i The Metaphysics of Space-time Substantivalism: A Methodological Study of the Structure of Space-time in the Context of General Relativity Sarah K. Grant A thesis submitted for the degree of Doctor of Philosophy at the University of Otago, Dunedin, New Zealand ii Abstract This thesis explores methodological approaches to studying substantivalism in metaphysics and the philosophy of science. My primary aim is to show that it is only when these two disciplines come together, that we can hope to make progress with respect to discovering the true nature of the structure of space-time. I begin by looking at Leibniz’s argument that substantivalism entails the possibility of qualitatively indiscernible worlds, which differ only with respect to how the material world is located in space-time. I explore whether there is anything more to the basis of this contention than non-naturalistic appeals to common sense and a priori metaphysical principles. I argue that there is, as a modern-day interpretation of Leibniz’s argument shows that substantivalism renders GTR indeterministic. I consider whether this problematic indeterminism arises as a result of a particular conception of substantivalism, or whether it follows from all formulations of substantivalism consistent with GTR, and to what extent it is our additional metaphysical commitments that are motivating the problem. iii Acknowledgements I would like to thank my primary supervisor Dr Heather Dyke for her comments, suggestions, criticisms and patience, and for providing models of what metaphysics can and should be. I would also like to thank my original supervisor at the University of Leeds, Professor Robin le Poidevin, without whose friendship, support and enthusiasm this thesis would not have been possible, and also Professor Alan Musgrave for his detailed comments on my work. I would like to thank my family for supporting me both financially and emotionally throughout graduate school, and my pup Cruiser for providing companionship, love and comfort during my writing up. A special thank you to my sister Emma and partner Kris, for their unwavering confidence in times of difficulty, and for always being there for me, regardless of distance. iv Table of Contents Chapter 1: Introduction 1 1.1 A Naturalistic Methodology 5 1.2 Thesis Outline 10 Chapter 2: Theories of Events 14 2.2 A Property Exemplification Account of Events 16 Identity Conditions for Events 16 An Argument from Common Sense 20 The Essentiality of Spatio-temporal Location 24 Modifying Kim’s Account 27 2.3 Davidson’s Theories of Events 30 A Casual Account of Event Identity 30 A Spatio-temporal Criterion 34 Conclusion 39 Chapter 3: Individuation and Discernibility in Space-time Metaphysics 41 3.1 The Argument from Indiscernibility 43 A Comment on Methodology 44 The Various Commitments of PII 46 Leibniz’s Argument 51 Primitive Identity for Space-Time Points 56 3.2 Forbes Against Primitive Identity 58 Forbes’s Argument 59 Theoretical Considerations 66 Conclusion 68 Chapter 4: On Explanatory Adequacy 70 4.1 The PSR and Necessitarianism 72 Leibniz’s Rationalist Model of Explanation 73 Avoiding Necessitarianism 79 4.2 The Legitimacy of Metaphysical Questions 84 The Brute Fact View 85 Sufficient Explanations 90 Conclusion 100 Chapter 5: Indeterminism and The Hole Argument 103 v 5.1 Space-time in General Relativity 104 An Illustration of the Mathematical Devices Used 105 The Hole Argument 108 Leibnizian Equivalence 110 5.2 The Threat of Determinism 113 The Failure of Determinism 114 The Threat from A Priori Metaphysics 117 5.3 An Alternative Conception of Determinism 119 A Lewisian Account 120 An Analysis of Thought-Experiments 123 Conclusion 126 Chapter 6: A Defence Against the Hole Argument 128 6.1 Is Manifold Substantivalism the Default View? 130 The Container Metaphor 131 A Cost/Benefit Analysis 133 The Linguistic Strategy 137 6.2 Manifold-plus-Metric Substantivalism 140 The Hole Argument Revisited 141 A Fine-Grained Ontological Distinction 145 6.3 Super-substantivalism 149 Varieties of Super-Substantivalism 150 Ontological Parsimony 153 Metaphysical Assumptions 156 The Modal Argument 161 Conclusion 164 Chapter 7: Denying Primitive Identity 166 7.1 Metric-field Substantivalism 167 Relationalism in Disguise 167 Responding to the Hole Argument 173 The Cost of Denying Primitive Identity 175 Conclusion 176 Concluding Remarks 178 Methodological Considerations 181 Summary and Future Directions 182 Bibliography 185 vi “Can philosophers really contribute to the project of reconciling general relativity and quantum field theory? Or is this a technical business best left to the experts? General relativity and quantum field theory are based on some profound insights about the nature of reality. These insights are crystallized in the form of mathematics, but there is a limit to how much progress we can make by just playing around with this mathematics. We need to go back to the insights behind general relativity and quantum field theory, learn to hold them together in our minds, and dare to imagine