivision?’22 Likewise, the central core of the theory is compatible with the material universe’s having a finite or infinite history, whether in the direction of the past, or in the direction of the future, or both.23 Leibniz’s reasons for espousing the infinite extent of the material universe but its finite history are again theological and speculative. In support of these views, Leibniz writes: Absolutely speaking, it appears that God is able to make the material universe fi24 nite in extent, but the contrary seems more consistent with his wisdom. ... If it is in the nature of things on the whole to grow uniformly in perfection, then the universe of creatures must have had a beginning. Thus there will be reasons for limiting the duration of things, even though there would be none for limiting 25 their extension.

Finally, R4 does not seem to be dictated by the central core either. Now many scholars have claimed that R4 must indeed be separable from the core of Leibniz's theory because it is in fact inconsistent with that core. The claim is that in admitting what he called true motion Leibniz was, in effect, retracting his official view that all motion is relative, a view which is entailed by R3. That was Clarke’s verdict in his Fifth Reply26 and many mod-

21

L.iv.21

22

L.iv.PS (Alexander, p. 44.)

23

Cf. Broad, op. cit., note 1, pp. 185-86.

24

L.v.30.

25

L.v.74.

26

At C.v.53: ‘Whether this learned author’s being forced here to acknowledge the difference between absolute real motion and relative motion does not necessarily infer that space is really a quite different thing from the situation or order of bodies, I leave to the judgement of those who shall be pleased to compare what this learned writer alleges with what Sir Isaac Newton has said in his Principia, Lib. I, Defin. 8.’ The reader is left in no doubt about Clarke’s own verdict. The text from Newton to which Clarke here

46

ern scholars have sided with him.27 But this is surely wrong. As was explained earlier, what Leibniz means by true motion is a species of relative motion; so it is sheer distortion to equate Leibniz’s true motion with Newtonian absolute motion, which requires the separate existence of absolute space and absolute time. But though R4 is quite consistent with the core of the theory, it is not logically dictated by that core. Leibniz does not seem to have intended the distinction between true motion and mere change of relative situation to have universal application; rather, he regarded the distinction as a matter of convenience which we may sometimes apply.28 3.5. Some recent objections We are now in a position to rebut some current objections to the relative theory, raised by Hugh Lacey and Cliff Hooker in two influential articles which were published in the early nineteen-seventies.29 Lacey says that on the relative theory, the continuity and infinite extent of space become empirical matters open to investigation, and not conceptual matters. 30Hooker agrees, and claims that these would depend on whether the occupants of space are in fact continuous and infinite in extent; he also claims that the theory does not admit the possibility of a vacuum.31 It seems to me that these claims do indeed constitute serious objections to what I have called the extreme version of the relative theory; but they do not apply to the moderate version to which I take Leibniz to be

refers is included in Alexander, pp. 150-60; the translations have now been superseded by the new one published in 1999. 27

A notable and specially unfortunate case is Alexander, who says in his editorial Introduction (p. xxvii), ‘There is ... no doubt’ that Leibniz’s admission of true motion ‘is inconsistent with his general theory of space’. For references to other modern scholars who have sided with Clarke’s verdict, see Parkinson, op. cit., note 5, p. 106, n. 68. 28

These points are ably argued by Parkinson; op. cit., note 5, pp.105 ff.

29

H. M. Lacey, ‘The Scientific Intelligibility of Absolute Space: A Study of Newtonian Argument’, pp. 317-42; C.A. Hooker, ‘The Relational Doctrines of Space and Time’, pp. 97-130.

30

Op. cit., p. 319

31

Op. cit., pp. 109-11.

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committed. Against what I regard as Leibniz’s own theory the objections before us rest on two mistaken assumptions. (i) The first is that according to that theory the reality of space requires that every place in it should be occupied (by something real); and (ii) the second is that the theory is committed to the view that the continuity of space can only be a consequence of the continuity of whatever occupies it. From (i) it would follow that the infinity of space would depend on the infinity of the material universe, and that a vacuum (unoccupied place) would be impossible. But (as we saw in section 3.4) neither of these assumptions is entailed by what I regard as the core of the theory, namely the conjunction of R1-R3. It is true that Leibniz himself believed that matter is both continuous and infinite in extent (R5); and, of course, this could only be the case if space too were continuous and infinite in extent. But while the continuity and infinite extent of matter would, if true, be matters of contingent fact discoverable by empirical investigation, the continuity and infinite extent of space would (as was pointed out in section 3) be matters of necessity, guaranteed a priori. Leibniz also believed that there is no vacuum (thesis R6), a belief which he supported by theological arguments; but he did not wish to deny the possibility of a vacuum, i.e. the possibility of an unoccupied place. The nonexistence of a vacuum is, for him, a contingent matter; but the possibility of a spatial world with unoccupied places is a matter of necessity, which is again guaranteed a priori. Hooker also asks: What are the relata of the relational theory of space? Extended physical objects? Physical fields? He argues that neither would do the job of securing the continuity and infinite extent that one wants geometrical space to have.32 But behind this objection is the same false assumption that we encountered before; namely, that according to the relative theory, the continuity and infinite extent of space can only be secured as consequences of the continuity and infinite extent of what is in it. Nor is it clear why the relata (or actual occupants) should all belong to one and the same category of spatial objects: why, in other words, they cannot be a mixed bag of physical objects, fields, shadows, etc., -- anything that requires a spatial position.

32

See sections 3-4 of his article.

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3.6. Armstrong on absolute and relative motion So far I have been dealing with objections generated by a failure to distinguish the moderate version of the theory from the extreme version. I now turn to two further objections which seem to me to rest on a mistaken reading of the theory on either version. In this section I will deal with one of these objections which was raised by David Armstrong in an article published over forty years ago. The article is worth discussing because it contains a subtle misunderstanding the dispelling of which will, I believe, cast a good deal of light on Leibniz’s actual theory. Armstrong distinguishes two components of the relative theory of space: (i) the first is its denial of the independent existence of absolute space as a container over and above what it contains; (ii) the second is its insistence that only the simultaneous can be spatially related, and its consequent denial that different objects (or different phases of the same object) existing at different times can be spatially related. He believes that the second component is logically separable from the first; so that while the second component entails the first, the converse does not hold, i.e. the first does not entail the second. One can, on his view, consistently deny the existence of space as a separate container without also denying that temporally separated objects can be spatially related; one can, in other words, reject the existence of absolute space as a container without insisting that only the simultaneous can be spatially related. But if so, then one can have absolute motion without absolute space; for if we allow that different phases of the same object can be spatially related, we do not need a frame of reference to specify a change of spatial position. We can then say (i) that an object has moved if its present phase is at a distance from its earlier phase (if it is no longer where it was); and (ii) that the object has been at rest if its present phase is at no distance from its earlier phase (if it is at no distance from where it was); and in either case the motion and the rest would be absolute and not relative. In what way does this constitute an objection to the Leibnizian theory that I am defending? In section 2, I isolated as the core of that theory the conjunction of the theses R1 - R3, which I regard as logically inseparable. In

33

D. M. Armstrong, ‘Absolute and Relative Motion’, especially pp. 215-18.

49

particular I regard R1 as entailing R3: R1 declares that the existence of space is derivative and dependent on the existence of physical objects; R3, which insists on the relativity of place, explains exactly how this dependence is to be understood. Now R3 requires that all motion is relative, in virtue of the relativity of place. But if Armstrong is right, then R1 need not have this consequence, and R3 would be separable from R1. Thus in terms of our exposition of the Leibnizian theory, Armstrong’s objection boils down to this: on my view R3 is not logically separable from R1, on his view it is. How does Armstrong defend his position? Basically he conducts his discussion in terms of two statements involving two physical objects, A and B, which for the sake of simplicity we will assume to be point-particles (or temporally distinct phases of point-particles).34 The two statements may then be expressed as follows: (1) A is now (at t1) three feet from B, and (2) A is now (at t1) three feet from where it was (at t2). Armstrong finds it strange that the relative theory should treat these two statements so very differently. For while the first statement is regarded as ‘complete’, the second is treated as ‘elliptical’, standing in need of completion by specifying the relevant frame of reference. But could we not regard the second statement as equally complete and non-elliptical, specifying a direct relation between temporally distinct phases of A? In both cases, ‘it would simply be a matter of objects [or phases of objects] having a certain relation to each other, without any question of absolute space’.35 To clear up the issues, let us first consider the above two statements from the standpoint of the absolute theory of space. Statement (1) would then express a derivative distance-relation based on a primary distance-relation of being permanently three feet apart obtaining between two point-regions of absolute space that are now occupied by A and B. It should be read as saying that there are two point-regions of absolute space, r1 and r2, which are per-

34

The reader is reminded that this simplifying device is intended to be existentially noncommittal.

35

Ibid., p.218; cf. p.215.

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manently three feet apart; and that these point-regions are now occupied by A and B. And statement (2) would express an equally derivative relation: it would be regarded as saying that there are two point-regions of absolute space, r1 and r2, which are permanently three feet apart, and that A now occupies r1 but occupied r2 at the earlier time. From the standpoint of the relative theory, statement (1) does indeed express a primary, non-derivative distance-relation between the pointparticles A and B. But not so with statement (2): it purportedly relates A to a place which is other than its own at the moment (t1) and which may not be occupied at that moment by anything else. My suggested rendering would be either of the following: (2*) If C were a point-particle distinct from A, and C were now (at t1) to occupy the same place as A did (at t2), then A would now (at t1) be three feet away from C.

Or (2**) If C were a point-particle distinct from A, and C had then (at t2) occupied the same place as A does now (at t1), then C would then (at t2) have been three feet away from A.

Either way we satisfy the requirement that only the simultaneous can be spatially related. But the two readings make it plain that statement (2) involves being at the same place at different times, a notion which on the relative theory can only make sense relatively to a frame of reference held to be fixed for a period of time that bestrides the earlier and later times, t1 and t2. So there is nothing surprising about treating statement (2) differently from statement (1), since statement (1) does not involve sameness of place at different times. My conjecture is that Armstrong misunderstands Leibniz’s theory because he approaches it from a four-dimensional framework, which is quite natural with many contemporary philosophers but quite foreign to the Leibniz-Clarke debate. R2, according to Leibniz, is an indivisible thesis which requires that spatial relations obtain primarily not just between material bodies, but between simultaneously existing ones. And what Armstrong’s argument has implicitly shown is that the four-dimensional approach permits the division of Leibniz’s thesis R2 into two separate subthe-

51

ses, namely (a) that the primary relata of spatial relations are material bodies, and (b) that only the simultaneous can be spatially related.36 3.7. On an alleged impurity I turn finally to an objection, raised notably by Graham Nerlich,37 to the effect that Leibniz’s theory is ‘impure’. The objection really concerns R2, the thesis that spatial relations are primitive and irreducible, but obtain primarily between material bodies and only derivatively between regions of space (for in reality there are no such things). Nerlich, and others, see in this admission of undefined spatial relations a weakness, an ‘impurity’, claiming that if the theory cannot dispense with spatial relations altogether then it has failed to deliver the goods and execute its reductionist programme. The same objection is implicitly made by Hooker who, as was pointed out earlier (in section 5), wrongly assumed that Leibniz’s theory made the continuity and infinite extent of space dependent on the continuity and infinite extent of matter. Proceeding on that assumption, Hooker argues that even if matter were indeed continuous and infinite in extent, it would not do the job; for a material object consists of parts that are spatially related, and spatial relations should not figure in the final analysis. Here he seems to be at one with Nerlich in assuming that the theory aims at analysing away spatial relations into something entirely non-spatial.38 Nerlich himself presses this objection by latching on to Leibniz’s famous ‘genealogical tree’ analogy, in which he likened what he took to be the correct account of place (or spatial position) to a genealogical place. Given a genealogical tree, the genealogical place of a person who figures in it is determined by certain blood relations (of conjugal and ancestral links) to certain ancestors taken as ‘fixed’. And allowing the possibility of metempsychosis, i.e. transmigration of souls, Leibniz suggests that we can conceive of one member of the family coming to occupy a different genealogical place in another life, and thus be in the same genealogical place as another member was: for example, ‘he who had been father or grandfather might become son or grandson’, and so

36

Here again I am indebted to John Bigelow.

37

See The Shape of Space, first edition, especially pp. 5-9 and 24-28.

38

Hooker, op. cit., note 29, pp. 3-4.

52

on.39 Nerlich finds this analogy ‘pretty thoroughly obscure’; but goes on to argue that ‘even if we were quite clear about how to understand the suggestion Leibniz made, we would be no better off’. For there is no way to make sense of a thing’s being extended in genealogical space, of filling up a region in the family system’. Moreover, with family space ‘we cannot construct a continuous space’, and ‘we have no foothold for many dimensions’. By now it should be pretty obvious that Nerlich fails to see the point of Leibniz’s analogy, and wrongly reads into it far more than it is intended to convey. I take the main point of that analogy to be the following. It is no less a mistake to think of spatial position (or place) as an independently existing entity than40 it is to think of a genealogical place as an independent entity. In both cases what we have is an ‘ideal thing’: the spatial position (or place) of a body is entirely analysable in terms of the spatial relations of that body to other coexistent bodies, just as the genealogical place of a certain individual is entirely analysable in terms of his blood relations to other individuals in his family tree. Leibniz is not here suggesting that spatial relations are themselves reducible to something entirely non-spatial, such as the blood relation of father and son. Nor is he suggesting that (in Nerlich’s words) ‘space is like a genealogical tree’: that, in other words, what is really behind what we call space is a non-spatial reality of which a genealogical tree is a more accurate model.41 And he drew the analogy, not between space as a whole and a genealogical tree, but between place as a location (or spatial position) and a genealogical place. Nor is it difficult to see why Leibniz is led to modify his analogy by introducing the fiction of metempsychosis. It is to secure an analogue of a change of place, of how a body A may come to occupy the same place as B did at an earlier time. The unmodified genealogical tree analogy has no room for that, but when we add the fiction of metempsychosis we can do better. Take Abraham, Isaac, and Jacob as constituting a genealogical line of 39

L.v.47, pp. 70-71.

40

Op. cit., pp.26, 28.

41

Of course at a deeper level Leibniz did want to do away with both space-occupants and their spatial relations; but that level does not belong to the relative theory which he put forward in the Clarke Correspondence; and it is with that theory that we are here concerned; cf. section 3.1 above.

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grandfather, father, and son (respectively), with Abraham as a ‘fixed’ ancestor. Then on the assumption of metempsychosis we would have to say that, although genealogically their bodies are unalterably related by ancestral links, what their names really refer to are their souls and not their bodies. Now suppose that upon the death of Isaac his soul comes to inhabit Jacob's body, and that the latter’s soul migrates elsewhere outside this genealogical line. Then we can say that Isaac has now come to occupy the same genealogical place (relatively to Abraham) as Jacob did before the migration of their souls, and that from being a son he has now turned into a grandson. I fail to see anything objectionable in this modified analogy, unless we totally misrepresent its purpose.42 3.8. Concluding remarks I believe I have shown that Leibniz’s relative theory of space, as reconstructed, is subtler and more interesting than it is commonly taken to be: it is not open to some of the allegedly serious objections that are currently held against it. Moreover, it would appear that none of the properties of space on which the Newtonians insisted need be surrendered by the Leibnizian relativist. Relative space, as much as Newtonian space, admits of unoccupied places, and is both infinite and continuous. And as the Newtonians would have it with regard to absolute space, the parts of relative space too are inseparable (or, as Clarke put it, ‘indiscerpible’); for relatively to a frame of reference, a spatial position is necessarily where it is. The identity of that position is constituted by the spatial relations which an actual or possible object has to the frame of reference; so that a change in these relations would mean a change of place. Hence, given two adjoining parts (or ‘re-

42

Contrast Nerlich: ‘It is hard to see how this really helps the problem of explaining motion by way of systems of relations’ (op. cit., p. 27). The Nerlich heresy seems to be spreading. Here is how a recent article on our subject begins: ‘Relational theories of space can be divided into two sorts, pure relational theories and impure relational theories. Pure relational theories reduce spatial things ... to non-spatial things; it is usually difficult to specify exactly what those non-spatial things are. Impure relational theories reduce some spatial things, such as spatial points, to other spatial things, such as the shapes of material objects and the spatial relations between them.’ (Andrew Newman, p. 200.) More recently I noticed that the same view had been expressed earlier in J.J.C. Smart, Between Science and Philosophy, pp. 208-9; so perhaps I should call this error the Smart heresy.

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gions’) of relative space, A and B, it is self-contradictory to suggest that they might be torn apart (or spatially separated), for that would destroy the identity of at least one of them.43 3.9. Appendix -- Leibniz on time44 As I mentioned in section 3.1, the discussion of time does not figure in Leibniz’s five letters to Clarke, but it is worthwhile to delve into his earlier writings to see what he thought a relative theory of time looks like. Leibniz did not elaborate his view of time as fully as he did with space. Some of his remarks make him a precursor of what is nowadays called the causal theory of time, which in my view is an extreme form of the relative theory of time. The main locus is a paper believed to be written some time after 1714. There Leibniz tells us: If a plurality of states of things is assumed to exist which involve no opposition to each other, they are said to exist simultaneously. Thus we deny that what occurred last year and this year are simultaneous, for they involve incompatible states of the same thing. If one of two states which are not simultaneous involves a reason for the other, the former is held to be prior, the latter posterior. My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore whatever exists is either simultaneous with other existences or prior or posterior. Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If magnitude of time is diminished uniformly and 45 continuously, time disappears into a moment, whose magnitude is zero.

43

Cf. L.v.51.

44

For this appendix I am indebted Vailati, pp. 120-21.

45

‘The Metaphysical Foundations of Mathematics’, in Loemker p. 666 = GM. viii, 17. This paper is believed to be written some time after 1714

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Leibniz here maintains that a procedure similar to that used in the case of space can be used in the case of time. All states of things, or events can be ordered on the basis of the temporal relations of priority, posteriority, and simultaneity. We can start with an actual event E as our point of reference, and treat all actual or possible events as simultaneous, if they are at the same temporal distance from E and in the same direction. Temporal positions are to be constructed out of the relation of simultaneity among events or states of things, and all temporal events, taken together, constitute time. Thus ‘time is the order of those things which are not simultaneous’. It may be thought that Leibniz’s attempt to construct time out of the temporal relations of priority, simultaneity and posteriority among events would make him an adherent of what McTaggart called the B-series (the static, untensed view of time); and one would expect him to believe that tensed statements (McTaggart’s A-series, or the dynamic view of time ) are reducible to untensed statements (the B-series). But such a construal of Leibniz’s position is doubtful. For Leibniz at times seems to adopt the tensed view that only the present instant, ‘the now’, is real, which would make him, in present-day parlance, a presentist. Moreover, a relative view of time as such does not entail an untensed view of time, because the relations of priority and posteriority may themselves be analysable in terms of tense determinations, i.e. in terms of past, present and future, as McTaggart himself famously suggests; ‘Event E is prior to event F’ is analysable into: ‘Either F is future when E is present or E is past when F is present’. As Peter Geach has put this weighty point, ‘grassroots temporal discourse just is framed in terms of past, present, and future tenses; and all temporal discourse appears to depend on this, logically and epistemologically.’46

46

Geach, 1979, p. 96.

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Chapter IV On Properties 4.1. Space, time, and individuals In his attack on Newtonian absolutism, Leibniz’s main weapon is going to be PII, which says that no two individuals can have all their properties in common. And before discussing and disambiguating PII we will devote this chapter to a preliminary classification of properties. At this point an important distinction is called for. We will assume, for the sake of simplicity, that the only things that count as individuals are: (a) spatio-temporal ‘occupants’ -- whether or not we adopt absolutism about space and time -- such as material bodies and events; (b) absolute space and absolute time and their parts; and (c) composites or, more accurately, mereological sums of any number of these two types of things. I borrow the term from David Lewis, who explains it as follows. ‘The mereological sum ... of several things is the least inclusive thing that includes all of them as parts. It is composed of them and of nothing more; any part of it overlaps one or more of them; it is a proper part of anything else that has all of them as parts.’1 Thus a billiard ball would count as an individual, and so would the region of absolute space that it now occupies; but there is besides a composite individual which is the mereological sum of that ball and that region, and this third individual exists so long as these two exist, and not just for the period when the ball actually occupies that region. The term ‘world’ (or ‘universe’) will be used simpliciter to denote the mereological sum of all the individuals that exist in it. By contrast, the term ‘material universe’ will be used to denote the mereological sum of all spatio-temporal occupants that exist in a particular world. A world (or universe) on this scheme is a maximal all-embracing individual, whereas a material universe is a maximal spatio-temporal individual. So that, on the Newtonian theory just sketched, no 1

See David Lewis, 1986, p. 69, n. 51.

material universe can constitute a whole world, since the maximal sum of spatio-temporal occupants would not include any part of absolute space or absolute time (not even the parts which they actually occupy). 4.2. Two classifications of properties The term ‘property’ will be used in the very large sense of anything that is true or false of an individual (or substance), however obvious or trivial. In this sense, beside having the important property of being a person, I have the obvious property of being not taller than myself, as well as the very trivial property of being identical with myself. Two classifications of properties are important for our purpose, and will be dealt with very briefly. The first is the traditional distinction between intrinsic (non-relational) properties, which individuals have in virtue of the way they themselves are, and relational properties, which individuals have in virtue of their relations or lack of relations to other individuals.2 Thus being green and being spherical are both intrinsic properties; whereas being a husband, being a teacher, not being a dentist, and being a pupil of Plato are all relational properties. Relational properties may be either positive or negative. A positive relational property is one which is true or false of individuals in virtue of their having relations to other individuals; whereas a negative relational property is one which is true or false of individuals in virtue of their lack of relations to other individuals. A more precise characterisation of a positive relational property is the following: (Definition 1) P is a positive relational property if and only if, for any individual, x, x’s having P consists in x’s having a certain relation to at least one indi3 vidual, y.

It is with positive relational properties that we are mainly concerned; and from now on I shall omit the qualification ‘positive’, as the context will 2

Cf. David Lewis, op. cit., p.61.

3

Note that, to make room for the pure-impure distinction which will be introduced in a moment, the right-hand side of this definition displays a deliberate scope ambiguity to cover the two different cases of a pure relational property and an impure one. In the impure case, the requirement is that there is a y such that x’s having P consists in having a certain relation to y; in the pure case, it is that x’s having P consists in there being a y such that x has a certain relation to y.

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make it clear whether the term ‘relational properties’ is meant to cover only the positive ones or the positive as well as the negative ones. The relations that are involved in various relational properties may be dyadic, triadic, tetradic, and so on; but to simplify matters, I shall confine my account to relational properties which involve only dyadic relations. And for the sake of convenience, I shall henceforth use the term ‘relation’ to mean a dyadic relation, and the term ‘relational property’ to mean a property involving a dyadic relation. The second distinction is that between what I shall call pure and impure properties. (I borrow these labels from Michael J. Loux, though he draws the distinction quite differently from the way I want to draw it. For him the pure-impure distinction coincides with the intrinsic-relational distinction, so that all relational properties are impure; whereas an important feature of our scheme is that some relational properties are pure.)4 An impure property is one that consists in the having of a relation to one particular individual. Or more precisely: (Definition 2) A property, P, is impure if and only if there is at least one individual, y, such that, for any individual, x, x’s having P consists in x’s having a certain relation to y.

Thus being a pupil of Plato is an impure property; for it consists in the having of a relation (being a pupil of) to one particular individual, namely Plato. Similarly being a son of Queen Elizabeth II, being married to Prince Philip, and being a resident of London, are all impure relational properties, involving a relation to a particular individual. Of course, an impure property may involve a relation to two or more particular individuals, but in that case the relation would have to be at least triadic. For example, the impure property (which a town may have) of being at an equal distance from Sydney and Melbourne consists in the having of a certain relation (namely, the triadic relation of being at an equal distance from x and y) to two particular individuals (Sydney and Melbourne). Such examples, however, are not covered by our restricted definition of impurity (Def. 2), which is tied down to dyadic relations. An unrestricted definition, which I owe to Lloyd Humberstone, would be the following: 4

See his Substance and Attribute, (1978), pp. 32-33.

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(Definition 2*) A property, P, is impure if and only if, for some number, n, there is some n-ary relation, R, and there are n-1 individuals, y1 , ..., y(n-1) such that, for any individual, x, x’s having P consists in x’s having R to y1,..., y(n-1)

A property is pure if and only if it is not impure. And by our second definition, all impure properties are relational properties. Whence it follows that all intrinsic (i.e. non-relational) properties are pure. Thus being green and being spherical are both intrinsic and therefore pure properties. However, relational properties may be either pure or impure. A pure relational property is one that satisfies our first definition but not the second: it is a property which consists in the having of a certain relation, not to one particular individual, but to some one or other of a group of individuals. For example, being a husband is a pure, though relational, property; unlike being the husband of Elizabeth II, which is an impure relational property. For the property which all husbands share is that of being married, not to a particular woman, but to some woman or other, i.e. some one individual or other among the class of women. Similarly, being a house-owner is a pure, though relational, property; for having that property does not depend on which particular house one owns. And again, the notion of being identical with oneself expresses two distinct relational properties, the one pure, the other impure. I have the impure property of being identical with the particular individual that I am (a property which I alone have); but I also have the property of being self-identical (a pure property which every individual has). The expression ‘consists in’, which I have used in characterising relational properties, is intended to mean something stronger than a necessary coextensiveness; and this is of crucial importance. To appreciate the difference, let us restate our definition of impurity (Def. 2) in terms of a necessary coextensiveness. The proposed definition would then read: (Definition 3) A property P is impure if and only if, there is at least one individual, y, such that, necessarily, for any individual, x, x has P if and only if x has 5 a certain relation to y.

5

This is essentially the definition of impurity given by Michael J. Loux, op. cit.,

p. 133.

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Then it is easy to see that this would not be tight enough. For suppose there is a necessary being, God (i.e. a being that exists in all possible worlds), who is also essentially omniscient (i.e. who, of necessity, knows every true proposition). And take any pure property, such as the intrinsic property of being green.6 Then, necessarily, x is green if and only if x is known by God to be green; so that (by Def. 3) being green is an impure relational property. Thus, if we abide by Definition 3, the necessary existence of such an essentially omniscient being would render all properties impure. But this result would not accrue if we abide by Definition 2; for we would not say that x’s being green consists in God’s knowing that x is green.7 To end this section with an overview. On the scheme which we have just sketched, properties fall into one or other of three basic types: (i) intrinsic and pure -- for I have so far presumed that all intrinsic properties are pure; (ii) relational but pure; and (iii) relational and impure. 4.3. Defining the intrinsic I say ‘so far’ for there are difficulties in the offing in the scheme that I have just sketched and propose to work with. So far I have equated the intrinsic with the non-relational. But this seems to clash with another approach to intrinsicality, popularised by the late David Lewis, whereby intrinsicality is equated with duplicity; or more precisely those properties are intrinsic which an individual shares with its exact duplicate or duplicates, on which more will be said in a moment. I propose to distinguish five different attempts at defining the intrinsic, which I will first enumerate and then consider in turn. The five attempts are: (i) to define intrinsic properties, as I have done, as properties that are not relational; (ii) to define intrinsic properties as pure but not relational,

6

Readers who, because of the problem of secondary qualities, would not agree that being green is in fact an intrinsic property may substitute in our example any other property which they consider undoubtedly non-relational, e.g. being made of pure iron. 7

Here I am indebted to Tom Karmo. Clearly, unlike the relation of a necessary coextensiveness, the relation of consisting-in is not symmetrical. But what is left over when we subtract the necessary coextensiveness from the relation of consisting-in? This question, which he calls the ‘Karmo problem’, is illuminatingly addressed by Humberstone, pp. 216-17 and 222-23.

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as again I have done; (iii) to define the intrinsic, as David Lewis has done, as duplicate-related; (iv) to define the intrinsic properties as the internal properties (or interior properties, as Humberstone prefers to call them); and finally (v) to equate the intrinsic with the purely qualitative. It would be extraordinary if all these approaches led to the same verdicts without exception. Of course they do not, but what is remarkable is the extent to which their verdicts coincide. But before considering each of these approaches individually, let us first consult and attempt to grasp the informal notion of intrinsicality. What exactly is it for a property to be intrinsic? Here is David Lewis’s answer: Some properties of things are entirely intrinsic, or internal, to the things that have them: shape, charge, internal structure. ... A sentence or statement or proposition that ascribes intrinsic properties to something is entirely about that thing; whereas an ascription of extrinsic properties to something is not entirely about that thing. ...The intrinsic properties of something depends only on that thing; whereas the extrinsic properties of something may depend, wholly or partly, on something else. If something has an intrinsic property, then so does any perfect duplicate of that thing; whereas duplicates situated in different surroundings will 8 differ in their extrinsic properties.

Our informal notion of intrinsicality is reinforced by our special epistemological stance towards it. As Dunn has pointed out, ‘epistemologically, an intrinsic property would be a property that one could determine by inspection of the object itself -- in particular, for a physical object, one would not have to look outside its region of space-time.’9 And now let us consider each of these approaches in turn. (i) The non-relational. On this approach we define the intrinsic in accordance with Definition 1, above, but equate it with the non-relational. What then becomes of the property that I alone have, of having Edward’s heart? Is not this a relation to a specific individual, Edward’s heart? Does not this feature make it an impure property by Definition 1? And should we not call it an impure intrinsic? My answer is that we should resist all these pressures, and insist that the property in question is impure but relational. 8

Lewis 1983, reprinted in Lewis 1999, pp. 111-12.

9

Quoted from Dunn in Humberstone, p. 240.

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(ii) The pure but not relational. In this approach we abide by Definition 2 of impurity, and equate the pure with that which is not impure. (iii)The duplicate-related. In this approach, which is associated with David Lewis, we abide by the following definition. (Definition 4) A property P is intrinsic if and only if no pair of distinct individuals that are exact duplicates can differ with respect to it.

Problem: If a and b are perfect duplicates, then each is a perfect duplicate of the other; so it seems that on this definition a relational property turns out to be intrinsic. A brilliant answer which is reported to have been given by David Lewis in conversation is that, ‘although the property [of a] is identified with reference to b, in itself it amounts to only an infinite conjunction of intrinsic properties of a, and hence is itself intrinsic.’10 (iv) The internal (or interior). In this approach we equate the intrinsic with the internal, and define the latter as follows:. (Definition 5) A property P is internal if and only if it is possible for some individual x to have P, although no contingent object wholly distinct from x actually exists.

What Lewis means by ‘wholly distinct from x’ is ‘having no part in common with x’, i.e. what is often expressed by the term ‘disjoint’. The reason why we find no contingent object wholly distinct from x’ is that if there are necessarily existing things (such God, space, time, numbers, etc.), then without the ‘contingent’ stipulation no property could be internal. Note that the intrinsic and the internal are not always the same. For example, ‘the property of not being six metres away from any rhododendron is certainly an extrinsic property, and not an intrinsic one; but it is a property that an unaccompanied (or lonely) object could possess, and indeed would have to possess, and so it counts as ‘internal’11 (v) The purely qualitative. This approach seems to me to equate the intrinsic with what I would call the class of the non-relational together with 10

Quoted from Dunn 1990, in Humberstone, pp. 249-50.

11

Humberstone, p. 230.

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the class pure relational properties, as laid down in Definitions 1 and 2 above. Thus being a husband as well as being made of iron are both purely qualitative properties on this scheme. This concludes my hasty survey of the present state of discussion on intrinsic properties. In the end, I am not persuaded that the three-fold classification summarised at the end of the last section should be abandoned or in any way modified; and in what follows I shall stick to it.

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Chapter V The Identity of Indiscernibles

5.1. Three grades of indiscernibility I shall take the principle of the Identity of Indiscernibles (PII) to be a necessary truth ranging over individuals (or substances). Now to say of two individuals, x and y, that they are indiscernible is to say that they agree in all their properties. But which properties count towards indiscernibility (and discernibility)? Using the above classification of properties, we may distinguish three grades of indiscernibility, giving rise to three different versions of PII. The weak version (PII.1) is quite unrestricted: it says that no two distinct individuals can share absolutely all their properties (whether pure or impure, relational or intrinsic). The middle version (PII.2) excludes impure (relational) properties, and says that no two individuals can share all their pure properties (whether intrinsic or relational). Finally, the strong version (PII.3) excludes all relational properties (pure and impure), and insists that no two individuals can share all their intrinsic properties. (I borrow the idea of distinguishing these three versions of PII from Robert M. Adams, but without using his awkward terminology. For historical reasons he works with a distinction between a ‘thisness’ and a ‘suchness’, and by the latter term he means what I call a pure property, whether intrinsic or relational.1)

1

See his ‘Primitive Thisness and Primitive Identity’, (January 1979), p.11.

Static Universes

The importance of clearly distinguishing these different versions of PII is brought out by the static universes depicted in Figure 1 above. We are to imagine four possible universes (U1, U2, U3, and U4), each consisting of a number of spheres made (let us say) of pure iron. The spheres in each universe are exactly alike in respect of all their intrinsic properties: they have the same diameter, the same kind of smooth surface, and so on. What distinguishes each of these static universes from the rest are certain impure relational properties, as specified in the following descriptions: Table I (‘Impure’ descriptions) U1: Consists of spheres A and B such that: A is 3 diameters away from B. U2: Consists of spheres C, D, and E such that: C is 3 diameters away from D; D is 3 diameters away from E; and E is 3 diameters away from C. U3: Consists of spheres F, G, and H such that: G and H are 2 diameters apart; and each is 3 diameters away from F. U4: Consists of spheres K, L, and M such that: K is 3 diameters away from L and 4 diameters away from M; and L and M are 5 diameters apart. Note that there is an upper limit on the number of spheres in each of the universes described. The limit is implicitly conveyed by the number of different names (A, B, C, ...) which figure in each description. Thus the U1 de-

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scription in Table I should be read as saying that ‘there are at most two spheres, A and B, such that ...’; and the U3 description as saying that ‘there are at most three spheres, F, G, and H, such that ...’. (Similar remarks apply to the descriptions to be given in Table II below.) Our problem is to find out how many spheres there are in each of these universes. How many spheres does each of the above ‘impure’ descriptions dictate? We will approach this question in two stages: first we will take each of the four descriptions in isolation, independently of any version of PII; then we will consider each of them in conjunction with each of the three versions of PII that we have distinguished. And at the start it is clear that, taken in isolation, the above description of U1 logically dictates that it consists of 2 spheres, while the remaining descriptions (of U2, U3, and U4) logically require that each of them contains 3 spheres. Next, let us deploy our three versions of PII on these four universes. Our task then is to determine the right number of spheres dictated by the combination of each of these descriptions with one or other of the three versions of PII. The twelve verdicts are tabulated at the bottom of Figure 1 in four columns; each column gives the required three verdicts for the universe under whose graphic representation it appears. Thus the third column says that the correct number of spheres that U3 has is: 3, according to PII.1; 2, according to PII.2; and 1, according to PII.3. Of course the last two verdicts, namely 2 spheres and 1 sphere, conflict with our graphic representation of U3, which shows it as having 3 spheres. But it must be emphasised that, strictly speaking, the verdicts for U3 have nothing to do with that representation of U3: the three verdicts are meant to be the logical results of applying the three different versions of PII to the ‘impure’ description of U3, as we have it in Table I. Whereas the graphic representation of U3 is meant to illustrate the ‘impure’ description of U3 when that description is taken in isolation, independently of any version of PII. Our present task, then, is to validate these twelve verdicts, taking each version of PII in turn. Starting with PII.1 we may argue for its four verdicts as follows. In our first universe, A has the property of being 3 diameters away from B, which B does not have. Hence A is not identical with B, and U1 consists of 2 spheres. In the second universe, C has the property of being 3 diameters away from D, which D does not have; hence C and D are distinct. But also, C and D both have the property of being 3 diameters away

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from E, which E does not have; hence both are distinct from E. Therefore U2 consists of 3 spheres. In the third universe, G has the property of being 2 diameters away from H, which H does not have; hence G and H are distinct. Moreover, both G and H have the property of being 3 diameters away from F, which F does not have; hence both are distinct from F. Therefore U3 consists of 3 spheres. (I leave out as obvious the supporting argument for the verdict that U4 has 3 spheres.) It will be noticed that invoking PII.1 has left our four universes undisturbed, and produced exactly the same results that we had earlier when we considered their ‘impure’ descriptions in isolation. Nor is this surprising. For PII.1 does not exclude impure relational properties, and in terms of these it can capture the distinctive features of our four universes to the full. PII.1 is in fact trivially true, and leaves everything as it is. On the other hand, if we abide by the stringent principle, PII.3, we would have to say that each of these universes contains just 1 sphere; for we have assumed that in each universe the spheres agree in all their intrinsic properties, and PII.3 is restricted to these. Relational properties are all excluded, but (according to Table I) the only things that make one sphere discernible from another are impure relational properties. Thus the overall effect of deploying PII.3 is to shrink each of our four universes into a single sphere, and declare their initial descriptions logically impossible. 5.2. The vagaries of PII.2 There remains the middle principle, PII.2. At first sight it might seem that PII.2 should yield the same result as PII.3, namely that each of our four universes contains only 1 sphere. For, it might be argued, according to Table I the spheres in each universe are discernible only in respect of impure relational properties; and these are excluded by PII.2 as well as PII.3. But that would involve a non sequitur: from the fact that all the properties that appear in Table I are relational and impure it does not follow that we have no pure relational properties to go by. The truth of the matter is that all these impure properties entail certain pure relational properties, and the latter should not be ignored when applying PII.2. Indeed every impure relational property that figures in Table I entails a corresponding pure relational property. For example, the impure relational property, which A has, of ‘being 3 diameters away from B’ entails the corresponding pure relational property, which A must also have, of ‘being 3 diameters away from some sphere’. By

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exploiting these entailments we can give alternative descriptions of what differentiates our four static universes, and specify their distinctive features in terms of pure relational properties. The corresponding ‘pure’ descriptions are given in the next table. Table II (‘Pure’ descriptions) U1: Consists of spheres A and B such that: each is 3 diameters away from some sphere. U2: Consists of spheres C, D, and E such that: each is 3 diameters away from some sphere, which is 3 diameters away from some sphere, which is 3 diameters away from it. U3: Consists of spheres F, G, and H such that: G and H are each 2 diameters away from some sphere and 3 diameters away from some sphere; and F is 3 diameters away from some sphere and not 2 diameters away from any sphere. U4: Consists of K, L, and M such that: K is 3 diameters away from some sphere and 4 diameters away from some sphere; L is 3 diameters away from some sphere and 5 diameters away from some sphere; and M is 4 diameters away from some sphere and 5 diameters away from some sphere. It will be noticed that, with each universe, both the ‘pure’ and ‘impure’ descriptions dictate the same result. Thus the ‘impure’ description of U1 (given in Table I) logically dictates that it should contain 2 spheres, but so does the ‘pure’ description (given in Table II). And likewise with the rest: the remaining descriptions in both tables logically require that U2, U3, and U4 should each contain 3 spheres. We are now in a position to work out the correct verdicts for the middle principle, PII.2. On the basis of the ‘pure’ descriptions given in Table II, we may argue for the four verdicts as follows. In our first universe, it is assumed that there are at most two spheres, A and B. Now besides sharing all their intrinsic properties, they both have the pure relational property of ‘being 3 diameters away from some sphere’. And given the ‘pure’ description for U1, this means that they agree in all their pure properties. Hence (by

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PII.2) A is identical with B, and U1 consists of 1 sphere. A similar argument applies to the second universe, where C, D, and E agree in all their intrinsic properties, and each of them has the pure relational property of ‘being 3 diameters away from some sphere, which is 3 diameters away from some sphere, which is 3 diameters away from it’. Given the ‘pure’ description for U2, this means that C, D, and E agree in all their pure properties. Hence (by PII.2) they are identical, and U2 consists of 1 sphere. And the supporting argument to the conclusion that U3 consists of 2 spheres would run as follows. F is distinct from both G and H; for they both have the property of ‘being 2 diameters away from some sphere’, which F lacks. But G and H, besides sharing all their intrinsic properties, both have the pure relational property of ‘being 2 diameters away from some sphere and 3 diameters away from some sphere’. Given the ‘pure’ description for U3, this means that G and H agree in all their pure properties. Hence (by PII.2) they are identical, and U3 has 2 spheres. Finally, the above ‘pure’ description of U4 makes it plain that each of the spheres K, L, and M differs from the other two in respect of a pure relational property; hence (by PII.2) they are distinct, and U4 contains 3 spheres. Thus the overall result of applying PII.2 to our four universes is that only U4 has survived intact; on the face of it, U3 has collapsed into 2 spheres, while U1 and U2 have both been reduced to single spheres. (I say ‘on the face of it’ because, as will be seen in a moment, the remaining verdicts for U1-U3 are not in fact viable.) Yet there is something capricious about these verdicts, especially the last three. Why should the ‘scalene’ configuration of the 3 spheres in U4 leave them undiminished, while their ‘isosceles’ configuration in U3 reduce them to 2, and their ‘equilateral’ configuration in U2 reduce them even more drastically to just 1 sphere? Surely the number of spheres in these universes cannot be affected by any spatial configuration they might assume. Moreover, one of the four verdicts that we have just reached may seem open to question. I have argued that the correct PII.2 verdict for U3 is that it consists of 2 spheres. But, it may be objected, the situation is by no means clear-cut; for there seem to be equally good arguments not only for a 2 sphere verdict but also for a 1- sphere verdict as well as a 3-sphere ver-

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dict.2 (a) The 2-sphere verdict may be defended by the argument given above, or by a variant on it which goes like this. (i) G and H, besides sharing all their intrinsic properties, both have the property of ‘being 2 diameters away from some sphere and 3 diameters away from some sphere’. Given the ‘pure’ description for U3, this means that they agree in all their pure properties. Hence (by PII.2) G and H are the same sphere (call it J). (ii) But J has the property of being 2 diameters away from some sphere, which F lacks. Hence (by PII.2) J and F are distinct; and so U3 has 2 spheres. (b) The argument for the 1-sphere verdict is again in two parts. (i) First we establish, as we did in argument (a), that G and H are one and the same sphere (call it J). (ii) Then, by a further application of PII.2, we reason as follows. F and J, besides sharing all their intrinsic properties, both have the property of being 3 diameters away from some sphere. Given the ‘pure’ description for U3, this means that they agree in all their pure properties. Hence F and J are identical; and so U3 has just 1 sphere. (c) Finally, we may argue for a 3 sphere verdict as follows. (i) F and H are distinct, since H is 2 diameters away from some sphere, while F is not. (ii) And since F is 3 diameters away from some sphere and H is 2 diameters away from some sphere, there must be a third sphere to which F and H are so related (it being assumed that there are at most three spheres). Hence U3 has 3 spheres. My reply to this objection is that I reject the second and third of the above arguments, but accept the first as the only one which correctly applies PII.2. It must be remembered that the sole function of these arguments is to ascertain the result of applying PII.2 to the ‘pure’ description of U3 which is given in Table II. This means that each part of the argument must constitute a correct application of PII.2 to the U3 description as it stands. Now in argument (b), the first part is faultless, but the second is not. The so-called ‘further application’ is in fact a misapplication of PII.2; for here the arguer ignores a relevant part of the U3 description, namely that J (which is either G or H) has the property of being 2 diameters away from some sphere, a property which F does not have. So F and J cannot be identical, and the argument fails. As for argument (c), it is clear that its first part is indeed a correct application of PII.2; but the second part is a distracting irrelevancy. It makes no use of PII.2, and has nothing to do with the argument’s allotted

2

Here I am particularly indebted to Aubrey Townsend.

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task, that of determining the result of applying PII.2 to the given U3 description. Rather it is an attempt to establish what that description, taken in isolation, logically requires, and of course it does require 3 spheres. Only in argument (a) is PII.2 correctly applied in both parts. I conclude that the only correct PII.2 verdict for U3 is that it has 2 spheres. But, the objector might retort, how can there be only 2 spheres, 3 diameters apart, if one of them has the property of being 2 diameters away from some sphere, which the other lacks? Now I agree that this is very odd; but what it shows is not that the 2-sphere verdict is incorrect, but that the correct verdict is incompatible with what the U3 description dictates when it is taken in isolation, namely that U3 has 3 spheres. Notice, however, that this oddity is not peculiar to U3: we encounter the same sort of oddity when we consider the PII.2 verdicts for both U1 and U2. For how can there be a universe consisting of just 1 sphere which is nonetheless 3 diameters away from some sphere?3 And how can there be a universe consisting of just 1 sphere which has the property of being ‘3 diameters away from some sphere, which is 3 diameters away from some sphere, which is 3 diameters away from it’ ? These oddities bring out a general feature of PII.2 which seems quite bizarre; namely, that in some cases it can deliver a verdict that is inconsistent with the description to which it is applied, though the given description is quite consistent internally and stated entirely in terms of pure properties. I say ‘in some cases’ for, as the example of U4 shows, this inconsistency does not always arise. (As far as I can see, no parallel situation can arise with PII.3. I can think of no concrete example in which PII.3 can deliver a verdict that is inconsistent with the description to which it is applied, though the description is logically coherent and stated entirely in terms of intrinsic properties.) Now such an inconsistency between the PII.2 verdict and the ‘pure’ description of a universe to which it is applied, is tantamount to declaring 3

It should be noted that throughout my discussion I am assuming, along with Leibniz and the Newtonians, that space has a three-dimensional Euclidean geometry and time has a linear chronometry, and that these are necessary truths. Following a suggestion made by Ian Hacking, Robert M. Adams rightly points out that in a very tightly curved non-Euclidean space, a sphere can be at a distance of 3 diameters from itself (op. cit., note 1, p.15; cf. I. Hacking, p.255. Leibniz argues that space is necessarily threedimensional in Theodicy, sec. 351 (= GP.vi.323).