a world more strange, more beautiful, but ultimately more reasonable than our current theories of it. For this daunting task, philosophical reflection is bound to be of help. Baez (2001, 177) Chap. 1 - Introduction 1 Chapter 1 Introduction “Clearly our first problem must be to define the issue, since nothing is more prolific of fruitless controversy than an ambiguous question” Russell (“Determinism and Physics”) Space-time substantivalism is a metaphysical theory. It is the thesis that space-time is something real: it is a robust entity with determinate properties which can exist independently of the material objects and events which may ‘occupy’ it.1 A consequence of this is that the whole material world might have existed just as it is, but shifted over in space three feet in any given direction. This means that the world could have been different in infinitely many ways, without those differences being discernible. To put the same point in the language of possible worlds, there are an infinite number of qualitatively identical possible worlds.2 Such worlds constitute a distinction without a difference: they do not differ in any empirically detectable way. We can reinterpret this as a problem regarding the spatio-temporal location of events: if substantivalism is true, then the whole history of events in the universe might have been located differently in space-time. This is an age-old Leibnizian problem, which is raised in the Correspondence between Leibniz and Clarke against Newtonian absolute space (H.G. Alexander: 1998). But is it a legitimate problem for substantivalists? Substantivalism is currently the most plausible theory of space-time consistent with the general theory of relativity (GTR). According to GTR, space-time affects and is affected by the presence and activity of matter. Gravitational energy, for example, can be transmitted at light speed in the form of gravitational waves between material 1 I shall later argue that the occupation relation is a mischaracterization of the relationship between space-time and material objects and events. 2 Throughout this thesis I intend to remain neutral between different accounts of possible worlds. I will therefore employ the language of Lewisian Modal Realism for the sake of clarity, but everything I say about concrete possible worlds can be translated into talk about abstract possible worlds. My talk of concrete possible worlds should therefore not be understood as committing me (or the substantivalist) to Modal Realism. Chap. 1 - Introduction 2 bodies and regions of matter-free space-time. This strongly suggests that space-time is substantival in nature, since it is extremely difficult for relationalism to explain the possibility of matter-free regions of space-time. As Dainton (2001) remarks, “How can something that can behave like this be anything other than substantival in nature?” (302) Relationalists also have difficulty explaining the odd behaviour of material objects as they move through regions of space with variable curvature (non-Euclidean holes) (Dainton: 2001). In such regions, objects alter their shape in a way that is not explained in terms of other forces, for example gravity or magnetism. Whereas substantivalists explain this behaviour straightforwardly in terms of alterations in the shape of space, relationalists have no such solution, since only material objects and spatial relations exist. The relationalist must therefore account for this odd behaviour in terms of relations between material bodies and other material bodies (object-object relations). However, given the various possible configurations of other material bodies, the laws involved in such an explanation, although possible, will be incredibly complex.3 While neither of these examples rules out relationalism altogether, it is clear that relationalism cannot compete with substantivalism in explaining natural phenomena in GTR. As Hoefer (1996, 1-2) remarks: “For this reason, just about everyone who writes on GTR… is broadly speaking, a substantivalist.” In light of this, to what extent should we take Leibniz’s musings seriously? When faced with strong (and observable) physical evidence in support of substantivalism, what weight should we assign Leibniz’s purely a priori metaphysical considerations? This is an instance of a much broader question regarding the overall value and significance of metaphysical inquiry: what can metaphysics contribute with respect to the pursuit of objective truth about reality, and in particular, about space- time? The question of the contribution of metaphysics to the pursuit of truth is a highly contentious issue, and is one of the central themes of this thesis. I argue that metaphysics can and does make a significant contribution to discovering the objective nature of reality, but metaphysics and science should not be pursued independently of one another, after all, their subject matter is the same. However, if metaphysics is to 3 For a further discussion of object-object relations in non-Euclidean space, see Nerlich (1991).