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that universe logically impossible. The matter may be argued in more detail with regard to U2. On the one hand, its ‘pure’ description in Table II logically requires that it should consist of 3 spheres. For each of the spheres is said to have a pure relational property which it can only have if it is distinct from two other spheres; and it is understood that there are at most 3 spheres; hence U2 consists of 3 spheres. On the other hand, PII.2 declares the spheres C, D, and E, indiscernible on the grounds that they share all their pure properties, including the above pure relational property which each of them can only have if there are at least two other spheres. Thus given that there are at most three spheres in U2, the effect of applying PII.2 to its ‘pure’ description is to declare that it has just 1 sphere, but that this single sphere has a property which it could only have if in reality there were 3 spheres. We thus end up with a contradiction, which is tantamount to declaring U2 logically impossible. It is perhaps worth pointing out that this conclusion partly depends on the assumption (made earlier in section 1) that PII, and hence PII.2, is to be taken as a necessary truth. If PII.2 were a contingent truth, our conclusion would have been weaker, namely that the U2 description is not logically impossible, but simply incompatible with PII.2. The final upshot, then, of deploying PII.2 on our four universes is that only U4 is admitted to be a possible universe; the remaining three are disallowed as logically impossible. In the last analysis, only the ‘scalene’ configuration of the three spheres is declared permissible, while both the ‘isosceles’ and the ‘equilateral’ configurations are prohibited, as is the symmetrical configuration of the two spheres in U1. But this is quite arbitrary, and doubly so. In the first place, it is arbitrary to admit a universe like U4 to be possible, but not universes like the remaining three. And it is arbitrary, in the second place, because the ‘scalene’ configuration of the three spheres can be made to approximate as closely as we like to an ‘equilateral’ or ‘isosceles’ configuration. Then, according to PII.2, so long as the configuration is ‘scalene’ we have a possible universe, however minutely it differs from either an ‘equilateral’ or ‘isosceles’ configuration; nevertheless, a really ‘equilateral’ or ‘isosceles’ configuration would at once yield an impossible universe. But it is sheer caprice to base such a fundamental distinction as that between a possible and impossible universe on differences of distance which could be infinitely minute.

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Dynamic Universe

Similar arbitrary restrictions would be imposed by PII.2 on a dynamic universe, by which I mean a universe in which there is always some sort of motion. For example, we can have a dynamic universe with 3 spheres constantly moving in all directions, provided they constantly preserve a ‘scalene’ configuration; and provided this condition is met, there would be no upper or lower limit on the successive distances between any two of them. But at no time in their career can these moving spheres assume an ‘equilateral’ or ‘isosceles’ configuration: that would be a logical impossibility. This means that innumerable types of dynamic universes, which are quite easy to imagine, would be discounted as impossible. A simple case is U5, the dynamic (or more precisely, kinetic) universe depicted above. Here we are to imagine a permanent universe inhabited by 3 spheres, N, O, and P, which oscillate at regular intervals between an ’equilateral’ and an ‘isosceles’ configuration. At time t1 the U2 description applies, so that O is 3 diameters away from P; at t2 the U3 description applies, so that O is now only 2 diameters away from P; at t3 they come apart again and assume the configuration they had at t1; and so on. At any time during a transitional period (e.g. the period between t1 and t2) the configuration of the 3 spheres will of course be ‘isosceles’; so that throughout their entire career the configuration will be either ‘equilateral’ or ‘isosceles’; which makes U5 impossible at all times. Of course, if the initial PII.2 verdicts for U2 and U3 were (per impossibile) viable, we would have to tell a different story about U5. We would have to say that at t1 there is only 1 sphere, but at t2 the number mysteriously increases to 2 spheres, and then at t3 it again shrinks to 1. Presumably the number of spheres would always be 2 except at those precise ‘odd’ moments (t1, t3, t5, ...) when their configuration is exactly ‘equilateral’, thus dictating a single sphere.

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To sum up: the middle principle (PII.2) has seemed to many writers less counter-intuitive and more defensible than its stringent rival (PII.3)4 But to my mind, the vagaries we have noticed in this section constitute a sufficient reason for rejecting it.5 5.3. Inter-world indiscernibility Deploying PII.3 or PII.2 on parts of worlds leads to odd and unacceptable results; but deploying PII.3 on whole worlds leaves them exactly as they are, or at least as they ought to be. I now want to argue that PII.3 is an acceptable inter-world principle, though it is to be rejected in its unrestricted form. The first point to note is that a whole world (as a maximal individual) is fully characterisable in terms of its intrinsic nature, i.e. the set of intrinsic properties that it has. This is based on a general principle: that any information we want to convey about a world by attributing relational properties to its parts can be equally conveyed by attributing intrinsic properties to that whole world. This principle is really a restricted version of a principle which David Lewis has applied to composites generally, whether they are

4

Most recent discussions of PII are implicitly concerned with the middle version. Thus Ian Hacking’s article is essentially a defence of PII.2. Likewise some well-known attempts to refute PII by imaginary counter-examples are aimed at this middle version, since they make use of what I have called pure relational properties. Thus in Max Black’s example of a two sphere universe, each is said to be two miles distant from the centre of some sphere. And in Ayer’s example of ‘an infinite series of sounds ... A B C D A B C D A ..., succeeding one another at equal intervals’, it is assumed that there is no difference between any two As in respect of their pure relational properties, since ‘each is preceded and succeeded by the same number of B C Ds’. See M. Black, p. 85, and Ayer, p. 32. A similar remark applies to Strawson’s chess-board universe example in the chapter on Monads, in his Individuals, p. 123. 5

Possibly some plausible restrictions on PII.2 could produce a middle version that is free from these erratic features, but I have no idea how this could be done. A restriction excluding what he calls ‘identity-properties’ (by which he means, I think, properties that are either identity-involving or diversity-involving) has recently been formulated by B. D. Katz, but without due attention to the important distinction between pure and impure properties. A similar neglect of this distinction vitiates a proposal by L. Frankel to exclude a number of relational properties, including properties arising from spatial relations. See Katz, pp. 37-44; and Frankel, pp. 197-99.

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whole worlds or parts of worlds.6 To illustrate, let us go back to our Static Universes. Taking the parts of U1 as our individuals, we note that each of the two spheres has the relational property of being 3 diameters away from the other. But the same information can be equally conveyed by taking the whole of U1 as an individual, and attributing to it the intrinsic property of having two spheres 3 diameters apart. Similarly in U3, we have the true proposition that each of the spheres G and H has the relational property of being 3 diameters away from some sphere; which proposition is equivalent to the proposition that U3 as a whole has the intrinsic property of containing two spheres each of which is 3 diameters away from some sphere. (Incidentally, the principle that I am here invoking gives a new edge to Leibniz’s well-known dictum that ‘there are no purely extrinsic denominations’; by which he seems to have meant that every relational property that an individual has is ‘grounded’ in the intrinsic properties of that individual along with some others. The dictum, I suggest, is true on the following interpretation: every proposition attributing a relational property to an individual which is part of a world is equivalent to some proposition attributing an intrinsic property or properties to the whole world of which that individual is a part.7) Secondly, relations between any two worlds supervene on their intrinsic natures taken separately. Thus given the intrinsic natures of U2 and U3 the relation of having an equal number of spheres supervenes. Hence the relational properties which consist in their having this relation to one another also supervene; namely, that U2 has the relational property of having the same number of spheres as U3, and conversely. Thirdly, whole universes are discernible from one another by their intrinsic natures. Thus given the intrinsic natures of our four static universes we can differentiate them from one another in exactly the way we would want to. It is an intrinsic property of U1 that it is inhabited by 2 spheres, of U2 that it is inhabited by 3 spheres in equilateral configuration, of U3 that it

6

See his discussion of a hydrogen atom and its parts in On the Plurality of Worlds, p.62.

7

Cf. Benson Mates, p. 219; and the relevant texts from Leibniz on this controversial subject, ‘strewn all through the Leibnizian corpus, many of them unpublished’, which Mates has conveniently appended to chapter XII of his book.

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has 3 spheres in an isosceles configuration, and of U4 that it has 3 spheres in a scalene configuration. For these reasons, the case for adopting PII.3 as an inter-world principle seems pretty strong. There can be no indiscernible worlds, though there can be indiscernible parts of worlds. On this point I am in disagreement with David Lewis, who writes: ‘As for indiscernibility, I have no idea whether there are indiscernible worlds; but certainly there are indiscernible parts of worlds ...’8 I suspect that what leads Lewis to leave the question open is his realism about possible worlds. Since for him the realm of existence contains all possible worlds, he is led to the view that there might be two indiscernible worlds, just as there might be two indiscernible spheres within the same world. But if we think of possible worlds as creatures of our own powers of conception, and confine the realm of existence to the one and only real world, then what distinguishes one possible world from another must always be some difference in their intrinsic natures. In the next chapter, where I discuss Leibniz’s PII-based objections to Newtonian absolutism) I shall, therefore, assume that as an inter-world principle, PII.3 is true, but that as intra-world principles both PII.2 and PII.3 are false. 5.4. Appendix: Leibniz’s derivation argument It will be remembered9 that at the start of the correspondence with Clarke, Leibniz treated PSR and PII as coordinate principles to both of which he adhered, and hoped that his opponent would do the same. In his Fourth Letter Leibniz says: ‘These great principles of a sufficient reason and the identity of indiscernibles change the state of metaphysics. That science becomes real and demonstrative by means of these principles.’10 But, seeing that Clarke denied PII while accepting PSR11, Leibniz changed his strategy and

8

Lewis 1986, p. 84.

9

See section 1.1 above.

10

L.iv.5.

11

Clarke’s fame rested on his long-winded elaboration of a form of the cosmological argument in his Demonstration.

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claimed, against Clarke, that if one accepts PSR one cannot deny PII; for he argued that PII is derivable from PSR, so that Clarke cannot deny PII without contradiction.. At the end of his last letter Leibniz declares triumphantly: ‘I believe reasonable and impartial men will grant me that having forced an adversary to deny this principle [sc. PSR] is reducing him ad absurdum.’12 In the course of his argument Leibniz drops a polite reminder: ‘I dare say that without this great principle [sc. PSR] one cannot prove the existence of God ...’13 ‘The principle in question is the principle of the need for a sufficient reason for anything to exist, for any event to happen, and for any truth to be valid.’14 ‘For the nature of things requires that every event should have beforehand its proper conditions, requisites and dispositions whose existence makes up the sufficient reason for that event.’15 And he goes on to declare: ‘I infer from this principle ... that there are not in nature two real absolute beings indiscernible from each other; because if there were, God and nature would act without a reason in ordering the one otherwise than the other.’16 The gist of his argument is repeated in several passages in his Fourth and Fifth Letters. For example: ‘It is a thing indifferent to place three bodies, equal and perfectly alike, in any order whatsoever, and consequently they will never be placed in any order by him [God] who does nothing without wisdom. But then, he being the author of things, no such things will be produced by him at all.’17 And again: ‘To suppose two things indiscernible is to suppose the same thing under two names. And therefore to suppose that the [material] universe could have had at first another position in time and space than that which it actually had, and yet that all the parts of the universe should have had the same situation -- such a supposition, I say, is

12

L.v.130.

13

L.v.126.

14

L.v.125.

15

L.v.18.

16

L.v.21, italics added.

17

L.iv.3.

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an impossible fiction.’18 And again: ‘To say that God can cause the whole [material] universe to move forward in a straight line, or in any other line, without making otherwise any change in it, is another chimerical supposition. For two states indiscernible from each other are the same state, and consequently it is a change without any change.’19 Against Leibniz’s derivation argument, stated in these several ways, I have two things to say. The first is that PSR, on which both Leibniz and Clarke put so much weight, is a spurious principle; and the second is that Leibniz’s account or accounts of PII are, in view of the full discussion in the previous sections, very muddled indeed. There is in fact a spurious self-evidence about PSR which is well worth pointing out. The apparent certainty of the proposition ‘Whatever exists or happens contingently has an external cause or reason’ is due to conflating two totally different senses of ‘contingent/ly’, the one relational, the other intrinsic and non-relational. In the relational sense to say of X that it exists or happens contingently is to say that it is causally dependent on something else. In the non-relational sense, to say that X happens or exists contingently is to say that, though it actually happens or exists, it might (as a matter of logical possibility) not have happened or existed. The result is that when we plug the relational sense into the sentence ‘Whatever happens or exists contingently, has an external cause or reason’, it will express a necessary, but trivially true proposition, to the effect that whatever is causally dependent on something else has an external cause or reason. But when we plug in the non-relational sense, that same sentence will express a substantive contingent proposition, namely that whatever actually exists or happens but might not have existed or happened, has an external cause or reason -- a principle which many philosophers have found no reason to accept. A parallel ambiguity infects the word ‘child’: in a relational sense it means ‘son or daughter’, in a non-relational sense it means ‘boy or girl’. And so the sentence ‘Every child has a parent’ can express a necessary but

18

L.iv.6;

19

L.iv.13, italics added.

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trivially true proposition on the one reading, but a substantive contingently true proposition on the other.20 As for the muddled state of Leibniz’s account or accounts of PII, I need not say much, in view of the full discussion and disambiguation of that principle in the previous sections. Suffice it to say that Leibniz pays little attention to the variety of properties that PII can range over and thus yield the different versions of PII which I sketched; and he vacillates between saying that PII applies to putatively different universes, putatively different individual members of the same universe, and putatively different states. One thing, however, is certain. If Leibniz regarded PSR as a necessary truth, and PII is validly derivable from PSR, then he was committed to regarding PII too as a necessary truth.

20

I owe this example to Lloyd Humberstone.

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Chapter VI The Nutcracker at Work

6.1. Introduction At length we are in a position to turn to Leibniz's PII-based objections to Newtonian absolutism. My plan will be to start (in sections 6.2 and 6.3) by unfolding the complete set of what I take to be Leibniz’s PII-based objections, without going into any textual matters at this stage. I have relegated discussion of the relevant texts in Leibniz to the end of this chapter (section 6.4). This way of proceeding in two stages will, I think, have several advantages. It will provide the reader from the start with a global sketch of what I take to be Leibniz’s total repertoire, uncluttered by matters of textual interpretation. It will also help us ‘locate’ the relevant passages in Leibniz in their proper places in the total scheme. An awareness of the total repertoire will also enable us to spot gaps in the extant Leibnizian texts, where there is a place for an argument of a certain type in the total repertoire but we know of no Leibnizian text that lends it support.1 But first, let me remind the reader of what I take to be the core of Newtonian absolutism, a view which I argued for in chapter II. The core of that theory consists of the conjunction of the following five theses: (A1) Space and time are absolute beings, each enjoying an autonomous existence apart from the things which we ordinarily regard as existing or happening in it. (A2) Space and time are continuous: there is no smallest region of absolute space and no smallest lapse of absolute time. (A3) In both cases the position of the part with respect to the whole is logically unalterable: the parts of absolute space are necessarily where they are and the parts of absolute time are necessarily when they are.

1

To anticipate a little, this is particularly true of Argument V in the sequel, and to a lesser extent, of Argument VI.

(A4) Absolute space is actually, and not just potentially, infinite in all directions; and similarly time is actually infinite, both with respect to the past and with respect to the future. (A5) We must distinguish between absolute and relative motion; absolute motion is defined by absolute distance over absolute time. Not all motion is relative.

To these five theses which constitute the core of their theory, the Newtonians, as we saw, tended to add another, modal thesis which I labelled A9 in chapter II, namely: (A9) Absolute space and absolute time both exist necessarily, in all possible worlds.

This thesis, I argued, is peripheral to the Newtonian theory, since it is not dictated by its core; but when A9 is added to the bare core constituted by A1-A5, we get what I have called the stronger, modal variety of Newtonian absolutism which is claimed to be true in all possible worlds; by contrast we may speak of the weaker theory that merely adheres to the bare core constituted by A1-A5 without adding A9, as the non-modal or contingent variety of Newtonian absolutism. Leibniz does not seem to have taken the modal variety of Newtonian absolutism seriously, and to have directed his attack on the weaker, contingent variety; and in this he was, I believe, being charitable. The Newtonians were in fact rather half-hearted about the modal thesis (A9), since it conflicted with their belief that God is the one and only necessary being.2 This distinction between the modal and non-modal varieties of absolutism will have to be borne in mind in evaluating Leibniz's objections. In all I have a scheme of seven objections to offer on Leibniz's behalf. And before expounding them I want to remind the reader that, for an argument of this sort to be successful, it would have to conform to the pattern dictated by our conclusion in the last chapter, namely that it would have to hinge on, or use as a major premiss, an inter-world application of PII.3; otherwise, it would not quite do, or at least it would not do as it stands. At this point I want to introduce an abbreviatory device, by using the term ‘interworld’ to qualify the type of argument to be discussed in this chapter in a rather special and complicated way. By an inter-world argument (or argu-

2

On which see section 2.4.

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ment in inter-world form), then, I shall mean a Leibnizian (or Leibnizinspired) argument which purports to refute the Newtonian absolutist view of space and/or time and which also purports to hinge on, or use as its major premiss, an inter-world application of PII.3.3 An argument which as it stands is not in inter-world form, in this special sense, may be capable of being equivalently turned into an inter-world argument, in which case I shall say of the original argument that it is statable in inter-world form, and successful on that account. An argument is to be rejected as a total failure only if it is neither in inter-world form nor statable in inter-world form. 6.2. Leibniz’s objections to absolute space Prima facie, an opponent of Newtonian absolute space has three types of spatial world to contend with. (i) The first is a world in which absolute space is only partially occupied, with a finite material universe in infinite absolute space; (ii) the second is a world in which absolute space is fully occupied by an infinite material universe; and (iii) the third is a world in which the infinite absolute space is completely empty and unoccupied. 4 And if the opponent of absolute space can establish that none of these three types of world is really possible, he would thereby show that absolutism with regard to space is not only false but necessarily false. Within this scheme, we will consider the following three arguments, all of them advanced by Leibniz, as attempts to block each of these three alternatives.

3

I realise, of course, that this is quite a heavy weight of meaning to load on to a single expression, but that only strengthens my resolve; for the introduction of this technical term will make the rest of this chapter so much briefer, and (I hope, from the reader's point of view) so much clearer. 4

There is yet another type of spatial world which neither party to the controversy would, I think, have allowed, though we of this age have no difficulty in accepting it; namely, a world with an infinite material universe that does not completely fill infinite space, so that there is room for the material universe to move; that is why I started with the qualification ‘prima facie’. (On this point, see next chapter, section 5.) Argument I* below is meant to fill this gap on Leibniz’s behalf; i.e. to supply the sort of objection he would have raised against this type of absolutist spatial world, if he were to agree with the present-day view that it is not incoherent.

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Argument I (spatial translation): Assume that in the actual world (AW) absolute space exists, with a finite material universe occupying a certain finite region of absolute space. Then there is a numerically distinct possible world (PW) in which the material universe is exactly the same except that it occupies a numerically different but exactly similar region of space. But PW is indiscernible from AW in respect of all intrinsic properties, and therefore, by PII.3, they are identical;. which yields a contradiction.

Argument II (spatial reversal): Assume that in the actual world (AW) space exists absolutely, with a material universe that is either finite or infinite. Then there is a numerically distinct possible world (PW) in which the material universe is the ‘reverse’ of ours: in which the same constituent bodies, while preserving all their mutual distances and relative positions, are ‘placed in reverse order (for example) by exchanging east and west’. But PW and AW agree in all their intrinsic properties; hence, by PII.3, they are identical; which yields a contradiction.

Leibniz’s characterisation of the reversal here is too terse and may be read in two different ways, so that his argument splits into two: one about spatial rotation, the other about spatial reflection. (a) On the first reading, the material universe in PW might have been the result of rotating our own material universe 180 degrees around an absolute north-south axis while keeping all the internal spatial relations the same. (b) On the second reading, the material universe in PW is a ‘mirror image’ of our own, which might have been the result of spatial reflection. To see the difference, imagine that our own material universe is outwardly shaped like an enormous right hand with its forefinger pointing to the east. Then on the ‘rotation’ argument, the material universe in PW is the same enormous right hand pointing to the west; but on the ‘reflection’ argument, the material universe in PW would be an enormous left hand pointing to the west. On the second reading, Leibniz here5 anticipates Kant’s famous claim that if the material universe consisted of nothing but a single hand, it would have to be a determinate matter whether

5

Most commentators have adopted the first reading; the most recent being Benson Mates, 1986, p.233. But Hermann Weyl perceptively adopts the second interpretation in his book Symmetry, pp. 20-21. More recently, John Earman, in his book (1989), p. 117, rightly points out that Leibniz's argument admits of both interpretations.

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it is a left hand or a right hand. The reply implicit in Leibniz’s argument is that a lone hand would be neither left nor right.6 Argument III (against empty space): Assume a possible world (PW) devoid of a material universe and in which only 7 empty absolute space (and time) exists. Now the whole of absolute space can be regarded as consisting of any number of finite ‘parts’ or regions all having the same shape and volume (‘perfectly alike and congruent’). However, such regions are indiscernible, in that they share all their intrinsic properties as well as their pure relational properties. Hence, by either PII.2 or PII.3, such ‘parts’ are not many but one; and the whole of infinite space can be regarded as identical with any finite part of it, however small. But this is a contradiction; hence PW is impossible.

Surveying these three objections, we must pronounce Argument III a total failure; it is neither in inter-world form as it stands, nor is it statable in inter-world form. Argument III relies on either PII.2 or PII.3 in their intraworld forms, which I regard as false; and this is a feature of the argument which is irremovable. Thus Leibniz does not succeed in refuting the hypothesis of an uninhabited world consisting of pure absolute space (and time). But Arguments I and II are both in inter-world form as they stand, and hence successful; they both rely on the inter-world version of PII.3 which I accept, and I can think of no intrinsic property which, in either case, is not shared by PW and AW. Nor does the fact that Argument II is capable of two different readings make a difference: for whether we take it the ‘rotation’ way or the ‘reflection’ way, the result would be the same, namely that PW would not differ from AW in any intrinsic property. Leibniz, then, is entitled to the conclusion that the absolute theory of space (in its non-modal, contingent variety) is false in any inhabited world, whether it contains a finite or an infinite material universe.

6

See Kant’s dissertation, ‘Concerning the Ultimate Foundation of the Differentiation of Regions in Space’, pp. 42-43. 7

The envisaged possible world could not consist of empty space alone, without time, since the notion of absolute space logically requires that all its parts or regions endure at all times. The converse hypothesis, that of absolute time existing without space, is perhaps more intelligible; on which see Argument VI below.

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This conclusion, be it noted, is established by Argument II alone (on either reading), since that argument envisages both kinds of material universe. But it could be equally established by Argument I together with another version of it (call it Argument I*). As it stands, Argument I refutes the combination of absolute space with a finite material universe. Argument I* would refute the combination of absolute space with an infinite material universe, and would proceed as follows: Argument I*: Assume that in the actual world (AW) absolute space exists with a material universe that is infinite and occupies a certain infinite region of absolute space. Then there is a numerically different possible world (PW) in which the same infinite material universe occupies a numerically different infinite region of absolute 8 space; etc.

Thus Leibniz fails to establish that (in its non-modal, contingent variety) the absolute theory of space is necessarily false, though he is entitled to the weaker, but quite substantial, conclusion that it is false in any inhabited world. However, had Leibniz directed his objections to the modal variety of the absolute theory of space, he would have successfully shown that it is necessarily false. The thesis under attack would then have been that it is necessarily the case that absolute space exists. But by either Argument I or Argument II, it is possible (and therefore necessarily possible) that absolute space does not exist. Hence it is necessarily the case that it is not necessarily the case that absolute space exists; which is tantamount to saying that the modal variety of the absolute view of space is necessarily false.

6.3. Objections to absolute time Turning next to Leibniz’s objections to absolute time, we would naturally expect the overall strategy to follow the same lines as the attack on absolute space. Prima facie, an opponent of absolute time might envisage three types of temporal world and attempt to undermine each of them. (i) The first is a

8

However, as I said earlier (in note 4 above), I do not think that either Leibniz or Clarke would have accepted as coherent the suggestion that the material universe should be infinite in extent without exhaustively occupying the whole of absolute space.

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world in which the material universe, whether finite or infinite in extent, has a beginning in absolute time, so that before it began there was nothing but unoccupied time (and, perhaps, space); (ii) the second is a world in which the material universe has no beginning in time, so that there was never a period of unoccupied time; and (iii) the third is a world containing absolute time (and space) but without any time occupants.9 And here again, if the opponent of absolute time can show that none of these types of world is really possible, he would thereby establish that absolutism with regard to time is not just false but necessarily false. Argument IV (temporal translation): Assume that in the actual world (AW) absolute time exists, with a material universe that had a beginning in time, so that it is of finite duration with respect to the past. Then there is a numerically different possible world (PW) in which the material universe has an earlier beginning (starting, say, a year earlier), without any internal difference; that is to say, its contents are exactly the same, and its parts are laid out, both temporally and spatially, in exactly the same positions relatively to one another as under the original assumption. Now by PII.3, PW and AW are identical, but on the Newtonian view they are not; which yields a contradiction.

It should be noted that, as with Argument I, there is a complementary argument to this one (call it Argument IV*) which is aimed at the combination of absolute time with a material universe that had no beginning in time. The complementary argument could be articulated as follows. Argument IV*: Assume that in the actual world (AW) absolute time exists, with a material universe that had no beginning in time. Then there is a numerically different possible world (PW) in which the material universe has exactly the same constituents

9

As in the case of spatial worlds, there is a fourth type of temporal world which neither party to the controversy would, I think, have allowed, though we in this age have no difficulty in accepting it; namely, a world with a material universe that has no beginning (or end) in absolute time but in which the same material universe occupies a different stretch of absolute time, e.g. one in which the constituent events occur exactly one year earlier than their counterparts in the actual world; that is why, here again, I started with the qualification ‘prima facie’. Argument IV* below is meant to fill this gap on Leibniz's behalf; i.e. to supply the sort of objection he would have raised against it if he were to accept the present-day view that such a supposition is not incoherent.

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but they all occur or exist earlier (by a year, say) than their counterparts in our own material universe; etc.

This argument would go through only if the absolutist does not insist on the objectivity of the present, and the distinction between the past and the future. However, an absolutist may insist that the distinction between past, present and future cuts across all possible temporal worlds. It is true that in the envisaged material universe all the ‘static’ temporal relations would be internally preserved: exactly the same events would be simultaneous with one another, and exactly the same events would succeed one another by exactly the same temporal distances. But bringing in the present as an objective intrinsic property of events, which they acquire in the fullness of time and then lose to be more and more deeply buried in the past, would introduce a difference that would effectively block this argument. Thus suppose our own material universe had a beginning, and started 6000 years ago (which is near enough to what the Jewish calendar says), and that the envisaged material universe started 6001 years ago. Then there is a distinctive intrinsic property that PW has, namely the property of ‘having a material universe in which every present event is later than some event that occurred 6001 years earlier than it’; and this is a property which AW lacks; hence they are not PII.3-indiscernible. Which leads me to a further, related, point. As a relativist, Leibniz thought that the only way in which we can suppose our material universe to have started earlier is by postulating a further chunk of events earlier than the first events in our own material universe.10 But this is unnecessary. Provided he takes seriously the distinction between past, present, and future, a relativist can consistently maintain that the material universe could have started earlier without any internal difference: the whole succession of events would be exactly the same, but what is present in our material universe would not coincide with what is present in the envisaged material universe. Argument V (temporal reversal): Assume that in the actual world (AW) absolute time exists, with a material universe that either had a beginning or had no beginning in time. Then there is an-

10

On which, see L.v.56

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other numerically different possible world (PW) in which the same material universe unfolds backwards: the same events occur as in the actual world, preserving all their mutual temporal distances and simultaneities, but they take place in reverse order. However, PW and AW agree in all their intrinsic properties, and hence by PII.3 they are identical. Which yields a contradiction. Hence the absolute theory of time is false in any inhabited world.

This argument is not to be found in Leibniz, but is here put forward as the temporal analogue of Argument II above (which envisages spatial reversal). It would appear that the only type of absolutist who could be swayed by this argument is one who believes that time has no objective direction, and takes a completely ‘bare’ view that allows temporal distance and temporal betweenness as the only objective temporal relations (simultaneity being the absence of temporal distance). Given this view, a protracted event (or process), and its reversed counterpart would be PII.3-indiscernible. But on a richer view of time, which bestows objectivity on the earlier-later relation, a protracted event and its reversed counterpart would be discernible, and the argument would fail. Argument VI (against empty time): Assume a possible world (PW) devoid of a material universe and in which only absolute time (and space) exists. Now the whole of absolute time can be regarded as made up of any number of finite ‘parts’ or lapses of equal duration. But these equal lapses are indiscernible in that they share all their intrinsic properties as well as all their pure relational properties. Hence, by PII.2 or PII.3, they are not many but one; so that the whole of infinite time can be regarded as identical with any finite part of it, however short. Which yields a contradiction; therefore PW is impossible.

Like its spatial analogue (Argument III above), this argument is a total failure, being neither in inter-world nor restatable in inter-world form. It fails because of an irremovable feature, that of relying on an intra-world version of either PII.2 or PII.3, both of which I take to be false. Argument VII (universal motion or rest): Assume that in the actual world (AW) absolute space and absolute time exist, with a finite or infinite material universe. Then there is a numerically different possible world (PW) with exactly the same material universe, except for one feature: in PW the material universe as a whole is at all times in its career moving at a constant speed along a certain straight line in a certain absolute direction,

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whereas in AW the material universe is in this respect always stationary. However PW and AW agree in all their intrinsic properties, and are therefore identical (by PII.3); which yields a contradiction. Hence in no inhabited world can the absolute theory of space and time be true.

It seems to me that this argument is unsuccessful. The envisaged material universe has the pure relational property of successively occupying different regions of absolute space, a property which the actual material universe lacks: it has, instead, the pure relational property of occupying the same region of space at all times. This difference can be equivalently stated (by the principle which we adopted in section 5.3) as a difference in intrinsic properties between PW and AW. Hence they are not PII.3-indiscernible. I will end by summing up Leibniz’s attempt to refute the absolute theory of time on the basis of the arguments examined in this section. Since both Arguments VI and VII undoubtedly fail, the question is: how strong are Arguments IV and V? We may distinguish three grades of temporal involvement that may be added to the core of the absolute theory of time (as outlined in section 1 above), giving rise to three different versions of the absolutism with regard to time. (i) At the lowest grade, we take a ‘bare’ view of time which assimilates it as far as possible to space; on this view, time has no intrinsic direction, and the only objective temporal relations are temporal distance and temporal betweenness. Against this version of the absolute theory of time, both Argument IV and Argument V go through; and Leibniz is entitled to the conclusion that the theory is false in any inhabited world. (ii) At the second grade, the absolutist insists on the objectivity of the earlier-later relation. And against this version of the theory Argument V (temporal reversal) fails, but not Argument IV (temporal translation); but Leibniz is entitled to the same conclusion as before, namely that the absolute theory of time is false in any inhabited world. (iii) At the third grade, the absolutist goes further and insists on the objectivity of the distinction between past, present, and future across all temporal worlds. And against this version of the theory Leibniz’s Argument IV also fails, so that none of his objections against temporal absolutism is successful.

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6.4. The relevant texts in Leibniz It is time to turn to Leibniz’s actual texts for these objections, and to justify my restatement of his arguments in the way I have just done, and the verdict that I have given in each case. As I have already said, two of the set of seven arguments that I have articulated and assessed in the last two sections, namely Arguments V and VI, receive no backing from anything Leibniz says in his letters to Clarke; and of these two arguments, Argument VI is only hinted at by Leibniz in a much earlier paper, from which I will quote the relevant passage in due course.11 But although there is no trace of Arguments V and VI in the letters to Clarke, I have recognised and included them among the set of seven objections in order to fill in obvious gaps in Leibniz's actual (as compared with his potential) repertoire. About the unwritten Argument V, I will say no more, and shall now go on to discuss the textual support for the remaining six arguments. With Argument I (spatial translation) such textual support as we possess is not exclusively relevant to that argument but backs up both it and its temporal analogue, i.e. Argument IV (temporal translation). I have in mind what Leibniz says at L.iv.6: which is relevant to both the spatial and the temporal arguments: To suppose two things indiscernible is to suppose the same thing under two names. Hence the hypothesis that the [material] universe should have had originally another position of time and place [space] than that which it actually had, and yet all the parts of the [material] universe should have had the same position [relatively to each other] as that which they actually have received, [such a sup12 position, I say,] is an impossible fiction.

The passage amalgamates both the temporal and the spatial translation arguments, in a form which is not quite acceptable, since Leibniz here applies

11

When I discuss in its turn Argument III (against empty space), which is the spatial counterpart of Argument VI (against empty time).

12

Here and elsewhere in the quoted passages from Leibniz, the words in square brackets have been added partly as an aid to understanding the text, partly to conform with my own preferred usage; on which see section 4.1.

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PII to material universes rather than whole possible worlds: the material universe of the actual (possible) world is compared with the material universe of an envisaged possible world, and pronounced to be indiscernible from the latter. But it should be remembered that, for a Newtonian absolutist, the material universe (by which I mean the totality of spatio-temporal occupants) can only constitute a part of a whole possible world; since, in any possible world in which there are spatio-temporal occupants, there must be absolute space as well as absolute time, and these constitute two further parts of that possible world. However, the reasoning employed in the above passage, though not in inter-world form, is capable of being restated in inter-world form; and the results I get are the two objections, Arguments I and IV, as stated earlier in this chapter, which I regard as successful. But if Argument I does not receive textual support in isolation, the case is different with its temporal counterpart, Argument IV. We have, from Leibniz himself, a full-blown articulation of this latter (temporal translation) argument at L.iii.6: The same is the case with regard to time. Suppose someone asks why God did not create everything a year sooner; and that the same person wants to infer from this that God did something for which there cannot possibly be a reason why he did it thus rather than otherwise. We should reply that his inference (son illation) would be correct (vraie) if time were something apart from temporal things [i.e. apart from the things that we ordinarily regard as existing or happening in time]; for it would be impossible that there should be reasons why things should have been created at (appliquées à) certain instants rather than others, while their succession remained the same.

This passage contains an unacceptable, theological version of the argument; in the passage PII is applied to the totality of what God creates in an envisaged possible world, as compared with the totality of what God creates in the actual world. But the totality of what God creates does not constitute a whole possible world: there is God besides; so we do not really have an inter-world application of PII here. But what Leibniz here says is, I believe, fully capable of being restated in inter-world form; and one possible result is what I have offered as Argument IV in section 3 above. And there is, in further support of Argument IV, a famous passage at L.v.55, where (surprisingly) Leibniz uses verificationist language: ... If anyone were to say that this same world, the world which has been actually created, could have been created sooner, without any other change, then he

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would say nothing intelligible. For there is no mark or difference whereby it would be possible to know that this world was created sooner. And therefore (as I have already said) to suppose that God had created the same world sooner is to 13 suppose something that is chimerical.

Argument II gets its textual support from the following passage at L.iii.5, where Leibniz uses both PII and (a theological version of) PSR as his starting-points: Space is something absolutely uniform, and without the things situated in it one point of space does not differ absolutely in any respect from another point of space [i.e. they are indiscernible]. Now from this it follows that if we suppose that space is something in itself other than the order of bodies among themselves [i.e. that space is an absolute being], then it is impossible that there should be a reason why God, preserving the same positions for bodies among themselves, should have arranged bodies in space in this way rather than otherwise, and why everything [the material universe] was not put the other way round, by (for example) exchanging east and west.

As for Argument III (against empty space), my main relevant text is the following passage from a short paper known as ‘Primary Truths’, which is believed to have been written by Leibniz some thirty years earlier than the date of his correspondence with Clarke:14 There is no vacuum. For the diverse parts of empty space would be perfectly similar and congruent with each other, and could not be distinguished from one another. And so they would differ in number alone, which is absurd. We can also prove that [empty] time is not a thing, in the same way as we have just proved it 15 for [empty] space.

13

Emphasis added to highlight the verificationist flavour of this passage.

14

It is an untitled paper, in Latin, which editors have baptised ‘Primary Truths’ because of its opening words, Primae veritates. Authorities believe that it was written somewhere between 1680 and 1686. First pub lished in Couturat, pp. 521-22; the relevant passage which is quoted below is translated in Loemker, p. 269, in Parkinson, p. 91, and in A & G, p. 33. 15

The mere existence of this passage, if the suggested dating is correct, refutes John Earman’s breathtaking conjecture that Leibniz had thought of no argument against Newtonian absolutism prior to the date of his correspondence with Clarke; on which see Appendix to this chapter, i.e. section 6.5.

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Also relevant to Argument III is L.iv.7, where Leibniz says: The same reason which shows that space outside the world is imaginary proves that all empty space is something imaginary; for they differ only as the great 16 from the small.

The relevant texts for Argument IV (temporal translation) are not scarce, and I have already detailed them, at the beginning of this section, in conjunction with the texts relevant to Argument I.17 But, as I have already said, there is no trace of Argument VI in Leibniz’s letters to Clarke. It is the obvious temporal analogue of Argument III (against empty space), and is hinted at in the same passage from ‘Primary Truths’ which was quoted above in connection with that argument. Finally, Argument VII is mainly supported by the following passage from L.iv.13, where the reasoning is based on both PII and PSR. Leibniz writes: To say that God can cause the entire [material] universe to move forward in a straight line or any other kind of line, without changing it in any other way, is another chimerical supposition. For two indiscernible states are the same state, and consequently it is a change that changes nothing. Moreover, there is no rhyme or reason in it. For God does nothing without a reason, and it is impossible that there should be a reason here. Besides it would be a case of acting without doing anything (agendo nihil agere), as I have just said, because of the indiscernibility.

Notice that, in so far as Leibniz bases his reasoning on PII, that principle he here applies to states rather than individuals or whole worlds. Leibniz here tells us that, at any time, the state of the material universe in the actual world (AW) is indiscernible from its state in the envisaged possible world (PW). But to get an inter-world argument out of this passage, I have had to restate its reasoning in terms of whole worlds, so as to make use of interworld indiscernibility. The same remarks apply to another relevant passage at L.v.31, where Leibniz says:

16

Cf. L.v.33.

17

The reader is reminded that Argument V is the unwritten objection about which I resolved, at the beginning of this section, to say no more.

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I do not allow that everything that is finite is movable (mobile). Indeed according to my opponents’ hypothesis, a part of [their absolute] space, though finite, is not movable. That which is movable must be capable of changing its position in relation to something else, and it must be possible for a new state discernible [i.e. ob18 servationally distinguishable] from the first state to arise: otherwise the change is a fiction. Thus a finite thing that is movable must be part of some other finite thing, in order that an observable change could take place.

This was in reply to Clarke’s unqualified assertion (at C.iv.5-6) that ‘nothing that is finite is immovable’, an assertion which should (I suggest) be charitably restricted to finite space-occupants, since Clarke was quite aware that finite regions of absolute space are immovable and ‘indiscerpible’.19 Also relevant to Argument VII, though I shall not quote it here (since it really belongs to the verificationist objection to be discussed in the next chapter), is what Leibniz says at L.v.52. This completes my survey of the Leibniz texts and my interpretation of them for the purpose of setting out the scheme of seven anti-Newtonian objections that I expounded and assessed earlier in this chapter. 6.5. Appendix -- John Earman on Leibniz In a book published in 1989, John Earman20 makes it plain that he thinks very poorly of Leibniz, both as a man and as a critic of Newtonian absolutism. In this note I wish to take him up on some of the things he says. Earman notes the passage in Leibniz’s third letter, with which he opens his attack on Newtonian absolutism, and reproduces the immediately following argument propounded by Leibniz, which I have labelled Argument II (against spatial reversal). There Leibniz wrote:

18

This gloss is added to alert the reader that Leibniz here uses ‘discernible’, not in the sense in which the term is relevant to PII, but in an epistemic sense.

19

As I hinted before, Clarke would not have allowed an infinite material universe to be mobile, since to his outmoded way of thinking, it would already pervade the whole of space, and so there would be no room for motion, or at least no room for rectilinear motion. This point is taken up in the next chapter, section 5.

20

World Enough and Space-Time, 1989.

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I have several proofs to refute the fantasy of those who take space to be ... an absolute being of some kind; but for the moment I will only make use of the one required by the present occasion. ... Space is something absolutely uniform, and without the things situated in it one point of space does not differ absolutely in any respect from another point of space [i.e. they are indiscernible]. Now from this it follows that if we suppose that space is something in itself other than the order of bodies among themselves [i.e. if we suppose that space is an absolute being], then it is impossible that there should be a reason why God, preserving the same positions for bodies among themselves, should have arranged bodies in space in this way rather than otherwise, and why everything [the material universe] was not put the other way round, by (for example) exchanging east and 21 west.

Commenting on this passage, Earman says that ‘Leibniz in typical fashion tries to turn Clarke’s argument upside down and use it’ against Newtonian absolutism with regard to space. But this is not all; Earman goes on to say: Leibniz intimates that his argument was drawn from a well-stocked arsenal of confutations of substantialism [i.e. Newtonian absolutism]... But as far as I can determine, there is nothing in the Leibniz corpus to indicate that Leibniz had other confutations up his sleeve. Nor is there any indication that this particular argument was explicitly constructed prior to the correspondence with Clarke, and the context strongly suggests that it was the product of opportunism and one22 upmanship.

I find these remarks astounding for many reasons. First, why should Leibniz be expected to have thought of this objection before there was occasion for it; and who in real life does that? Secondly, Earman does not seem to have done enough home-work competently to determine the issue; for there is at least one passage in the Leibniz corpus in which Leibniz sketches two PII-based arguments (one against absolute space, the other against absolute time), and which occurs in a short essay competently believed to have been written some thirty years before the date of the correspondence with Clarke; and I have quoted the passage, in the last section, as the textual basis for Arguments III and VI in my reconstruction of the Leibnizian repertoire. Thirdly, Earman, in true Newton-Clarke style, reads the

21

L.iii.5. The opening part was quoted and commented on in chapter I, section 1.1; the ‘proof’ that follows was quoted in the last section as the textual basis for Argument II in my reconstruction. 22

Op. cit., pp.116-17.

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quoted passage in a peculiarly blinkered way: in saying ‘I have several demonstrations’ Leibniz can hardly be taken to mean ‘I have already written out, and even published, several demonstrations (which anyone who doubts my honesty is invited to inspect)’; but rather ‘I can produce several demonstrations’. The context strongly suggests that the particular argument which Leibniz proceeds to propound is meant to be one among several arguments which he rightly expects the unprejudiced intelligent reader to be able to fill out. And, finally, what sort of shameful misdeed is Leibniz here being accused of ? Who in the world of honest controversy has ever heard of the fallacy of turning your opponent’s argument ‘upside down’ (or ‘inside out’) ? These are heavily loaded terms, suggesting (without any backing) that Leibniz’s use of the same arguments against the Newtonians is illegitimate, because in his own version they are not right-side-up (or 23 right-side-out). What is uncontroversial is that Leibniz attempted to turn the tables, to turn Clarke’s own argument (or arguments) against him. Our crucial question should therefore be whether Leibniz succeeded in doing so. And I should have thought that if you do (really) succeed in turning your opponent’s argument against him, then you have achieved an honest victory by honest labour, and not, as Earman here insinuates, shamefully or shamelessly blackmailed your opponent and plagiarised his own argument by using it against him.

23

I say ‘the same argument’ rather than ‘Clarke’s argument’, because neither Newton nor Clarke has proprietary rights over the arguments in question, which are common property. These arguments hinge on the same basic principle, namely that mobility is an essential property of any material object, however large or small; a principle, which, I believe, can (with a small but legitimate proviso) be accommodated by Leibnizian relativism. The point is argued above in section 2.3.

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Chapter VII Leibniz’s Verificationist Argument

7.1. The argument The last PII-based objection that we have encountered (Argument VII in chapter VI) is elsewhere given by Leibniz a very different twist, which I find interesting enough to be worth a chapter on its own. For his wording of this new objection suggests what would nowadays be called a verificationist argument; and this is rather puzzling, because it seems alien to his general epistemological stance as a rationalist. The immediate target of Leibniz’s objection is a possibility allowed by the Newtonians which he dismissed as a ‘chimerical supposition’.1 The Newtonians (as we saw earlier in chapter II) held that the created material universe is finite in extent and occupies a particular region of infinite absolute space, so that it is possible that the entire material universe should have occupied a different region of absolute space from the one which it actually occupies at any given time. Hence Clarke allowed the following suggestion (which I shall call Clarke’s hypothesis) to be a coherent logical possibility: that the material universe as a whole should, for any period of time and without any internal alterations, be in a state of uniform rectilinear motion along a certain absolute direction.2 A uniform motion of the universe as a whole would not be perceived by anyone; but that does not make it unreal; for, Clarke insists, a state of motion is different from a state of rest, and this is as true of the material universe as a whole as it is true of any part of it.3

1

L.iv.13.

2

Clarke would seem to disallow the possibility of an infinite material universe moving forward in absolute space, for a reason to be discussed later. 3

Cf. C.iv.13, quoted below

Leibniz rejected Clarke’s hypothesis as incoherent, and some of his remarks in his Fifth Letter suggest what I have called a verificationist objection: that hypothesis is unverifiable, and therefore meaningless (a mere ‘fiction’), because, Leibniz claimed, the alleged rectilinear motion of the entire material universe would involve no observable change. Thus we find him saying, early in his Fifth Letter: I have demonstrated that space is nothing else but an order of the existence of things, which is observed in their simultaneity. Hence the fiction of a finite material universe as a whole moving about in an infinite empty space cannot be admitted. ... It would produce no change that could be observed by any person 4 whatsoever.

He grants that motion is independent of actual observation, and that a ship may be in motion while no one perceives it to be moving; but he insists that motion must be at least capable of being observed: there can be no motion in the absence of observability. Leibniz sums up the position as follows: In order to prove that space without bodies is an absolute reality, the author [i.e. Clarke] raised it as an objection to my view that a finite material universe might move about in space. I answered that it does not appear reasonable that the material universe should be finite; and even if we were to suppose it to be finite, it is unreasonable that it should have any motion except in so far as its parts change their situation among themselves: because such a motion would produce no observable change. ... It is another thing when its parts change their situation among themselves; for then we recognise a motion in space, but it consists in the order of relations which are changed. The author now replies that the reality of motion is independent of its being observed, and that a ship can go forward without a man who is in it perceiving the motion. I answer that motion is independent of its being observed, but it is not independent of observability. There is no motion when there is no observable change. And indeed, when there is no observable 5 change, there is no change at all.

(In the same vein Leibniz dismissed the supposition that the created universe might have had an earlier beginning in absolute time, ‘without any other change’. If anyone were to say this, ‘he would say nothing intelligible,

4

L.v.29; cf. L.v.31.

5

L.v.52.

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for there is no mark or difference whereby it would be possible to know that it had been created earlier’.6) 7.2. A proposed interpretation These remarks have a very modern ring, and might have been made in our times by a logical positivist. Yet coming from Leibniz, they may strike us as incongruous for two reasons. First, we may find it ironical that Leibniz should invoke verifications against an opponent, since by the same token some of his own fundamental doctrines would have to be discarded as paradigm cases of unverifiable propositions. That everything consists ultimately of mind-like monads, that this is the best of all possible worlds, that there is a pre-established harmony between the mind and the body, are all propositions which seem quite immune to any possible observational verification or falsification. But the fact of the matter is that there is no genuine incongruity here. Leibniz regarded his own fundamental doctrines as a priori necessary truths, and he claimed to be able to deduce them from other necessary propositions as startingpoints which he took to be self-evident; so they are verifiable in the sense that they are provable in this fashion. Nor would he have been disturbed by an objection to the effect that these doctrines are observationally unfalsifiable: on the contrary, he would not have allowed anything that can be observed to tell against them, any more than we would allow anything observable to tell against the proposition that two and two make four.7 More interesting is a second apparent incongruity. Let us call observational verificationism the principle that any factual proposition is meaningless unless it is capable of being verified on the basis of some observational evidence. I do not think that Leibniz would have disowned this principle provided it is confined to what he regarded as genuine factual propositions. Now Leibniz’s verificationist objection presupposes that Clarke's hypothesis is a would-be factual proposition, but Leibniz’s own relativism commits him to regarding that hypothesis as a necessary falsehood. How is this discrepancy to be explained?

6

L.v.55.

7

Cf. Benson Mates, 1986, pp. 234, 242-4.

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One way out would be to adopt a trivial reading of the two passages that I have just quoted. Leibniz, it might be suggested, is not here advancing a verificationist argument at all, but merely pointing out a consequence of accepting his own relative view of motion, which he takes himself to have proved: he is not arguing that, because the alleged motion in Clarke's hypothesis is unverifiable, it is unreal, but rather that it is unverifiable because it is unreal; and that it is unreal is guaranteed by his own relative theory of motion. His argument would then be articulated as follows: All motion, whether observed or not, is necessarily relative. But under Clarke's hypothesis there is nothing left over relatively to which the entire material universe may be said to move. Hence there is no motion at all, and a fortiori there is no observable change caused by any motion.

But if this is what Leibniz’s argument amounts to, it is plainly questionbegging, and would carry no weight against an opponent like Clarke who upholds absolutism. But I can think of a more favourable interpretation of Leibniz’s argument which ties up with his overall strategy. It has been noticed by some commentators that on a number of points in the Clarke controversy Leibniz met his opponents half-way, adopting what I would call an ad hominem stance. By this I mean that he granted for the sake of argument certain assumptions upheld by his opponents which he himself regarded as false. In particular, he granted the reality of extended bodies, though his considered opinion was that extended bodies are not irreducibly real but merely ‘wellfounded phenomena’ whose basis is the unextended monads. And he granted the irreducible reality of relational properties (particularly those involving spatial and temporal relations), though his official view was that relational properties cannot stand on their own but are always parasitic upon non-relational (intrinsic) properties.8 Now I want to suggest that Leibniz is here adopting a similar stance with regard to the status of Clarke’s hypothesis. Leibniz, on the proposed interpretation, did not himself regard that hypothesis as constituting a factual proposition, but his opponents did; and so they are committed to the view that, to be meaningful, that hypothesis should be observationally verifiable. In short, my suggestion is that Leibniz’s verificationist objection should be regarded as an ad hominem argu8

See, for example, C.D. Broad, 1952, p. 187.

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ment based on an assumption which he did not accept but to which he rightly took his opponents to be committed, namely that Clarke’s hypothesis constitutes a factual proposition. The argument may then be stated as follows. Argument VIII: 1. A factual proposition such as Clarke’s hypothesis is meaningless unless it is observationally verifiable. 2. The proposition that the universe as a whole is in rectilinear motion is verifiable only if the alleged global motion would involve an observable change that would count as evidence for it. 3. But no global rectilinear motion of any kind (whether permanent or otherwise, uniform or otherwise) would involve an observable change. 4. Hence Clarke’s hypothesis is unverifiable, and therefore meaningless. Leibniz probably intended the argument to cover non-rectilinear motion as well, but I shall set this point aside.9 And since I am here exclusively concerned with rectilinear motion, I shall henceforth drop the qualification and restrict my use of the expressions ‘motion’ and ‘movement’ to rectilinear motion or movement. 7.3. Clarke’s reply Now what was Clarke’s reaction to this argument ? It is significant that at no time did he deny either premiss (1) or premiss (2); to my mind this shows his tacit adherence to observational verificationism and to the factual character of his own hypothesis. What he denied was premiss (3). Against Leibniz’s assertion that in no case whatever would a global motion involve an observable change, Clarke argued that there must be instances in which it would. Clarke saw an exact parallelism between the movement of the universe as a whole and what happens within the universe when, to use his own example, a ship is in motion. A sudden increase or decrease in the velocity of the ship would produce a ‘sensible shock’, with an observable effect on the basis of which a man who is locked up in a windowless cabin could make certain inferences about its state of motion. Similarly, Clarke argued, 9

Cf. L.iv.13, where Clarke’s hypothesis is dubbed a ‘chimerical supposition’, whether the global motion is taken to be ‘in a straight line or any other line’.

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a sudden increase or decrease in the velocity of the universe as a whole would produce an observable ‘shock’ among ‘all the parts’, and from our discovery of such an effect we could infer certain things about the state of motion of the whole. But what about uniform motion? There, too, Clarke saw an exact parallelism. The uniform motion of the ship would not be perceivable by the man in the windowless cabin as long as it stays uniform, but that does not make it unreal; for it ‘has real different effects’ which distinguish it from a state of rest. And similarly with the uniform motion of the universe as a whole, though the movement would not be perceivable by anyone as long as it is uniform. Says Clarke: 10

If the world be finite in dimensions, it is movable by the power of God ... Two places, though exactly alike, are not the same place. Nor is the motion or rest of the universe the same state, any more than the motion or rest of a ship is the same state, because a man shut up in a cabin cannot perceive whether the ship sails or not so long as it moves uniformly. The motion of the ship, though the man perceives it not, is a real different state and has real different effects, and upon a sudden stop it would have other real effects; and so likewise would an indis11 cernible [i.e. unperceivable] motion of the universe.

Such, then, was Clarke’s reply to Leibniz’s verificationist objection as I have restated it. He rejected it by denying its third premiss, and put forward an argument which purports to establish that there are counterinstances to what Leibniz says in that premiss. Of that argument Leibniz did not take any notice, and Clarke was certainly justified when, in his final reply, he complained: ‘It is affirmed that the motion of the material universe would produce no change at all, and yet no answer is given to the argument I alleged that a sudden increase or stoppage of the motion of the whole would give a sensible shock to all the parts’.12 He is more emphatic a few sections later: Leibniz’s merely repeating his assertion that no global motion would involve an observable change cannot settle the issue in his favour, ‘unless he could disprove the instance which I gave of a very great change

10

For Clarke this is tantamount to saying that the movement of the universe as a whole is logically possible, since he believed that the power of God does not range over logical impossibilities.

11

C.iv.13; cf. C.iii.4, and C.v.29.

12

C.v.29; emphasis added.

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that would happen, viz. that the parts would be sensibly shocked by a sudden acceleration or stopping of the motion of the whole; to which instance he has not attempted to give any answer’.13 7.4. An answer to Clarke But perhaps we can attempt an answer on Leibniz’s behalf. I believe there are two decisive objections to Clarke’s argument which would settle the matter in Leibniz’s favour. First, however, we need to get clear about what Clarke meant by a ‘sensible shock’ and how he thought it could be detected. Let us take a concrete example: when a car14 in motion suddenly stops, a passenger in it is momentarily ‘thrown forward’. This would be our ordinary way of putting it, but from a Newtonian point of view, it is very misleading to say that the passenger is thrown forward. For that would imply that some force has pushed the passenger forward -- a piece of Aristotelian or folk physics which Newtonian physics claimed to supersede: the forward movement of the passenger’s body is due to the absence of a force, combined with its inertia, its tendency to continue to move at the previous speed of the car. Now according to Newtonian physics what happens in this case is that a ‘shock’ (or force) is applied to the car (through the action of the brakes), which causes a sudden reduction of speed (or deceleration). And for a few seconds, the passenger’s body (or at least the unattached parts of it) continues to move at the previous speed of the car, until it catches up with the deceleration; so that at the height of the ‘shock’ the passenger’s head, for example, is in a different position relatively to the rigid inner shell of the car. This change of relative position is taken to be causally related to the shock undergone by the car (and all its rigid parts), and enables us to infer that such a shock has occurred. The shock in this case is rendered ‘sensible’ because there is a perceivable (‘sensible’) relative motion, a perceivable change of relative position between certain parts of the passenger’s body and the inner shell of the car; if every part of the passenger’s body were to move in perfect unison with the sudden reduction of speed we would not be able to detect any shock because there would be no change of relative position. 13

C.v.52-3; cf. C.iv.13.

14

Or a horse-driven carriage, in Clarke’s day.

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Now Clarke claims that a similar ‘effect’ would occur if the entire universe in motion were to come to a sudden stop, and that the sudden deceleration would give a sensible shock to all the parts. Now it is true that Newtonian physics requires that, if the material universe as a whole were suddenly to decelerate then each part of the universe would be subjected to the force which causes this deceleration, and that this force could be called a global shock. But, given Newtonian physics, such a global shock would not have any observable effects, and so would not assist Clarke. Clarke does not seem to realise that a necessary condition of his envisaged global movement is that it should occur without any internal change: that is to say, that so long as the material universe as a whole is moving, its entire internal history would be exactly what it would have been if the universe were not moving. This entails a more specific necessary condition regarding spatial relations: that so long as the universe as a whole is moving, the relative spatial positions among all its parts at any given time should be exactly the same as they would have been at that time if the universe as a whole were not moving. Whence it follows that a mere change in global motion cannot by itself generate a change of spatial positions among the parts. But this leaves no room for any changes in relative position that would be the effect of a sudden global acceleration or deceleration: they are excluded by the very notion of global movement. If a force could be detected by some means other than that of observing relative motions, then conceivably we could all detect a force acting on each of us even if our relative positions remained constant; but in fact nothing in Newtonian theory gives us any means of detecting a force other than that of observing relative accelerations and decelerations. This, to my mind, is a decisive point which effectively undermines Clarke’s position. Clarke argued, by analogy with movements that take place within the actual universe, that there must be instances in which a global movement would produce an observable change; but it turns out that, in the nature of the case, the sort of change that he had in mind could not occur. Thus the short answer to Clarke’s argument is that the analogy on which he relied is bound to break down at the crucial point.15 I conclude that, viewed as an ad hominem argument, Leibniz’s verificationist objection to the Clarke hypothesis is quite successful. But one can go further and maintain that, even if Clarke were right in thinking that there is an exact parallelism between the

15

Here I am indebted to John Bigelow.

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movement of the ship and the movement of the material universe as a whole, this would not secure the verifiability, and hence meaningfulness, of his hypothesis about global motion. To show this, let us assume (contrary to what I have so far argued) that globally generated shocks could occur, and that were they to occur there would be no difficulty in recognising them and distinguishing them from locally generated shocks, such as what happens to a passenger in a car or a man in a ship’s cabin. And let us grant Clarke, merely for the sake of argument, that there is a complete parallelism between the movement of a ship and the movement of the material universe as a whole. A man who is locked up in a windowless cabin may notice a sudden jolt whereby loose items in the cabin moved in the direction A to B (where A and B are identifiable fixed locations within the cabin). How much can he infer from his observation of the jolt inside the cabin about the state of motion of the ship? He may know, on independent grounds, that if the ship were moving it would be moving in the direction A to B. And on the basis of these two things (the jolt within the cabin and the direction in which the ship would be moving if it were) he could infer that the observed shock was the result of a sudden decrease in the ship’s velocity, and that before the jolt the ship was travelling at a uniform speed. But what can he infer about the state of the ship after the jolt? Nothing definite: the ship may now be either stationary or travelling at a lower uniform speed. And conversely, if the man knew on independent grounds that the direction of the ship's movement, if it were moving, would be from B to A, he could infer that the experienced jolt was the result of the ship’s sudden increase of speed in that same direction (B to A), so that after the shock the ship must be moving at a higher uniform speed. But nothing definite could be inferred about its state of motion before the experienced shock: prior to the shock, the ship may have been either stationary or moving at a lower uniform speed. But now for the analogy with the entire material universe to be at all plausible, we must deny the man in the cabin any knowledge of the direction in which the ship would be moving if it were moving. This means that he would be in no position to tell whether what went on inside the cabin was the result of deceleration of the ship in the direction A to B, or the result of acceleration of the ship in the opposite direction (B to A). All that he would be entitled to infer is that there has been a sudden change of velocity in the

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one direction or the other: from which nothing definite could be inferred regarding the state of motion of the ship either before or after the shock. Hence in the supposedly parallel case of global motion, all that an observer can infer from the occurrence of globally produced shocks is that a sudden change of velocity has occurred in the global motion, but nothing definite could be inferred about the state of motion of the entire material universe either before or after the global shock: for all we know, the material universe could be either at rest or in uniform motion in the relevant direction or its opposite. Now Clarke is committed to the view that the material universe as a whole may move in the absolute direction X to Y (say east to west), and that this is a different possibility from the equally meaningful possibility of its moving in the opposite absolute direction Y to X. But the relevant shocks, if they were to occur and be recognised as global rather than local shocks, would not enable us to distinguish the one hypothesis from the other. And what this shows is that, in so far as Clarke’s hypothesis is absolute-direction-specific, it is unverifiable. But setting aside absolute direction, there is another dimension to Clarke’s hypothesis, namely with respect to duration; and here we may distinguish a weak from a strong version. Clarke is committed to the meaningfulness, and hence verifiability of both versions. The weaker version of his hypothesis is that the material universe is moving uniformly for a limited period of time, with a beginning and/or end to that uniform motion; the stronger hypothesis is that it is moving at a uniform velocity permanently, at all times. So far we have been considering the weaker version of that hypothesis with the global shocks generated by the sudden acceleration or deceleration which it involves. Epistemic access to these global shocks makes the weaker hypothesis verifiable. But what about the stronger hypothesis ? If the material universe is permanently moving at a uniform speed then no globally produced shocks would ever occur, and the hypothesis would be permanently unverifiable. To say, with Clarke, that the stronger hypothesis is verifiable because if the global motion were suddenly to stop there would be globally generated shocks that we can observe, is paradoxical: for the global shocks would falsify rather than verify the stronger hypothesis. In a possible Newtonian world in which the strong hypothesis is true there would be no available evidence for its truth: the epistemic access adduced by Clarke is incompatible with the state of affairs to be verified. But with the weaker thesis, the epistemic access is compatible with the state of affairs to

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be verified. So if global shocks provide our only epistemic access, then the weaker thesis can be verified but cannot be falsified, whereas the stronger thesis can be falsified but cannot be verified.16 Summing up the above reply to Clarke, I have argued, on behalf of Leibniz, for two main conclusions. The first is that Clarke’s hypothesis is unverifiable because global shocks of the kind envisaged by Clarke are by their very nature undetectable; and the second is that, even if global shocks were detectable and distinguishable from local shocks, Clarke’s hypothesis in its strong form would be at best falsifiable but not verifiable. Argument VIII, with which we are concerned in this chapter, is really the unverifiability counterpart of the PII-based Argument VII which was discussed in the last chapter; for they both envisage the hypothesis of a finite material universe moving as a whole in absolute space. And if I am right, then we have reached an interesting overall conclusion: that though Leibniz’s objection fails as a PII-based argument, it is successful if we see it, as some of the Leibniz texts certainly suggest, as an unverifiability argument. Some commentators17 have gone as far as to suggest that Leibniz’s appeal to PSR in the Clarke Correspondence ‘might be taken’ as an appeal to the principle of verifiability or ‘testability’18, but this is to ride rough-shod over what Leibniz thought he was doing. I have suggested that his appeal to verifiability is better taken as ad hominem; for as a rationalist he did not, and could not, abide by this principle. Leibniz was an honest and consummate controversialist, who knew that if his arguments are to have any swaying power he had to meet his opponents half-way, by granting for the sake argument certain principles to which he did not in fact adhere; in the present instance he appeals to the principle of verifiability, which he rightly thought his opponents could not totally deny.

16

A similar difficulty has surfaced in recent discussion of Michael Dummett’s antirealism. Crispin Wright uses the following example. On the assumption that a man is truly a master criminal only if he never leaves a trace of any of his criminal activities, a statement such as ‘Disraeli was a master criminal’, if true, will have no evidence for it. It is, like Clarke’s strong hypothesis, falsifiable but not verifiable. See C. Wright, 1987, pp. 309-16. I am indebted to Lloyd Humberstone for bringing this point to my attention. 17

See, e.g., J.J.C. Smart.

18

Cf. Smart, p. 215.

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7.5. An unhelpful God At this point Clarke could have turned to God for help. He could have claimed that, as an omniscient being, God would know that the hypothesis of global motion is true if it were true, and he would know this directly and not by any kind of inference. Now the Newtonians had very definite views about how God achieves omniscience. To recapitulate and anticipate a little, they held that God is omnipresent in the sense of literally existing at all points of space; God is a spatial but immaterial substance who pervades the whole of infinite space in much the same way as (let us say) a magnetic field pervades a limited region of space, and by his omnipresence he directly perceives whatever happens anywhere in space.19 Newton also adopted a peculiar form of the representative theory of perception according to which what happens when we human beings perceive external objects is that the mind, which is taken to be in space, perceives pictures of them which are imprinted on the brain by our sense-organs. It is the brain and not any particular sense-organ which is regarded by this theory as the sensorium. The theory also assumes that the mind of any person is literally present at all points of his brain where these pictures can be imprinted, so that the mind’s perception of these pictures is always unmediated. Newton’s famous dictum that space is ‘as it were’ the sensorium of God should be seen in the context of this theory as its background; it is intended to bring out two points of similarity between divine and human perception. The first is that the objects of God’s perception are all the things that lie in space, just as the objects of perception of a person's mind are all the pictures that appear on his sensorium. And the second point is that the manner of perceiving is in both cases unmediated, and is achieved through the ‘immediate presence’ of that which perceives to what is perceived; God is bound to perceive whatever lies in space just as the mind of a person is bound to perceive whatever appears on his sensorium.20 19

The reader will remember that what follows is a reconstruction of the Newton-Clarke account of omniscience; officially, the view was rather that space and time are attributes of God, on which see chapter II, sections 2 and 3.

20

Cf. Clarke at C.i.3; and Newton’s Opticks, end of Query 28 (included in Alexander’s edition, p.174). Leibniz raised powerful objections to this Newtonian account of omniscience, which I discuss in chapter IX.

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Armed with these convictions, Clarke could claim that God’s way of verifying his hypothesis of global motion would be immune from the difficulties attending its verification by human beings. God would immediately perceive the steady motion of our finite material universe because it would successively occupy different regions of absolute space, each of which is permanently and completely permeated by a different part of his own spatially extended substance. So he would be aware of the global motion directly, in something like the way in which I would be aware of a tickling sensation moving down my arm. But notice that what God would then perceive would not be absolute motion but motion relatively to his own spatially extended substance; it is not absolute space but God’s own infinitely extended substance that now acts as the ultimate frame of reference. Now Clarke might concede this point, but insist that it makes no difference to his claim since the divine substance is immovable: every part of that substance is permanently tied down to a particular region of absolute space; hence to move relatively to those parts is eo ipso to move in absolute space. But why should the divine substance be immovable? Clarke would say (I think) that, being infinitely extended, it already pervades the whole of infinite space, and so there is no room for motion (or at least rectilinear motion) because there is no unoccupied space to move into. For the same reason he would have rejected the possibility of an infinite material universe moving forward in absolute space; that is why he confined his hypothesis to a finite material universe.21 But there is a fallacy here, based on an outmoded way of thinking of the infinite as if it were finite. For us there is no difficulty in conceiving an infinite material universe moving forward as a whole in infinite absolute space without ever exhausting it: every finite part of the infinite universe would successively occupy different finite regions of space in an endless process in which there is no last step. Thus the Newtonian view that God pervades the whole of infinite space does not preclude his mobility. We should therefore widen the scope of Clarke’s original hypothesis and include the Newtonian God along with material objects as mobile space-occupants. Clarke would now have to declare the following (revised) hypothesis to be a genuine possibility: that the totality of space-occupants, 21

Cf. C.iv.5-6, and C.v.52-3.

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including God, should be in a state of uniform rectilinear motion for any length of time. But how would God verify this if it were true? It seems to me that, confronted with the revised hypothesis, the Newtonian God would be no better off than ordinary mortals. No longer a stationary spectator, he would not then perceive the global movement directly, nor could he legitimately infer its occurrence from any change of scenery that he might perceive. Earlier I quoted Leibniz’s assertion, in his Fifth Letter, that the alleged motion of the entire universe in Clarke’s hypothesis would involve ‘no change that could be observed by any person whatsoever’. In a draft of that letter which has come down to us Leibniz was tempted to put the point more strongly by using the 22 words ‘not even by God’, which in the end he crossed out. Implicit in these words is another ad hominem objection which he could have raised and which I have tried to articulate. The Newton-Clarke account of divine omniscience has this consequence, that not even God could know whether or not the entire universe of space-occupants is moving in absolute space.

22

Included in Robinet, p. 148.

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Chapter VIII A Digression on Boethius: Eternity and Omniscience

8.1. Preliminaries So far I have set aside as far as possible the theological dimensions of the Leibniz-Newton controversy, in the belief that these are logically independent of the metaphysical core of the rival theories of space and time. The two exceptions were at the end of chapter II (section 2.4) where I tried to tidy up the Newtonian views regarding the relation of God to space and time, and the end of the last chapter (section 7.5), where it seemed appropriate to pursue Leibniz’s envisaged objection at a theological level. In the remaining three chapters I propose to take up a single theological dimension in that controversy, and focus on the rival accounts of God's omniscience that were given by Leibniz and the Newtonians. Both Leibniz and the Newtonians had very definite views about God's omniscience and how it is achieved. The Newtonians assimilated God's knowledge to human perception, and were led to maintain that God must exist everywhere and at all moments in absolute space and time in order to be aware of what is going on. For his part Leibniz found the Newtonian account inappropriate, and believed that God's knowledge is not spectatorial but active: God knows how things are not by inspection but by consulting his own will, since what he wills is the same as what exists. And holding the existence of space to be parasitic upon the existence of their created occupants, Leibniz was led to the view that God exists outside space and time altogether. I shall begin with a digression on Boethius who was also concerned with the question: How is omniscience achieved? But though, like the Newtonians, he assimilated God's knowledge to human perception, he was led to

place God outside time (and space). The digression will serve a dual purpose. A detailed examination of Boethius will, I hope, point up the danger inherent in trying to give too definite an account of God's knowledge; and it will force us to make what I regard as an important distinction between two questions regarding omniscience. That distinction will be worth bearing in mind in the sequel when discussing the rival accounts of omniscience which are to be found in Newton and Leibniz. 8.2. Boethius on eternity Many philosophers and theologians have used the term ‘eternity’ in a sense which was meant to be incompatible with being in time, occurring at a time, or enduring throughout the whole of time. One such notion of eternity was connected, particularly by the rationalists, with necessity: it was held that whatever is necessary is somehow outside time and is in this sense eternal. Thus Leibniz at one stage thought that whatever exists is in time except God, because God is a necessary being, and necessity excludes being in time. Another non-temporal view of eternity, which goes back to Plato and Aristotle, is connected with immutablity or complete absence of change: it was held that time requires change, so that whatever is changeless is necessarily non-temporal, and is in this sense eternal. I do not intend to discuss either of these classical theses, but rather to consider in some detail a puzzling notion of non-temporal eternity which is given somewhat poetic expression by Boethius in the final sections of The Consolation of Philosophy.1 For his peculiar doctrine Boethius seems to claim the ancestry of Plato, and indeed at times conveys the impression that he is merely expounding or giving more precision to Plato’s view. I hope to show, however, that the two views of eternity in Boethius and Plato are quite different -- that despite the fact that they are both non-temporal notions of eternity, the philosophical motivation behind each of them is of a different order. It is important to bear in mind that Boethius is led to discuss eternity in the context of investigating the nature of knowledge, in order to distin1

Book V, sections 4-6. I am very grateful to the late Harry Stainsby for his generous help in collating the quotations from these sections with the Latin text, which for the most part resulted in fresh translations of his own.

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guish divine knowledge from human knowledge. His starting point is the principle that ‘everything which is known is known, not according to its own nature but rather according to the cognitive faculties of the knowers’2 and he goes on to argue that ‘according to this principle various and different substances know things in different ways. Some living things incapable of movement, such as shell-fish ... which live by clinging to rocks, have no other mode of cognition than that of sensation ... But reason belongs to the human race alone, just as pure intelligence belongs to God alone ...’.3 There is thus a hierarchy of ways or grades of knowledge ranging from the simple sensation of shell-fish to the ‘pure intelligence’ or omniscience that belongs to God alone. Armed with this principle, Boethius goes on to expound his own doctrine of eternity. I quote the relevant passages in full.4 Since whatever is known is known not according to its own nature but according to the nature of the knowers, let us consider ... the nature of the Divine Being so that we may also discover the nature of his knowledge. It is the common judgment of all who live by reason that God is eternal. So let us consider what eternity is, for this will reveal to us both the nature of God and the nature of divine knowledge. Eternity then is the perfect possession of endless life all at once -- a notion which becomes clearer by comparison with things temporal. For whatever lives in time proceeds as something present from the past to the future, and there is nothing situated in time which can encompass the whole span of its life without distinction: it has not yet attained tomorrow and has already lost yesterday; and even in this life of today you do not live more fully than in that fleeting and transitory moment. Whatever, therefore, is subject to the condition of being in time, even if it never has a beginning or an end, and its life extends through the infinity of time, ... is still not such that it may properly be considered eternal. For though its life is infinite, it does not include and encompass the whole span of that life all at once (totum simul); rather, it does not yet have the future nor does it still have the past. Thus it is that which encompasses and possesses all at once the whole fullness of endless life, which lacks no part of the future and has lost no part of the past, which may be properly called eternal ...

2

Ibid., section 4.

3

Ibid., section 5.

4

Ibid., section 6.

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Therefore, those philosophers are wrong who, when they hear that Plato held that 5 this world did not have a beginning in time and would never come to an end, think that, on this view, the created world is co-eternal with the creator. For it is one thing to be carried through an endless life, which is what Plato attributed to the world, and another thing to have embraced all at once the complete presence of an endless life; and the latter is clearly the special property of the divine mind. Nor should God be thought of as prior to his creation in respect of time, but rather in respect of the peculiar simplicity of his nature. For the infinite change of temporal things [merely] imitates this presentness of changeless life ... And since it cannot possess the whole fullness of its life without distinction, it seems to emulate to a certain extent that which it cannot fully attain and express, and does this by somehow never ceasing to be. It ties itself to a certain kind of present in this narrow and transient moment which, because it bears a certain likeness to that abiding present, confers upon whatever it touches a semblance of existence. ... Thus if we wish to call things by their right names, we should follow Plato in 6 saying that God indeed is eternal, but the world is everlasting. Since, therefore, every judgment comprehends the objects falling under it according to its own nature, and since God is always in an eternal present, his knowledge too transcends all temporal change and abides in the simplicity of its present: it embraces all the infinite regions of the past and the future, and in its simple knowledge views all things as if they were happening now. Thus if you consider the immediate confrontation by which God discerns all things, you will realise that it is not foreknowledge of something as future, but rather knowledge of a never failing present. That is why it is called not prevision (praevidentia) but 7 providence (providentia), because it is far above the lowly details of the world and sees all things as though in a prospect from the highest summit. ...

5

Boethius would seem to have in mind commentators who consider this view in isolation, without taking into account Plato’s distinction between time and eternity. As far as I know this view about the world is nowhere explicitly stated in Plato. He does, however, affirm that ‘time came together with the heaven’ and that ‘they may be dissolved together, if ever a dissolution of them should come to pass’ (Timaeus, 38B). From which we can infer that for Plato the heaven at least is sempiternal, having no beginning or end in time, unless we are prepared to say of what begins and ends ‘together with time’ that it has a beginning and an end ‘in time’. (By ‘sempiternal’ I mean ‘existing at all moments of time’; cf. footnote 12 below.)

6

Timaeus, 38C. It seems clear that Boethius is here claiming Plato’s ancestry for his own view of eternity, as if he were merely explaining Plato’s famous doctrine that time is, ‘as it were, a moving likeness of eternity’ (Timaeus, 37D).

7

I shall argue later that this should not be taken as a straightforward appeal to etymol-

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And if the human and divine present may be aptly compared, it may be said that just as you see things in this your temporary present, so God discerns all things in his eternal present. Therefore, this divine foreknowledge does not change the nature and properties of things; it simply sees future things present before it exactly as they will later turn out to be in time. ... Those future events which proceed from free will God sees as present. Hence relatively to God's vision, they become necessary because of the nature of divine knowledge; but in reality, considered in themselves, they lose nothing of the absolute freedom of their own na8 ture.

In a paper published in 1961 William Kneale rejected this notion of eternity as self-contradictory. Boethius ... seems to me to be running together two incompatible notions, namely that of timelessness and that of life. ... Perhaps it is not essential to the notion of life that a living being should produce changes in the physical world. But life must at least involve some incidents in time, and if, like Boethius, we suppose the life in question to be intelligent, then it must involve also awareness 9 of the passage of time.

But the notion of life does not, as far as I can see, play a significant role in Boethius’ characterisation of eternity. The words ‘life’ and ‘live’ are indeed used throughout, but only as poetic substitutes for ‘existence’ and ‘exist’, which can for the most part be read into the text without alteration of meaning. Boethius in fact might have less misleadingly expressed the essence of his doctrine by saying that whatever exists in time cannot have the entire span of its existence ‘all at once’, for at any moment of its existence, it eiogy to support Boethius’ own view of divine knowledge. 8

The last four sentences announce a major result claimed by Boethius: namely, that his doctrine of eternity resolves the apparent conflict between divine omniscience and human freedom. He would grant that if God knew in advance the outcome of a future action of mine, that would deprive me of the ability to act differently, and the outcome would be necessary and inevitable. But Boethius maintains that in relation to God that outcome is not in the future but in the present, and so his knowledge of that outcome does not deprive me of my freedom; for if someone observes what I am doing contemporaneously with my doing it, it by no means follows that I could not have acted differently. This, however, is a claim which I shall not try to assess; I am concerned with the intelligibility of the Boethian doctrine of eternity, rather than its application to the problem of human freedom and divine foreknowlege. 9

W. Kneale, p. 99.

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ther lacks some past portion or some future portion of this entire span, or both; but not so with what exists in eternity. It is strictly speaking the existence of God, and not his ‘life’, which is said to be eternal. But if so, the objection loses its force: the notion of life in its ordinary employment is unquestionably bound up with being in time, but the notion of existence need not be. Nor is it at all obvious that God’s awareness of the passage of time requires that he should be in time. A human being indeed must be in time in order to be aware of the lapse of time through the perception of change; but God, being omniscient, need not be in time in order to know of the passage of time. A more cogent argument intended to show that the Boethian notion of eternity is self-contradictory was put forward by Martha Kneale in a paper published eight years later. Events in time, she points out, happen either successively or simultaneously, and simultaneity logically excludes succession; but to say with Boethius that the whole course of events somehow coexist involves saying that successive events are not successive. To contrast eternity with time by saying that it is tota simul is self-defeating because simul is itself a temporal notion. Things in time happen either successively or together (simul) and to say that parts of time, past, present and future happen 10 together is to deny the necessary condition of simultaneity.

She reinforces this point by examining Boethius' favourite analogy, in which God’s providence is likened to seeing things in prospect from the highest mountain. Presumably what he means is that the man on the top of the mountain can see the bends and ups and downs of a road all at once whereas the traveller on the road sees only a limited stretch at a given time. But when we come to think it out, the simile does not help. The spectator on high sees the road all at once but he does not see the traveller in all positions at once. This would be a contradiction. His 11 perceptions must be as successive as the positions themselves.

It seems to me that this way of dealing with the analogy is too literal, and hence unfair; for unless God’s ‘vision’ is assimilated to human perception, 10

Martha Kneale, p. 227.

11

It should be remembered that the book was written in prison while Boethius was awaiting execution

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and God is taken to be in time, the requirement that he must have successive perceptions in order to know of the traveller’s successive positions on the road is out of place. I will try to show in the sequel that there are at least two non-literal ways of interpreting this analogy which would make it both appropriate and free from contradiction. But first I want to deal with Martha Kneale’s own version of the claim that the Boethian notion of eternity is self-contradictory in that it makes simultaneity compatible with succession. I can imagine Boethius providing a successful rebuttal of this charge. ‘Your objection’, he might have replied, ‘would have been telling had I been discussing time; for I agree that in the case of time simultaneity excludes succession. But I was out to characterise eternity, and it is only with respect to the latter that I wanted to maintain, for the reasons given in my book, that simultaneity is compatible with succession. Indeed, had I written my book in less unfortunate circumstances,12 and had I been in a more purely argumentative and less poetic mood, I might have reasoned as follows. In the case of time and things temporal, past, present and future are mutually exclusive, and simultaneity is incompatible with succession; but in the case of eternity, past, present and future coalesce, and so simultaneity is not incompatible with succession. Hence to be eternal is incompatible with being in time. I did deny, in the case of eternity, that simultaneity excludes succession; but I did it with my eyes open, in order to distinguish eternity from time, even endless time. That is why I said, "it is one thing to be carried through an endless life ... and another thing to have embraced all at once the complete presence of an endless life". This was my way of saying that the notion of eternity, in my sense, is different from being in time (even the notion of being sempiternal, in your sense;13 but had I

12

In her article (p. 223) Martha Kneale defines sempiternity as follows. ‘A sempiternal object is one which exists at all moments of time. This definition holds whether we believe time to be finite in one or both directions or infinite in both.’ It will be noticed that throughout this chapter, I avoid using the word ‘timeless’ because it is ambiguous between the sense in which ;it means on-temporal’ (being outside time altogether) and the sense in which it means ‘sempiternal’ (as here defined). It is also worth noting that, strictly speaking, the sempiternal differs from the everlasting in that the later is compatible with having a beginning in time, but once born it does not cease to exist, or rather it continues to exist at all moments of time ever after. 13

Ibid.

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been more precise I should have said that the two notions are incompatible. That simultaneity is compatible with succession is for me a defining property of eternity, and I am no more disturbed by your objection than Georg Cantor would have been disturbed if someone were to point out to him that his notion of infinity commits him to denying that the whole is greater than its part.’ 8.3. Boethius’ account of omniscience Martha Kneale’s way of dealing with Boethius is, to echo John Wisdom, ‘legalistic’: she deals with a symptom but ignores the cause. To get at the root of the matter we must face this question: what attracts an intelligent philosopher (Boethius is not alone) to such an incoherent notion of eternity? And a satisfactory answer to this question must explain why the ‘highest summit’ analogy seems appropriate, though taken strictly it is of no help. Here I should like to offer two suggestions. One is that the Boethian view of eternity is perhaps the result of spatialising time. More particularly, it is the result of a failure to distinguish between the temporal and the spatial senses of 'in': things or events are mistakenly taken to be in time in the same sense as bodies are in space; and just as space contains all bodies at once, time contains all events (past, present and future) at once. In short the fallacious assimilation of the temporal sense of ‘in’ to the spatial sense leads to the view of time as a container, like space, but containing all events ‘at once’; and the suggestion is that it is this view of time that Boethius calls eternity. The point of the ‘highest summit’ analogy is that God sees the whole course of events in this manner; to see successive events as the eternal being does is to see them sub specie spatii. The peculiarity of this view of time is that it denies any privileged status to the present; and the point of saying that events (past, present or future) are in time ‘all at once’ is not to deny the reality of succession, but to convey that they are equally real, that in some tenseless way they are all there. Any two events within the temporal container would still be either simultaneous or successive, but not both. But a much more plausible suggestion is that the Boethian notion of eternity is the result of a mistaken assimilation of knowledge, particularly

14

Ibid.

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God's knowledge, to (human) perception.15 This is a direct consequence of his belief, mentioned earlier, that the various forms of cognition are on an ascending scale, with simple sensation at the bottom and pure all-embracing knowledge at the top. In veridical perception the object perceived must be simultaneous, or nearly simultaneous, with the act of perceiving it; and it is mistakenly assumed that a similar condition applies to knowledge, that when I know something what I know about must be simultaneous with my knowing about it. But simultaneity is obviously not a necessary condition of knowing: if I know at this moment something about an event X, my present knowledge is perfectly compatible with X’s having in fact happened in the past, or X’s being due to happen in the future, however remote. Now a theist may wish to hold that God is omniscient, meaning thereby that God knows everything simultaneously, ‘all at once’. And assuming God's existence to be sempiternal, the theist is entitled to say that God's knowledge is actual and complete at all moments of time. But since the knower and that which he knows about need not be coexistent, God’s sempiternity is not to be regarded as a consequence of his omniscience but as an additional attribute of his. Omniscience and sempiternity are in fact logically independent attributes, and it is not impossible to conceive a shortlived being attaining omniscience during the whole or a certain portion of his life. The appropriateness of Boethius’ favourite analogy becomes clear: to see things from the highest summit is to see them as an omniscient being does, sub specie omniscientiae. God, at any moment of time, knows all the positions of the traveller's journey; but this does not mean that in being thus known they lose their successiveness, neither does it require that in order to achieve this all-embracing knowledge he should enjoy an ‘eternal’ but nontemporal existence. It therefore seems that Boethius’ primary aim in evolving his peculiar notion of eternity was to explain how omniscience is possible. Thus his motivation was epistemological, whereas Plato's distinction between time and eternity was made in order to answer a different, ontological, problem concerning the changelessness or immutability of the Platonic Forms. It is not

15

This may have been inspired by Aristotle, with whose writings Boethius was very familiar. In the De Anima (429a 10-18) it is taken for granted that knowledge and perception are closely parallel.

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surprising therefore to find that, though they are both non-temporal, the Platonic and Boethian notions of eternity are not quite the same. Had Plato been forced to argue for his distinction more precisely he would, I imagine, have produced one or both of the following arguments.16 He might have reasoned that, whereas past, present and future are necessary components of time, they are not applicable to what is eternal; hence eternity is incompatible with being in time. Or he might have said that whatever is in time is necessarily subject to change, but what is eternal must be changeless; hence what is eternal is necessarily non-temporal. Both of these arguments, it will be noticed, are isomorphic with the argument I ascribed earlier to Boethius; but they draw the same conclusion from two different sets of premisses. Plato denied that past, present and future applied to what is eternal, but the Boethian doctrine requires that past, present and future always apply in the realm of eternity; only (and this is what is crucial in Boethius), in the sense in which they do apply to what is eternal, they are not mutually exclusive. It also appears that when Boethius says that divine knowledge is more properly called ‘providence’ than ‘prevision’ he is not to be taken as adducing etymological support for his own doctrine. The Latin prefix pro- may convey the idea of being directed towards the future, as in ‘progress’, ‘proceed’; or it may signify the spatial idea of being directed towards the front, as in ‘protrude’, ‘prospect’ (in the sense of view). Boethius should rather be taken as saying that we are philosophically committed to interpreting the prefix in ‘providence’ (i.e. the divine knowledge) in a non-temporal sense; for God’s knowledge extends to the past and the present as well as to the future. But since God’s knowledge, by analogy with human perception, requires that what he knows about should be contemporaneous with his knowing about it, it would be less misleading to assimilate the ‘pro-’ in ‘providence’ to the spatial sense; for spatial relations too can only hold between things that coexist. However, the assimilation is then based on a partial analogy which should not be pressed: the requirement of simultaneity does not mean that God is in a spatial relation to what he knows about, unless we assume that he is literally in space and that whatever he knows about is also always located in space. These assumptions (as I shall show in the next chapter) were later explicitly made by some philosophers in the empiricist

16

Cf. Timaeus, 37C-38C.

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tradition. But there is nothing in Boethius to suggest that he would have granted them. Indeed, though Boethius does not explicitly make this rather obvious point, placing God outside time is eo ipso placing him outside space too, since on any view of space, the existence of space presupposes the existence of time. 8.4. The refutation of Boethius If the above diagnosis of Boethius’ error is the right one, then he has produced no case for holding that an omniscient being must exist outside time in ‘eternity’. His insistence that knowing and that which is known must be contemporaneous would rule out the seemingly coherent notion of an omniscient but temporal being existing either sempiternally or for only a part of time. And the same principle would land him in a difficulty with regard to ordinary piecemeal human knowledge. For if the principle of contemporaneity is valid, how is it that we know now about certain events that have happened in the past or will happen in the future? Given Boethius’ doctrine, a human being cannot at this moment attain knowledge of future or past events unless he somehow detaches himself from the present and ranges over the past and the future events that he knows about, so that in each case the event becomes simultaneous with his present knowledge of it. Thus Boethius is committed to the view that in so far as we achieve knowledge about the past and the future (if we ever do), we participate in eternity; and hence he was mistaken in assuming that eternity in his peculiar sense belongs only to God’s manner of existence. However, these objections do not amount to a refutation. The false assumption about knowledge certainly seems to have supplied Boethius with a strong motive for framing his puzzling notion of eternity, but it may not have been his only motive; many commentators have read him differently. I now wish to present an argument intended to be a refutation of any philosopher who for any reason whatever maintains that omniscience requires that the omniscient being should not be in time.17 Knowledge is indeed like perception, but not in the respect that Boethius thought it was. They are analogous in that, as ordinarily understood, they can only be ascribed to agents

17

The idea of a refutation along these lines was suggested to me by the late Professor A.C. Jackson.

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who are taken to endure in time, and whose knowing or perceiving is datable. Knowledge is datable in the sense that, if A knows that p, then (assuming that A endures through both t1 and t2) it always makes sense to say that A knows that p now (at t2) or that he knew, or did not know, that p then (at t1). Understood in this special sense, datability is a necessary condition of knowledge; hence to say with Boethius that God knows everything but is not in time is self-contradictory. This argument is plainly of the same form as Martha Kneale’s objection, but the terms are different: Boethius seems to me to be quite aware that in the realm of eternity, as he conceived it, simultaneity is not incompatible with succession; but what he did not realise is that by placing God outside time he had denied him a necessary condition of knowledge, so that talk of omniscience with regard to such a being is completely out of place. It will be plain that datability, as defined above, does not preclude sempiternal knowledge: A’s knowing that p may last for any period within his life span, and provided he exists sempiternally, he may have this knowledge at all moments of time. Nor does the datability condition commit one to accepting any one of the many and controversial views about knowledge that have been put forward. Thus it has been held that knowledge always requires belief, a mental state which may be dispositional as well occurrent; but that knowledge is something more than true belief (though opinions differ as to what more is needed), since true belief may be had accidentally, in which case we do not really have knowledge. But I know of at least one philosopher who has recently defended the view that knowledge is no more than true belief.18 None of these views is precluded by the datability condition here insisted on, which I take to be uncontroversial. That its denial cannot be seriously entertained is confirmed by the fact that, controversial though they are, all the above-mentioned views about knowing are committed to it--that it does not entail but is entailed by each of them. For dispositional as well as occurrent mental states are datable in precisely the same way; and so one who is convinced that knowing always involves a state of belief is bound to maintain that knowing is datable by reason of the datability of this state of belief.

18

See Crispin Sartwell, 1991, pp. 157-65; and his subsequent article, 1992, pp. 167-80.

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But why, it may be objected, should God’s knowledge be the same as human knowledge? Before we can meet this objection we need to articulate and distinguish two questions regarding knowledge: (i) What is knowledge ? Or: What is it for an agent A to know that p (where p stands for some proposition)? (ii) How is it achieved? Or: How does an agent A come to know that p? In a human context these are two very different questions. The first question asks for a definition of knowledge, for an explanation of the meaning of the word ‘know’, whoever the agent may be. Whereas the second question asks for a recipe (or recipes) for knowledge, for a description of the various procedures by means of which a human agent may attain knowledge; and it may well receive the answer, ‘Well, it depends on what the proposition in question is, and sometimes on who the agent is, and the circumstances he happens to be in’. But a parallel distinction applies in a divine context. What is divine omniscience ? And how does God achieve it ? Only, here the how-question has to be treated with extreme caution. For besides being omniscient, God is also traditionally endowed with omnipotence, which places him above the reign of cause and effect; theists generally believe that, unlike the powers we ordinarily speak of, God’s powers are not sustained, or restricted, by causal laws. Now when a theist says that God knows everything, he cannot escape using the word ‘know’ in its ordinary sense. The what-question regarding knowledge does not admit of a different answer in a divine context. If the theist asserts that God knows everything, but goes on to qualify this statement by saying that the sense in which God knows something is quite different from (and may even be incompatible with) the sense in which we ordinary mortals may know something, then the qualification would simply cancel the initial assertion; and he might as well have said nothing. Alternatively, we may regard this self-cancelling statement as a symptom of his failure to observe the distinction between the what-question and the howquestion. How God, if he exists, knows everything may (for all we know) be quite different from any of the ways in which we human beings come to know of anything; for we know nothing about the way in which God attains his all-embracing knowledge, which is a good reason for remaining silent on

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this issue. But once we choose to speak, and assert (perhaps as a matter of religious conviction) that he knows everything, we must take the consequences, on pain of seeming to say something while saying nothing.

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Chapter IX Omniscience: Leibniz versus Clarke

9.1. How is omniscience possible? With few exceptions, theistic philosophers have often failed to distinguish the what-question from the how-question with regard to God’s knowledge, and have often put forward views about the ‘nature’ of his knowledge which are no more than bold speculations as to how he knows. Not surprisingly, those in the empiricist tradition -- and Boethius can be regarded as one of them -- have tended to assimilate his way of knowing to perception, whereas the rationalists have assumed that it is purely conceptual. Within this scheme I propose (in this chapter and the next) to consider as further illustrations the relevant doctrines of Berkeley, Newton, and Leibniz. Now once we assimilate knowledge to perception we are apt to equate an omniscient being with an omniperceiver; but since perception is such that the notion of a person perceiving everything at once makes no sense, some dehumanising of perception becomes necessary. In Boethius this is achieved by endowing the omniperceiver with a special kind of vision which, unlike its human analogue, is not affected by distance or confined to a single direction at a time. The point of placing the arch-observer on the highest summit is presumably to convey that the most distant object in the vista is as clearly visible to him as the most near. Nor is his field of vision confined to what lies before him at a certain time: like a thousand-eyed Argus, he surveys at any moment all the ‘lowly details’ of the whole surrounding territory in every direction all at once. 9.2. The Newtonian account of omniscience With Berkeley the assimilation of knowledge in general to ordinary human perception would appear to be full and complete, and it is made to seem obvious by exploiting an ambiguity; for ‘to perceive’ does in one sense mean ‘to know’, as when I say ‘I perceive that equilateral triangles are equiangu-

lar’. This leads Berkeley at one place to claim that what his famous principle about the existence of ‘sensible things’ asserts is that ‘their being is to be perceived or known’.1 The identification of knowledge with perception is also implicit in the following famous passage where the existence of God as an arch-perceiver is introduced in order to secure the continued existence of ‘sensible things’ when they are not in fact perceived by any human beings. When I deny sensible existence out of the mind, I do not mean my mind in particular, but all minds. Now it is plain they have an existence exterior to my mind; since I find them by experience to be independent of it. There is therefore some other Mind wherein they exist, during the intervals between the times of my perceiving them: as likewise they did before my birth, and would do after my supposed annihilation. And, as the same is true with regard to all other finite created spirits, it necessarily follows there is an omnipresent eternal Mind, which knows 2 and comprehends all things, and exhibits them to our view

Now in order to fulfil the function assigned to him by Berkeley, God cannot have powers of perception that are totally different from ours; otherwise, when perceiving objects that are not being perceived by human beings at the time, he would not be perceiving the same ‘ideas’ as human beings would have perceived had any been present. But then, how is omniperception (with which omniscience is to be identified) at all possible? The quoted passage, which purports to be a proof of the existence of God, would suggest the following answer on Berkeley’s behalf. A perceiver whose powers of perception are no different from ours would, in the required sense, be an omniperceiver provided he is sempiternally present at all points of space (‘eternal’ and ‘omnipresent’). The view of omniscience that I am attributing to Berkeley is quite explicit and fully articulated in the writings of Newton and of Clarke in the latter’s correspondence with Leibniz. It was the belief that God is literally present at all points of space which led Newton to regard space as the ‘sensor-

1

George Berkeley, The Principles of Human Knowledge, section 6; I have emphasised the last three words. See Berkeley’s Philosophical Writings, edited by D. M. Armstrong, p. 63. 2

Three Dialogues between Hylas and Philonous, Third Dialogue, in Berkeley’s Philosophical Writings, edited by D. M. Armstrong, p.193. Note the slide from ‘perceiving’ to ‘knowing and comprehending’.

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ium’ of God, a doctrine which to Leibniz seemed theologically unacceptable. For, Leibniz argued, ‘if God stands in need of any organ to perceive things by, it will follow that they do not altogether depend upon him, nor were produced by him’. In his reply Clarke points out that Newton did not mean to identify space with an organ, but was merely making use of an analogy, saying that space is ‘as it were’ the sensorium of God. Newton’s analogy, according to Clarke, was in fact intended to convey the opposite view, namely that God perceives things ‘without the intervention or assistance of any organ or medium whatsoever’.. Newton adopted a peculiar form of the representative theory according to which what happens when we human beings perceive external objects is that the mind, which is taken to be in space, perceives pictures of them which are imprinted on the brain by our sense-organs. It is the brain and not any particular sense-organ which is regarded by this theory as the sensorium. The theory also assumes that the mind of any person is literally present at all points of his brain where these pictures can be imprinted, so that the mind’s perception of these pictures is always unmediated. With this theory as its background, Newton’s assertion that space is ‘as it were’ the sensorium of God was intended (it would seem) to bring out two points of similarity between divine and human perception. The first is that the objects of God’s perception are all the things that lie in space, just as the objects of perception of a person’s mind are all the pictures that appear on his sensorium. And the second point is that the manner of perceiving is in both cases unmediated, and is achieved through the ‘immediate presence’ of that which perceives to that which is perceived: God is bound to perceive whatever lies in space, just as the mind of a person is bound to perceive whatever appears on his sensorium. As Clarke puts it, God being omnipresent ‘sees all things by his immediate presence to them, by being actually present to the things themselves, to all things in the universe, as the mind of man is present to all the pictures of things formed in his brain’.5

3

L.i.3.

4

C.i.3; cf. Newton, Opticks, end of Query 28 (Alexander, p.174.)

5

C.i.3.

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9.3. Leibniz’s objections To many readers Leibniz’s scornful dismissal of Clarke’s detailed explanation of how Newton meant his analogy to be taken must have made it appear that Clarke got the better of him. Dr. Johnson’s remarks, addressed to a rare, Scottish admirer of Leibniz, may be cited as a typical reaction from the kind of reader I have in mind: Leibniz persisted in affirming that Newton called space sensorium numinis [‘the sensorium of God’], notwithstanding he was corrected, and desired to observe that Newton’s words were QUASI sensorium numinis [‘as it were the sensorium of God’]. No, Sir; Leibniz was as paltry a fellow as I know. Out of respect for 6 Queen Caroline, who patronised him, Clarke treated him too well.

But this is sheer distortion. Leibniz certainly took notice of Clarke’s explanation of the point of Newton's analogy, and right from the start he raised objections to the peculiar representative theory of perception which the analogy took for granted.7 And in his fifth paper (which was to be his last) he remarks tersely: ‘The author alleges it was not affirmed that space is God’s sensorium but only as it were his sensorium. The latter seems as inappropriate [peu convenable] and as little intelligible as the former.’8 Thus his final verdict seems to have been that the analogical version of the sensorium doctrine fares no better than the literal version which he mistakenly imputed to the Newtonians in his first paper. It is not hard to give Leibniz’s accusation a precise content, and to see that he was quite right. Clarke certainly underestimated the force of Leibniz’s initial objection, whose main point was that any view about God’s knowledge which made him subject to causal conditions must be rejected because it involves the absurdity of binding God the creator by causal laws 6

Johnson’s remarks are reported in James Boswell, Journal of a Tour of the Hebrides, under Tuesday, 5th October, 1773. They were made in conversation with the Rev. Hector McLean, the Minister of Coll and Tiree, who was an enthusiastic supporter of Leibniz in this controversy. Johnson, relying on his memory, here misquotes Newton’s own words in the Latin original of his Opticks, which were ‘tanquam sensorio suo’ but their meaning is the same. 7

See, e.g., L.ii.4, L.iii.12, and L.iv.30.

8

L.v.78.

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which are properly of his own making. Clarke naively assumes that having exposed Leibniz’s initial misrepresentation of the Newtonian doctrine, he has thereby disposed of Leibniz’s initial objection. Hence his smug final reply to Leibniz’s revised objection just quoted, that it contained nothing new, and that ‘the aptness and intelligibleness of the similitude made use of by Sir Isaac Newton, and here excepted against, has been abundantly explained in the foregoing papers’.9 Clarke, however, has no reason to be selfsatisfied; for it is easy to see that the analogical version of the Newtonian doctrine involves the same absurdity as the literal version. The peculiar representative theory of perception to which Newton subscribed makes it a causally necessary condition for the mind of man to be literally present at all points of space occupied by the pictures of things formed in his brain in order to perceive them; and the point of Newton’s ‘similitude’ as explained by Clarke is that God’s knowledge (or perception) of things is subject to precisely the same causal condition. This, Clarke maintains, is a condition of human perception which applies equally to God’s knowledge;10 for, he argues, ‘nothing can any more act or be acted upon where it is not present’. (The words ‘act or be acted upon’ clearly imply that this condition of God’s knowledge is to be taken as causally necessary.) Thus the fact that Leibniz persisted in raising the same objection does not mean that he persisted in putting on the Newtonian doctrine the wrong interpretation. He rightly thought the analogical interpretation to involve the same absurdity, and found the analogy inappropriate because it assimilated God’s knowledge of his own creation to that of a mere spectator. (Besides, Clarke’s insistence on ‘immediate presence’ as a necessary condition of God’s knowledge clearly implies that God exists in space in precisely the same sense as the things he knows about are in space, just as the human mind (according to the Newtonian theory of perception) is in space in the same sense as the pictures of things on the brain are in space. But this is incompatible with Clarke’s official view that God does not really exist in space, because space is really a property of God, the one which theologians call his immensity.11 Thus there is a crack in Clarke’s overall 9

C.v.78.

10

C.ii.4; cf. C.i.3.

11

On which see section 2.4.

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theological position: his official view that space is merely a property of God does not cohere with his account of how God achieves omniscience. But we will set this point aside.) Newton’s idea of omniperception through omnipresence could, however, be rejected on other grounds which do not seem to have occurred to Leibniz. The gist of the latter’s criticism of the Newtonian doctrine was that endowing God with such a mode of cognition is theologically unacceptable. But he could have gone further and argued that the doctrine is incoherent since the notion of unmediated perception on which it relies seems quite unintelligible. Newton and Clarke characterise God’s mode of cognition mainly in language appropriate to visual perception; but normal human vision is necessarily causally conditioned by such factors as distance, the presence of light, the laws of perspective, and the point of view of the observer. Now it seems meaningful to suggest that one or other of these causal conditions of vision might have been different. For example, although normally we do not see objects when in direct contact with our eyes, the idea of vision by contact is not unintelligible: our faculty of vision might have been slightly different, so that we could see things when they are in direct contact with our eyes as well as when they are at some distance from them; or more drastically, our vision might have been subject to the same condition as our sense of touch, so that we could see things only when they were in contact with our eyes. But this latter visual analogue of the sense of touch is not quite the mode of cognition with which God is endowed by the Newtonians: he is not properly speaking in contact with the objects that he ‘sees’ because, though he permeates the whole of space, he is completely immaterial and does not see with eyes. Perhaps ‘vision by cohabitation’ would be a more appropriate label for what Newton had in mind: by this mode of cognition God (in Newton's own words) ‘sees the things themselves intimately, and throughly12 perceives them, and comprehends them wholly by their immediate presence to himself.13 Put more precisely, the view is that, with 12

An archaism for ‘through and through’.

13

Newton, Opticks, end of Query 28 (Alexander, p. 174). Cf. Clarke (C.iv.30): ‘God discerns things by being present to and in the substances of the things themselves, by being continually omnipresent to everything which he created at the beginning’.

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regard to any object existing in space at any time, God actually perceives, and perceives all at once, everything about that object which could possibly be perceived, and that he is able to do this because he completely pervades the very same region of space which the object occupies. God knows, for example, that I have a key in my pocket because he is where the key is; and he also knows everything about the internal structure of the key because he is literally present at all points in that region which is at any moment occupied by the key. Finally, it may be urged that even if the Newton-Berkeley version of omniperception were perfectly intelligible, it could not fulfil the function of omniscience. An omnipercipient being would perceive (and thus know about) everything that was happening at present; but this would not make him omniscient, since to be the latter he must also have an all-embracing knowledge of the past and of the future. Yet the notion of literally perceiving now what is in the past or in the future does not seem to make sense. This difficulty does not seem to have occurred to either Newton or Berkeley; nor could they have met it by endowing God with an all-embracing memory and precognition, without abandoning their central claim that God must be omnipresent (and sempiternal) in order to be omniscient. For if God can know the present and the future without being present, why should he be present to the present in order to know what is happening at present? It seems to me that it was partly to meet this difficulty that Boethius was led to place God the omniperceiver outside time, in an eternity in which simultaneity and succession are not incompatible: he thereby showed at least a greater awareness of the problems involved in trying to equate omniscience with omniperception than either Newton or Berkeley.

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Chapter X Omniscience and Omnipotence: Clarke and Arnauld against Leibniz

10.1. Leibniz’s account of omniscience To return to Leibniz himself, the view which he put forward against the Newtonians was that ‘God knows things because he produces them continually’.1 This was taken literally by Clarke, who replied that ‘God discerns things, ... not by producing them continually (for he rests now from his work of creation), but by being continually omnipresent to everything which he created at the beginning’.2 Leibniz was then forced to explain his position more fully: God himself cannot perceive things by the same means whereby he makes other things perceive them. ... He perceives them because they proceed from him, if one may be allowed to say that he perceives them: which ought not to be said unless we divest that word of its imperfection, for else it seems to signify that things act upon him. They exist and are known to him because he understands 3 and wills them, and because what he wills is the same as what exists.

This did not satisfy Clarke who, in his final reply, complained that he found Leibniz's account of omniscience unintelligible.4 Nor is this surprising. For Leibniz's account cannot be fully appreciated without a proper explanation of the logical doctrines on which it was based, but which received no mention in what he wrote to Clarke. Those views took shape some thirty years earlier, when Leibniz wrote his Discourse on Metaphysics, and were de-

1

L.iv.30.

2

C.iv.30.

3

L.v.87.

4

C.v.83-88.

fended at length in some of his letters in the correspondence with Arnauld.5 Of particular relevance here is his famous doctrine of ‘the complete individual notion’: for every possible individual, Leibniz maintained, there is a notion which includes an exhaustive description of everything that would happen to it, if it were to exist. God's act of creation consists in choosing to actualise the corresponding individual notion; afterwards the created individual unfolds its own history as laid down in its complete notion. So Leibniz’s assertion that God ‘continually produces’ things should not be taken literally to mean that God is continually engaged in creating things, but rather that in deciding to actualise a certain set of individuals he thereby produces all that happens to them at any time, because whatever happens to them is a logical consequence of having that set. To my mind the doctrine of the complete individual notion is so difficult to accept that the interesting issue is not whether it is at all viable, but why Leibniz persisted in clinging to it despite his clear realisation that it conflicted with other important features of his system which he could not easily abandon, especially his need to secure an absolute distinction between necessity and contingency. It seems to me that the motivation was largely theological -- that the doctrine forced itself upon him in the course of trying to give an answer, compatible with his thoroughgoing rationalism, to the question ‘How does God achieve omniscience?’ God, he thought, achieves knowledge not by experience but always by a pure exercise of his understanding, in the same way as we come to know necessary truths. And since God knows every true proposition, every truth must be capable of being known, at least in principle, in this purely conceptual way. Leibniz also adhered to the view that in every proposition there is a subject and a predicate; or more precisely, that every proposition is either explicitly of the subjectpredicate form or reducible to an equivalent proposition of that form. Hence he was led to maintain that the truth of a proposition consists in the fact that

5

Both the Discourse on Metaphysics and the correspondence with Arnauld date back to 1686, but were not published in Leibniz’s lifetime. References to the correspondence with Arnauld are to the standard edition contained in the second volume of C.I. Gerhardt, ed., Die Philosophischen Schriften von G.W. Leibniz, 7 volumes (Berlin, 1875-90); here and elsewhere abbreviated as GP. The English translation by H.T. Mason (The Leibniz-Arnauld Correspondence, Manchester, 1967) gives Gerhardt’s pagination on the margin.

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its predicate notion is contained in its subject notion. The theory of the complete individual notion is simply a consequence of these doctrines about the nature of truth and the structure of propositions, when they are applied to singular propositions, that is, propositions whose subject-term refers to an individual. For, since every true proposition about the individual in question rests on the fact that its subject notion includes the notion of its predicate, the notion of that individual must include all that can be truly predicated of it. This enabled Leibniz to provide a recipe for ‘infallible knowledge’, that is to say a rule for deciding whether one has attained such knowledge. We know a proposition infallibly, according to Leibniz, when we clearly see that its predicate is included in its subject; and assuming that every proposition is of the subject-predicate form, or at least reducible to a proposition of this form, God knows every proposition in this way.6 That Leibniz's theory of truth was theologically inspired seems clear from the following passage, taken from a short semi-autobiographical paper which was written soon after the correspondence with Arnauld came to an end: I saw that in every true affirmative proposition, whether it is universal or singular, necessary or contingent, the predicate is in the subject, that is, that the notion of the predicate is in some way contained in the notion of the subject. I also saw that this is the basis (principium) of infallibility in every kind of truth in a being who knows everything a priori. ... [But] God alone knows contingent truths a 7 priori, and sees their infallibility otherwise than by experience.

6

A similar recipe (or ‘rule’) appears in the Port-Royal Logic, where it is confined to axioms. The rule is stated as follows: ‘If moderate attention ... suffices to show that the attribute is truly contained in the idea of the subject, then we have a right to take as an axiom the proposition joining the attribute to the idea of the subject’. Leibniz’s recipe for divine knowledge extends this ‘rule’ to all true propositions. See Antoine Arnauld, Logique ou l’Art de Penser, sixth edition (Amsterdam, 1685), Part IV, Chapter 6, Rule 1; cf. English translation as The Art of Thinking by J. Dickoff and P. James (New York, 1964), p.321; the emphasis is mine.

7

‘On Freedom’, editors’ title, first published by A. Foucher de Careil in his Nouvelles Lettres et Opuscules inédits de Leibniz (Paris, 1857), pp. 178-85; the quotation is from pp. 179, 181. There are translations of this paper in Loemker, Item 29; Parkinson, pp. 106-11; and A & G, pp. 94-8. Loemker believes that the paper was written in or before 1679, but Grua’s suggested date, 1689, seems more plausible. See G. Grua (ed.), Leibniz: Textes inédits (Paris, 1948), p. 326.

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In this paper Leibniz makes use of the notion of infinity in an unsuccessful attempt to preserve an absolute distinction between necessary and contingent propositions without abandoning his doctrine of truth.8 But what is remarkable about the quoted passage is the unhesitating answer that is given to the question ‘How does God achieve omniscience?’. Leibniz was too steeped in his rationalism and too convinced of his logical doctrines to be in the slightest doubt: God knows every true proposition a priori by seeing that the predicate notion is included in its subject notion, because this is what truth is. And despite the great respect in which he held Arnauld, he did not even heed the latter’s warning, in a letter which he must have received three years earlier, against this kind of speculative dogmatism. The wise remarks of that great philosopher, with which I will end this section, contain a warning which has too often been neglected in philosophical theology:9 I find it hard to believe that it is good philosophising to seek in God’s way of knowing things what we are to think of either their specific notions or their individual notions. ... For what do we know at present [i.e. in this life] concerning God’s knowledge (science)? We know that he knows all things, and that he knows them all by a single act. ... When I say we know this, I mean we are assured that it must be so. But do we understand it? And should we not recognise that, however assured we may be that it is so, it is impossible for us to conceive how this can be achieved (comment cela peut être)?

8

The point is well argued in L. Couturat, La Logique de Leibniz d’après des Manuscrits inédits (Paris, 1901), pp. 212-3. Briefly, the view advanced by Leibniz is that, in reality and in the eyes of God, all true propositions are analytic (or as he would say ‘identical’); and that to say of a true proposition that it is necessary or contingent is merely to speak of our ability or inability to demonstrate its analyticity in a finite number of steps. We human beings cannot demonstrate the truth of a contingent proposition, because to do so would involve going through an infinite series of steps, which is a logical impossibility; only God can see the analyticity of a contingent truth, not by going through the whole series, but by grasping whatever is in the series in a single intuition. Evidently, on this view the distinction between necessity and contingency becomes merely epistemic and relative to human knowledge. 9

The letter, dated 13 May 1686, is part of Leibniz-Arnauld Correspondence; the following passage is from GP.ii.31.

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10.2. ‘The paradox of omnipotence’ In the remainder of this chapter I shall dwell on a relevant feature of the Leibniz-Arnauld Correspondence, connected with the so-called paradox of omnipotence, which has not, to my knowledge, been noticed before. Can an omnipotent being bring into existence an object which he cannot control? If the answer is ‘No’, then he is not omnipotent. And if the answer is ‘Yes’, then he is not omnipotent either; for in that case he could not bring it about that: such an object should come into existence and still remain under his control. This problem, which has been much discussed in recent years, has come to be known as ‘the paradox of omnipotence’.10 Many writers have seen the apparent contradiction as demonstrating the incoherence of the notion of omnipotence. Nor is this view confined to critics of theism. A notable example is the eminent Roman Catholic philosopher Peter Geach; in an influential and much discussed article he has argued, largely on the basis of variants on this paradox, that a Christian must not believe that God is omnipotent, since no graspable sense can be attached to this proposition. Geach is therefore led to distinguish omnipotence, the power to do everything, from what he calls "almightiness", which he takes to mean having power over all things. The basic belief of Christianity, he maintains, is not that God is omnipotent but rather that God is almighty.11 I believe that this paradox is specious. However, my main object here is to show that the paradox looms large, but unnoticed, in the LeibnizArnauld Correspondence. The paradox-question assumes the form: Can God create an individual, or indeed a whole universe, with whose future history he cannot interfere? Leibniz held that God is not merely able to do this, but that this is in fact what happened in creation. In creating the universe that he did create God thereby chose the whole sequence of events peculiar to it, so 10

The label is due to J.L. Mackie, who initiated recent discussions of this problem in his influential article ‘Evil and Omnipotence’, published in 1955 (see Mind, Vol. 64, pp. 210-13). It is worth pointing out that Mackie's statement of the alleged paradox was not a new discovery. Writing in 1906, McTaggart put forward a variant on this paradox, to be discussed later in this section, which he described as an ‘old question’. 11

P.T. Geach, ‘Omnipotence’, Philosophy, Vol. 48 (1973), pp. 7-20; reprinted in his book Providence and Evil (Cambridge, 1977), ch. 1.

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that the unfolding of these events is no longer under his control.12 Arnauld opposed this doctrine in the belief that it set severe limitations on God's power (or ‘freedom’, as both parties to the dispute preferred to call it). Taking the example of Adam and his posterity, Arnauld remarked: How many men have come into the world only through the very free decrees of God, such as Isaac, Samson, Samuel and so many others? When therefore God knew them conjointly with Adam, that was not because they were contained in the individual notion of the possible Adam independently of God's decrees. It is therefore not true that the individual persons of Adam's posterity were contained in the individual notion of the possible Adam, since they would have had to be 13 contained in it independently of the divine decrees.

And yet, as Leibniz was quick to point out, God’s power would be equally limited if he could not create the universe in this fashion.14 Thus whereas Leibniz upheld the affirmative answer to the paradox-question, Arnauld unhesitatingly took the negative answer to be the correct one. (Of course, I am not suggesting that either Leibniz or Arnauld was aware of the paradoxquestion: my claim is that the paradox is implicit, but never quite surfaced, in their Correspondence.) Before discussing the Leibniz-Arnauld version of the paradox I wish to offer a definition of omnipotence, and to sketch a solution to the paradox in some of its contemporary forms. The definition presupposes a distinction

12

See GP.ii.12, 17-19, and 42; and compare Discourse on Metaphysics, sections 8 and 13 (GP.iv.432-33 and 436 ff.). 13

GP.ii.29; cf. GP.ii.15.

14

Thus Leibniz wrote in a letter addressed to the Landgrave Ernst of Hessen-Rheinfels, who acted as his intermediary: ‘Can God, who sees everything perfectly, ... have chosen a particular Adam without also considering and deciding everything that is connected with him? Consequently it is ridiculous to say that this free decision of God deprives him of his freedom. Otherwise, in order to be free one [God?] would have to be always undecided’(GP.ii.23). By ‘freedom’, as I said before, both parties to the dispute seem to mean ‘(range) of power’. Now to be ‘undecided’ about a certain action implies refraining from exercising the power to perform it; so we may take it that what Leibniz finds ridiculous is the idea that God has the power to create the universe in this fashion only if, and as long as, he refrains from exercising that power. 15

For a more detailed discussion, see E.J. Khamara, ‘In Defence of Omnipotence’,

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which we commonly draw between the outcome of a particular exercise of power and the act which constitutes that exercise, between the result of a certain action and the manner in which that result is achieved. Plainly the same result may often be achieved in different ways: for instance, one can travel from Paris to Hanover by taking different routes or by using different means of transport. It seems to me that this distinction is crucial in the present context. We must distinguish between what an omnipotent being can do and his manner of achieving it. We may succeed in defining the range of his power in terms of its possible outcomes without being able to say how he is able to accomplish a certain feat, or even when he should act in order to do so. Now there are two things which no agent can rationally be expected to accomplish: logical impossibilities and logical necessities. A logical necessity holds no matter what. It cannot be the outcome of any action, and so bringing about a logical necessity is logically impossible. And to speak of a logically impossible situation (or state of affairs) is a mere façon de parler; it amounts to saying that we have a form of words which simply fails to describe any situation (or state of affairs). Clearly the maximal range of what might be brought about must include every situation that is neither logically necessary nor logically impossible. So I am led to offer the following definition: an omnipotent being is one who has the power to bring about any logically contingent situation. Or, to put it differently: an omnipotent being is one for whom it is practically possible to bring about any logically contingent situation. Whence it follows that inability to bring about a logically non-contingent situation, whether necessary or impossible, does not tell against omnipotence. I believe that these restrictions on omnipotence would have been quite acceptable to Leibniz as well as Arnauld. For they both opposed the view that a truly omnipotent being is not bound by logical truths, and Leibniz was pleasantly surprised to find that Arnauld dissociated himself from the Cartesian doctrine of a supra-logical God.16 With this definition in hand, let us 1978, pp. 204-19, where other variants on this paradox are also dealt with. 16

Arnauld, GP.ii.15, 27 and 29; Leibniz, GP.ii.38, 49 and 51. Arnauld maintained, for instance, that God could not create a human being incapable of ‘thought’, or rather consciousness, since he took it that the task is logically impossible.

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return to the paradox in its general contemporary form: Can an omnipotent being bring into existence an object O which he cannot control? It is important to realise that the paradox involves two questions, each having to do with a distinct ability. The main question concerns creation: it is whether an omnipotent being is able to bring into existence an object O which he cannot control. But there is also a subordinate question concerning control: namely, whether, assuming that an object such as O is already in existence, an omnipotent being is able to control it. As regards the subordinate question it is clear that, since O is taken to be an object which an omnipotent being cannot control, the envisaged outcome is that he should control an object which he cannot control. But whoever an agent may be, it is logically impossible that he should control something which he cannot control. Thus the negative answer to the subordinate question is quite satisfactory, since, on our definition, inability to bring about logical impossibilities does not tell against omnipotence. However, this is perfectly compatible with an affirmative answer to the main question, which I regard as the correct answer. For it seems clear that there are many things whose control in a certain respect is logically impossible, and therefore beyond the power of an omnipotent being as defined. I say ‘in a certain respect’, for to speak of controlling an object is incomplete unless we make clear in what respect it is to be controlled. The examples I have in mind make use of the Aristotelian distinction between essential and accidental attributes, which may be briefly stated as follows. A property is an essential attribute of a certain object if, and only if, the object has that property and could not have existed without having it; whereas a property is an accidental attribute of a certain object if, and only if, the object has that property and could have existed without having it.17 If an object loses an essential attribute it ceases to exist, though (as Aristotle put it) it may "turn into" another. Thus when a log of wood is burnt up in a fireplace it ceases to exist, having lost one or more of its essential attributes; but we may say that it has turned into ash. The crucial point is that whatever

17

The way in which the distinction is here drawn may not be quite true to Aristotle, but this is irrelevant to my purpose. Cf. Baruch A. Brody, ‘Why Settle for Anything Less than Good Old-Fashioned Aristotelian Essentialism?’, pp. 351-65; incorporated later in his book Identity and Essence (Princeton, 1980), ch. 4.

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survives the loss of an essential attribute must be regarded as a (numerically) different object from the one that lost it. Now let us suppose a creature C with consciousness as an essential attribute. Then controlling C in the form of depriving it (permanently) of its consciousness is logically impossible, and so is controlling it in the form of endowing it with consciousness. The former task involves bringing about a logical impossibility, the latter a logical necessity.18 Hence to control C in either of these ways is not within the power of an omnipotent being as defined. The creature C may indeed lose its consciousness, but in doing so it would turn into another creature D. So while an omnipotent being can destroy C by making it lose its consciousness, he cannot control it in the form of depriving it of its consciousness; for whatever is controlled must be at least logically capable of persisting thereafter. Yet it is certainly possible that a creature such as C should come into existence, and so an omnipotent being can bring it into existence. In general, an omnipotent being as defined cannot control any object in respect of an essential attribute, but this does not rob him of his ability to bring into existence objects with essential attributes. And this shows that the affirmative answer to the main question is quite satisfactory. I conclude that, considered in its general form (in terms of ability to control), Mackie's paradox is specious. For while the answer to the subordinate question is ‘No’, the answer to the main question is ‘Yes’; but these are perfectly compatible, and so there is no paradox. However, there are two further variants on the paradox, each with its own peculiar features, which I should now like to discuss. The first (advanced by Geach) argues that the notion of omnipotence is selfcontradictory in terms of a specific example: making an object that its maker cannot destroy. This is certainly a possible task, a task which some people have performed. And yet (the argument goes) a contradiction results when we press the question: Can an omnipotent being perform the task or can he not? If he cannot, there is already some logically possible task which he cannot perform. If he can perform the task, let us suppose that he does. 18

It may sound odd to speak of control in such cases; but this should be regarded as a convenient façon de parler, on a par with our talk of a logically impossible, or logically necessary, situation.

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Then we are supposing an omnipotent being to have brought about a situation in which he has made an object which he cannot destroy; and in that situation destroying this object is a logically possible task which he cannot accomplish, for (to quote Geach) ‘we surely cannot admit the idea of a creature whose destruction is logically impossible’.19 But here again the paradox is specious. In this example the envisaged outcome contains a reference to the agent, and this makes a crucial difference. The task is indeed logically possible for a human being, but where the agent is taken to be an omnipotent being as defined the relevant task becomes logically impossible. For then our main question will be: Can an omnipotent being bring about an object which he cannot destroy? On our definition of omnipotence, he will be unable to destroy the object only if it is logically incapable of ceasing to exist. Now the object is taken to be made (or created), and therefore one that comes to be; but whatever comes to be must be logically capable of ceasing to be. Therefore an object which is made but cannot cease to be is a contradiction in terms. Thus where the agent is an omnipotent being, the relevant object is logically impossible; so that the negative answer to the main question is correct but innocuous. And the subordinate question will amount to this: Can an omnipotent being destroy an object which cannot exist? I think the question should be dismissed, since the idea of destroying something which is necessarily non-existent is nonsensical. It is worth noticing that McTaggart, from whom Geach’s example appears to be borrowed, used it rather differently.20 He did not, like Geach, regard it as a logically possible task which a supposedly omnipotent being cannot perform. On the contrary, he offered it as an instance of a logically impossible task, which he thought a truly omnipotent being should nonethe-

19

P.T. Geach, op. cit., note 11, p.14. This is a close paraphrase of Geach’s argument with a minor amendment. I speak of an omnipotent being throughout, and have refrained from calling the latter ‘God’ where Geach does so. The reference to God is irrelevant to the argument, which has to do solely with the notion of an omnipotent being.

20

McTaggart states the paradox as follows: ‘Could God create a being of such a nature that he could not subsequently destroy it? Whatever answer we make to this question is fatal to God’s omnipotence. If we say that he could not create such a being, then there is something that he cannot do. If we say that he can create such a being, then there is still something that he cannot do -- to follow such an act of creation by an act of destruction.’ (J. McT. E. McTaggart, p.204.)

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less be able to perform. For here as elsewhere McTaggart wanted to have it both ways. He insisted that omnipotence should be understood in an absolute sense -- as the ability to do anything whatever, including the logically impossible; but he also found it meaningless to suggest that a logical impossibility might be actualised. And while it was irrational of him to insist on absolute omnipotence, I believe he was quite right in regarding the task involved in this particular example as logically impossible.21 The second variant on this paradox which I want to discuss has to do with controlling a free agent. In Mackie’s famous article the paradox-question arises in connection with the so-called free will defence, a solution to the problem of evil which absolves God of any responsibility for the moral evil in this world. All moral evil, according to the ‘free will’ solution, is committed by human beings of their own free choice, and so they alone are responsible for it. Against the ‘free will’ solution, Mackie says that there is a fundamental difficulty in the notion of an omnipotent God creating men with free will, for if men's wills are really free this must mean that even God 22 cannot control them, that is, that God is no longer omnipotent.

The paradox-question then assumes this form: Can God create (really) free human agents with whose future free actions he cannot interfere? If God cannot do this then he is not omnipotent; and if he can do this then, once he creates such free agents, he is no longer omnipotent. I believe that this ‘free will’ version of the paradox is equally specious. Here again we must distinguish the main question from the subordinate question. The main question concerns God’s ability to create free agents: Can an omnipotent being do this? The answer is uncontroversially ‘Yes’. The subordinate question concerns God’s ability to manipulate free agents and dictate the outcome of their free actions: Can an omnipotent being do this? To answer this last question, we need to take into account the distinction between compatibilism and incompatibilism regarding the nature of free action. Let F stand for the thesis of freedom and D stand for the thesis of determinism; the two theses may be spelled out as follows: 21

See McTaggart, pp. 202-4, 207, and 217.

22

Mackie, op. cit., pp. 209-10.

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F - Some human actions are free D - Whatever happens is the result of antecedent (causal) conditions given which it cannot fail to come about. The distinction between compatibilism and incompatibilism has to do with the logical relation between F and D, without regard to their actual truth or falsity. An incompatibilist believes that F and D are inconsistent, i.e. that they cannot both be true; whereas a compatibilist believes that they are consistent. With this distinction in mind, let us return to the subordinate question: Can an omnipotent being dictate the outcome of a free action in a human agent while leaving that action free? (i) Given compatibilism, the answer is ‘Yes’, for then the fact that an action is free does not rule out its outcome's being determined or dictated by some external power, such as an omnipotent being. (ii) But given incompatibilism, the answer is ‘No’: that a human action should issue in an outcome that is dictated by some external power is inconsistent with its being free. The envisaged outcome would then be logically impossible, and failure to bring it about would not tell against omnipotence. In sum, the ‘free will’ version of the paradox of omnipotence is no less specious than the general version discussed above. Here again the paradox disappears once we distinguish the main question from the subordinate question. The answer to the main question is ‘Yes’. But to the subordinate question the answer is ‘Yes’ if compatibilism is correct, but ‘No’ if incompatibilism is correct. And either way there is no paradox. Mackie’s own discussion of free action shows him to be a compatibilist, and as a compatibilist he is committed to saying that an omnipotent being can dictate the outcome of a free action while leaving it free; or, to put it my way, he is committed to the ‘Yes’ answer to the subordinate question. But this is inconsistent with his saying (in the passage I have just quoted) that ‘if men’s wills are really free this must mean that even God 23

23

Against incompatibilism he raises the familiar objection that it would equate freedom with randomness, and thus absolve free agents of any responsibility for their actions, since their actions would not then be determined by their characters (op. cit., p.209).

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cannot control them’; this assertion would be true only if we adopted incompatiblism, but then God’s inability to control men’s free wills would not tell against his omnipotence. It seems then that, to get his ‘free will’ version of the paradox going, Mackie would have to have it both ways. On the whole it does not seem that the notion of an omnipotent being, as defined above, is paradoxical. But this does not settle the question whether God is omnipotent. For God is traditionally endowed with other necessary (i.e. essential) attributes besides omnipotence, such as being wholly good; and this seems to reduce the range of his powers. The difficulty (raised by Geach24 and others) may be stated as follows. There are things which the God of Christianity and Judaism cannot do; for example, telling a lie or breaking a promise. But telling a lie or breaking a promise is a contingent matter: it is something which men and women certainly can (and often in fact) do, and which therefore an omnipotent being as defined can also do. It thus appears that there are things which an omnipotent being as defined can do, but which God cannot. Does this show that a God who is wholly good cannot be omnipotent? I do not think so. It seems to me that the God of Christianity and Judaism can be omnipotent in exactly the sense defined, and that the apparent counter-examples just mentioned carry no weight. To show this I will employ what may be called the method of resolution and composition.25 Let A be an omnipotent being, in the sense defined, and B a being with perfect goodness as one of his essential attributes; whether A is essentially or only accidentally omnipotent does not matter for our present purpose. Let us also assume, for the sake of argument, that perfect goodness is logically incompatible with telling a lie or breaking a promise under any circumstances. Then let us ask: Can A, who is omnipotent, bring it about that B should tell a lie or break a promise? The answer to which I am committed is ‘No’; but

24

Op. cit., note 11.

25

I believe that several other items on the list of what God cannot do may be dealt with in the same way, so as to show that they are not incompatible with the doctrine of omnipotence. Cf. Aquinas, Summa Contra Gentiles, II.25; English title, On the Truth of the Catholic Faith, Book II, trans. by James F. Anderson, New York, 1956; chapter 25, pp. 73-76.

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since the envisaged outcome is ex hypothesi a logical impossibility, A’s inability to bring it about does not tell against his omnipotence. Now there is nothing in this reasoning to preclude A and B from being identical. So let us assume that they are in fact one and the same being, and give him the proper name ‘God’. We may then restate the question we have just asked in terms of A and B as follows: Can God, as an omnipotent being, bring it about that he, as a being whose perfect goodness is an essential attribute, should tell a lie or break a promise? The answer must be again ‘No’, though this negative answer does not tell against God’s omnipotence. Alternatively, we may take it that B is wholly good but only accidentally, so that although he can tell a lie or break a promise, he consistently refrains from doing so. In that case A, who is omnipotent, can bring it about that B should perform such misdeeds. And assuming as before that A and B are in fact one and the same being, named God, our composite question will be: Can God, who is omnipotent and accidentally wholly good, tell a lie or break a promise? To which the answer is that, qua omnipotent, God can certainly perform either of these deeds, but being wholly good, he consistently refrains from doing so. 10.3. The Leibniz-Arnauld version We may now restate the Leibniz-Arnauld version of the paradox as follows: Can God, as a necessary being who is essentially omnipotent, create an individual (or a whole universe) which he cannot control in any respect? As with the previous forms of the paradox, the subordinate question is unproblematic. So I shall concentrate on the main question. The answer depends on whether it is possible to have a created (and hence contingent) individual whose entire history contains no feature that is not essential to it. For only then would an omnipotent being, as defined, be incapable of controlling it in any respect. Arnauld insisted that at least some features of an individual's history are not essential to it,26 and therefore would have rejected the idea of a created individual which an omnipotent being cannot control in any respect. So he would have upheld the negative answer.

26

See GP.ii.32-33.

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But Leibniz was committed to the affirmative answer by his doctrine of the ‘complete individual notion’, which was expounded in section 1 of this chapter. For every possible individual, he thought, there is a notion which includes everything that would happen to it as well as its life-span, if it were to exist. God’s act of creating an individual consists in bringing into existence the corresponding complete notion. Afterwards the created individual simply unfolds its own history as laid down in its complete notion. To control a created individual in any respect is logically impossible, since for Leibniz every element in its actual history is essential to it; nor is it possible to destroy it before it has gone through its proper life-span or to prolong its existence beyond that period. However, God’s inability to do any of these things does not mean that he ceases to be omnipotent once he creates; for, as Leibniz saw it, the envisaged outcome would in each case be a logically non-contingent situation, and thus fall outside the domain of omnipotence as defined. Arnauld was therefore wrong in concluding that Leibniz’s position was incompatible with God’s omnipotence. The source of their disagreement lies deeper in Leibniz’s bizarre doctrine of the complete individual notion, which, if true, makes the realm of contingency (and consequently the domain of omnipotence) much narrower than we take it to be. Ordinarily we want to say that at least some episodes in my life, such as my taking a certain journey on a particular date, are logically contingent situations;27 but given Leibniz’s doctrine of the complete individual notion, no episode in my actual history is logically contingent, so that whatever happens to me is outside the domain of omnipotence. As we saw earlier (in section 1 of this chapter), the Leibnizian recipe for divine knowledge leaves no room for genuine contingency: in reality the domain of logical contingency is empty, and so God’s omnipotence in the sense defined is reduced to total impotence. However, to secure a realm of absolute contingency, Leibniz more often than not adopted a less radical position. His usual manner of expounding his doctrine is to say that a proposition such as ‘Alexander died a natural death’ is necessary not absolutely but ex hypothesi.28 That is to say, given 27

See Arnauld, GP.ii.33.

28

Cf. GP.ii.16-20. The example about Alexander is taken from the Discourse on Meta-

149

that God resolved to create him, it necessarily follows that Alexander would in time die a natural death; for this is included in the complete notion of him. On this interpretation, the proposition ‘Alexander died a natural death’ is logically necessary but existentially neutral, for its subject-term refers not to the actual Alexander but to the complete notion of him as a possible individual (‘sub ratione possibilitatis’).29 It is only the proposition ‘Alexander existed’ that is contingent; and God’s knowledge of the latter is based on his knowledge that he created Alexander, or rather the universe to which Alexander belongs. In general, all true existential propositions are contingent, and they all follow logically from God’s free decision to create this universe. God is still omnipotent in the sense defined, but the only thing he can do is to create a certain universe; having done that, he has no power over what happens afterwards. In this way omnipotence is reduced to unipotence followed by total impotence. The main source of trouble is, of course, Leibniz’s misguided attempt to give a recipe for divine knowledge. To my mind Arnauld’s greatest contribution in that unfinished correspondence is to have marked the line between philosophising proper and pontificating, between legitimate conceptual analysis and mere guesswork based on inappropriate models. Unfortunately, his clear warning against this kind of dogmatism went unheeded by Leibniz, and is too often neglected even today. How God achieves omniscience (or omnipotence) is a question on which the wisest course is to advance no hypothesis, and openly profess learned ignorance.

physics, section 8 (GP.iv.432-33). 29

GP.ii.52.

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